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Multifaceted information fusion : a decision support system for online monitoring in water distribution… Aminravan, Farzad 2014

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MULTIFACETED INFORMATION FUSION: A DECISION SUPPORT SYSTEM FOR ONLINE MONITORING IN WATER DISTRIBUTION NETWORKS  by Farzad Aminravan B.Sc. Mechanical Engineering, Azad University, 2005 M.Sc. Mechanical Engineering, Tehran Polytechnic, 2008  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  DOCTORATE OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Mechanical Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA (Okanagan) January 2014 ? Farzad Aminravan, 2014  ii Abstract Online monitoring of distributed infrastructure systems such as water distribution networks is necessary to assure the safety and security of our modern societies. Often various patterns of uncertainty and information deficiency about spatial relationships and interdependence of distributed observations impede effective monitoring of distributed systems. As a result, spatiotemporal monitoring of complex infrastructure systems needs a unified framework to deal with various sources of information deficiency induced by subjective and objective information. This thesis investigates the problem of combining deficient spatiotemporal evidential sources for the purpose of quality monitoring and relative risk analysis. Spatiotemporal monitoring of water distribution networks based on surrogate water quality parameters is formulated in terms of integrated quality monitoring and relative risk analysis. Distributed quality assessment is performed through an extended fuzzy evidential reasoning scheme that employs the fuzzy interval grade and interval-valued belief degree (IGIB). The former facilitates modeling of uncertainties in terms of local ignorance associated with expert knowledge whereas the latter allows for handling the lack of information on belief degree assignments. Local multi-sensor fusion for relative risk analysis is based on a proposed extended fuzzy belief rule-based (BRB) system that employs the IGIB structure in its rule consequent. The proposed extended BRB system can handle nonlinear input-output relationships and inconsistencies in the inference process as well as epistemic uncertainties including ambiguity, vagueness, and interval uncertainty in the knowledge base. A multi-level information fusion framework capable of modeling interdependency and uncertainty is proposed for distributed online monitoring. The multi-level information fusion framework encompasses a networked fuzzy belief rule based (NF-BRB) system and a high- iii level BRB system. The NF-BRB system is employed for knowledge elicitation from the expert about locally perceived relative risk associated with drinking water, while the high-level BRB system with hybrid learning enhances the event detection performance. Extensive simulated water quality degradation events, originated from real dataset of the Quebec City?s main water distribution network and contamination tests in a pilot distribution facility, were exploited to validate the efficacy of the proposed framework for integrated water quality monitoring and relative risk analysis.  iv Preface I carried out the primary design, analysis and writing of this thesis under the supervision of Dr. Homayoun Najjaran. Valuable feedback was provided by Dr. Homayoun Najjaran, Dr. Rehan Sadiq on the methodology aspects and by Dr. Manuel. J. Rodriguez and Dr. Mina Hoorfar on the water quality modeling case studies. Different parts of Chapters 3, 4, and 5 were published for dissemination of the results in different journal papers and conference proceedings as follows. Chapter 3 is partly published in the International Journal of Intelligent Systems and the proceedings of 2011 IEEE conference on Systems, Man and Cybernetics coauthored by F. Aminravan, R. Sadiq, M. Hoorfar, M. J. Rodriguez, A. Francisque, and H. Najjaran. The methodology presented in Chapter 4 and part of the results are based on the manuscript submitted for publication in the Applied Soft Computing and a conference proceeding in 2012 IEEE conference on Systems, Man and Cybernetics. Chapter 5 is based on a manuscript submitted to Expert Systems with Applications and a conference proceedings published in 2013 joint conference of Information Fuzzy Society Association/North American Fuzzy Information Processing Society coauthored by the same authors. Helpful consultation and feedback on the characteristics of the Quebec City?s main water distribution network were provided by Dr. Alex Francisque, and Mr. Christian Plettier from Quebec City water treatment plant management group.     v  Table of Contents Abstract .................................................................................................................................... ii	 ?Preface ..................................................................................................................................... iv	 ?Table of Contents .................................................................................................................... v	 ?List of Tables ........................................................................................................................... x	 ?List of Figures ....................................................................................................................... xiii	 ?Acknowledgements ............................................................................................................ xviii	 ?Dedication .............................................................................................................................. xx	 ?Chapter  1: Introduction ...................................................................................................... 21	 ?1.1	 ? Research objectives ................................................................................................ 22	 ?1.2	 ? Proposed research method ..................................................................................... 23	 ?1.3	 ? Research case study: Decision support system for water distribution network ..... 28	 ?Chapter  2: Background Review ......................................................................................... 31	 ?2.1	 ? Uncertainty modeling and management ................................................................ 32	 ?2.1.1	 ? Different types of uncertainty ......................................................................... 33	 ?2.1.1.1	 ? Aleatory uncertainty ................................................................................. 33	 ?2.1.1.2	 ? Epistemic uncertainty ............................................................................... 34	 ?2.1.1.3	 ? Amalgamtic uncertainty ........................................................................... 35	 ? vi 2.1.2	 ? Uncertainty modeling within belief function theory ....................................... 36	 ?2.1.3	 ? Uncertainty management in monitoring systems ............................................ 38	 ?2.1.3.1	 ? Fuzzy and neurofuzzy systems for monitoring ........................................ 40	 ?2.1.3.2	 ? Belief function theory employed for monitoring ..................................... 42	 ?2.1.3.3	 ? Belief rule-based system for monitoring .................................................. 43	 ?2.1.3.4	 ? Uncertainty management via learning approaches .................................. 44	 ?2.2	 ? Multifaceted information fusion in complex systems ............................................ 45	 ?2.2.1	 ? Information fusion for event detection and online monitoring ....................... 45	 ?2.2.2	 ? Information fusion for risk analysis ................................................................ 46	 ?2.3	 ? Case study background: Online drinking water quality monitoring ...................... 49	 ?2.3.1	 ? Online water quality parameters ..................................................................... 49	 ?2.3.2	 ? Early warning systems .................................................................................... 53	 ?Chapter  3: Multifaceted Information Fusion for Decision Support ................................... 60	 ?3.1	 ? Introduction ............................................................................................................ 60	 ?3.2	 ? Basics of belief function theory ............................................................................. 61	 ?3.3	 ? Basics of fuzzy Dempster-Shafer theory ............................................................... 66	 ?3.4	 ? Extended fuzzy Dempster-Shafer theory ............................................................... 67	 ?3.4.1	 ? Fuzzy interval grade and interval-valued belief degree (IGIB) structure ....... 70	 ?3.4.2	 ? Extended FDST for combining evidence with fuzzy IGIB structure ............. 77	 ? vii 3.4.3	 ? Preference ranking using the interval-valued assessments ............................. 82	 ?3.5	 ? Extra uncertainty patterns in belief degree assignment ......................................... 85	 ?3.5.1	 ? Extra uncertainty ............................................................................................. 87	 ?3.6	 ? Local water quality assessment for Quebec City main distribution network ........ 90	 ?3.7	 ? Conclusions .......................................................................................................... 101	 ?Chapter  4: Local information fusion for risk analysis ..................................................... 103	 ?4.1	 ? Introduction .......................................................................................................... 103	 ?4.2	 ? A novel fuzzy evidential belief rule-based system .............................................. 104	 ?4.2.1	 ? Fuzzy composition using matching function ................................................ 113	 ?4.2.2	 ? Fuzzy composition using similarity measure ................................................ 113	 ?4.2.3	 ? Determining the degree of firing ................................................................... 115	 ?4.3	 ? Enhanced rule aggregation algorithm .................................................................. 117	 ?4.4	 ? Numerical simulation using the local fusion algorithm ....................................... 129	 ?4.5	 ? Local water quality information fusion ................................................................ 132	 ?4.6	 ? Conclusions .......................................................................................................... 142	 ?Chapter  5: Spatiotemporal monitoring: Distributed information fusion .......................... 144	 ?5.1	 ? Introduction .......................................................................................................... 144	 ?5.2	 ? Problem statement ................................................................................................ 148	 ?5.3	 ? Sensor data preprocessing .................................................................................... 149	 ? viii 5.3.1	 ? Adaptive transformation ............................................................................... 149	 ?5.4	 ? Feature extraction for anomaly detection ............................................................ 151	 ?5.4.1	 ? Primary feature extraction ............................................................................. 151	 ?5.4.2	 ? Secondary (parametric) feature extraction .................................................... 152	 ?5.5	 ? Distributed multi-level information fusion: Hierarchical belief rule-based system ............................................................................................................................. 155	 ?5.5.1	 ? The structure of the NF-BRB system for spatiotemporal monitoring .......... 155	 ?5.5.2	 ? The structure of the high-level BRB system ................................................. 157	 ?5.5.3	 ? Dynamic fuzzy evidential fusion for improved temporal risk analysis ........ 158	 ?5.6	 ? Hybrid learning for the high-level BRB system .................................................. 165	 ?5.6.1	 ? Discussion of sensor fault tolerance ............................................................. 167	 ?5.6.2	 ? Belief rule parameter tuning with optimal learning ...................................... 168	 ?5.6.3	 ? Rule base structure updating with belief rule utility ..................................... 170	 ?5.7	 ? Problem setting for validation .............................................................................. 171	 ?5.8	 ? Validation: Distributed online water quality monitoring ..................................... 176	 ?5.8.1	 ? Distributed monitoring .................................................................................. 176	 ?5.8.1	 ? Testing strategies with spatiotemporal data .................................................. 176	 ?5.9	 ? Conclusions .......................................................................................................... 187	 ?Chapter  6: Conclusions .................................................................................................... 189	 ?6.1	 ? Summary .............................................................................................................. 189	 ? ix 6.1.1	 ? Summary of the contributions ....................................................................... 190	 ?6.2	 ? Main conclusions ................................................................................................. 191	 ?6.3	 ? Future work .......................................................................................................... 194	 ?Bibliography ..................................................................................................................... 198	 ? Appendix A: Statistics of water quality parameters in the Quebec City?s main WDN ...... 224  x 	 ?List of Tables Table 2.1 Commonly employed algorithms for different modules of distributed monitoring systems (Schwabacher & Goebel, 2007) .............................................. 40 Table 2.2 Common WQPs, their online availability and indications for online water quality monitoring .......................................................................................... 51	 ?Table 2.3 The available information fusion algorithms in the EWS for water distribution ............................................................................................................... 56	 ?Table 3.1 The IGIB structure probability distribution and its corresponding belief and plausibility for Example 3.4 .............................................................................. 82	 ?Table 3.2 The IGIB structure belief degree distribution derived from the generalized fuzzy sets of Example 3.4 and their corresponding belief and plausibility for the point estimate ?0.6? on the universe of discourse .................................................................................................................. 90	 ?Table 3.3 The belief and plausibility intervals and maximum uncertainty of water quality levels for sampling point ?QC317? in the main distribution network .................................................................................................................... 96 Table 3.4 The belief and plausibility intervals and the maximum uncertainty of aggregated water quality levels for sampling point ?QC317? in the main distribution network ........................................................................................ 98	 ?Table 4.1 An overview of different modeling features of the proposed IGIB-RB  xi system, conventional evidential reasoning and BRB (i.e., RIMER) frameworks ............................................................................................................ 110 Table 4.2 Distributed belief, plausibility and pignistic probability intervals of the output fuzzy grades and their comparison with the belief degree approximation using the matching function and the similarity measure ............... 132	 ?Table 4.3 Some rules of the designed rule base of the proposed IGIB-RB system for local relative risk analysis ................................................................................ 136	 ?Table 4.4 Proportional rule contribution (PRC) measure for increasing number of    aggregated rules ..................................................................................................... 137	 ?Table 4.5 The distributed interval assessment of belief, plausibility, and betting commitments of relative risk associated with drinking water at SPs ?QC117? and ?QC324? ........................................................................................... 138	 ?Table 4.6 Comparison of distributed assessment of average pignistic probability using the proposed extended BRB system and RIMER approach for fifty-two sampling points in Quebec City?s main water distribution network .................................................................................................................. 140 Table 5.1 The information of the estimated RT from treatment plant (node v1) for different nodes of the Quebec City's main WDN .................................................. 173	 ?Table 5.2 Initial spatiotemporal rules between water treatment plant (v1) and Venturi (v2) nodes (RT~ 1 hours) .......................................................................... 175	 ?Table 5.3 Relative changes of FRC  signals as the composit response to injection of eight biological and chemical contaminants in a testing facility ....................... 183  xii Table A.1 The statistics of microbial WQPs at fifty-two SPs of the Quebec City?s main WDN  ............................................................................................................ 224 Table A.2 The statistics of physicochemical WQPs at fifty-two SPs of the Quebec City?s main WDN ..................................................................................... 225  xiii  List of Figures Figure 1.1 Overview of the proposed nontraditional decision support and online monitoring framework ................................................................................ 25	 ?Figure 2.1 A simplified relationship between risk analysis, risk assessment, and risk management (Liu et al., 2013) ............................................................. 47 Figure 2.2 The schematic of a water distribution network with optimally selected online monitoring locations with connection to the SCADA architecture .................................................................................................. 55	 ?Figure 3.1 Representation of (a) individual grade sets, and (b) the corresponding individual and interval evaluation grade in the IGIB structure ................... 74	 ?Figure 3.2 The fuzzy IGIB structure with their associated interval-valued belief degree for both individual and interval fuzzy subsets of the frame of discernment ................................................................................................. 78 Figure 3.3 The membership functions for two assessment grades in Example 2 ......... 82	 ?Figure 3.4 The generalized fuzzy sets for defining the interval-valued belief degrees provided by two experts in Example 3.5 ....................................... 90	 ?Figure 3.5 Generalized fuzzy sets representing water quality grades: (a) for microbial parameters, (b) for physicochemical parameters ........................ 94	 ?Figure 3.6 Final hypotheses representing the water quality levels based on aggregated water quality parameters ........................................................... 96	 ? xiv Figure 3.7 The upper and lower bounds of cumulative Bel and Pls functions for Scenario 1 at sampling point ?QC317? ........................................................ 97	 ?Figure 3.8 The generalized fuzzy sets for interval-valued belief degree assignment to individual and interval water quality grades: (a) for microbial parameters, (b) for physicochemical parameters ........................ 99	 ?Figure 3.9 The upper and lower bounds of cumulative Bel and Pls functions for Scenario 2 at sampling point ?QC317? ...................................................... 100	 ?Figure 4.1 The schematic of the IGIB-RB inference engine ...................................... 107	 ?Figure 4.2 The schematic of grades and the belief degree assignment in the consequent of the Rule 1: a) the general case modeled using both the RIMER approach and the IGIB-RB system, b) the vagueness and local ignorance case modeled using the proposed IGIB-RB system, c) simultaneous existence of vagueness, local ignorance, and delimited belief degrees modeled using the proposed IGIB-RB system .................. 112	 ?Figure 4.3 Two trapezoidal fuzzy sets as the incoming evidence Eq and the fuzzy grade Af ...................................................................................................... 115	 ?Figure 4.4 Individual and interval fuzzy subsets of frame of discernment for Example 4.2 .............................................................................................. 121	 ?Figure 4.5 The fuzzy sets representing the evaluation grades for the input and output variables. ........................................................................................ 130	 ?Figure 4.6 The fuzzy grades representing different levels of the antecedent parameters of the IGIB-RB system and the incoming fuzzy inputs for  xv SPs ?QC117? and ?QC324? ....................................................................... 137	 ?Figure 4.7 Different measures of conflict for the increasing number of aggregated rules for (a) SP ?QC117?, and (b) SP ?QC324? ......................................... 137	 ?Figure 4.8 The distributed assessment of microbial risk of drinking water for SP ?QC117? using (a) the normalized betting commitment intervals, and (b) the belief distributed assessment based on the RIMER approach ....... 139	 ?Figure 4.9 The distributed assessment of the microbial risk of drinking water for SP ?QC324? using (a) the normalized betting commitment intervals, and (b) the belief distributed assessment based on the RIMER approach .................................................................................................... 139	 ?Figure 5.1 The schematic of the multi-level information fusion framework for spatiotemporal monitoring ........................................................................ 147	 ?Figure 5.2 The schematic of two nodes of the WDN, the calculated average retention time, and the contamination slug in the water flow ................... 149	 ?Figure 5.3 The schematic of multi-level information fusion for spatiotemporal monitoring ................................................................................................. 156	 ?Figure 5.4 The schematic of six sampling locations in the Quebec City's main WDN and their connectivity in the spanning tree G={V,E} .................... 172	 ?Figure 5.5 The probability density function of residual Cl concentration at four nodes of the Quebec City's main WDN. ................................................... 174	 ?Figure 5.6 The unprocessed sensor measurements of residual Cl concentration and TUR at treatment plant and Venturi nodes during three months of  xvi 2011 ........................................................................................................... 177	 ?Figure 5.7 The distributed assessment of water quality grades betting commitments in January 2011 (normal condition without any added anomaly) ................................................................................................... 177	 ?Figure 5.8 Anomaly patterns with positive and negative strength of 1. ..................... 178	 ?Figure 5.9 The FRC concentration signals and their adaptive transformation at 6 nodes of the WDN during three months of the year 2011 ........................ 179	 ?Figure 5.10 The FRC concentration signals at the treatment plant (v1) and Ventri (v2) nodes with six U-shape anomaly patterns added to the Venturi measurements ............................................................................................ 180	 ?Figure 5.11 The estimated betting commitment at each time window using the hierarchical BRB system with static fusion and without learning ............ 181	 ?Figure 5.12 The estimated betting commitment at each time window using the hierarchical BRB system  with dynamic fusion and hybrid learning ....... 181	 ?Figure 5.13 The simulated and estimated relative change of FRC concentration at Venturi (v2) node for six contamination experiments ............................... 184	 ?Figure 5.14 The aggregated water quality assessment (solid line) and annotated  water quality events (dotted time samples) using the hierarchical BRB system without learning (static assessment) ............................................. 184	 ?Figure 5.15 The aggregated water quality assessment (solid line) and annotated water quality events (Dotted time samples) using the hierarchicsl BRB system with hybrid learning (dynamic assessment) .................................. 185	 ? xvii  Figure 5.16 The ROC curves of water quality event detection results using a) hierarchical BRB system (static fusion), b) hierarchical BRB system (dynamic fusion), c) hierarchical BRB system (static fusion with learning), and d) hierarchical BRB system (dynamic fusion with learning) .................................................................................................... 186 Figure A.1 Distribution of fifty-two SPs within the Quebec City?s main WDN ........ 223  xviii  Acknowledgements Completing this thesis was possible only through the generous help of several people. I would like to deeply thank my supervisor Dr. Homayoun Najjaran, who always motivated me to simplify and see through the fundamentals. Many of the ideas in this thesis was originated during discussions with Homayoun. His imagination and technical insight have deeply influenced the way that I think about research problems and I am sure this will serve me well in the future. To my supervisory committee; Drs. Rehan Sadiq, Mina Hoorfar, and Manuel J. Rodriguez, thank you for you helpful feedback and guidance during the years. The interdisciplinary nature of this thesis led to many fascinating and helpful discussions. I specially enjoyed the conversations with Rehan during the early stages of this thesis. Also, Thanks to Dr. William Melek for reviewing an early draft of this thesis and providing comprehensive comments that really helped in improving it. I?d like to give special thanks to Dr. Carl Carlson and Dr. Ronald Yager for invaluable guidance and discussions. I?d also particularly like to thank Dr. Alex Francisque who helped me a great deal in the beginning to learn about the field of water quality monitoring. I?d also like to thank Vedad Famourzadeh and Tomas Veloz for very interesting discussions that helped me expand some of the ideas in Chapters 4 and 5. Thank you to the graduate students in ACIS laboratory, specifically Biddut Battacharjee, Nima Farrokhsiar, Hadi Firouzi, Mohammed Salem, Nikolai Kummer, Fuhad Hassan, Ali Mohandes, Elaheh Aghaarabi, Miguel Murran, and David Kadish. You?ve been great help from logistical advice, technical help, and even emotional support.  xix Thanks to my parents Talieh and Abbasali who thought me anything is possible and that anything worth doing is worth doing right. To my sister Mojgan and my brother Sina, you?ve helped in several ways more than what you can imagine during the years. Thank you and remember to never stop dreaming. Last, but by no means least, thanks to my all supportive friends especially Gale, Neil, Mouhebat, Vedad, Kat, James, Clair (especially for all hidden chocolate around the 746 house), Logan, Mina, Kuri, Hadi, Golriz, Alireza, Saheb, Toni, Hamid, Wendi, Julia, Ali, Sasha, Parnia, Nima, Alex, Gustavo, Nadine, Reddi, Maria, Kevin, Priscila, Kiano, Mona, Roger, Josh, Jasmine, Amin, Iman, Atousa, Mohammad, Kami, Ehsan, Siavash, Farshad, Hesam, Damoon, Marzi, Hamed, Faezeh, Mazi, Salma, Mehdi, and Mandana. You are all wonderful.  xx Dedication        To my beloved parents   21 Chapter  1: Introduction Informed assessment of the state of distributed infrastructure systems, such as water distribution networks (WDN), is vital for safety and security of our modern societies. Since water quality can deteriorate after leaving the treatment plant, monitoring policies are required at the distribution network level. Failure events at the distribution network level can be extremely critical due to the high impact public health consequences affecting millions of people. Moreover, drinking water is directly distributed among consumers who seldom have safety barriers before consumption, except sometimes a filtration device at the consumers? tap. Distribution networks are considered as a type of complex systems, which are characterized by dynamic parameters observed at spatiotemporal dimensions of the system. Online monitoring of distributed systems is an effective approach to issue warnings when and where anomalies are detected in the system. In a broader view, the methods offered by safety instrumented systems have been employed for estimating the reliability or remaining useful life of various distributed engineering systems. However, although handling uncertainty is one of the most important elements of safety of complex systems, it is also its least mature element. Thus, still there are many undiscovered opportunities that can be offered by enhanced prognostic systems as game-changing technologies to push the boundaries of complex system health monitoring. As a sensitive application, the early detection of water quality anomalies in a water distribution network is mainly followed by risk-based or reliability-based water quality failure analysis to allow for timely preventive public health interventions. One major shortcoming of the current safety instrumented technologies, mainly known as early warning systems (EWS) in the water quality monitoring community, is the use of ineffective inference  22 mechanisms that do not provide interpretable and intuitive operational information. In essence, the existing EWSs cannot take into account all sources of uncertainties involved in event detection, diagnostics, and risk-based analysis. Consequently, their employed inference mechanisms do not provide an efficient and reliable EWS. Thus, more effective paradigms are required to analyze multifaceted and spatiotemporal monitoring data in the WDN. The realization of this goal drastically reduces the public health and economic impacts of false negative and false positive results of EWSs. 1.1 Research objectives This research focuses on the development of an effective decision support system (DSS) for EWS with an application in online water quality monitoring. The main challenge of spatiotemporal monitoring of distributed systems is the estimation of the uncertainty associated with predicting the reliability or the relative perceived risk once an anomaly is detected. In this research, novel uncertainty modeling and management techniques are proposed to design a reliable spatiotemporal monitoring system. Several multifaceted information fusion frameworks, which can capture historical uncertainty-based information at both local and distributed levels, are introduced to identify the potential failure events through flexible knowledge elicitation frameworks. In the meantime, the online assessment, which characterizes the current state of anomalies, provides the incremental information that can be used for updating the knowledge base. The proposed frameworks for local and distributed information fusion will be tested through historical datasets available for spatiotemporal monitoring in a real water distribution network. The main contributions of the conducted research are  23 1) Development of a local information fusion module for decision support based on heterogeneous sensor data that features a) Flexibility in handling various patterns of information deficiency b) Effective local alarm generation in case of warrantable anomalous sensor data  2) Development of an expert local information fusion module for quality assessment that uses partial and deficient knowledge elicited from experts or the imperfect information collected in one monitoring location  3) Development of a distributed information fusion module that can handle different aspects of information uncertainty and sparsity in the spatiotemporal monitoring of distributed systems  4) Development of a comprehensive DSS that can handle uncertainties and inconsistencies in online monitoring of distributed systems  5) Validation of the previous objectives using actual case study problems and simulations for each of the developed modules in this research through design and testing of an EWS for the WDN. 1.2 Proposed research method A comprehensive DSS that can integrate all spatiotemporal monitoring features of a distributed system for the purpose of anomaly detection, regardless of the system under study, has not been designed yet. Besides, since the modern diagnostics and prognostics has engendered from the reliability engineering and condition-based maintenance with major applications in health monitoring of aerospace structures and manufacturing processes, most employed algorithms in these fields are based on prediction models such as general path model to estimate the remaining useful life (RUL) of a population or an individual component in a distributed system (Schwabacher & Goebel, 2007; Garvey & Hines, 2008). However, decision support for the EWS as part of online monitoring of distributed infrastructure systems such as WDNs should be designed considering specific facets of  24 information deficiency in these systems. In online spatiotemporal monitoring systems, sufficient temporal data samples at one monitoring location are available. Thus, in one monitoring location, the problem does not suffer from sparsity in data in the temporal domain. However, still there is a need to aggregate subjective information elicited experts or originated from available regulations about the incommensurate and heterogeneous sensor data. On the other hand, due to the limitations in increasing the number of monitoring locations, the data samples in the spatial domain are highly sparse. This condition implies that data-driven aggregation algorithms can be employed to provide online assessment at one sampling location owing to more frequent samples. However, for a set of monitoring locations, due to the sparsity of spatial data, the data-driven information fusion approaches are likely to fail. In the mean time, the complex dynamics of the system reduces the direct applicability of model-based detection and diagnostic approaches. As these properties are shared among a large class of distributed systems equipped with sensor network monitoring architecture, the proposed solution to this problem is quite generic. In order to provide a solution to the aforementioned limiting conditions, we propose a nontraditional DSS that features several sub-modules for online spatiotemporal monitoring. At the core of the nontraditional DSS are local and distributed information fusion frameworks that use uncertainty-based information processing and management upon sparse data sampling space. The proposed architecture of the DSS for spatiotemporal monitoring is illustrated in Figure 1.1. The proposed DSS consists of four main sub-modules: the preprocessing unit, the anomaly detection unit, the local multifaceted information fusion unit, and the distributed multi-level information fusion unit.  25  Figure 1.1 Overview of the proposed nontraditional decision support and online monitoring framework The main contributions of this research are modeling broader uncertainty manifestation and management in online monitoring of distributed systems. Thus, methods that can handle various types and patterns of uncertainty are chosen as the basis for development of the nontraditional DSS. Based on the involved uncertainty in spatiotemporal monitoring, the architecture of the proposed DSS is based on the fuzzy belief function theory and several extensions of the belief rule-based (BRB) system. The former can model subjective and Anomaly Detection Multifaceted Information Fusion Preprocessing Unit Anomaly? Sensor Network Local Quality Assessment Unit Local water quality level at SP1?SPN Multi-level Information Fusion Uncertainty Management Unit Perceived levels of risk at SP1?SPN Local water quality Outlier? Signal smoothing  Adding to historical patterns Yes No Detection Unit Local Relative Risk Assessment Unit Spatiotemporal Aggregation High-level BRB System Yes No Distributed water quality Human Interface Networked BRB System  26 objective uncertainty while the latter handles nonlinearity and inconsistencies in the inference process. Different sub-modules of the DSS are built upon previously proposed architectures for fuzzy evidential reasoning, connectionist-based frameworks, and neurofuzzy systems with embedded evidential reasoning algorithms (e.g., Basir et al., 2005; Yager, 2009). The proposed algorithms for different sub-modules of the proposed nontraditional DSS for online monitoring are explained in the rest. In the proposed architecture, the anomaly detection unit identifies the episodes of probable anomaly events and the multifaceted information fusion unit provides an assessment of water quality in one sampling location through aggregating heterogeneous sensor measurements and subjective knowledge (Chapter 3). If the inferred local assessment of water quality falls below the acceptable ranges a specific procedure using several extended BRB systems for relative risk analysis, based on statistics of the monitored water quality parameters (WQP) in each time window will be performed (Chapter 4). If online spatiotemporal measurements of WQPs are available the multi-level information fusion module, which is based on hierarchical BRB systems, is activated for online spatiotemporal data aggregation at all recent time windows. The multi-level information fusion module comprises the networked fuzzy belief rule-based (NF-BRB) system and the high-level BRB system. The NF-BRB system employs various sources of information including the expert knowledge about spatiotemporal variations of WQPs, and approximate retention time (RT) between different monitoring locations, and then performs distributed information fusion using an efficient rule aggregation algorithm. The high-level BRB system performs inference about water quality based on the assessments provided by the NF-BRB system and the analysis of extra parametric features extracted from online WQP signals. A hybrid learning method is  27 introduced to improve the water quality assessment and event detection performance. Besides, a dynamic information fusion framework for the high-level BRB system provides online assessments of the water quality through employing appropriate uncertainty management techniques based on availability of expert knowledge on the spatiotemporal characteristics of the WDN and objectively measured WQPs. If the expert knowledge is not available but training samples are provided the learning algorithm for the high-level BRB system can be employed to build the rule base. Finally, local and distributed assessments of water quality considering the associated uncertainty and sensor fault tolerance will be communicated with system operators or policy makers. The validation of different modules of the proposed nontraditional DSS is performed after presenting the basics of each module throughout Chapters 3 to 5. Several validation strategies are employed to demonstrate the advantages of the proposed DSS including 1) Modeling broad uncertainty manifestation based on uncertain subjective information and incomplete objective data 2) Intuitive rule aggregation with the assumption of independence between the rule consequents and analysis of the applicability of fuzzy evidential rule combination 3) Proper handling of conflict and inconsistency among belief structures through flexible knowledge representation and information fusion 4) Propagation of interval uncertainty to the distributed quality assessments using extended fuzzy evidential reasoning 5) Efficient manifestation of extended fuzzy evidential reasoning for rule aggregation through assessing the consistency and conflict among the rule consequents 6) Generic modeling for spatiotemporal monitoring through constructing locally optimum and compact rule bases 7) Effective dynamic information fusion and hybrid learning algorithms for online water quality assessment and contamination event detection    28 1.3 Research case study: Decision support system for water distribution network The proposed frameworks for local information fusion for decision support and spatiotemporal monitoring are introduced through Chapters 3 to 5 to design a comprehensive EWS for WDNs. In the literature of the EWS for WDNs, some specific implications are found which are helpful in designing the different modules of the proposed nontraditional DSS for online water quality monitoring. In the WDN, due to the complex nature of water quality degradation process, various hypotheses can be associated with water quality degradation alarms. The set of potential water quality failure scenarios implies different consequences that should be tolerated (Glasgow et al., 2004; USEPA, 2005; 2009; 2010). Thus, a major monitoring trend in this field is focused on estimating the relative risk associated with drinking water. Currently, the detection of a potential anomaly and predicting the relative risks based on expert knowledge is obtained solely based on individual signals of surrogate WQPs. Normally, a significant and persistent deviation of the signal from the suggested regulatory range for each sampling location warrants an anomaly. However, combining data across signals and backward in time can provide richer sets of features for both event detection and relative risk analysis with the possibility of reducing the false negative and positive results. Thus, a set of primary and secondary features are extracted to train the extended BRB system for local and distributed information fusion. Then, the perceived probabilities associated with anomalies that may relate to water quality failure are communicated by the designed nontraditional DSS. The primary features are based on the amplitude of WQPs at each time step, after some initial preprocessing of the WQP signals. The preprocessing is required because of inherent  29 noisy signals and rapid changes in the signals due to the operational actions in the network. Besides, the accurate calibration of online WQP sensors is very important to get accurate results. Thus, a credibility factor for assessing the sensor reliability and sensor fault tolerance is considered for the aggregation of incommensurate sensor data for local relative risk analysis. Secondary features are extracted through considering a recorded time window of recent measurements. Based on several experimental studies in Sandia National Laboratories (Yang et al., 2007b; 2009), it was observed that simulated anomaly scenarios in a pilot water distribution facility can induce various temporal signal patterns in commonly monitored WQPs. Specifically, free residual chlorine (FRC), pH, and turbidity (TUR) showed specific types of geometries in their contamination slug during laboratory contamination event simulations (Yang et al., 2009). These results imply that the algorithms that can adapt the temporal WQP signal shape such as matching pursuit algorithm can be employed to extract more insightful and quantified information that are sensitive to specific water quality degradation events. In Chapter 5 a signal modeling and feature extraction method using parametric Gaussian functions to model the slug geometry in non-stationary WQP signals is introduced. Extensive informative features that can quantify the transient water quality events are used to design the high-level BRB system for distributed monitoring whenever frequent temporal data are available. The next chapters of the thesis are organized as follows. Chapter 2 provides a background of multifaceted information fusion, and previous EWS for event detection in the WDN. Chapter 3 introduces the extended fuzzy evidential reasoning framework for decision support and a case study of water quality assessment using local information fusion at fifty-two sampling  30 locations of the Quebec City?s main WDN. The validation is based on the available statistical data at individual sampling locations of the network to test the performance of the multifaceted information fusion module. Chapter 4 introduces the extended BRB system and uses the same dataset as in Chapter 3 to test the performance of the extended BRB system in relative risk analysis. Chapter 5 introduces the structure of the proposed multi-level information fusion for online spatiotemporal monitoring based on extended and hierarchical BRB systems along with the learning and dynamic fusion algorithms. Besides, a novel validation method for EWSs used for online monitoring of the WDN is introduced. The available online monitoring data for the Quebec City?s main WDN in the year 2011, water quality events generation using CANARY software based on experimental data, and manual contamination slug modeling were employed to perform the validation. Chapter 6 provides some concluding remarks of the thesis with outlining the major contributions and future research directions.  31  Chapter  2: Background Review The current trend in the existing and emerging online monitoring systems for detection and diagnosis of anomalous events is driven forward to provide a reliable tool for predicting the potential failure events in distributed systems. Early warning systems (EWS) for water quality monitoring are one the emerging solutions which is advancing toward the spatiotemporal data aggregation (Hall & Szabo, 2010). However, no integrated spatiotemporal monitoring tool has been employed at the readiness level, for online water quality monitoring. Besides, it has been reported that many of the existing online monitoring technologies have not been effectively used because of a) ignoring the real implementation needs of the utilities, b) the low reliability and high cost of the off-the-shelf sensor technologies, and c) the inefficient information handling units that cannot provide intuitive predictions of water quality (Storey et al., 2011). Hence, there are still various unexplored opportunities that the online systems can offer to water safety and security management and public health agencies. Three main approaches have been followed to resolve the challenging issues in online water quality monitoring systems: the approaches that concentrate on sensor development technology, the approaches which focus on distribution network hydraulic modeling and optimal sensor placement, and the approaches which utilize more effective information fusion and inference schemes for appropriate interpretations of water quality parameters. Since this research focuses on effective information fusion for water quality monitoring, an up-to-date review of EWSs used in existing and emerging technologies for online water  32 quality monitoring and the EWS constituents will be presented. Specifically, the review on sensor and information fusion approaches and other emerging trends that can be employed in online water quality monitoring is emphasized. Then, the literature review on information fusion and uncertainty modeling frameworks that can be adopted or modifies to design a more effective EWS will follow. Due to the lack of statistical data, the performance evaluation of most emerging technologies in this field is rarely available. Considering this fact, the background that follows is concentrated on the technologies that address a novelty in information fusion or practical use for online water quality monitoring. In a glance, this chapter reviews the background of four main areas as ? Uncertainty modeling and management for online monitoring ? Multifaceted information fusion in complex systems  ? Online drinking water quality monitoring ?? Online water quality parameters (WQP) ?? Early warning systems (EWS) 2.1 Uncertainty modeling and management The presence of various types of uncertainty introduces a challenge to the prediction of states of complex and distributed systems. In this section, the issues related to characterization and modeling different types and patterns of uncertainty are discussed. Probability theory tries to quantify the grade expressing the likelihood of happening specific events. Although this does not provide a definite answer about the occurring event, in the perspective of a probability theorist knowing something is better than nothing. Thus, expert-informed probability assessment has been extensively studied due to high reliance on expert opinion and engineering judgment in addition to historical datasets and experiments collected on a  33 problem. Since there exist other types of uncertainty which the probability theory cannot account for this section is dedicated to a review on various types and patterns of uncertainty in complex systems. The frameworks in the literature of uncertainty modeling and distributed sensor fusion for the purpose of online monitoring are presented. 2.1.1 Different types of uncertainty In this section, different types of uncertainty that can exist in both quantitative and subjective information are briefly presented. Mainly, different types of uncertainty that can be modeled using probability theory, the belief function theory and fuzzy set theory are reviewed. First, three different types of uncertainty that may exist in information fusion problems for distributed monitoring including aleatory, epistemic and amalgamtic uncertainties are introduced in the following. 2.1.1.1 Aleatory uncertainty Aleatory uncertainty stems from randomness in a representative sample, which refers to the variability in a known population. This type of uncertainty is insensitive to additional information or observations once it is completely characterized. This implies that additional knowledge cannot decrease this type of uncertainty. The standard error in sampling is defined as   SE =sn (2.1) where  n  is the sample size and  s  is the sample standard deviation. The standard error drastically increases for small values of  n  while for larger values of  n  the deviation in the  34 standard error is small. This property is in line with frequentist?s view of probability theory that states if a large enough sample size is used, aleatory uncertainty is objective and insensitive to additional information. The stochastic property of aleatory uncertainty is explained with the Bayes theorem that considers the probability as a function of knowledge where extra information can drastically reduce the uncertainty. The complementary type of uncertainty that is related to the individual?s state of knowledge is called epistemic uncertainty. 2.1.1.2 Epistemic uncertainty Epistemic uncertainty is an approximation to represent the subjective knowledge that can predict the behavior of a complex system. Since this type of uncertainty is based on the modeler?s understanding of the system it can be improved through more observation or knowledge elicitation from experts. The reduction of epistemic uncertainty can be modeled via Bayesian updating according to the Bayes rule.  It is believed that physical processes and events can be explained by scientific laws or theories based on the current state of our knowledge. These scientific characterizations are subjective and thus some scholars have questioned the foundation of these characterizations. In my view, this characterization is approached in a better way if it is seen as a function of one?s understanding of the system rather than limiting one?s inference model to a specific framework that uses predefined objects of different uncertainty types to express the knowledge. Thus, we note that a more complete type of uncertainty may exist in information that can be called amalgamtic uncertainty.  35 2.1.1.3 Amalgamtic uncertainty Amalgamtic uncertainty exists whenever both aleatory and epistemic uncertainty components are present in a problem (Inkabi, 2009). The probabilistic nature of amalgamtic uncertainty has elements of randomness and variability, but by itself has no meaning. The question that arises is whether Bayesian methods are adequate for handling this type of uncertainty or there is a need to employ other methods. One approximate answer to this problem is to estimate the dominant type of uncertainty whenever the presence of amalgamtic uncertainty is guaranteed. For instance, if aleatory uncertainty is negligible with a reasonable approximation Bayesian updating can be used to reduce the uncertainty. In this approach, the expert?s judgment and his ability to make this reduction in the type of uncertainty is the key to proper management of uncertainty. On the other hand, in many engineering systems, it has been stated that it is impossible or impractical to characterize the uncertainty in two separate groups of aleatory and epistemic uncertainties. Obviously, this view is not shared by this author, however, this author tries to popularize a more flexible framework for modeling uncertainty that allows the decision maker to understand what sources of uncertainty can be reduced (for instance through Bayesian updating or various belief structure combination rules) and what sources need different uncertainty management techniques. Meanwhile, a more detailed distinction between different types and pattern of uncertainty assures that they are properly modeled through introducing extended uncertainty modeling frameworks. Since in reality a mixture of aleatory, epistemic and inseparable amalgamtic uncertainty are present in distributed engineering systems, this research focuses on the development of an integrated framework for uncertainty characterization and management for the purpose of  36 decision support, risk analysis, and online monitoring. However, since we mainly deal with systems with low failure likelihood and catastrophic consequences in case of a failure, the expert knowledge is highly relied upon. The collective knowledge from the experts has an epistemic nature even if the process itself has elements of randomness. As a result, the uncertainty types discussed in this thesis are mostly epistemic. In this regard, a background on belief function theory that can be used for handling the simultaneous existence of epistemic uncertainty and variability is followed. Then, production systems such as fuzzy rule-based systems that can model nonlinear input-output relationships are reviewed. 2.1.2 Uncertainty modeling within belief function theory This section introduces different patterns of uncertainty that can be modeled under the belief function theory. Dempster-Shafer theory (DST) of evidence is the basis for most uncertainty modeling methods that are categorized under belief function theory. Then, the extra types and patterns of uncertainty that original DST does not account for are defined. The evidential reasoning provides a framework for representing and managing the epistemic uncertainty, including granularity, nonspecificity, and conflict. The core of most evidential reasoning approaches is the Dempster-Shafer theory (DST) (Dempster, 1967; Shafer, 1976), which has been successfully used in different applications. To-date, several evidential reasoning algorithms using distributed modeling frameworks based on the DST have been introduced (e.g., Yang & Singh, 1994). A more recent, well-known method called evidential reasoning (ER) for multi-attribute decision analysis (MADA), which can be derived in both analytical and recursive formulations, was proposed for managing uncertainty while it has the advantage of lower computational cost compared to the DST (Yang & Xu, 2002a; 2002b). Ambiguity and global ignorance are two types of uncertainty that can be modeled in  37 the original DST framework. These parameters are defined as follows. Ambiguity- Ambiguity is defined as the condition that two different types of uncertainty namely nonspecificity and strife coexist in the information. Strife refers to the conflict in the assignment of partial belief degrees among various sets of alternatives in a multi-source information fusion problem. Nonspecificity is related to imprecise cardinalities of the sets of alternatives.  Global ignorance- Global ignorance is a more distinctive term for the ignorance in the original DST theory that refers to the undistributed belief degree that is assigned to the referential set. The vicious belief structure is defined when the only focal element is the referential set (i.e., complete ignorance with  m(?) = 1). The following extra types of uncertainty cannot be modeled in the original DST but may exist in the process of knowledge elicitation from the expert. The extensions to account for these extra types of uncertainties in an extended fuzzy evidential reasoning and in a fuzzy evidential rule-based framework will be introduced in Chapters 3 and 4.  Vagueness- In the context of production (rule-based) systems, vagueness is defined as the linguistic imprecision in the antecedent and consequent grades that are used in building the rule base. Grades refer to ordered subsets of the referential set. As a result, grades can represent different levels of antecedent attributes (inputs) or consequents (outputs). The grades can be either fuzzy or crisp. A crisp grade is a special case of the fuzzy grade where the membership is always 1. In conventional fuzzy rule-based systems, both antecedent and consequent grades are modeled with fuzzy sets. However, in belief rule-based (BRB) systems due to the incorporation of the Dempster?s combination rule, which is only valid for disjoint  38 focal elements, the grades in the consequent are crisp. Vagueness in the consequent grades can be considered through defining ordered fuzzy subsets of the referential set. Interval uncertainty- In the literature of belief functions, interval uncertainty mainly refers to two different patterns of uncertainty which are known as (a) Local ignorance- Local ignorance is a pattern of interval uncertainty which refers to the belief degree that cannot be distributed with certainty among some individual adjacent grades. However, it is assigned to a new interval grade that represents the local ignorance focal element that covers all those adjacent grades. Since interval grades are defined to represent the local ignorance, this type of uncertainty is considered as a pattern of interval uncertainty (Guo et al., 2008; 2009). (b) Interval-valued belief degree- Another pattern of interval uncertainty is due to belief degrees delimited to an interval. The interval-valued belief degree in knowledge representation systems refers to the state where the belief degrees associated with focal elements in the consequent are not precisely known (Den?ux, 1999). Interval-valued belief degrees are used to model the imprecise belief degrees expressed by the expert or derived based on historical datasets (Yager, 2001). In the rest, a brief review of different approaches previously employed for uncertainty management in monitoring systems will follow. 2.1.3 Uncertainty management in monitoring systems Spatiotemporal monitoring systems are required to provide anomaly alarms; diagnostic, and prognostic outputs based on distributed sensing and decision fusion techniques. In detection and diagnostics, uncertainties such as signal to noise ratio of the extracted features and false  39 alarm mitigations have been historical issues that need specific solutions. Due to the fact that prognostic systems are developed based on the current measurements, current diagnostic results, and absence of the future measurements, they include considerable uncertainties propagated to the prediction model (Tang et al., 2007).  In the case of sparse datasets, the role of confounding factors as well as inferring unobserved parameters of the complex systems becomes considerably important. Besides, it is often observed that data samples do not perfectly measure the real target parameters; thus, there is a need to use more robust methods to characterize the deficiency of measurements and the dispersion from the real states (Deb et al., 1998). Indeed, accurate uncertainty representation and decision fusion approaches for managing various patterns of uncertainty (e.g., imprecision, ambiguity, interval uncertainty, and vagueness) are at the core of designing distributed online monitoring systems.  Table 2.1 presents a categorization of commonly-used frameworks for detection, diagnostics and prognostics in distributed monitoring systems. Primary information fusion systems were mainly centered on the classical logic theory. Bayesian and belief function frameworks showed their value for handling various structures of contingencies in the shade of non-classical logic theory. In order to handle the stochastic uncertainty, the Bayes theorem was utilized with inference and updating the probability associated with propositions (Saha & Goebel, 2008). However, in a large class of objective functions related to monitoring of distributed systems other patterns of uncertainty may exist in the information obtained from a distributed sensor network; hence, epistemic and possibilistic approaches were introduced to handle these types of uncertainty.   40 Table 2.1 Commonly employed algorithms for different modules of distributed monitoring systems (Schwabacher & Goebel, 2007) Framework Detection Diagnostics Prognostics Physics-based System theory N/A Failure propagation models AI model-based Expert systems Finite states machines N/A Conventional numerical Linear regression Logistic regression Kalman filter and its variants Machine learning Clustering Decision trees Neurofuzzy systems  2.1.3.1 Fuzzy and neurofuzzy systems for monitoring In the possibilistic framework, fuzzy set theory and its numerous extensions have been used for uncertainty modeling in monitoring systems. For instance, a fuzzy point-wise similarity measure was proposed for condition monitoring in a nuclear plant (Enrico & Francesco, 2010). A library of different patterns of evolution was made and a pattern matching technique based on fuzzy similarity measures was presented to determine the remaining life of individual elements of the plant. Fuzzy rule-based (FRB) systems also became popular for knowledge representation and uncertainty management in expert systems (Zadeh, 1983). Fuzzy logic provided a language with syntax and local semantics to incorporate the qualitative knowledge of the expert about a complex system. Due to the remarkable interpolation properties of the FRB system, high robustness to the variations in the system parameters and disturbances can be obtained. Togai and Watanabe (1986) first reported the design of an FRB system for real-time implementation of approximate reasoning on a very-large-scale integrated (VLSI) chip, which could be used for monitoring purposes. FRB systems can be used to design prognostic systems through modeling various sources of  41 uncertainty in measurement, process noise, and the uncertain process parameters. However, other sources of uncertainty may arise during the design of the prognostic system based on FRB systems. In particular, nonspecificity in assessing the relative value of each observation to the consequent state, as well as the conflict imposed by sparse datasets collected in the distributed sensing system, are important factors to be considered. This necessity resulted in growing interest in uncertain FRB systems, which allow the deficient knowledge to be incorporated into the knowledge base. One of the major extensions to account for the uncertainty in FRB systems was accomplished by introducing type-2 FRB systems (Karnik et al., 1999). However, type-2 FRB systems were not extensively incorporated in monitoring systems in practical applications mainly due to their computational complexity and dependency on extra training for the field experts. Meanwhile, the FRB systems when solely designed based on human expertise may not result in an accurate and adaptive model for complex distributed systems. Therefore, neurofuzzy modeling, which acquires knowledge from a set of input-output data, has been actively investigated (Jang, 1993; Juang & Lin, 1999; Ryu & Won, 2001). Adaptive network-based fuzzy inference system (ANFIS) (Jang, 1993) was extensively employed for diagnostics and for uncertainty bound reduction in prognostic systems (Dragomir et al., 2008, Samanta & Nataraj, 2008). Although it may seem that the neurofuzzy system is an AI model-based prognostic method, almost no AI model-based prognostic scheme exists in the literature, as it is shown in Table 2.1. The reason is that, most of the time, the neurofuzzy frameworks that are used for uncertain knowledge representation are encapsulated in another machine learning algorithm (Schwabacker & Goebel, 2007). Therefore, it is more appropriate to consider these methods as machine learning methods, which have extra capabilities for  42 handling the uncertainty owing to a more comprehensive knowledge representation model. Besides, due to the overfitting of the data in neural network (NN) methods, it is essential to choose the appropriate structure of the neurofuzzy model to assure acceptable generalization ability. Particularly, the state-of-the-art support vector machine (SVM) (Vapnic, 1995) has been used in order to find the number of network nodes or fuzzy rules, based on the given error bounds (Chan et al., 2001; Chu et al., 2004). The support vector neural network (SVNN) was also presented to select the best structure of radial basis function networks for a predefined precision (Chan et al., 2001). It should be noted that a successful design for monitoring of a distributed and complex system at least should provide interpretability in its outputs as well as the proper handling of involved uncertainty in distributed measurement and subjective expert knowledge. In the literature, there are many data-driven methods that do not satisfy the mentioned requirements to design a distributed monitoring system with interpretable outputs. Thus, a main part of this thesis focuses on specific knowledge representation systems that provide improved interpretability through employing belief function theory and belief rule-based (BRB) systems. A review of these approaches employed for monitoring is presented in the rest. 2.1.3.2 Belief function theory employed for monitoring Belief function theory provides a framework for representing and managing aleatory and epistemic uncertainty, including granularity, ambiguity (i.e., a situation when nonspecificity and conflict coexist). The core of most evidential reasoning approaches is the Dempster-Shafer theory (DST) (Dempster, 1967; Shafer, 1976), which has been successfully used in different applications. Different interpretations of the DST such as transferable belief model (Smets & Kennes, 1994) have been used in monitoring systems (Smets, 1988; Goebel et al.,  43 2006; Kallappa & Hailu, 2005). Extensions such as pignistic transformation (Smets, 1989) or plausibility function transformation (Cobb & Shenoy, 2006) to manage the uncertainty and to provide more interpretable DSS have been proposed. The concept of prognostic fusion was recently presented, which is based on the idea of designing individual prognostic systems and aggregating their predictions to reduce the combined uncertainty (Goebel et al., 2006). 2.1.3.3 Belief rule-based system for monitoring A class of AI-based monitoring systems that work based on BRB systems has been recently presented. The classic BRB system is used to model a facet of uncertainty when the expert cannot make strong judgments. Alternatively, imprecise and incomplete data may be used to make the rule base that governs the BRB system. The BRB system allows deficient information to be used for building a knowledge base when the input-output relationship is not highly assured. Thus, the BRB system is appropriate for representing the uncertainty in subjective and analytical elements of knowledge. The classic BRB systems (Liu et al., 2004; Yang et al., 2005; Yang et al., 2006) claim to be able to model ambiguity (i.e., a situation where two different types of uncertainty including conflict and nonspecificity coexist). Some learning algorithms were implemented for constructing BRB systems after initial setting based on the expert knowledge. Yang et al. (2004) presented an optimal learning algorithm using nonlinear multi-objective optimization to minimize the difference between the output of the BRBs and the given data. In their proposed formulation, parameter specific limits and partial expert judgments are formulated as constraints. Yang et al. (2006) proposed a generic rule-based inference methodology using the evidential reasoning (RIMER) which has been employed in various applications. An integrated inference and learning methodology, which is developed for tuning the parameters of RIMER approch, was  44 proposed by Xu et al. (2007). Specifically, in designing the rule bases, manual adjustment of the parameters of the rule base can be time-consuming and approximate. Liu et al. (2008; 2010) also investigated the learning algorithms which result in a consistent belief rule base considering that the consistency most of the time is violated. The proposed learning algorithm is developed based on the construction of a set of constrained nonlinear optimizations. Chen et al. (2013) consider the classic BRB system as a distributed approximation process where the combined belief degrees are obtained through aggregating the belief structures in the consequent of all activated rules. They have shown that employing learning algorithms can improve the approximation properties of BRB systems. 2.1.3.4 Uncertainty management via learning approaches Several Bayesian learning algorithms have been proposed for learning in FRB systems. For instance, the Bayesian Ying-Yang (BYY) learning (Xu, 2002; 2004) was used for learning in FRB systems. BYY learning is based on the harmony of two representations: the mapping of the input data into an inner representation cluster, and the generation of the input data from an inner representation cluster. Besides, the support vector machine (SVM) has delivered good performance in machine learning for monitoring. SVM learning mechanisms for FRB systems were presented by Chen and Wang (2003) and Chiang and Hao (2004). However, the SVM showed a number of practical limitations such as working with non-probabilistic parameter estimation, necessity to estimate the error/margin trade-off parameters and the requirement of a specific kernel function. Later, relevance vector machine (RVM) learning mechanisms, which are not suffering from the limitations of SVM, were proposed for modeling of nonlinear dynamic FRB systems (Kim et al., 2006). The proposed algorithm uses the RVM to design a FRB system that simultaneously performs system optimization and  45 regression generalizations. The structure of the FRB system is the same as that of the Takagi?Sugeno FRB system. However, in the proposed method by Kim et al. (2006), the gradient ascent method is employed to reduce the number of rules through optimizing a marginal likelihood and tuning the parameter values of kernel functions. 2.2 Multifaceted information fusion in complex systems There are many areas within a distributed monitoring system where information fusion techniques play a contributing role. Specifically, many of the feature extraction, estimation and knowledge representation techniques can be employed for event detection, diagnostics and prognostics (Vachtsevanos et al., 2006). Although detection and diagnostics have been deployed for many engineered systems, prognostics remains an almost unexplored area. Schwabacher (2005) reported that prognosis of a distributed system is extremely difficult, and while some prototypes have been presented, a prognostic system that takes advantage of measured parameters of the system is not readily available. However, for complex distributed system with a low likelihood of failure and high-impact consequences risk analysis has been employed as a means to evaluate potential failures in a similar way to population-based prognostics. In the case study of this thesis, distributed sensing network for online monitoring of complex infrastructure systems such as water distribution networks is considered. Thus, we mainly focus on anomaly detection and diagnosis followed by specific risk analysis techniques that highly rely upon expert knowledge and uncertainty modeling. 2.2.1 Information fusion for event detection and online monitoring The model-based monitoring schemes for distributed systems are classified into three main architectures: centralized, decentralized, and hierarchical (Gertler, 1998). Although the centralized architecture is the most popular one, it has the drawbacks of high memory  46 requirement and computational cost. Besides, a failure in the computational units of the system may result in the total failure of the diagnostic system. On the other hand, decentralized monitoring systems make the event detection and diagnosis based on local models without the need for the global model computations (Pencole & Cordier, 2005). Decentralized architecture can be shown to be as optimal (based on Bayesian decision logic) as centralized fusion. However, it is not possible to infer about the states of all distributed systems with a high level of confidence just by processing information at individual spatial locations. A trade-off between the centralized and decentralized architectures can be achieved by a hierarchical monitoring architecture (Harris et al. 2002). Roychoudhury et al. (2009) proposed two algorithms, which are capable of analyzing the time and space characteristics of local diagnosers to design hierarchical detection and diagnostic system. One of their proposed algorithms is based on the predefined subsystem structure of the local diagnosers. Whereas their other proposed algorithm develops the subsystem models and the diagnosers at the same time. A hierarchical structure seems to be the best monitoring architecture that can be employed for a distributed system equipped with a distributed sensing network. In a distributed infrastructure system such as the WDN, information on hydraulic connectivity of the nodes in local strata when coupled with the surrogate WQP sensor measurements can provide more operational information for early event detection. 2.2.2 Information fusion for risk analysis In short, risk analysis is a variety of techniques and processes including monitoring and predicting unexpected hazards for the overall goal of management of risks. Risk assessment is defined as the collection of all activities that should be performed before taking any decision in regard to a proper risk reduction strategy (Frosdick, 1997; Liu et al., 2013). A  47 simplified relationship between risk analysis, risk assessment and risk management adopted from IEC/EN 61511 and IEC/EN 61508 standards is presented in Figure 2.1.   Figure 2.1 A simplified relationship between risk analysis, risk assessment, and risk management (Liu et al., 2013) The general scope of risk assessment is vast and various techniques are employed to assess the risk of different processes. In the IEC/EN standards, the concept of risk is defined as the probable rate of ? Occurrence of a hazard causing harm ? The degree of severity of harm  48 Thus, risk can be seen as the product of ?incident frequency? and ?incident severity?. Handling uncertainty is one of the main issues in the risk analysis of complex systems. Uncertainty in the risk analysis problem mainly stems from two resources including randomness due to inherent variability in the system and imprecision due to deficient information or knowledge (Apostolakis, 1990; Helton, 2004). In this perspective, the risk analysis problem falls in the category of amalgamtic uncertainty modeling and management where both aleatory and epistemic uncertainties coexist.  Traditional approaches to risk analysis are mainly based on probability theory. However, often the types of uncertainty encountered in hazards are not completely modeled using probability theory. Specifically, the uncertainty is introduced by vagueness in meaning or nonspecific assessments. For instance, quantitative risk analysis (QRA) including probabilistic risk analysis, fault tree analysis (FTA) and failure mode, effects, and criticality analysis (FMECA) are widely used, but often lack the capability of modeling subjective knowledge and vagueness in the meaning of parameter estimates. This research mainly focuses on the development of relative risk analysis techniques based on more comprehensive expert knowledge representation systems. These models are used in cases where accurate mathematical or statistical models are difficult to obtain or relying just on the qualitative knowledge-based systems lead to inaccurate predictions. The developed algorithms in this thesis are mainly based on belief function theory and extended BRB systems for complex distributed systems that were reviewed in Section 2.1.3.   49 2.3 Case study background: Online drinking water quality monitoring In this section, a brief review of the case study problems for validation of this research which involve the online water quality monitoring at local and distributed levels of the WDN is presented. First, a brief review of commonly monitored WQPs and their indications is presented. Then, current and emerging technologies for online water quality monitoring are reviewed. 2.3.1 Online water quality parameters The WQPs monitored in a distribution network should comply with the regulatory requirements and provide municipalities with the ability to assess the water quality and determine the sources of water quality failure events (National Guide to Sustainable Municipal Infrastructure (NGSMI), 2004). The quality of the source water and the treatment process will also affect the WQPs in the water distribution network. This section provides a non-exhaustive list of drinking WQPs which are more commonly used for drinking water quality monitoring in chlorinated water distribution networks. The background review on WQPs that their online measurement is practical and can be employed in EWSs for distributed online monitoring systems is emphasized.  The WQPs for monitoring purposes have been studied previously and there is a wealth of literature in this area (e.g., Swamee & Tyagi, 2000; Sadiq et al., 2010). Microbial WQPs are specifically used to represent the microbial contaminants of drinking water. The importance of monitoring microbial WQPs is associated with immediate potential public health consequences that can result from the consumption of water contaminated by pathogenic bacteria. Another set of indicators represents the physicochemical properties of drinking water. The use of physicochemical WQPs is more common in online water quality  50 monitoring systems (Hall et al., 2007; Yang et al., 2008) since their online measurement is much easier than that of microbial WQPs. However, the degree of reliance on microbial measures, if available, is higher as they provide more insightful information about the water quality. We need to deal with on various WQPs whose ordered levels could be related to water quality or relative risk associated with drinking water with a linear or nonlinear relationship. Comprehensive descriptions and different state-of-the-art multi-parameter water quality monitors for EWS are reviewed in USEPA reports (USEPA, 2005; 2009). Journal (2009) also studied the commercially available and emerging EWSs for continuous online water quality monitoring in the WDN and provided a comprehensive list of WQPs used in these systems. The list of common WQPs measured for drinking water and the possibility of their online measurement is presented in Table 2.2. Free residual chlorine (FRC) protects the system through deactivating the existing bacteria in distribution networks and also added bacteria when an intrusion occurs in the system. FRC can be used as an indicator of residence time in the distribution network. Total organic carbon (TOC) provides an indication of the potential for the regrowth of heterotrophic bacteria (Payment et al., 2003). If the online measurement device for this parameter is not available, UV_254 nm can be used since it gives an indication about natural organic matter available in the finished water (Kirmeyer et al., 2002). Water turbidity is considered as a microbial measure since an increase in water turbidity can provide a surrogate for the possible microbial contamination (Payment et al., 2003; USEPA, 2004). Turbidity is also useful from the operational perspective because it can exert chlorine demand (NGSMI, 2004). The heterotrophic plate count (HPC) bacteria can also be used for microbial water quality monitoring; however, its online measurement  51 remains is not available in online monitoring systems and thus it does not appear in Table 2.2. Table 2.2 Common WQPs, their online availability and indications for online water quality monitoring Parameters Available Current Online Monitoring Parameter Indications Color Online equipment is available, Broad applications. Indicator of aesthetics and dissolved solids. Color could be correlated to colored organic matter, metals such as iron or colored industrial wastes. Monitoring of color could be very useful. 1 Conductivity Online equipment is available, Broad monitoring applications, Conductivity can be related to color, taste, and corrosion. Online measurements of total dissolved solids such as salts can be estimated based on conductivity (USEPA, 2005). Corrosion and scale forming potential of water can be estimated based on conductivity (National Guide to Sustainable Municipal Infrastructure (NGSMI), 2004). Free residual chlorine (FRC) Online equipment is available, Specific parameter.  FRC protects water against new microbial contamination and regrowth of bacteria in the distribution network. FRC levels can provide information regarding microbiological activity in the distribution network and its integrity. Oxidation reduction potential (ORP) Online equipment is available, Broad monitoring applications, ORP can be related to suspended solids, aesthetics and turbidity. ORP is an indicator of dissolved oxidizing and reducing agents (e.g., metal salts, chlorine, and sulfite ion) (USEPA, 2009). pH Online equipment is available but the sampling rate is not very high, pH level can indicate potential microbial growth or some adverse health effects due to pH adjustment during treatment process.  Indicator of hydronium ion concentrations in water. Many chemical and biochemical processes are pH dependent (USEPA, 2009).  Changes in pH often accompany a contamination event and a rapid pH change may indicate a chemical over-feed at the treatment plant (NGSMI, 2004). The pH affects the chemistry of water (Payment et al., 2003). For example, low pH water can increase corrosion rates in metal pipes (Sadiq et al., 2007).                                                 1 Changes in color from that normally seen can provide warning of possible quality changes or maintenance issues and should be investigated. The change in color may, for example, reflect degradation of the source water, corrosion problems in distribution systems, changes in performance of adsorptive treatment processes (such as activated carbon filtration) and so on (Payment et al., 2003). 	 ? 52 Temperature (Temp) Online equipment is available, Specific parameter.  Biological and chemical processes are heavily temperature dependent (Payment et al., 2003; Francisque et al., 2009b). Nitrification most commonly occurs at temperatures greater than 15oC (NGSMI, 2004). Total organic carbon (TOC) Online equipment is available, Broad monitoring applications, Present technology is costly and involves high maintenance.  Disinfection by-products can form with the presence of high TOC and FRC (NGSMI, 2004). TOC can exert chlorine demand. With the availability of TOC measurement at the point of entry, its level should not significantly change within the distribution network.  Total trihalomethanes (TTHM) Online equipment is available. Application at monitoring locations. The formation of an important class of chlorinated disinfection by-products is monitored (NGSMI, 2004). TTHM have long-term impacts on human health and thus their online monitoring is not of high priority. Turbidity (TUR) Online equipment is available, Application at point of entry and monitoring locations. Indicator of suspended solids and potentially present microbes in drinking water. Turbidity is not associated specifically with faecal material, but increases in turbidity are often accompanied with increases in pathogen numbers, including cysts or oocysts (Payment et al., 2003). High turbidity can exert chlorine demand (NGSMI, 2004). UV_254 nm Online equipment is available, Broad monitoring applications, A surrogate of natural organic carbon. High levels of UV_254 nm could indicate the increased presence of biodegradable organic matter that promotes bacteria regrowth (Le Chevallier et al., 1991; van der Kooij, 1992). Indicator of suspended solids and microbes.  Furthermore, physicochemical WQPs including water temperature, pH, and color are considered. Biological and chemical processes are heavily temperature dependent (Payment et al., 2003; Francisque et al., 2009b). Many of the chemical and biochemical processes are also pH dependent. For instance, disinfection processes based on chlorine-based chemicals are sensitive to the pH of water. Changes in pH can indicate the occurrence of contamination events or chemical over-feed at the treatment facility (NGSMI, 2004). In order to make more  53 informative relationship between the WQPs and water quality degradation events, other parameters such as conductivity, oxidation-reduction potential (ORP) and total trihalomethanes (TTHM) may also be considered based on their availability. TTHM is a family of four commonly occurring chlorinated disinfection by-products and its formation is dependent on the availability of TOC and chlorine disinfectant in the distribution network. Monitoring disinfection by-products is important as they are considered as carcinogenic compounds; however, due to their long-term health impact their online monitoring is not of high priority. 2.3.2 Early warning systems The first step of designing an early warning system (EWS) inherently involves considering the current level of sensor development technology and optimal sensor placement methods. In the rest, a brief review the available and commonly used technologies at the time of performing this research are presented. However, since the main focus of the case study problem is to provide a more effective DSS for the EWS, a more comprehensive review of the existing approaches for the EWSs is presented.  A major class of software employed by municipalities is used for optimal sensor placement. Most algorithms are independent of the sensitivity or the type of employed sensors and current available technology. Software such as optiMQ-S using genetic algorithm optimization techniques are developed for optimal sensor placement (Ostfeld & Salomons, 2005). The optiMQ-S is costly in computational implementation and is limited to the short intrusion scenarios in simulated contaminant intrusion scenarios. Another sensor placement optimization tool (SPOT) is a product of the Threat Ensemble Vulnerability Assessment (TEVA) project of the USEPA laboratories and Argonne National Laboratory named TEVA- 54 SPOT. TEVA-SPOT is open source software that uses various optimization algorithms including heuristic, Lagrangian, and integer programming (Janke et al., 2009) for sensor placement. The location of sensors in a WDN can be found based on different objective functions such as detection likelihood, reduction of affected population, or protection of high-risk areas that have sensitive facilities, such as hospitals, due to their sensitivity to public health. A report by Ostfeld et al. (2008) summarizes the results obtained in a competition among various sensor network designs for online contaminant monitoring. The results provide further discussions and directions on designing distributed sensor networks and challenges faced in optimal sensor placement. Since the validation of this research is through assessing the effectiveness of the proposed DSS for online monitoring and early warning in the WDN in Chapters 3, 4 and 5, it is assumed that the required online sensor technology, the optimal number of sensors, and their location in the WDN are known. The schematic of a WDN, some monitoring locations, and supervisory control and data acquisition (SCADA) architecture is presented in Figure 2.2. It is notable that in our validation strategy through designing a DSS for the EWS, the emphasis is mainly on historical WQP dataset in different sampling locations in the network and the elicited expert knowledge on the relationship of water quality degradation and the observed patterns in surrogate WQPs. Current emerging and commercially available EWSs take advantage of data handling and processing units to identify water quality degradation events. The summary of the existing approaches used in current EWSs for online water quality monitoring is presented in Table 2.3.  55  Figure 2.2 The schematic of a water distribution network with optimally selected online monitoring locations with connection to the SCADA architecture Main leading research teams in this area are the US Environmental Protection Agency (USEPA) and Sandia National Laboratories that focus on real-time change detection algorithms to find water quality degradation events. These research groups have presented three detection algorithms including time-series increments with linear filter (McKenna et al., 2006), multivariate nearest neighbor or k-means clustering (Klise & McKenna, 2006a; 2006b), and binomial event discriminator (McKenna et al., 2008). These algorithms are implemented in the noncommercial software called CANARY (Hart et al., 2009). Each proposed algorithm predicts the anticipated levels of the WQPs at the current time window based on the previous observations. A change in the water quality is detected by determining a large difference between the anticipated values and the current measured values. The performances of these algorithms were compared in a recent USEPA report (USEPA, 2010).  56 Table 2.3 The available information fusion algorithms in the EWS for the WDN  Information fusion method Advantages Limitations References Binomial event discriminator ? Anomaly detection ? Baseline treatment   ? Detection only at one sampling station ? Lag time between the true onset of the event and the detection time ? McKenna et al. (2008) Linear filter ? Anomaly detection ? Baseline treatment ? Detection only at one sampling station  ? McKenna et al. (2006b) Multivariate nearest neighbor distance ? Anomaly detection ? Reduced number of false positive alarms ? Detection only at one sampling station  ? Klise and McKenna (2006a) ?  Klise and McKenna (2006b) READiw and kinetic relation analysis ? Anomaly detection and diagnosis ? Robustness to the baseline variations of the system ? Reduced false alarms using the analysis of kinetic relations  ? The proposed approach for contamination slug geometry detection is not fully automated ? Offline monitoring is presented at the current stage  ? Yang et al. (2007b) ? Yang et al. (2009) EDA algorithms : SOM, Sammon?s mapping, Projection pursuit ? Potentially can be used for online monitoring  ? Provide desirable data representation outputs ? Sensitivity to the baseline water quality ? Postolache et al. (2005) ? Mustonen et al. (2008) Model-based detection methods ? Transient failure detection ? Promising solution for the network level failure isolation ? The isolation of intrusion or breakage ? The complexity of the model  ? Unresolved issues on practical use in real distribution networks ? Difficulty in defining the anomaly criteria ? Shang et al. (2008) ? De Sanctis et al. (2010) Simulated signals to represent bimodally distributed WQPs were employed to test these algorithms. The tunable parameters of these event detection algorithms were set for background water quality parameters using nominal datasets in a way that the number of false positive results is reduced. Among the three algorithms, the multivariate nearest neighbor algorithm showed the lowest number of false positive results during the set point  57 assignment. Exploratory data analysis (EDA) algorithms, which in contrast to hypothesis testing approaches place more emphasis on data characteristics rather than the hypotheses, have also been for online water quality monitoring. Postolache et al. (2005) investigated online water quality monitoring using self-organizing map (SOM), a well-known unsupervised learning algorithm of the neural networks. Multilayer perceptron neural networks and SOM for multi-dimensional data representation and validation were employed to investigate the possibility of forecasting the distributed water quality. Another algorithm based on multivariate data exploration methods that uses SOM and Sammon?s projection, a nonlinear dimension reduction mapping that aims to maximally preserve the original structure of the measurements, was also proposed (Mustonen et al., 2008) for water quality monitoring. The pressure shocks were used to model the water quality degradation events. Since the variations in water quality due to the changes in water flow are highly transient, the possibility to design an EWS using temperature, turbidity and conductivity measurements has been discussed. Zhang and Dong (2009) proposed the projection pursuit technique, a dimension reduction method based on certain reconstruction rules, using genetic algorithm for water quality anomaly detection and diagnosis. Compared to other common classification methods, the value added by their approach is that the objective index weights can be easily determined using the projection direction vector. However, the main drawback of the EDA algorithms is that the irregular baseline variations for WQPs in the water distribution network can influence the trigger method used for detection and will dictate whether a contaminant can be detected (Hall & Szabo, 2010). In CANARY open-source software, the algorithm that is employed to resolve the baseline  58 variation issue is the water quality trajectory clustering, which supports learning all the recurring patterns at one monitoring location of the WDN (Vugrin et al., 2009). The trajectory clustering technique is based on low order polynomial regression models and the fuzzy multivariate clustering of the regression coefficients. The algorithm reduces the number of false positive results, which are due to the operational variations of the system. However, large sets of statistical data for each monitoring location of the distribution network are required to learn the nominal patterns. On the other hand, other approaches are proposed, which are more robust to the changes in the baseline of WQPs. Yang et al. (2007b) proposed a real-time event adaptive detection, identification, and warning (READiw) technique that can be used for detection and classification of various contaminant intrusions in the distribution network. An adaptive sensor output treatment using time-series and signal processing techniques was proposed to separate the signals from the background variations. Numerous common WQPs were used to test the detection and identification performance of the proposed system in case of different intrusion scenarios. The concentration-time pattern matching was proposed for anomaly detection, and the analysis of the kinetic relations in sensor responses was later added for detection verification and contaminant identification (Yang et al., 2009). Model-based methods form another class of current online water quality degradation detection algorithms that more common in the distributed water quality monitoring. In the model-based approach, a model to estimate the WQP signature is developed, and model predictions are compared to the measured WQPs. Shang et al. (2008) proposed a filtering method to detect the inconsistent changes in the error signal calculated based on the difference between the model-predicted baseline and the estimated WQP values. Their  59 network model is based on EPANET-MSX, which is the multispecies extension of the EPANET (USEPA, 2011) for water quality modeling. However, in general, due to high complexity of the dynamics of the WDN, the exclusive use of model-based approaches are not highly reliable for practical applications of online water quality monitoring. Thus, they are often coupled with data-driven or knowledge-based approaches. As it is seen in Table 2.3, most algorithms for the EWS are limited to a single sampling location and mostly do not incorporate operational information such as residence time between different sampling locations, and the status of reservoirs or valves in the network. In Chapter 5, which is focused on the development of a spatiotemporal monitoring framework, it is discussed how these types of information can be incorporated for distributed online water quality monitoring.  60  Chapter  3: Multifaceted Information Fusion for Decision Support 3.1 Introduction Decision support systems (DSS) have been widely used as a mean for condition monitoring in multi-objective system engineering in which a final assessment can be made only through considering different sources of information. For reliable condition monitoring, two important factors should be considered: 1) the best system condition indicators, and 2) the most appropriate fusion technique for the selected indicators. The data revealing the system condition or involved risks are inherently uncertain. The ability to deal with uncertain information is as important as the information itself. Thus, a fusion technique is required to handle these uncertainties in an online monitoring procedure. This chapter presents a comprehensive and flexible knowledge representation framework when various facets of information deficiency are present at local units of a complex system. The subjective knowledge is very important in interpreting the quantitative data such as measurements of the surrogate parameters that describe the behavior of a complex system. The belief function theory is suitable for modeling both aleatory uncertainty and epistemic uncertainties including conflict and nonspecificity. While belief function theory can model a wide range of epistemic uncertainty, several examples of knowledge elicitation are provided to demonstrate that the original belief function theory cannot model all facets of information deficiency such as interval uncertainty about the state of a distributed complex system. First, the basics of belief function theory are reviewed and the required extensions to take into account the extra patterns of interval uncertainty about the state of a complex system are  61 considered. Then, an extension of the fuzzy evidential reasoning theory is introduced to perform local information fusion when in addition to uncertainty that can be modeled using belief function theory, extra patterns of interval uncertainty and vagueness exist. The extended fuzzy Dempster-Shafer theory (FDST) based on the simultaneous use of fuzzy interval grade and interval-valued belief degree (IGIB) is proposed. The latter facilitates modeling of uncertainties in terms of local ignorance associated with expert knowledge whereas the former allows for handling the lack of information on belief degree assignments. Also, generalized fuzzy sets can be readily transformed into the proposed fuzzy IGIB structure. Throughout this chapter, it is shown how the proposed extended FDST provides flexibility in the knowledge elicitation process from the experts. Finally, a case study problem of decision support for water quality assessment in the Quebec City?s main water distribution network (WDN) based on the datasets available between the years 2003 and 2005 is presented. The obtained results demonstrate the effectiveness of the extended FDST for multifaceted information fusion for decision support in complex systems. 3.2 Basics of belief function theory The epistemic uncertainty, which includes both ignorance and subjectivity, is traditionally handled by the Bayesian approach. Dempster-Shafer theory (DST) (Dempster, 1967; Shafer, 1976) provided a framework for handling nonspecificity and conflict, and it has been successfully used in many different applications (Parikh et al., 2001; Sadiq & Rodriguez, 2005; Sadiq et al., 2006; Yang & Kim, 2006; Basir & Yan, 2007). Thus, the DST can be considered as an extension of the Bayesian approach that allows probabilities to be assigned to any subsets rather than only mutually exclusive singletons. This allows modeling  62 ignorance due to the lack of information. Besides, the DST provides a convenient framework for combining two or more pieces of evidence under certain conditions. In this section, basic ideas of the DST are presented. The DST is based on a set-valued mapping from a probability space ?  to a probability space ? , which is called the frame of discernment. The mapping : P?? ??  associates each element in ?  with a set of elements in .?  The multi-valued mapping can be considered as a compatibility relationships between ?  and ?  (Shafer, 1976). The granule of an element s  in ? , denoted as ( )G s , is the set of all elements in ?  that are compatible with s , ( ) { | , }.G s t t sCt= ??  A belief structure of space ?  is a mapping : [0,1]m P? ? such that  m( A) =p(si)G(si)=A?1? p(si)G(si)=?? (3.1)  Ai  is called a focal element if  m( Ai) > 0.  let ?  be a finite list of n  non-null focal elements of ,? that is  ? ={Ai | Ai ? ?, i = 1,...,n}. The belief structure : ( , )m? = ?  is introduced to indicate the association of belief degree to the elements on the list ?  such that  m( Ai)i=1n?= 1.  (3.2) The probability distribution of ?  is constrained by belief structures but not uniquely determined by them. Shafer (1976) dropped the notion of probability and showed that for anyB??, there is a range ( ) [ ( ), ( )],R B Bel B Pls B=  where Bel  and Pls  denote the belief (Bel) and plausibility (Pls) functions, respectively. The belief of event B denoted as ( )Bel B is  63 defined as  Bel(B) = m( Ai)Ai?B; Ai???,  (3.3) and the plausibility of event B denoted as ( )Pls B  is defined as  Pls(B) = m( Ai)Ai?B???. (3.4) Another important concept in belief function theory is the pignistic probability transformation of a belief structure. The pignistic probability transformation for the crisp subset B  of ?  is defined as  BetP(B) =B? AiAim( Ai)1? m(?)Ai???. (3.5) where .  denotes the cardinality of the set. The pignistic probability indicates the probability that an ideal decision maker assigns to a proposition when required to make a final decision or take an action. The next example shows how belief, plausibility and pignistic probability transformation of a classic belief structure are calculated. Example 3.1- Assume that the frame of discernment ?  includes five elements as 1 2 3 4 5{ , , , , }.t t t t t? =  The belief structure 1m  with three focal elements 11{ },A t= 2 1 2 3{ , , }A t t t= , and ?  is defined as 1 21 1 1 1 1 1 1 2 3 1: ( ) ({ }) 0.2, ( ) ({ , , }) 0.3, ( ) 0.5.m m A m t m A m t t t m= = = = ? =  The belief and plausibility of some subsets of ?  through evaluating Equations (3.3) and  64 (3.4) are found as 1 2 3 1 2 31 2 3 1 2 3({ }) 0.2, ({ }) 0, ({ }) 0, ({ , , }) 0.3, ( ) 0.5,({ }) 1, ({ }) 0.8, ({ }) 0.8, ({ , , }) 0.8, ( ) 1.Bel t Bel t Bel t Bel t t t BelPls t Pls t Pls t Pls t t t Pls= = = = ? == = = = ? = The pignistic transformation of the belief structure 1m  through evaluating Equation (3.5) for each element of ?  is found as 1 2 3 4 5({ }) 0.4, ({ }) 0.2, ({ }) 0.2, ({ }) 0.1, ({ }) 0.1.BetP t BetP t BetP t BetP t BetP t= = = = =  Another important procedure of the DST is known as the Dempster?s combination rule, which is used for aggregating multiple belief structures. Let 1m  have the focal elements  {Ai,i = 1,...,n1} and 2m  has the focal elements  {Bj, j = 1,...,n2}.  The conjunction of these two belief structures is a new belief structure 1 2m m m= ?  whose focal elements are subsets of  ? = HkHk=Ai?Bj??{ }. The associated belief degrees are  m(Hk) =m1( Ai)m2(Bj)Ai?Bj=Hk?1? KK = m1( Ai)m2(Bj)Ai?Bj=????????????? (3.6)  (3.7) where  K  shows the belief degree assigned to the empty set. The same formulation can be extended for the case of multiple belief structures. The Dempster?s rule of combination for  L  belief structures is  65  m1? ...?mL(Hk) =1(1? K )mi( Ai)i=1L??i=1LAi=Hk?K = mi( Ai)i=1L??i=1LAi=??????????? (3.8)  (3.9) In the belief function literature, there are examples of cases in which the Dempster?s combination rule gives counterintuitive results. This problem is known as the conditioning problem in the Dempster?s combination rule (Zadeh, 1986; Sentz & Ferson, 2002). Some researchers suggest that for successful application of the DST it is crucial to assure that prior to employing the Dempster?s combination rule specific conditions are met (e.g., Liu, 2006). Other researchers have proposed modifications of the original DST to solve the conditioning problem, for instance using different averaging (compromising) operators (e.g., Yager, 1987; Sentz & Ferson, 2002). Ignoring some uncertainty such as vagueness in complex systems may result in less uncertain assessments but can also result in serious underestimation of the uncertainty that should be propagated to the target space. The fuzzy set theory has been presented as a basis to handle vagueness in information (Zadeh, 1975; 1978). The coexistence of vagueness and ambiguity in many engineering and decision analysis problems engendered the fuzzy extension of evidential reasoning paradigms (Dubois & Prade, 1982; 1985; 1989; Yager, 1982; Yen, 1990; Zhu & Basir, 2005; 2006; Zadeh, 2005). Among these extensions, an important group focuses on the FDST and its application in fuzzy inference, data fusion, and classification (e.g., Binaghi & Madella, 1999; 2000; Zhu & Basir, 2005). The FDST makes it possible to combine several pieces of evidence when the evaluation grades are fuzzy subsets of the frame of discernment. The basics of the FDST follows in the next section.  66 3.3  Basics of fuzzy Dempster-Shafer theory This section presents the basics of FDST as the core concept of evidential reasoning for combining fuzzy evidence. Different schemes for fuzzy generalization of DST have been presented and gradually improved (Yang et al., 2003; Yager, 1982; Yen, 1990; Yager, 1995; Zhu & Basir, 2006; Den?ux, 2000; Miao et al., 2008). Issues arising in generalization of the DST to FDST have been addressed by Tsiporkova-Hristoskova et al. (1997). The core of FDST presented here for extending to a more general fuzzy belief structure is based on Yen (1990) approach which uses linear programming formulation for computing belief (Bel) and plausibility (Pls) functions of the fuzzy hypotheses. The classic fuzzy belief structure is represented by a mapping of the fuzzy proposition to the belief degree as  m :{?Ai|?Ai? ?}?{mi| mi?[0,1]}, i = 1,...,n,  (3.10) where  ?Ai  denotes a fuzzy subset of the frame of discernment  ?,  and  mi is the corresponding belief degree assigned to ?Ai. The minimum and maximum probability masses assigned to a fuzzy hypothesis  ?H  from the crisp set  Ai  are defined as  m*(?H : Ai) = m( Ai)? infu?UAi??H(u)m*(?H : Ai) = m( Ai)? supu?UAi??H(u)??????? (3.11) (3.12) Yen (1990) proposed to distribute the belief degree associated with a fuzzy focal element  67 among its resolutions defined by  ?j?  cuts. As each resolution of  ?Ai can be considered as a crisp set, the minimum and maximum contribution of fuzzy focal element  ?Ai  to the probability mass of fuzzy hypothesis  ?H  is  m*(?H :?Ai) = m*?? ( ?H : ?Ai? )m*(?H :?Ai) = m*?? ( ?H : ?Ai? )??????? (3.13) (3.14) where  ?A?i  is a resolution under level .?  Applying Equations (3.13) and (3.14) to all fuzzy subsets  ?Ai  of the frame of discernment ? gives the Bel and Pls functions of the fuzzy hypothesis  ?H  as  Bel(?H ) = m*(?H :?Ai)i=1n?= m*(?Ai) (?j??j?1) infu?U?A?j(u)?j??Ai?,   Pls(?H ) = m*(?H :?Ai)i=1n?= m*(?Ai) (?j??j?1) supu?U?A?j(u)?j??Ai?.  (3.15)  (3.16) This formulation of the FDST will be extended in the next section. Three steps are necessary to be utilized for comprehensive fuzzy evidential reasoning approaches (Zhe & Basir, 2006): formulating a fuzzy evidence structure, combining fuzzy evidence structures of different sources, and decision making based on the combination results. 3.4 Extended fuzzy Dempster-Shafer theory This section introduces an extended fuzzy evidential reasoning framework that can model various facets of information deficiency. The FDST presented in Section 3.3 can model  68 ambiguity and vagueness in a problem. The extended FDST is introduced to model two extra patterns of interval uncertainty during the knowledge elicitation process.  Extensive research focused on adapting the Dempster?s combination rule to handle interval uncertainty that is different from ambiguity in information. In the literature, there are two facets of interval uncertainty considered under belief function theory, namely interval-valued belief degree and local ignorance. Den?ux (1999; 2000) presented a general model for the interval-valued belief degree to handle the first aspect of interval uncertainty. The evidential reasoning (ER) approach was also extended to deal with interval-valued belief degree. An extension of the ER approach for the interval-valued belief degree assignment to crisp propositions was provided (Lee & Zhu, 1992; Nguyen et al., 1997; Wang et al., 2006a; 2007). In order to model local ignorance in the ER approach interval grades were introduced. First, belief degree assignment to crisp interval grades (Xu et al., 2006) was proposed to model local ignorance and later was extended to fuzzy interval grades by Guo et al. (2008; 2009). These extensions refer to two different patterns of interval uncertainty. Hence, a model with the ability to model both aforementioned patterns of interval uncertainty (i.e., interval-valued belief degree and local ignorance) in a united formulation will be more comprehensive. Providing this unified formulation for fuzzy evaluation grades is the main focus of this chapter. Two patterns of interval uncertainty are demonstrated using a simplified water quality example. The problem is faced when the expert cannot justify a specific belief degree allocation to a fuzzy grade. This situation is widely faced when expert knowledge is incorporated into a condition assessment of a complex system.  69 Example 3.2- An example of different possible belief structures can be illustrated by the expert belief model for water quality assessment. Three grades indicating the drinking water quality as ?Low {L}?, ?Medium {M}?, and ?High {H}? are considered. Imagine that three experts provide their assessment of water quality as Expert 1: The water quality is ?Low? with the probability of 10 percent and ?Medium? with the probability of 50 percent and ?High? with the probability of 40 percent. The assessment provided by Expert 1 can be represented as 1 {({ },0.1), ({ },0.5), ({ },0.4)}.E L M H=  Expert 2: The water quality is ?Low or Medium? with the probability of 60 percent and ?High? with the probability of 40 percent.  This assessment can be expressed as  2 {({ },0.6), ({ },0.4)}.E L M H= ?  Compare to Expert 1, Expert 2 makes a less specific judgment on water quality. However, he still can make a specific assessment for the interval grade {L-M} and individual grade {H}.  Expert 3: The water quality is ?Low or Medium? with the probability of 50 to 70 percent and ?High? with the probability of 30 to 50 percent. Similar to Expert 2, Expert 3 is unable to make specific assessments about {L} and {M} grades. Besides, he is uncertain about a specific allocation of the belief degree to the interval grade {L-M} and individual grade {H}. He provides less specific assignments using the interval-valued belief degrees as  70 3 {({ },[0.5, 0.7]),({ }, [0.3, 0.5])}.E L M H= ?  Provided that the grades are mutually exclusive and crisp, the DST presented in Section 3.1 can model both cases described by Expert 1 and Expert 2. However, since the water quality grades are better modeled with fuzzy grades, the above representations of the expert knowledge may not provide a realistic representation of the expert judgment. Under fuzzy grade assumption, the FDST explained in Section 3.2 can be employed to represent the assessment by Expert 1. The FDST can also deal with the assessment provided by Expert 2, only if an appropriate definition for the interval fuzzy grade is introduced. However, the FDST presented in Section 3.2 cannot be employed to represent the assessment provided by Expert 3, which involves interval-valued belief degrees. In the rest, the fuzzy interval grade and interval-valued belief degree (IGIB) structure, which allows the representation of both patterns of interval uncertainty, is introduced. In order to combine several pieces of evidence represented on the IGIB structures, the extended FDST that is based on an optimization approach is proposed. Finally, the preference relation approach for local monitoring and state classification in the proposed framework is presented. 3.4.1 Fuzzy interval grade and interval-valued belief degree (IGIB) structure In practical data fusion problems the performance of the fusion algorithm depends on the proper modeling of expert knowledge and uncertainties involved. Although the classic fuzzy belief structure as in Equation (3.10) considers the vagueness in grades, it is still restricted to the condition that the expert can only assign the belief degrees to individual grades. The individual grades are represented by predefined fuzzy subsets of the frame of discernment.  71 The fuzzy interval grade and interval-valued belief (IGIB) structure can be used to represent ambiguity, local and global ignorance, interval-valued belief degree, and vagueness in the evaluation grades within a unified framework. The development of the fuzzy IGIB structure is based on (a) building a seed list of continuous fuzzy evaluation grades of ,?  (b) extending the seed list to build interval fuzzy evaluation grades, and (c) extending the belief structure to an interval setting, and (d) generalizing the compatibility relation in the DST. In the rest, for simplicity in language the fuzzy IGIB structure is referred to as the IGIB structure. Definition 3.1. Assume a list of finite numbers of subsets of the frame of discernment ?  with an ordering relationship among the subsets is defined such that  ? ={Ai| i = 1,...,n, A1? A2? ...? An} where the ordering relationship between two subsets is defined as  X ,Y ? ?, X ? Y iff ti?X ? ?tj?Y , ti< tj.  Each element on this list is called an evaluation grade that can be defined through a crisp or a fuzzy subset of  ?. A crisp evaluation grade is a special case of the fuzzy grade where the membership of all of its elements are 1. If evaluation grades are defined based on specific fuzzy subsets of ?  the ordering relationship between the fuzzy sets can be used for simpler mathematical notations. A finite list of fuzzy subsets of ?  as  ? ={?Ai| i = 1,...,n,?A1??A2? ...??An} is generated. Each element on this list is called a fuzzy evaluation grade. The evaluation analysis model is defined to represent uncertain subjective judgments using evaluation grades in a hierarchical structure of attributes. For a simple hierarchical structure of two levels with a general attribute at the top denoted by y  and a set of basic attributes as { | 1,..., }.ie i l=  Let the set of importance weights of basic attributes be defined as  72 { | 1,..., }.i i l? =  Given all  evaluation grades on the list  ? ={Ai| i = 1,...,n, A1? A2? ...? An} used for assessing each attribute, a distributed assessment for ie  can be mathematically represented through an assignment  ?(ek) := (?,?k) ={( Ai,?i,k) | i = 1,...,n}, for k = 1,...,l,  where ,i k?  denotes a belief degree such that ,11.ni ki?==?  k?  is the list of all belief degrees associated with the elements on the list .?  The fuzzy evaluation analysis model is defined if ,i k? are assigned to fuzzy evaluation grades which are on the list  ? ={?Ai| i = 1,...,n,?A1??A2? ...??An}. Although this ordering relationship can be defined for any convex fuzzy subsets of  ?,without loss of generality, we present the algorithm for fuzzy triangular and trapezoidal fuzzy sets. These fuzzy sets satisfy the conditions of being both convex and normal. This assumption let us represent the ordering relationship between the elements on the seed list with simpler mathematical notations. In order to model interval-valued belief degree and local ignorance, the IGIB structure is proposed through a multi-valued mapping   ?m :{?Aij??? ,i ? j}?{[mij?,mij+] | mij?,mij+?[0,1],mij?? mij+}, i = 1,...,n, j = i,...,n, (3.17) such that (a) mij?j=in??1i=1n? and (b) mij+j=in??1i=1n?.  Therefore, it is assured that there exists a true belief structure constrained by interval-valued belief structure .m?   ?Aij denotes the fuzzy evaluation grades represented on an extended list ??  defined as  73  ?? =?A11?A12... ...?A1n?A22?A23... ...?A2n... ... ... ...?Ajj...?Ajn... ...?Ann????????????????????.  (3.18) The interval-valued belief degree  [mij?, mij+] denotes the delimited cardinality of the belief degree assigned to the fuzzy evaluation grade  ?Aij. The belief degree assigned to the interval grade  ?Aij, i = 1,...,n, j = i +1,...,n,  represents the local ignorance in assigning the belief degree to the individual fuzzy grades  ?Aii  to  ?Ajj.  With the assumption of trapezoidal fuzzy subsets of  ?,  individual and interval grades are represented in Figure 3.1. The interval grades were initially suggested for modeling local ignorance by Xu et al. (2006) and were extended to the fuzzy interval grades by Guo et al. (2008; 2009). The approach in Xu et al. (2006) cannot model the interval-valued belief degrees and vagueness of the evaluation grades. The formulation in Equation (3.17) compared to Xu et al. (2006) and Guo et al. (2009) is more flexible for knowledge representation since it can model interval-valued belief degrees and vagueness in the grades. Besides, in Equation (3.17) the vagueness pattern of the grades for different pieces of evidence can be arbitrarily defined, while the approach in Guo et al. (2009) is based on a very restrictive assumption that requires the fuzzy grades under all pieces of evidence be similar.  74  Figure 3.1 Representation of (a) individual grade sets, and (b) the corresponding individual and interval evaluation grade in the IGIB structure In the proposed fuzzy IGIB structure, interval-valued belief degree models the nonspecific cardinality of the belief degree. Whenever associated with an individual grad set, interval-valued belief degree represents the nonspecific belief degree assignment to a predefined fuzzy subset of the frame of discernment while when it is associated with an interval grade set, it affirms the unknown cardinality of the local ignorance parameter. Several researchers have tried different belief structures to model interval-valued belief degree assignments to the crisp propositions (Den?ux, 1999; 2000; Yager, 2001; Wang et al., 2006b; 2007). The interested reader can find more details and the limitations of some of the previous approaches in Wang et al. (2007). The approach presented in this chapter is the extension of Wang?s optimality approach to the FDST for combining the IGIB structures. Definitions 3.2 and 3.3 which will follow are the generalized forms of the representations in Wang et al. (2007). Definition 3.2. Let  A ={?Aij| i = 1,...,n j = i,...,n} be the extended list of individual and interval fuzzy evaluation grades and {[mij?,mij+], i = 1,...,n, j = i,...,n} be the set of interval-valued belief  75 degrees assigned to the corresponding elements of  A  which satisfy the following conditions: 0 1,ij ijm m? +? ? ?  ,ij ij ijm m m? +? ?  11n niji j im?= =???    and  11.n niji j im+= =???  (3.19) (3.20) (3.21) The IGIB structure is said to be valid if Equation (3.21) is satisfied. A valid IGIB structure ensures that a set of belief assignments are found which satisfy  mij(?Aij) = 1j=in?i=1n?, mij(?Aij)?[mij?,mij+]}.  (3.22) Definition 3.3. Let  m  represents a valid fuzzy IGIB structure. The IGIB structure is said to be normalized if the following conditions are satisfied: ( )11, {1,..., }, { ,..., },n nst ij ijs t sm m m i n j i n+ + ?= =? ? ? ? ?? ?  ( )11, {1,..., }, { ,..., }.n nst ij ijs t sm m m i n j i n? + ?= =+ ? ? ? ?? ?  (3.23)  (3.24) If these conditions are not met then the structure is non-normalized. A non-normalized IGIB structure means that some of the assigned interval-valued belief degrees are too wide to be reached (Wang et al., 2007). Thus, they need to be modified before the evidence combination.  76 By considering that  mij(?Aij) = 1j=in?i=1n?, the belief degree assigned to each fuzzy proposition is derived as  mij(?Aij) = 1j=in?i=1n?, mij(?Aij)?[mij?,mij+]}.  (3.25) Considering the condition in Equation (3.20) in Definition 3.2 and Equation (3.22), one obtains ( ) ( , ) ( ) ( , )1 , 1 .ij ij ijs i t s t j s i t s t jm m m+ ?? ? ? ? ? ?? ?? ?? ? ?? ?? ?? ? ? ?  (3.26) As  mij should also satisfy  mij?? mij(?Aij) ? mij+ , it is implied that ( ) ( , ) ( ) ( , )max (1 ), , min (1 ), .ij ij ij ij ijs i t s t j s i t s t jm m m m m+ ? ? +? ? ? ? ? ?? ?? ? ? ?? ?? ? ? ?? ? ?? ?? ? ? ?? ? ? ?? ?? ? ? ?  (3.27) It is essential that the interval-valued belief degree assigned to each fuzzy subset of the frame of discernment to be normalized using Equation (3.27) before the evidence combination. Example 3.2- Consider the simplified water quality assessment case in Example 3.1. Assume that microbial water quality is represented on an IGIB structure with more linguistic labels as {VL}, {L}, {M}, {H}, {VH} associated with the fuzzy grades defining the water quality as 1 11 1({ }) [0.1,0.6], ({ }) [0.3,0.5],({ }) [0.4,0.7], ({ }) [0,0.1].m VL m L Mm M H m VH= ? =? = =  This is a valid IGIB structure, but since the Equation (3.24) in Definition 3.3 is not met, the  77 IGIB structure is non-normalized. The structure can be normalized using Equation (3.27). The normalized structure is 1 11 1({ }) [0.1,0.2], ({ }) [0.3,0.4],({ }) 0.4, ({ }) 0.m VL m L Mm M H m VH? ? ?? = =  Example 3.3- Assume that three individual fuzzy subsets of the frame of discernment are as depicted in Figure 3.2. Three triangular and trapezoidal continuous fuzzy sets represent these individual fuzzy subsets (These sets are denoted by  ?A11,?A22,  and  ?A33 ). All sets of possible interval fuzzy subsets of the frame of discernment are shown by  ?A12,?A13,  and  ?A23 . The IGIB structure is defined as  ?A:={(?A11,[0.1, 0.15]),(?A12,[0.05, 0.2]),(?A13,[0.05, 0.1]),(?A22,[0.2, 0.3]),(?A23,[0.1, 0.2]),(?A33,[0.4, 0.5])}. The belief degree assigned to  ?A13denotes the global ignorance parameter the same as the original FDST (Yen, 1990). The belief degrees assigned to  ?A12  and  ?A13  represent the local ignorance parameters. It is notable that the summation of lower bounds of the interval-valued belief degrees associated with all individual and interval grades is smaller than 1 while the same summation for the upper bound is greater than 1 (i.e., Equation (3.21) is satisfied and thus the structure is valid). 3.4.2 Extended FDST for combining evidence with fuzzy IGIB structure The FDST can combine only the precise belief degrees in a classic fuzzy belief structure and needs to be modified to handle the interval-valued belief degrees and local ignorance. In order to combine the evidence from different sources presented on the IGIB structure, an optimality approach on FDST is developed.  78 0 0.2 0.4 0.6 0.8 100.510 0.2 0.4 0.6 0.8 100.511.5 (A13,[0.05,0.1])(A33,[0.4,0.5])(A11,[0.1,0.15])(A23,[0.1,0.2]) (A12,[0.05,0.2])(A22,[0.2,0.3]) Figure 3.2 The fuzzy IGIB structure with their associated interval-valued belief degree for both individual and interval fuzzy subsets of the frame of discernment Provided with the definition of interval grades in Definition 3.1 on the frame of discernment, FDST can handle the interval grade fuzzy sets with precise belief degrees. The belief and plausibility of a fuzzy hypothesis  ?H  are found by solving  min / max mij(uk:?Aij)j?i?i?uk?U?H?  (3.28) with the constraints  m(uk:?Aij) ? 0, i ={1,...,n}, j ={i,...,n},m(uk:?Aij) = 0, for uk?U?Aij,m(uk:?Aij)uk?= mij(?Aij), i ={1,...,n}, j ={i,...,n}. (3.29) (3.30) (3.31) For an IGIB structure with specific belief degrees, the belief and plausibility of the fuzzy hypothesis  ?H  are defined as  79  Bel(?H ) = m*(?H :?Aij)j=in?i=1n?= m(?Aij) (?l??l?1) infu?U?Aij?l(u)?l??Aij?   Pls(?H ) = m*(?H :?Aij)j=in?i=1n?= m(?Aij) (?l??l?1) supu?U?Aij?l(u)?l??Aij? (3.32)  (3.33) where  m*  and *m  functions are defined through Equations (3.12) and (3.13). In contrast to the IGIB structure with specific belief degrees, nonlinear optimization with extra constraints should be formulated to calculate the belief and plausibility of a fuzzy hypothesis on a set of general IGIB structures. However, as the belief degrees assigned to the fuzzy subsets of the frame of discernment are not independent they cannot be optimized separately to reach a global optimum. In other words, different possible point estimates of interval-valued belief degrees can contribute to the final belief and plausibility of a fuzzy hypothesis; thus, the final results are non-specific. The set of functions of the form of Equation (3.28) can be extended to a larger nonlinear optimization problem with additional constrained optimization variables (i.e., interval-valued belief degrees) used in the extended FDST in the rest. The proposed extended FDST is consistent with the optimization model addressed by Wang et al. (2006a; 2007) in postponing the optimization procedure after the combination and normalization of all bodies of evidence (i.e. IGIB structures). This procedure is followed in order to incorporate the true interval-valued belief degrees associated with all normalized fuzzy propositions (i.e., individual, interval, and intersection fuzzy subsets) in the optimization model. Besides, the normalization step of the fuzzy propositions in FDST can be postponed to the final step prior to optimization without changing the combination results  80 (Yen, 1990). The subnormal fuzzy subsets are normalized in two steps: 1) The membership function of subnormal fuzzy intersection subset  ?Apq?st  is scaled up by  k = max??Apq?st(u). 2) The interval-valued belief degree is modified by  ?mpq?st(?Apq?st) ?1k[mpq?st?,mpq?st+].   The minimum and maximum values of belief or plausibility of a fuzzy hypothesis  ?H  are found by solving the following optimization problem for all fuzzy subsets  ?Aij  of the frame of discernment  min / max mij(uk:?Aij)j?i?i?uk?U?H?  (3.34) with the constraints  m(uk:?Aij) ? 0, i ={1,...,n}, j ={i,...,n},m(uk:?Aij) = 0, for u ?U?Aij,m(uk:?Aij)uk?= mij(?Aij), i ={1,...,n}, j ={i,...,n},mij?? m(?Aij) ? mij+, i ={1,...,n}, j ={i,...,n},m(?Aij)j?i?i?= 1 (3.35) (3.36) (3.37) (3.38) (3.39) where two last constraints are extra compared to (3.28) optimization and are due to the interval-valued belief degree assignment in the IGIB structure. Solving (3.32) gives the lower and upper bounds of belief as Bel(?H )?[Bel?(?H ), Bel+(?H )] , while solving (3.33) gives the  81 lower and upper bounds of plausibility as  Pls(?H )?[Pls?(?H ), Pls+(?H )].  The bounds of belief and plausibility are related with each other by  Pls?(?H ) = 1? Bel+(?Hc)Pls+(?H ) = 1? Bel?(?Hc) (3.40) (3.41) where  ?Hc  denotes the complement of the fuzzy hypothesis  ?H .  Example 3.4- In order to illustrate the proposed procedure, consider the following IGIB structures: 1 11 12 22 2({ }) [0.05,0.09], ({ }) [0.3,0.35],({ }) [0.4,0.5], ({ }) [0,0.1],({ }) [0.15,0.25], ({ }) [0.35,0.55],({ }) [0.1,0.2], ({ }) [0,0.05],m VL m L Mm M H m VHm VL m L Mm H m VH? ? ?? ? ?? ? ?? ? where 1m  and 2m  are two independent belief structures on microbial water quality proposed by two experts. Both structures are valid and non-normalized. As a result, normalization using Equation (3.27) is necessary. The set of individual grades in each IGIB structure is assumed to be triangular fuzzy subsets of the frame of discernment and only two adjacent individual grades are allowed to intersect. These two sets individual fuzzy grades are shown in Figure 3.3.The results of combination using the extended FDST are presented in Table 3.1. Note that im  denotes the normalized interval-valued belief degree of the  ith  IGIB structure. This example can be easily extended to the case of multiple IGIB structures to find the belief and plausibility of any arbitrary fuzzy hypothesis.  82 0 0.25 0.5 0.75 101Membership0 0.25 0.5 0.75 101Universe of DiscourseMembershipVL L-M M-H VHVHHL-MVL Figure 3.3 The membership functions for two assessment grades in Example 2 Table 3.1 The IGIB structure probability distribution and its corresponding belief and plausibility for Example 3.4 G  1( )m G  2( )m G  1 2 ( )mBel G?  1 2 ( )mPls G?  VL [0.05, 0.09] [0.2, 0.25] [0.025, 0.044] [0.200, 0.251] L 0 0 [0.018, 0.026] [0.512, 0.573] M 0 0 [0.200, 0.229] [0.738, 0.796] H 0 [0.15, 0.2] [0.072, 0.102] [0.487, 0.537] VH [0.06, 0.10] [0, 0.05] [0.002, 0.013] [0.075, 0.107] LM [0.31, 0.35] [0.5, 0.55] [0.364, 0.400] [0.823, 0.865] MH  [0.46, 0.50] 0 [0.303, 0.342] [0.853, 0.905] 3.4.3 Preference ranking using the interval-valued assessments The presented fuzzy evidential reasoning framework provides the distributed belief and plausibility assessments on the frame of discernment. The decision maker?s goal is to report a hypothesis as the most appropriate representative of the system condition level. This problem  83 is handled by introducing a method for ranking the intervals of belief (Bel) and plausibility (Pls) associated with the hypotheses. The distributed aggregation results can be represented by cumulative Bel and Pls functions. A more refined assessment can be obtained by refining the universe of discourse. The theory for this approach has been proved for DST. The detailed formalism for defining the cumulative Bel and Pls functions was provided by D?motier et al. (2006). The extension of their approach for the FDST will follow. The refinement of universe of discourse is denoted by  Ur with the discrete refinement mapping  d  as 1({ }) [ , )k k kd u ? ??=  (3.42) where d generates the real intervals as the propositions of rUm . In the rest, the same idea is extended for representing the cumulative Bel and Pls functions of interval belief and plausibility of fuzzy hypotheses. The cumulative belief and plausibility of a fuzzy focal element are defined by  CBelUr(u) = mU(?Ai){?Ai??|sup d (?Ai)?u}?,  CPlsUr(u) = mU(?Ai){?Ai??|inf d (?Ai)?u}?.  (3.43)  (3.44) The upper bound and lower bounds of cumulative Bel and Pls functions will be computed by the extended FDST as addressed in Section 3.3.2 for the above definition of cumulative Bel and Pls functions.  84 Decision making at the credal level can be performed using the Bel and Pls functions on U  or rU  using additive utility assumption. Once the belief and plausibility intervals of the target hypotheses are obtained decision making about the system state can be performed using one of the following rules: (a) Absolute maximum belief rule: this rule chooses the hypothesis with maximum average belief provided that its difference with the next largest average belief is less than average belief uncertainty of both hypotheses. (b) Absolute maximum plausibility rule: this rule chooses the hypothesis with maximum average plausibility provided that its difference with the next largest average plausibility is less than average plausibility uncertainty of both hypotheses. (c) Absolute maximum belief and plausibility rule: only the hypothesis is chosen when both (a) and (b) suggest the same hypothesis. No doubt, using different rules can result in different decision making results. This case is more probable when local ignorance and interval-valued belief degrees are considerable. For the general case, the rule (c) seems to provide the most comprehensive preference relation at the credal level. The definition for degree of preference for two hypotheses (Liu et al., 2005; Wang et al., 2005; Hua et al., 2008) is extended for rule (c) when dealing with interval belief and interval plausibility assessments. Definition 3.4. The minimum and maximum possible degrees of preference of  ?Hi over  ?Hj  are defined as  P?Hi??Hjmin= max 0,max[0, Pls?(?Hi) ? Bel+(?Hj)]? max[0, Bel?(?Hi) ? Pls+(?Hj)][Pls?(?Hi) ? Bel?(?Hi)]+ [Pls+(?Hj) ? Bel+(?Hj)]??????,  (3.45) and  85  P?Hi??Hjmax= min 1,max[0, Pls+(?Hi) ? Bel?(?Hj)]? max[0, Bel+(?Hi) ? Pls?(?Hj)][Pls+(?Hi) ? Bel+(?Hi)]+ [Pls?(?Hj) ? Bel?(?Hj)]??????. (3.46) The following rules for preference relation between two possible hypotheses of the frame of discernment are applied: 1) Hypothesis  ?Hi is absolutely superior to hypothesis  ?Hj  (i.e. ?H i ? ?H j ) if  P?Hi??Hjmin> 0.5.  2) Hypothesis  ?Hi is absolutely inferior to hypothesis  ?Hj  (i.e. ?Hi??Hj ) if  P?Hi??Hjmax< 0.5.  3) Hypothesis  ?Hi is indifferent to hypothesis  ?Hj  if  P?Hi??Hjmin= P?Hi??Hjmax= 0.5.  4) The preference relation of  ?Hi and  ?Hj is not fully determined if  P?Hi??Hjmin< 0.5  and  P?Hi??Hjmax> 0.5.  Applying these rules on the all sets of defined hypothesis will determine the most likely quality level on the universe of discourse. 3.5 Extra uncertainty patterns in belief degree assignment Fuzzy sets have been previously used for belief degree assignment (Sadiq & Rodriguez; 2005). When a precise belief degree assignment cannot be justified, the generalized fuzzy sets can be used to represent the non-specific belief degrees. Generalized fuzzy sets mainly refer to two paradigms which both were extended for handling higher types of fuzzy information and bi-lattices (Cornelis et al., 2003), namely interval-valued fuzzy sets (Gorzaczany, 1987; Turksen, 1986; Dubois & Prade, 2005), and intuitionistic fuzzy sets (Atanasov, 1989). In our perspective, both presentations of the degree of certainty in the generalized fuzzy sets are helpful as they convey more local information for the belief degree assignment. In  86 essence, interval uncertainty is a special case of generalized fuzzy sets. Several reasoning methods based on generalized fuzzy sets have been developed (Bustince, 2000; Cornelis et al., 2004). Two membership functions known as true and false membership functions for a proposition are defined.  t?Aij(uk)  denotes the true membership function and  f?Aij(uk) refers to the false membership function. The precise membership of  uk is in the subinterval  [t?Aij(uk),1? f?Aij(uk)].  Here, instead of incorporating the generalized fuzzy set arithmetic into our approach we propose a procedure to transform the imprecise information presented on generalized fuzzy sets to the distributed interval-valued belief degree assignments in the IGIB structure. It is noted that the fuzzy propositions representing the vagueness between different quality grades can possess a complete different membership pattern from those defining the belief degrees. However, here we assume that the pattern of belief degrees and the vagueness in the grades can be represented in a unified framework using generalized fuzzy sets. The underlying assumption behind this formulation is that the upper and lower bounds of membership in a generalized fuzzy set are related to the interval-valued belief degrees while the general membership pattern of fuzzy propositions will represent the vagueness pattern among different quality grades. The interval-valued belief degrees are defined based on the following calculations. For the general case of fuzzy evidence  ?eqthe lower belief degree is found by  87  mij?=?( ?eq,t?Aij)12[?( ?eq,t?Aij)+?( ?eq,1? f?Aij)]j=inq?i=1nq? (3.47) where  t?Aij and  1? f?Aijare the lower and upper bounds of memberships of generalized fuzzy sets. Similarly, the upper belief degree is found by  mij+=?( ?eq,1? f?Aij)12[?( ?eq,t?Aij)+?( ?eq,1? f?Aij)]j=inq?i=1nq?. (3.48) In both Equations (3.47) and (3.48), ?  is the matching function of the fuzzy incoming evidence defined as  ?( ?eq,??Aij) = maxu[min(?eq(u),??Aij(u))]  (3.49) where ?  can be  t?Aij or  1? f?Aij.  This function can have other structures which will result in different matching degrees (Zimmermann & Zysno, 1980; Zimmermann, 2001). The use of this transformation for interval-valued belief degrees is shown by an example at the end of this section. 3.5.1 Extra uncertainty In order to provide a comprehensive knowledge representation model, extra patterns of uncertainty such as incompleteness and partially credible input information are considered in the proposed extended FDST framework. Incomplete inputs: The input features of a DSS can be incomplete as the data provided to the expert is itself incomplete. As an example, they can be the readings of the unreliable sensors. If any of the input features is incomplete then the interval-valued belief degree of  88  ?Aij  defined by Equations (3.47) and (3.48) will be modified as  mij(?Aij) ?[?qmij?(?Aij),?qmij+(?Aij)]where  ?q  is the measure of incompleteness for the input feature corresponding to the quality indicator .q  q?  can have a value between 0 and 1 ( ?q= 1 means that the data is complete for feature q ). Credibility factor: The weights of different quality indicators can be assigned based on the expert opinion on the relative importance of them. These weights are subjective and context dependent. The credibility factor for each indicator was suggested by Yager (2004) for the first time. Alternatively, the credibility factors can be tuned using a training algorithm based on objective data. Assume that there are  p  indicators and  p  weights assigned to them denoted by ?q, q = 1, As suggested by Tesfamariam et al. (2009), credibility factors can be assigned in a way that the weight of the most significant indicator is 1 and other indicators take the relative weights based on that. Assume that the IGIB structure by assigning the interval-valued belief degrees using Equations (3.47)-(3.49) is normalized. The upper and lower bounds of the belief degree are modified by applying the credibility factor as ( )( )1( ) ., { , | 0},1( ) .qqq qij q q ij iji j iijq qij q q ij iji j im m mni j mm m mn??? ?? ?? ?? ?? ? ?? ++ + +?? ? ??? ? ?= +? ? ??? ? ?? ? ? ?? ? ?? ? ? ?? = + ? ?? ? ? ?? ? ?????? (3.50) where n?  is the total number of individual and interval fuzzy grades whose upper bound of belief degree calculated in (3.48) is greater than zero. In other words, Equation (3.50) allocates the discounted degree of belief to all grades that originally have a non-zero belief  89 degree. The IGIB structure obtained through (3.47)-(3.50) for system attribute q  is shown as  ?q:={(?Aij,[(mij?)(?q), (mij+)(?q)]) |?Aij??? }, i = 1,...,nq, j = i,...,nq.  (3.51) Example 3.5- Assume that generalized fuzzy sets for assigning the interval-valued belief degree to microbial water quality grades are elicited from two experts. The generalized fuzzy sets are shown in Figure 3.4. The fuzzy water quality grades are the same as Example 3.4. The interval-valued belief degrees are assigned based on (3.47)-(3.50). For instance, if the observed evidence is a point estimate whose corresponding value on the universe of discourse is 0.6, the interval-valued belief degree assignments by two experts are 1 11 12 22 2({ }) 0, ({ }) [0.371, 0.391],({ }) [0.586, 0.651], ({ }) 0,({ }) 0, ({ }) [0.557,0.619],({ }) [0.412, 0.412], ({ }) 0.m VL m L Mm M H m VHm VL m L Mm H m VH= ? ?? ? == ? ?? =  In this example, it is assumed that the input data is complete and the credibility of the experts is the same, thus the above belief degrees remain the same. However, the IGIB structures presented above are non-normalized and need to be normalized using Equation (3.27). It is notable that after normalizing the second IGIB structure there will be only one specific value for the belief degree. The results of combination for a point estimate identical to 0.6 on the universe of discourse as presented in Figure 3.4 are presented in Table 3.2.   90 0 0.25 0.5 0.75 101Membership0 0.25 0.5 0.75 101Universe of DiscourseMembershipM-H VHVLH VHL-MVLL-M Figure 3.4 The generalized fuzzy sets for defining the interval-valued belief degrees provided by two experts in Example 3.5 Table 3.2 The IGIB structure belief degree distribution derived from the generalized fuzzy sets of Example 3.4 and their corresponding belief and plausibility for the point estimate ?0.6? on the universe of discourse G  1( )m G  2( )m G  1 2 ( )mBel G?  1 2 ( )mPls G?  VL 0 0 0 [0.114, 0.120] L 0 0 0 [0.430, 0.438] M 0 0 [0.225, 0.230] [0.832, 0.835] H 0 0 [0.164, 0.167] [0.487, 0.537] VH 0 [0.410,0.412] 0 [0.075, 0.107] LM [0.371,0.391] [0.585,0.588] [0.353, 0.356] [0.823, 0.865] MH [0.609,0.629] 0 [0.430, 0.435] 1  3.6 Local water quality assessment for Quebec City main distribution network In this section, an application of the proposed extended FDST for simultaneous use of fuzzy  91 interval grade and interval-valued belief degree (IGIB) structures is investigated. Specifically, the capabilities of the proposed approach are demonstrated using a problem of drinking water quality monitoring in a distribution system which has been previously addressed in other studies using evidential reasoning techniques (Sadiq & Rodriguez, 2005; Sadiq et al., 2006). The application suffers from both nonspecificity and local ignorance in communicating the water quality. Also, vagueness in water quality levels is a characteristic property of water quality assessment. The qualitative terms or continuous functions for characterizing water quality have been studied previously and there is a wealth of literature in this area (Swamee & Tyagi, 2000; Sadiq et al., 2010). Different parameters are suggested which can be used to assess the quality of drinking water. Comprehensive descriptions and state of the art of different multi-parameter water early warning systems (EWS) for the detection of drinking water quality degradation are presented in two USEPA reports (USEPA, 2005; 2009). Different technologies to measure microbial contaminants and chemical contaminants are presented in both reports. One set of indicators is specifically used to represent the microbial contaminants of drinking water. Another set of indicators represents the physicochemical properties of the drinking water. The use of physicochemical parameters is more common in an online monitoring system (Hall et al, 2007; Yang et al., 2008) as they can be measured much easier than the microbial measures. However, the degree of reliance on microbial measures, if available, is higher as they provide more insightful information about the water quality. Immediate potential public health consequences can result from the consumption of water contaminated by pathogenic bacteria. The purpose of this section is to design and test an expert system for fusion of all the data acquired from the multiple incommensurate water  92 quality parameters (WQP) at local sampling units. We focus on the parameters which demonstrate an ordered relationship between the estimated water quality levels and the parameter values as addressed by Francisque et al. (2009a). The set of two indicators including heterotrophic plate count (HPC) bacteria and water turbidity (TUR) are selected as microbial measures. An increase in water TUR can provide a surrogate for the possible microbial contamination (Payment et al., 2003; USEPA, 2004). Moreover, two physicochemical parameters including water temperature (Temp) and total trihalomethanes (TTHM) are considered. The experimental data between the years 2003 and 2005 for fifty-two sampling locations in the Quebec City?s main WDN, are available (Francisque et al., 2009a). The average value and the standard deviation of parameters in a season at local sampling locations will be used to test the proposed algorithm. Two scenarios of expert knowledge elicitation will be considered. These scenarios are provided to emphasize the utility of the proposed IGIB structure. The vagueness in quality grades and nonspecific assessments based on single parameters are elicited from the expert using generalized fuzzy sets. The true and false memberships for assigning the interval-valued belief degree associated with quality grades very low ({VL}), ?low {L}?, ?medium {M}?, ?high {H}?, and ?very high {VH}? are defined based on the expert knowledge about the Quebec City?s main WDN and the literature on surrogate WQPs available in the dataset (See Chapter 2 for indications of each WQP). Scenario 1: interval-valued belief degree assignment In Scenario 1, the expert knowledge is represented through eliciting the true and false  93 memberships of generalized fuzzy sets that represent the belief degree assignment based on the value of four WQPs, i.e. TUR, HPC, TTHM and Temp. The common characteristic of these measures is that their value can be consistently related to the water quality level as long as they are independently evaluated. For all of the above WQPs, the lower the value the better the water quality. It is not a common practice for experts to represent their knowledge on ordered generalized fuzzy sets of the WQPs. However, the presented approach helps in modifying the regulations for water quality assessment in the WDN. In Scenario 1, assume that the expert cannot provide specific belief degree assignments to the defined grades. This scenario can be modeled using generalized fuzzy sets representing the interval-valued belief degrees and vagueness among different grades as depicted in Figure 3.5. Expert will assign the upper and lower memberships are assigned since he is uncertain about the belief degree associated with the individual quality grades given a value for each WQP. The interval-valued belief degrees are calculated using Equations (3.47)-(3.49). However, deciding about the local water quality levels just by looking at a single parameter cannot be a complete and totally credible assessment for decision making. Thus, the credibility of the quality assessments using each parameter must be considered. The suggested credibility factors for aforementioned WQPs are calculated using analytic hierarchy process (AHP) as (Francisque et al., 2009a) 1, 0.33,0.083, 0.17.Turb HPCTTHM Temp? ?? ?= == =  Here, the details of calculations for one sampling location are provided. The mean values of all WQPs for the sampling point ?QC317? in the WDN are considered. It is assumed that all the inputs are complete ( 1, 1,..., 4q q? = = ).  94 0 0.2 0.4 0.6 0.8 1 1.201Turbidity (NTU)Membership 0 100 200 300 400 50001HPC (cfu/ml)MembershipVH H M L VLVLLVH MH (a) 0 20 40 60 80 100 12001TTHM (microg/l)M e m b e r s h i p0 5 10 15 20 2501Temp (degree C)M e m b e r s h i p VH M L VLH M L VLHVH (b) Figure 3.5 Generalized fuzzy sets representing water quality grades: (a) for microbial parameters, (b) for physicochemical parameters  95 The average value of parameters for sampling point ?QC317? in the distribution network during a season are found as2 TUR= 0.49 NTU, HPC= 39.29 Cfu/mL, TTHM= 34.65 ?g/L, Temp= 11.25 ?C. The credibility factors are used to modify the interval-valued belief degrees for microbial parameters as ({ }) 0, ({ }) [0.132,0.135],({ }) 0, ({ }) [0.132, 0.135],({ }) [0.05, 0.053], ({ }) [0.133, 0.135],({ }) [0.899, 0.999], ({ }) [0.285, 0.291],({ }) 0, ({ }) [0Turb HPCTurb HPCTurb HPCTurb HPCTurb HPCm VL m VLm L m Lm M m Mm H m Hm VH m VH= ?= ?? ?? ?= ? .309, 0.315], and for physicochemical parameters as ({ }) [0.182,0.186], ({ }) [0.161,0.169],({ }) [0.182,0.186], ({ }) [0.161,0.169],({ }) [0.194,0.199], ({ }) [0.331,0.348],({ }) [0.248,0.254], ({ }) [0.161,0.169TTHM TempTTHM TempTTHM TempTTHM Tempm VL m VLm L m Lm M m Mm H m H? ?? ?? ?? ? ],({ }) [0.182,0.186], ({ }) [0.161,0.169].TTHM Tempm VH m VH? ?  The expert defines the fuzzy hypotheses representing the aggregated quality as in Figure 3.6. The resulting belief and plausibility intervals through combining the IGIB structures using the extended FDST are shown in Table 3.3. The ranking is performed through applying the preference relation presented in Section 3.3.3 to the all defined hypotheses. For instance, the minimum and maximum values of the preference degree of {L} and {M} grades are                                                 2 NTU: Nephelometric turbidity unit, Cfu: colony forming units  96 calculated as  P{ M }?{L}min= max[0,(0.785? 0.175)? (0.468? 0.329)(0.785? 0.468)+ (0.392? 0.175)]= 0.882,P{ M }?{L}max= min[1,(0.798? 0.163)? (0.480? 0.370)(0.798? 0.480)+ (0.370? 0.163)]= 1.  0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 101Universe of DiscourseM e m b e r s h i pVH H M L VL Figure 3.6 Final hypotheses representing the water quality levels based on aggregated water quality parameters Table 3.3 The belief and plausibility intervals and maximum uncertainty of water quality levels for sampling point ?QC317? in the main distribution network G  ( )WQBel G  ( )WQPls G  max ( )WQU G?  Rank VL [0.004,0.0045] [0.0256,0.0271] 0.0231 4 L [0.163,0.175] [0.370,0.392] 0.229 2 M [0.468,0.480] [0.785,0.798] 0.329 1 H [0.0292,0.0333] [0.130,0.142] 0.113 3 VH [0.0012,0.0015] [0.0044,0.0053] 0.0041 5 Hence, the hypothesis {M} is preferred to the hypothesis {L}. The preference relations between all pairs of hypotheses are investigated that will result in the ranking shown in Table 3.3. The upper and lower bounds of cumulative Bel and Pls functions for the refinements of universe of discourse are shown in Figure 3.7.  97 The lower and upper bounds of the cumulative Bel and Pls functions in Figure 3.7 and Table 3.3 are mainly determined through the constraints imposed by interval-valued belief degrees. Meanwhile, the maximum uncertainty defined with the range  ?UWQmax= max(Pls)? min(bel) is induced by interval-valued belief degrees, vagueness in the grades, conflict among different WQPs, and local and global ignorance. Thus, it is observed that in most quality levels propagated uncertainty due to interval-valued belief degrees is significantly smaller than uncertainty imposed by other types of uncertainty. 0 0.2 0.4 0.6 0.8 100.10.20.30.40.50.60.70.80.91Aggregated Water Qualityb e l /  m a x ( Pls ( d <  u ) )  min Plsmax Plsmin belmax bel Figure 3.7 The upper and lower bounds of cumulative Bel and Pls functions for Scenario 1 at sampling point ?QC317? Scenario 2: local ignorance and interval-valued belief degree In Scenario 2, the local ignorance in belief degree assignment and interval-valued belief degrees coexist as the expert is uncertain about the cardinality of both local ignorance and the  98 belief degree assignment to individual water quality grades. The generalized fuzzy sets as shown in Figure 3.8 are utilized to assign the interval-valued belief degree to individual and interval grades. In this case, the expert cannot justify assigning a portion of the interval-valued belief degree to the adjacent individual water quality grades but he can justify an interval-valued assignment to an interval quality grade. The interval-valued belief degrees for sampling point ?QC317? (after applying the credibility factors) are as ({ }) 0, ({ }) [0.220,0.224],({ }) 0.951, ({ }) [0.220,0.224],({ }) [0.0476,.0501], ({ }) [0.550,0.561],Turb HPCTurb HPCTurb HPCm VL L m VL Lm M m Mm H VH m H VH? = ? ?= ?? ? ? ? ({ }) [0.304,0.310], ({ }) 0.275,({ }) [0.317,0.322], ({ }) 0.450,({ }) [0.371,0.377], ({ }) 0.275.TTHM TempTTHM TempTTHM Tempm VL L m VL Lm M m Mm H VH m H VH? ? ? =? =? ? ? =   Table 3.4 The belief and plausibility intervals and the maximum uncertainty of aggregated water quality levels for sampling point ?QC317? in the main distribution network G  ( )WQBel G  ( )WQPls G  max ( )WQU G?  Rank VL [0.0031,.0032] [0.0170,0.0175] 0.0144 4 L [0.100,0.105] [0.307,0.317] 0.217 2 M [0.524,0.530] [0.860,0.864] 0.340 1 H [0.0089,0.0092] [0.135,0.139] 0.130 3 VH [0.0183,0.0194] [0.0273,0.0285] 0.0102 5   99 0 0.2 0.4 0.6 0.8 1 1.201Turbidity (NTU)M e m b e r s h i p 0 100 200 300 400 50001HPC (cfu/ml)M e m b e r s h i pH-VH M VL-LM VL-LH-VH (a) 0 20 40 60 80 100 12001TTHM (microg/l)M e m b e r s h i p0 5 10 15 20 2501Temp (degree C)M e m b e r s h i p H-VH M VL-LH-VH M VL-L (b) Figure 3.8 The generalized fuzzy sets for interval-valued belief degree assignment to individual and interval water quality grades: (a) for microbial parameters, (b) for physicochemical parameters   100 The belief and plausibility intervals, maximum uncertainty and the ranking of the hypotheses are presented in Table 3.4. The ranking of the hypotheses representing the quality grades for the communication to the operators and managers is performed as presented in Section 3.3.3. Note that using all decision making rules mentioned in Section 3.3.3 result in selecting the fuzzy quality grade ?medium {M}? since the interval uncertainty in the belief degree assignments is much lower than global ignorance, conflict, and vagueness effects. Figure 3.9 depicts the cumulative Bel and Pls functions of the water quality for the refinements of the universe of discourse. 0 0.2 0.4 0.6 0.8 100.10.20.30.40.50.60.70.80.91Aggregated Water Qualityb e l / p l s ( d < u )  min Plsmax Pls min belmax bel Figure 3.9 The upper and lower bounds of cumulative Bel and Pls functions for Scenario 2 at sampling point ?QC317? The obtained results for local information fusion in both scenarios indicate that incommensurate and multiple sources of information can be incorporated in the inference process. It is also demonstrated that the proposed local fusion framework is efficient when in  101 addition to the vagueness and global uncertainties in information, cardinality of the belief degrees assigned to quality grades and cardinality of the local ignorance are both unknown. The evidence combination using the extended FDST for multiple fuzzy IGIB structures can provide us with a more comprehensive local quality assessment that can be used for appropriate decision making or modifying the existing regulations through incorporating a wider variety of the existing uncertainties to the inference process. 3.7 Conclusions The chapter provided a unified formulation for handling two patterns of interval uncertainty in addition to vagueness in evaluation grades and ambiguity for the purpose of local information fusion. A novel fuzzy interval grade and interval-valued belief degree (IGIB) structure was presented. The aggregation was performed through the extension of the FDST using an optimization approach. Interval-valued belief degrees were used for modeling the delimited belief degree in the knowledge elicitation process. The generalized fuzzy sets and its close connection with interval uncertainty motivated the introduction of a method for interval-valued belief degree assignment using generalized fuzzy sets. Two local water quality assessment scenarios were considered to show the advantages of the proposed scheme. The first scenario included interval-valued belief degrees while the second scenario included delimited belief degrees assigned to both individual and interval grades (i.e., interval-valued local ignorance and interval-valued belief degrees assigned to the predefined grades). Both cases were applied to a water quality assessment example and were investigated through the data obtained from the Quebec City?s main WDN, available at several sampling locations for two years. The results indicated that the additional patterns of interval uncertainty represented in the proposed approach are effectively propagated  102 throughout the complex structure of distributed quality assessment. Thus, the final model for decision making based on the distributed assessment of the quality level is more comprehensive when compared to the previous models. The next chapter presents an extension of this method to incorporate WQPs that introduce inconsistency and nonlinearity to water quality evaluation and relative risk analysis processes.  103  Chapter  4: Local information fusion for risk analysis 4.1 Introduction This chapter introduces a local information fusion framework for relative risk analysis that uses an extended belief rule-based (BRB) system to model the uncertain expert knowledge, inconsistencies in the inference process, and nonlinear input-output relationship. The concept of relative risk analysis refers to providing an assessment of the risk levels without identifying the hazard associated with the severity of the consequences. Extra patterns of uncertainty that can exist during the knowledge elicitation about the perceived levels of relative risk at local strata of a distributed system are modeled using the proposed extended BRB system. In order to add a flexible sub-module for relative risk analysis to the decision support system (DSS), the fuzzy interval grade and interval-valued belief degree (IGIB) structure presented in Chapter 3 is embedded in the rule consequent of the extended BRB system. The rule combination based on the extended FDST presented in Chapter 3 and a stopping criterion for rule aggregation are employed to obtain a computationally efficient rule aggregation with interpretable results. The proposed extended BRB system is called interval grade and interval-valued belief degree rule-based (IGIB-RB) system. In a spatiotemporal monitoring problem, inputs to the relative risk analysis engine for local information fusion are observed or estimated features of the system. The output is normally an anomaly alarm or a justifiable degree of a target property of the system such as likelihood or severity of an anomaly, reliability of the overall system, or the probability of occurrence of a potential hazardous event in the system.  104 A numerical example is employed to show that the proposed IGIB-RB system can provide intuitive results when the set of input-output training data are known. The parameters of the IGIB-RB system are tuned using a training algorithm to assign the initial belief degrees to the evaluation grades in the consequent. The validation of the proposed IGIB-RB system is performed through a case study example for relative risk analysis at individual sampling locations of the water distribution network (WDN). 4.2 A novel fuzzy evidential belief rule-based system In many applications, the knowledge used to make the rule base of an inference engine is uncertain. The uncertain rules may be elicited from experts or through training based on deficient and incomplete datasets. This chapter considers the former case where the expert knowledge (i.e., subjective judgment) is incorporated into the inference engine. In this case, ignorance may occur due to the weak implications of the expert in assigning a certain relation between a set of antecedent attributes and the consequent terms of a rule (Yang et al., 2006). Generally, inference through subjective judgment of human beings is based on an implication on a few numbers of features. Most rules provided by experts are nonlinear mappings of the antecedent attributes onto the consequent basic probabilities of the events. The nonlinear mapping in a classic fuzzy rule-based system is modeled using fuzzy implications. The basic probability of the implied fuzzy subset in the consequent of a classic fuzzy rule-based system is always 1. In most scenarios that there are adequate information the rules provided by the expert have the basic belief degree of 1. However, when the inference through expert judgment based on several antecedent attributes is inconsistent, a rule-based system that can model uncertainty due to incompleteness seems necessary.  105 The detailed explanation of the individual-based fuzzy implication and the procedure to define firing degree of the rules in an IGIB-RB system follows. Similar to the classic rule-based system, antecedent attributes that indicate the inputs of the IGIB-RB system are evaluated by predefined fuzzy evaluation grades. However, the rules of the IGIB-RB system are defined as the mapping of different combinations of antecedent attributes to the IGIB structures in the consequent. First, the required definitions to explain the IGIB-RB system are introduced. Then, the detailed explanation of the individual-based fuzzy implication and the procedure to define firing degree of the rules in the IGIB-RB system follows. Definition 4.1. A function  F : [0,1]n? [0,1]n??? is called an aggregation operator if it has the following properties: a)  F(x1,...,xn) ? F( y1,..., yn)  whenever  xi? yi for all  i?{1,...,n},  b)  F(x) = x  for all  x ?[0,1],  c)  F(0,...,0) = 0,  and  F(1,...,1) = 1.  Definition 4.2. Triangular norm (t-norm) operator is a binary operation on a unit interval as  [0,1]n? [0,1]  which is commutative, associative, non-decreasing in each component, and has  1 as neutral element. Triangular norms are binary, associative, and symmetric aggregation operators with 1 as neutral element. For instance, minimum, product, Lukasiewicz, and drastic product are four basic t-norms. Definition 4.3. s-norm (t-conorm) operator is a binary operation on a unit interval,  106  [0,1]n? [0,1]  when given  (x1,...,xn)  as input, returns one minus the t-norm of  (1? x1,...,1? xn).  Similar to the classic rule-based system, different antecedent attributes that indicate the inputs of the IGIB-RB system are represented by predefined fuzzy evaluation grades. However, the rules of the IGIB-RB system are defined as the mapping of different combinations of antecedent attributes to the IGIB structures in the consequent. Since more facets of uncertainty can be modeled in the IGIB-RB system compared to the classic BRB system, the IGIB-RB system is a more flexible framework for representing the expert knowledge. The schematic of the IGIB-RB inference engine is shown in Figure 4.1. In the general case, the set of incoming fuzzy pieces of evidence  Ex={?e1,?e2,...,?eq,...,?ep}  with the associated set of membership function  ?Ex={??e1,??e2,...,??eq,...,??ep} is the input to the proposed IGIB-RB system. The proposed IGIB-RB engine uses the Mamdani?s inference procedure to determine the degree of firing of each rule. In the Mamdani?s inference system, each rule is initially fired and then the rules are aggregated under the assumption of independence. Different initial weights for each antecedent attribute and each rule are considered. In the proposed IGIB-RB system, weighting with t-norm operators is used to account for the importance of each antecedent attribute. The fuzzy composition operation on the membership of the fuzzy grades of antecedent attributes and the incoming input will result in various degrees of firing of the rules  R1, R2,..., Rk,..., RL.The degree of firing of the rule kR and its initial weight k?  are used to define the global degree of firing k? . The global degree of firing k?  will be used for either discounting the belief structures or modifying the  107 membership of the fuzzy evaluation grades in the consequent of the rule .kR  The detailed procedure of fuzzy composition and determining the global degree of firing of the rules in the proposed IGIB-RB system will be explained in Sections 4.2.1 to 4.2.3.  Figure 4.1 The schematic of the IGIB-RB inference engine It should be emphasized that rule combination and defuzzification are not followed in the same fashion of the classic fuzzy rule-based system. The consequent of each rule is an IGIB structure with subnormal fuzzy focal elements after accounting the degree of firing. The IGIB structure in the consequent of fired rules will be aggregated using the extended fuzzy Dempster-Shafer theory (FDST) developed in Chapter 3. The result of rule combination can be represented as a distributed assessment of belief, plausibility or betting commitments intervals. When it is necessary to make a decision about an output grade, the rational behavior is that some regular betting commitments are assigned to the consequent evaluation grades (Smets, 2005). The validation of the final output grade selection at the pignistic level is performed through comparison with a conventional BRB system (i.e., RIMER approach) in Section 4.4.  108 To explain the structure of IF-THEN rules in the proposed IGIB-RB system, we assume that p  antecedent attributes, { , 1,..., },=qE q p are used in the rule base and each antecedent attribute has qL  propositions { ,qfF  1,..., }.=f Lq  Assume that n individual evaluation grades on the seed list  ? ={?Ai,i = 1,...,n}are defined to represent different levels of the consequent of each rule. Interval grades are also defined as additional elements on the extended list  ?Aij? ?? , i = 1,...,n, j = i +1,...,n.  By defining r rules in the rule base of the proposed IGIB-RB system as , 1,..., ,kR k r=  the general structure of the thk  rule is  Rk: IF 1E is 1fF  and 2E is 2fF ? and pE is ,pfF  THEN the consequent is  {(?A11,[m11?,m11+]),...,(?A1n,[m1n?,m1n+]),(?A22,[m22?,m22+]),...,(?A2n,[m2n?,m2n+]),...,(?Ann,[mnn?,mnn+])},   (4.1) with initial weight of rule Rk  asv k?  and weight of feature qE  (i.e., the thq  antecedent attribute) as ?q. The rule consequent is a fuzzy IGIB structure presented in Definition 3.2. The IGIB structures in the consequent of the rules elicited from the expert may be non-normalized. The rule weights  ?k and the antecedent attribute weights  ?q are also defined by the expert or using historical data. In the case study of this chapter,  ?q weights are assigned by the expert through forming comparison matrix similar to the analytical hierarchy process (AHP) (Saaty, 1980). The initial rule weights  ?k  are assigned through a frequency-based approach by evaluating the number of times that each rule is activated during testing with the historical training dataset. In knowledge elicitation for local risk-based decision making that this chapter focuses on, the  109 flexibility of the knowledge representation framework provides a great advantage in effective uncertainty propagation (Dubois & Guyonnet, 2011). In order to show the flexibility of uncertain knowledge representation in the proposed IGIB-RB system, we compare its structure with the ?rule inference methodology using the evidential reasoning? (RIMER) approach that is a well-known BRB system (Yang et al., 2006). This approach has been used in various multi-criteria decision making (MCDM) problems such as pipeline leak detection (Zhou et al., 2009) and online monitoring of smart homes (Augusto et al., 2008). The proposed approach in this chapter is more flexible in modeling the uncertain expert knowledge in various aspects compared to the RIMER and other conventional evidential reasoning frameworks (See Table 4.1). Since the proposed method is a rule-based system more specific comparison with the RIMER approach will be provided. First, the RIMER approach does not take into account vagueness in the consequent grades. However, it is more intuitive to assume that the consequent grades are not crisp and to model them with fuzzy subsets of the frame of discernment. The proposed IGIB-RB system can be used to represent the vagueness in the consequent grades. Second, the RIMER approach only allows the precise belief degrees to be assigned to individual evaluation grades in the consequent of each rule. The lack of knowledge on belief degree assignment is only accounted by global ignorance. On the other hand, the IGIB-RB system can model both local and global ignorance. Third, the expert can provide interval-valued belief degrees in the consequent of the IGIB-RB system. These improved aspects are better illustrated in a simplified relative risk analysis example.    110 Table 4.1 An overview of different modeling features of the proposed IGIB-RB system, conventional evidential reasoning and BRB (i.e., RIMER) frameworks. Model Features Method  Parameter DST (Dempster, 1967;Shafer, 1976) ER Approach (Yang and Xu,2002a; 2002b) FDST (Yen, 1990) Extended ER Approach (Wang et al., 2007; Guo et al., 2009) RIMER Approach (Yang et al., 2006) IGIB-RB System (proposed method)     Uncertainty Types Randomness Included Included Included Included Included Included Nonspecificity Included Included Included Included Included Included Conflict Included Included Included Included Included Included Global ignorance Included Included Included Included Included Included Local ignorance Not included Not included Not included Interval grades Not included Interval grades Vagueness Not included Not included Arbitrary fuzzy subsets of ?  Specific fuzzy subsets of ?  Only in antecedent Both in antecedent and consequent Interval-valued belief degree Not included Not included Not included Included Not included Included    Structure  Evaluation analysis model  Not included  Fixed grades under all attributes  Not included  Fixed grades under all attributes  Fixed grades for all rule consequents Arbitrary grades  on extended list ?? for each rule consequent Rule based Not included Not included Not included Not included Included Included  Example 4.1 Assume that the risk associated with drinking water can be assessed based on the level of two water quality parameters (WQP), namely free residual chlorine (FRC) and turbidity (TUR), each represented by three levels as ?low {L}?, ?medium {M}?, and ?high {H}?. During the knowledge elicitation process the expert is asked to provide their judgment about the risk associated with drinking water by assigning belief degrees to three disjoint crisp  111 sets ?low {L}?, ?medium {M}?, and ?high {H}? that represent the risk levels. Assume that the measurements for FRC and TUR are {L} and {H}, respectively. The elicited rule can be represented as  1 :R IF FRC is {L} and TUR is {H} THEN the microbial risk of drinking water is {(L, 0.1), (M, 0.3), (H, 0.6)}. This consequent representation may not be necessary if there is no lack of information. However, when only incomplete and deficient information are available this type of rule can be used to express the nonspecificity in the expert judgment. This type of rule can be modeled using both the RIMER approach and the proposed IGIB-RB system (Figure 4.2.a). However, a more realistic situation is when the propositions which define the relative risk grades in the consequent are fuzzy subsets of the frame of discernment. We assume that triangular and trapezoidal fuzzy sets are used to represent the evaluation grades in the consequent. For simplicity, triangular and trapezoidal fuzzy sets are shown using their lower and upper least possible and most possible points on the universe of discourse. Thus, the triangular fuzzy evaluation grade is shown by{[ , , ] | , , , }a b c a b c a b c?? < < and the trapezoidal fuzzy evaluation grade is represented by {[ , , , ] | , , , , }.a b c d a b c d a b c d?? < < <  Assume that three fuzzy evaluation grades {L}, {M} and {H} are defined as L= [0, 0.2, 0.4], M= [0.2, 0.4, 0.8], H= [0.4, 0.8, 1]. In the meantime, the expert may not be able to assign a distinctive belief degree to {L} and {M} evaluation grades, but can still assign a belief degree to a new trapezoidal fuzzy evaluation grade ?low to medium {L-M}? defined as  L-M= [0, 0.2, 0.4, 0.8],  112 with the rule represented as (Figure 4.2.b) 1 :?R IF FRC is {L} and TUR is {H} THEN the microbial risk of drinking water is {(L-M, 0.4), (H, 0.6)}. In another case, the expert might not be certain how to assign a precise belief degree to evaluation grades {L-M} and {H}. However, he can provide his judgment using belief degrees delimited to intervals. The elicited rule can be shown as (Figure 4.2.c)  1 :??R IF FRC is {L} and TUR is {H} THEN the microbial risk of drinking water is {(L-M, [0.3, 0.55]), (H, [0.5, 0.8])}. Obviously, in the above cases the expert has an opportunity to express their less specific judgments, that is an advantage of the proposed IGIB-RB system. All these types of uncertain judgments can be accepted and represented using the proposed IGIB-RB system.  Figure 4.2 The schematic of grades and the belief degree assignment in the consequent of the Rule 1: a) the general case modeled using both the RIMER approach and the IGIB-RB system, b) the vagueness and local ignorance case modeled using the proposed IGIB-RB system, c) simultaneous existence of vagueness, local ignorance, and delimited belief degrees modeled using the proposed IGIB-RB system  113 4.2.1 Fuzzy composition using matching function Different methods for fuzzy composition can be used to determine degree of firing of the rules. Fuzzy composition using the matching function (Wang, 1997) is explained in this section. Assume that the data is complete for an incoming feature vector  Eq={?e1,...,?eq,...,?ep}.   The matching degree of a new feature vector for attribute qE  that has qL  propositions can be found through the use of ?sup-star? operation and normalization as  ?qf=sup{(??eq*?(?Af)q)}?f =1Lqsup{(??eq*?(?Af)q)} (4.2) where qf? is the matching degree that evidence is  ?eqassociated with proposition  ?A f .  The symbols * and ?  represent t-norm and t-conorm operators, respectively. Here, the minimum is chosen as the t-norm *, and the algebraic sum is selected as the t-conorm .?  For the singleton input,  eq, the supremum operator is the maximum operator acting on a set of crisp numbers. 4.2.2 Fuzzy composition using similarity measure An alternative to the matching function to perform fuzzy composition is the use of a similarity measure based on geometric and set theoretic approaches (Gottwald & Pedrycz, 1988; Chen et al., 1995). Several similarity measures are available and can be employed for fuzzy composition. More recently Deng et al. (2004; 2011) proposed a new geometric similarity measure based on the mixture of distance measure, set theoretic approach, and the radius of gyration concept from mechanics. The employment of this similarity measure for fuzzy composition improves the flexibility of the multifaceted information fusion in the BRB  114 system. Using other similarity measures is also possible and results in different fuzzy compositions. However, when the similarity measure is employed for fuzzy composition, most fuzzy grades in the antecedent have non-zero similarity degrees and thus more rules are fired. Therefore, appropriate algorithms for rule aggregation are required. The summary of this similarity measure follows in Definition 4.4. Definition 4.4. Assume that the trapezoidal fuzzy set  ?Eq  represents an incoming body of evidence and  ?Af  represents a fuzzy evaluation grade in the antecedent. The similarity measure between these two trapezoidal fuzzy sets which are schematically presented in Figure 4.3, is defined as  ?qf=?(??eq,??Af) = 1?| ei? ai|i=14?4??????????1? | ry??eq? ry??Af|???????(S??eq,S??Af)min(ru??eq,ru??Af)max(ru??eq,ru??Af) (4.3) where  S? is equal to 1 when the membership for some elements of the set is smaller than 1 and otherwise is zero and  ?(S??eq,S??Af)  is defined as  ?(S??eq,S??Af) =1, S??eq+ S??Af> 0,0, S??eq+ S??Af= 0,??????? (4.4) and  ?  will guarantee that the middle term in Equation (4.3) gets a value rather than 1, only when at least one of the  ??eq or  ??Af has a fuzzy membership. If  u  is defined as the axis of universe of discourse and  y  as the axis of membership, the radius of gyration for the fuzzy sets from mechanics is defined as  115  ru=IuA, Iu= y2A?dA,ry=IyA, Iy= u2A?dA. (4.5) (4.6)   Figure 4.3 Two trapezoidal fuzzy sets as the incoming evidence Eq and the fuzzy grade Af This similarity measure was recently adopted for belief degree assignment (Deng et al., 2011) and was further extended for more flexible belief degree assignment using a normal distribution-based method (Xu et al., 2013). In Section 4.4, the Deng?s method will be utilized to estimate the normalized betting commitment of output fuzzy evaluation grades where the input-output function is known. This extension will allow us to validate the betting commitments obtained using the proposed IGIB-RB system for decision making at the pignistic level. 4.2.3 Determining the degree of firing After finding the matching weight  ?qf based on the matching function or the similarity measure, the global matching weight is calculated based on all additional uncertainty parameters (i.e., initial rule weight and attribute weight) for any fuzzy set  ?Af  in the antecedent part of the rule  k  as  116  ?fk= ?q=1p ?q(?qf)k?qq=1p?????????????????????1??1??q=1p1??q(?qf)k?qq=1p??????????????????????  (4.7) 1( )kk fkf rlk fl? ??? ?==? (4.8) where k? is the initial rule weight, q?  is the attribute weights of all antecedent features and kf? is the modification factor of the global weight of each fuzzy set in the antecedent of rule .k  ? represents a modification factor known as ? -model to compromise between the conjunctive ( ? = 0 ) and disjunctive ( ? = 1 ) operators in aggregating the matching degrees (or similarity measures) found through fuzzy composition. Finally,  ?k= [?k1*?k2*...*?kFq]  determines the global degree of firing of the rule  k. ?  and * are both t-norms that can be different. The memberships of all referential sets of the consequent in the rule  k  are modified by  ??Aks(t) = supt??[?k??Ek?Gk(t)], i = 1,..., N , j = i,..., N .  (4.9) where the upper index s emphasizes the subnormal membership function of the evaluation grade  ?Aij. Two commonly used t-norm operators, namely minimum and product operators are considered. By applying any of these operators a set of subnormal fuzzy focal elements are obtained. Then, the subnormal fuzzy focal elements are normalized in two steps: 1) The membership function of subnormal fuzzy focal element  ?Aij is scaled up by  117  ? = max??Aijs(u).  2) The interval-valued belief degrees are modified by  ?mij(?Aij) ?1? [mij? ,mij+ ].  In order to perform the rule aggregation, any fired rules whose IGIB structures have at least a non-zero upper bound of belief degree can be a candidate for combination using the extended FDST. However, different from the classic BRB system all fired rules are not combined. A more sophisticated algorithm that will follow in the next section is used to aggregate the fired rules. 4.3 Enhanced rule aggregation algorithm Due to the use of evidential reasoning for rule aggregation, a challenge in the IGIB-RB system is to deal with the higher computational overhead compared to the classic rule-based systems. Thus, different approaches to design parsimonious inference engines can be employed to reduce the computational complexity (Zhou & Gan, 2007). Meanwhile, several researchers addressed the issues attributed to the normalization of the DST in the case of highly conflicting belief structures that obtain irrational combination results (Zadeh, 1984; 1986; Smets, 1988). These issues can also arise when the extended FDST is used to combine IGIB structures. Thus, the applicability of the extended FDST should be checked when the IGIB structures are combined. This is more important when similarity measures are employed for fuzzy composition since more rules are fired. The idea of combining all activated rules in large rule-based systems is more likely to be followed within the classical logic that presumes all belief structures are true (Yager, 1996). However, it is not the best approach to combine all fired rules. There are two main  118 drawbacks to this approach. First, combining highly conflicting rules with the currently combined rules reduces the distinctiveness among the hypotheses at the decision making level. Second, adding more rules when they are not introducing high conflict increases the computational complexity of the rule combination without making a marked change in the final distribution of betting commitments of the hypotheses. Defining a measure to assess the consistency among IGIB structures follows two main objectives. The first objective is to justify the applicability of the extended FDST for rule combination. The second objective is to define a stopping criterion for rule aggregation. A measure to assess the consistency among IGIB structures can be introduced by defining a comprehensive measure of conflict among IGIB structures. Liu (2006) addressed the idea of identifying the conflict among classic belief structures using a two-element measure of conflict. The first element is based on the distance between betting commitments and the second element is based on the belief assigned to the empty set in the combined belief structure. More recently, Fu and Yang (2011) extended the Liu?s conflict measure for applicability analysis of the DST combination rule to the interval-valued belief structures. The DST combination for the interval-valued belief structures uses the optimality approach proposed by Wang et al. (2006a; 2007). It should be noted that defining the conflict measure even for the classic belief structures is still an open problem and there are different views on defining a comprehensive and qualitatively intuitive conflict measure (Smets, 2007; Daniel, 2009; 2010). The following definitions extend the Liu?s viewpoint on conflict in classic belief functions (Liu, 2006) as well as Smets? definition of pignistic transformation (Smets, 2005). These new extensions  119 allow us to check the applicability of the extended FDST combination rule and to define a stopping criterion for the rule aggregation step in the IGIB-RB system. Definition 4.5. The pignistic probability transform for the fuzzy proposition ?H on the frame of discernment?is defined as  BetP(?H ) =?H ??Aij?Aijm(?Aij)1? m(?)?Aij???H??Aij???.  (4.10) Definition 4.6. If  ?A?lij represents an  ?l? cut of the fuzzy evaluation grade  ?Aij  in a fuzzy belief structure, each  ?l? cut is considered as a crisp proposition. Thus, the pignistic probability of an element  t ??  is found through  BetP(t) =(??????1)m(?Aij)?A??ijt??A??ij?.  (4.11) Finally, the pignistic probability of the fuzzy hypothesis  ?H  can be calculated as  BetP(?H ) =?H ??Aij1? m(?)(??????1)m(?Aij)?A??ijt??A??ij??Aij???H??Aij???. (4.12) In the rest, Definitions 4.7 to 4.12 are introduced to provide the background for designing a more computationally efficient IGIB-RB system. At the end, Definition 4.13 will demonstrate how the defined concepts can be used to define a stopping criterion for rule aggregation in the proposed IGIB-RB system.   120 Definition 4.7. The possible interval of the distance between the betting commitments of two IGIB structures is defined through the optimization problem formulated as  difBetP(m1,m2)?min/ max?H??max BetP1(?H )? BetP2(?H )( )( ) (4.13) with the constraints  BetP1(?H ) =?H ??Aij?Aijm1(?Aij)1? m1(?), i = 1,..., N , j = i,..., N?Aij???H??Aij???,BetP2(?H ) =?H ??Aij?Aijm2(?Aij)1? m2(?), i = 1,..., N , j = i,..., N?Aij???H??Aij???(mij?)1? m1(?Aij) ? (mij+)1, i = 1,..., N , j = i,..., N ,(mij?)2? m2(?Aij) ? (mij+)2, i = 1,..., N , j = i,..., N ,m1(?Aij)j?i?= 1,i?m2(?Aij)j?i?= 1.i?  (4.14)  (4.15)  (4.16) (4.17) (4.18)  (4.19) Example 4.2- Consider two IGIB structures  m1 and  m2 with fuzzy focal elements presented in Figure 4.4. The IGIB structures with specific interval-valued belief degree assignments are defined as  m1:{(?A11,[0.2, 0.3]),(?A12,0),(?A13,[0.1,0.3]),(?A22,[0, 0.1]),(?A23,0),(?A33,[0.4, 0.6])},m2:{(?A11,[0.1, 0.2]),(?A12,[0.2, 0.3]),(?A13,0),(?A22,[0.2, 0.3]),(?A23,[0.3, 0.5]),(?A33,[0.1,0.2])}. The pignistic transformations of two IGIB structures through Equation (4.12) are found as  121  BetP1={(?A11,[0.052, 0.125]),(?A12,[0.11,0.217]),(?A13,[0.22,0.333]),(?A22,[0.061,0.133]),(?A23,[0.154,0.275]),(?A33,[0.102, 0.217])},BetP2={(?A11,[0.054, 0.118]),(?A12,[0.188, 0.307]),(?A13,[0.25,0.373]),(?A22,[0.152, 0.255]),(?A23,[0.196, 0.32]),(?A33,[0.06,0.127])}. It should be noted that even if the original belief degrees assigned to local ignorance grades in  m1 (i.e.,  ?A12  and  ?A23 ) and global ignorance grade in  m2  (i.e.,  ?A13 ) are zero, the interval-valued pignistic transformation of these belief degrees associated with these grades is delimited to intervals. The possible interval of the distance between the betting commitments of  m1  and  m2 is found by evaluating Equation (4.13) as  difBetP(m1,m2)?[0.019,0.194].  Figure 4.4 Individual and interval fuzzy subsets of frame of discernment for Example 4.2 Liu (2006) considers different importance values for  m(?)  and  difBetP  in order to define a two-element measure of conflict for the classic belief structures.  Fu and Yang (2011) have argued that  m(?)  and  difBetP  stem from two different intrinsic natures.  m(?)  is dependent on the structure of focal elements as well as distribution of the belief degrees, while difBetP  is only dependent on the belief degrees assigned to the focal elements. Thus, they proposed  122 that the uncommitted belief degree be calculated after applying the pignistic transformation to each belief structure and subtract the effect of the equal-pignistic-valued focal elements. This belief degree is shown by  pm?(?) . The extension of this view to define the interval-valued uncommitted belief degree for the IGIB structures is elaborated in Definition 4.8. Definition 4.8. Let 1m  and 2m  represent two IGIB structures and 1BetP and 2BetP represent their individual pignistic transformation and 1,2EP  be the set of their equal-pignistic-valued focal elements as  EP1,2={?Aij|?Aij? ?? , BetP1(?Aij) = BetP2(?Aij)}. The uncommitted belief, which is calculated without the contribution from equal-pignistic-valued evaluation grades, is defined as  pm?(m1,m2)?min/ max BetP1(?Aij)BetP2(?Ast)?Aij,?Ast??,?Aij??Ast=???BetP1(?Aij)BetP2(?Ast)?Aij,?Ast?EP1,2,?Aij??Ast=??.  (4.20) with the constraints  BetP1(?Alm) =?Alm??Aij?Aijmk(?Aij)1? m(?),?Aij???Alm??Aij???l = 1,..., N , m = l,..., N ,i = 1,..., N , j = i,..., N ,k = 1,2,(mij?)1? m1(?Aij) ? (mij+)1, i = 1,..., N , j = i,..., N ,(mst?)2? m2(?Ast) ? (mst+)2, s = 1,..., N , t = s,..., N ,m1(?Aij)j?i?= 1,i?m2(?Ast)t?s?= 1.s?  (4.21)   (4.22) (4.23) (4.24) (4.25)   123 Definition 4.9. Assume that  m1,...,mrrepresent the IGIB structures for combination. The parameters ( )pm? ?  and difBetP  are the uncommitted belief and the distance between betting commitments, respectively. The two-element measure of conflict for these IGIB structures is defined as  cf (m1,...,mr) = pm?(?),difBetP .  (4.26) Example 4.2 (continued)- The possible interval of uncommitted belief  pm?  for the two belief structures defined in Example 4.2 is found as [0.19,0.5].pm? ?  It is notable that in solving the optimization problem in Equation (4.20), most of the time, the set of equal-pignistic-valued focal elements is empty. Thus, the two-element measure of conflict between the two IGIB structures using Definition 4.6 is found as  cf (m1,m2) = pm?(?),difBetP = [0.19, 0.5], [0.019, 0.194] .  Definition 4.10. It has been shown that  pm?(?)  and difBetP  have no intrinsic difference (Fu & Yang, 2011). Thus, these two parameters can be used to define a general disapproval measure  dm = f ( pm?(?),difBetP)  as  dm(m1,...,mr) = ?1pm?(?)+ ?2difBetP,?1+ ?2= 1, 0 ? ?1?1, 0 ? ?2?1.  (4.27) As it was mentioned earlier,  m(?)and difBetP  in the DST applicability rules provided by Liu (2006) have different importance weights, yet no specific disapproval measure is provided. For our application, it is assumed that ( )pm? ?  and difBetP  have the same level of contribution to the disapproval measure dm  among all the belief structures. Thus,  124  ?1pm?(?) = ?2difBetP.  (4.28) Through the elimination of 1?  and 2?  in Equations (4.27) and (4.28), the disapproval measure for classic belief structures is found as  dm(m1,...,mr) =2 pm?(?)difBetPpm?(?)+difBetP.  (4.29) The disapproval measure for the IGIB structures is defined in the rest. Definition 4.11. Let Let 1,..., rm m  represent the set of IGIB structures that should be combined and 1,..., rBetP BetP  represent their individual pignistic transformation which are used to find ( )pm? ?  and difBetP . The possible interval of the disapproval measure for the IGIB structures is defined as  dm(m1,...,mr)?min/ max2 pm?(?)difBetPpm?(?)+difBetP,  (4.30) with the constraints  pm?(?) = BetP1(?Aij)...BetPr(?Ast)?Aij?...??Ast=???BetP1(?Aij)...BetP1(?Ast)?Aij?EP1,2?,difBetP = argmax?H???s,t={1,...,r}, s?tBetPs(?H )? BetPt(?H )( ),BetPk(?H ) =?H ??Aij?Aijmk(?Aij)1? m(?), i = 1,...,n, j = i,...,n, k = 1,...,r,   (4.31)  (4.32)  (4.33)   125   (mij?)k? mk(?Aij) ? (mij+)k, i = 1,..., N , j = i,..., N , k = 1,...,r,mk(?Aij)j?i?= 1,i?k = 1,...,r. (4.34) (4.35) In the rest, we assume that 1dm  and 2dm  represent the solution to minimization and maximization in Equation (4.30), respectively and 1 2[ , ]dmI dm dm=  represents the interval of disapproval measure. Definition 4.12. Let 1,..., rm m  represent the IGIB structures that are combined and 1 2[ , ]dmI dm dm=  represents the interval of disapproval measure among the IGIB structures. The measure to assess the consistency among the IGIB structures is defined through a mapping cm  such that 11 2[( ) / 2] .cm dm dm?= +  (4.36) The higher the value of  cm  the more consistent the IGIB structures are. Example 4.2 (continued)- Since disapproval measure dm  is a nonlinear function of  pm?  and  difBetP , it is not feasible to directly find the possible interval of disapproval measure for  m1  and  m2  IGIB structures through the ranges obtained for these two parameters (i.e.,  pm?(?)?[0.19, 0.5] ,difBetP?[0.019, 0.194] ). If one followed direct interval calculation, they would have obtained the incorrect interval [0.0104,0.928] which is wider than the correct possible interval of .dm  The correct possible interval of the disapproval measure dm  for the belief structures 1m  and 2m  is found through solving the optimization problem in  126 Equation (4.30), which obtains [0.096,0.335].dmI = The consistency measure between 1m  and 2m  is found through evaluating Equation (4.36) which obtains a consistency measure of 4.64.cm =  It should be noted that in most BRB systems including the RIMER approach (Yang et al., 2006), all activated rules are aggregated without checking the applicability of the Dempster?s combination rule. The implied assumption in these BRB systems is that the belief structures aggregated using the Dempster?s combination rule are not highly conflicting. Discounting the belief structures based on the calculated global degree of firing for each rule can reduce the conflict among belief structures. However, the next example illustrates that the problem regarding the not applicability of the Dempster?s combination rule cannot be always alleviated by discounting the belief structures. Example 4.3- Assume that the following belief structures should be combined based on their global degrees of firing 1 2,? ?  and  ?3:  m1(?A1) = 1, m1(?A2) = 0, m1(?A3) = 0,m2(?A1) = 0, m2(?A2) = 1, m2(?A3) = 0,m3(?A1) = 0, m3(?A2) = 0, m3(?A3) = 1. If the degree of firing for the three rules are as 1 2 3 1,? ? ?= = =  the qualitative interpretation of the conflict among these belief structures shows that they are totally incompatible. The two-element measure of conflict also confirms the total incompatibility as   cf (m1,m2,m3) = pm?(?),difBetP = 1,1 .   127 However, if the degrees of firing of the rules are  ?1=?2= 0.4  and  ?3= 0.2 , discounting the belief degrees based on the degrees of firing will result in the following belief structures:  m1(?A1) = 0.6, m1(?A2) = 0.2, m1(?A3) = 0.2,m2(?A1) = 0.2, m2(?A2) = 0.2, m2(?A3) = 0.6,m3(?A1) = 0.267, m3(?A2) = 0.466, m3(?A3) = 0.267, which now have to be combined. Thus, the two-element measure of conflict is found as  cf (m1,m2,m3) = pm?(?),difBetP = 0.3091,0.4 . In this case, discounting the belief structures using the global degrees of firing reduces both measures of conflict. However, the problem arises when true discounting weights of evidential sources are not precisely known. This condition is a common problem in multi-sensor fusion frameworks in dynamic environments. If the discounting weight of an evidential source is not correctly estimated, incompatible rules can appear in the set of rules with high degrees of firing. For instance, assume the third belief structure is incorrectly estimated as completely credible (i.e., 3 1? = ). As a result, by applying the rule weights 1 2 0.4,? ?= =  and 3 1? =  the following belief structures are obtained and now need to be aggregated  m1(?A1) = 0.6, m1(?A2) = 0.2, m1(?A3) = 0.2,m2(?A1) = 0.2, m2(?A2) = 0.2, m2(?A3) = 0.6,m3(?A1) = 0, m3(?A2) = 0, m3(?A3) = 1, Thus, the two-element measure of conflict between these three structures is found as  cf (m1,m2,m3) = pm?(?),difBetP = 0.5803,0.8 .   128 The large value of difBetP  implies that the applicability of Dempster?s combination rule can be questioned and thus, employing it for rule aggregation without checking the two-element measure of conflict may result in counterintuitive results. Definition 4.13. In order to select the best number of rules for combination in the set of fired rules without unnecessary increase of the computational cost, the proportional rule contribution (PRC) is defined as ( ) 11121,( 1)rrNk kNr k kkcm cmPRCN BetP BetP???=?=? ??  (4.37) where rN represents the number of combined rules with the highest global activation weights. kcm is the consistency measure (See Equation (4.36)) between the discounted consequent belief structures of the first  k  fired rules with the highest global degrees of firing. kBetP  represents the vector of pignistic transformation of the combination of the same belief structures. The number of combined fired rules that obtains the maximum value of  PRCNrin Equation (4.37) defines the stopping criterion for combining the IGIB structures in the consequent of fired rules. The number of fired rules to be combined is determined through ( ) 11121argmax .( 1)rrNk kSCN r k kkcm cmNN BetP BetP???=? ??? ?=? ?? ?? ??  (4.38) The application of the PRC measure in reducing the complexity of rule aggregation in the IGIB-RB system is shown in a case study of drinking water local risk analysis in the next section.  129 4.4 Numerical simulation using the local fusion algorithm A numerical simulation in which the input-output relationship is known, the function approximation results obtained will be compared versus the betting commitments estimated using the matching function or the similarity measure defined in Sections 4.2.1 and 4.2.2, respectively.  The following numerical example is presented to evaluate the performance of the IGIB system as a general function approximation tool before using it for the local information fusion for the case study example. Assume that a nonlinear function of three input variables  x1,x2  and  x3 is generated as 2 3 1[ ( . )]. ( ).y C x x Sin x?= ?  (4.39) For running the numerical simulation assume that 140,C =  and the input parameters are allowed to take values in the ranges  x1?[0.2,0.75], x2?[0,140], x3?[0,1].  Thus, the output function y  will have values between 0 and 140. The fuzzy sets that represent the evaluation grades of the input and output variables are defined as shown in Figure 4.5.  130 0.2 0.3 0.4 0.5 0.6 0.700.51x1Membership0 20 40 60 80 100 120 14000.51x2Membership0 0.2 0.4 0.6 0.8 100.51x3Membership0 20 40 60 80 100 120 14000.51yMembershipL ML HMVL L M H VHMHHL Figure 4.5 The fuzzy sets representing the evaluation grades for the input and output variables. Since the mapping between the input parameters and the output is known the rule base can be generated using the known function in Equation (4.39). The rules for the IGIB-RB system are automatically generated using Monte Carlo simulation for each rule. In each simulation, the input parameters are limited to the specific ranges imposed by the least possible values of the fuzzy grades. The probability density function (PDF) of the output for each rule is found. The interval-valued belief degrees associated with the fuzzy grades of output y are defined based on the PDF obtained. The lower bound of the interval-valued belief degrees is assigned by integrating the PDF between the most possible values of the output evaluation grades while the upper bound of the interval-valued belief degrees is assigned by integrating the PDF between the least possible values of the output evaluation grades. A normal distribution for each input variable is assumed and input-output samples are generated through the Monte Carlo simulation. In order to test the IGIB-RB system, the input samples are used as the incoming body of evidence to the designed IGIB-RB system. The performance of the designed IGIB-RB system is evaluated by finding the betting  131 commitments intervals. The betting commitment intervals are compared with the estimated normalized betting commitments of the output fuzzy evaluation grades. The normalized betting commitments of the fuzzy evaluation grades are approximated using the evaluation of matching function or the similarity measure explained in Sections 4.2.1 and 4.2.2. As an illustration, the results for the case where  x1= 0.281, x2= 36.458,  and  x3= 0.8  are reported in Table 4.2. Since the mapping between the input variables and output is known, the value of the output function is found through evaluating Equation (4.39) as  y = 85.617. This value for the output can be considered as a crisp set to approximate the betting commitments of the output evaluation grades. The estimation of the normalized betting commitment of each fuzzy evaluation grade of the output using the matching function and the similarity measure are compared with the betting commitment intervals obtained using the IGIB-RB system. The pattern observed in the estimated betting commitments using the similarity measure is consistent with the betting commitment intervals suggested by the IGIB-RB system. Using the hypothesis preference ranking method presented in Chapter 3, both IGIB-RB system and the similarity measure suggest the fuzzy evaluation grade ?low to medium {L-M}?. However, using the matching function approximation results the fuzzy evaluation grade ?medium {M}? is selected. It is known that the similarity measures provide more intuitive estimates of the belief degrees (Deng et al., 2011) or normalized betting commitments than the matching function. Thus, the observations in Table 4.2 indeed emphasize the intuitive results obtained when the IGIB-RB system is used for function approximation. The IGIB-RB system provides intuitive results that when the uncertain expert knowledge is employed to build the rule base. This scenario and further discussions on it will be presented in local drinking water risk analysis case study example in the next section.  132 Table 4.2 Distributed belief, plausibility and pignistic probability intervals of the output fuzzy grades and their comparison with the belief degree approximation using the matching function and the similarity measure Output Grade Bel Interval Pls Interval BetP Interval Similarity Approx. Matching Approx. VL [0.012, 0.025] [0.0145, 0.268] [0.035, 0.069] 0.1306 0 L [0.237, 0.382] [0.640, 0.812] [0.523, 0.751] 0.2256 0.384 M [0.139, 0.267] [0.561, 0.705] [0.341, 0.566] 0.2253 0.615 H [0.015, 0.037] [0.093, 0.225] [0.053, 0.130] 0.209 0 VH 0 [0.006, 0.015] [0.002, 0.005] 0.209 0  4.5 Local water quality information fusion This section illustrates the utility of the proposed IGIB-RB system for local information fusion for relative microbial risk analysis of drinking water in the WDN. The knowledge on the implications of monitored WQPs are elicited from the experts to build the rule base of the IGIB-RB system and the RIMER approach (Yang et al., 2006). The distributed relative risk analysis results using the IGIB-RB system for individual monitoring locations of Quebec City?s main WDN are compared versus the results obtained using the RIMER approach. The problem of microbial risk analysis of drinking water has been previously studied using a set of evidential reasoning approaches (Sadiq et al., 2006). One of the main challenges in quantifying risk in this field is dealing with incommensurable and uncertain information that need rational aggregation schemes (Francisque et al., 2009a). Assume that some incommensurable attributes are available that can indicate the microbial risk of drinking water. The microbial water quality indicators considered in this case study are free residual Chlorine (FRC), heterotrophic plate count (HPC) and turbidity (TUR). High HPC may indicate some failure in the water treatment process and contamination events within the  133 distribution network (Allen et al., 2004; Sartory, 2004). Turbidity provides a surrogate for possible microbial contamination. Maintaining adequate levels of FRC reduces the potential for microbial contamination of water. The value of these indicators can be represented by fuzzy evaluation grades. Different levels of each attribute are defined based on the expert judgments found in the literature (e.g., Sadiq & Rodriguez, 2005; Francisque et al., 2009a). The experts were provided with the statistics including the mean value and the standard deviation of each water quality indicator in the Quebec City?s main WDN during different seasons of the years 2003 to 2005. The output fuzzy evaluation grades are defined to represent the relative risk associated with drinking water based on the value of each indicator in the main distribution network. For the value of attribute FRC, an expert may choose from the set of fuzzy grades ?very low {VL}?, ?low {L}?, ?medium {M}?, ?high {H}?, and ?very high {VH}?. Attribute HPC is defined by three fuzzy grades ?not frequent {NF}?, ?frequent {F}?and ?highly frequent {HF}?. Attribute TUR is defined by three fuzzy grades ?low {L}?, ?medium {M}?, and ?high {H}?. The IGIB-RB system output, which is defined as the risk associated with drinking water, is characterized by five fuzzy evaluation grades ?very low {VL}?, ?low {L}?, ?medium {M}?, ?high {H}?, and ?very high {VH}?. An IGIB-RB system with a specific rule structure is designed to model expert knowledge about the relative microbial risk of drinking water. Using the proposed framework, it is feasible to model nonspecific and uncertain assignments provided by the expert for a set of possible scenarios. Other parameters showing the rule uncertainty such as the weight of each attribute and the initial weight of each rule are considered as well. Since the membership of all evaluation grades are fixed, the initial rule weights are assigned by following the suggestions  134 presented by Ishibuchi and Nakashima (2001) and Ishibuchi and Yamamoto (2005) through utilizing a frequency-based rule activation weighting based on the training dataset. The weights of three WQPs are assigned using the comparison matrix method elicited from the expert similar to the analytic hierarchy process (AHP). The weights of these attributes remain unchanged in the designed IGIB-RB system as 0.63, 0.09, 0.27.= = =FRC HPC TUR? ? ? The attribute weights do not require to sum up to 1. Some sample rules of the designed IGIB-RB system and their initial weights are presented in Table 4.3. The statistics of the datasets collected between the years 2003 to 2005 at fifty-two sampling locations of Quebec City?s main WDN are provided in Appendix A. The final results will be presented in the form of distributed assessment of belief interval, plausibility intervals, and betting commitment intervals for the relative microbial risk of drinking water using the designed IGIB-RB system. The sampling points (SP) ?QC117? and ?QC324? of the Quebec City?s main WDN are chosen to illustrate the distributed risk analysis results using the IGIB-RB system. The obtained distributed assessments are compared versus the RIMER approach (Yang et al., 2006) results. It should be noted that the rule bases elicited from the expert are not exactly identical in both methods since the RIMER approach cannot account for all the uncertainty patterns which are modeled in the proposed IGIB-RB system. However, the closest possible rule bases to be used in the IGIB-RB system and RIMER approach are elicited from the same experts. In order to provide an assessment of the relative risk associated with drinking water during one season, the mean and standard deviation of the microbial parameters between the years 2004 and 2005 are used to build the incoming fuzzy inputs to either the IGIB-RB system or the RIMER approach. The statistics of water quality indicators for SPs ?QC117? and  135 ?QC324? are       SP ?QC117?: FRC=0.02 ? 0.002 mg/L, HPC= 39.65?16.87 Cfu/mL, TUR= 1.11?0.46 NTU, and  SP ?QC324?:  FRC=0.13?0.04 mg/L, HPC= 107?26.4 Cfu/mL, TUR= 0.26?0.07 NTU. The triangular and trapezoidal fuzzy propositions representing the evaluation grades of the antecedent parameters and the incoming fuzzy inputs to the designed IGIB-RB system for SPs ?QC117? and ?QC324? are shown in Figure 4.6. The similarity measure is used for fuzzy composition and all rules are activated with different activation weights. The average values of different measures of conflict for increasing number of aggregated rules for SPs ?QC117? and ?QC324? are depicted in Figure 4.7. These measures include the distance between the betting commitments, ,difBetP  uncommitted belief,  pm?(?) , and disapproval measure,  dm. Since  pm?(?)  and difBetP  have no intrinsic difference, they are both used to define the disapproval measure  dm. The disapproval measure always has a value between two other measures of conflict. In order to assure the applicability of the extended FDST rule combination, it is possible to set an upper threshold for both difBetPand ( )pm? ?  or alternatively a threshold for .dm  If the upper threshold for dm  is set to  ? = 0.75,  Figure 4.7.a suggests that the use of two to four rules for SP ?QC117? does not violate the applicability of the extended FDST combination rule. Figure 4.7.b shows that for SP ?QC324?, even with a higher number of combined rules (i.e., up to six rules) the extended FDST combination rule is still applicable. However, the higher the number of combined rules, the higher the computational complexity  136 of the rule combination step. Table 4.4 presents the proportional rule contribution (PRC) measure based on Definition 4.10 for the increasing number of aggregated rules. The maximum value of this measure is obtained for the first two activated rules with the highest global degrees of firing for both SPs ?QC117? and ?QC324?. Thus, it is suggested that combining the first two fired IGIB structures in the consequent part can provide the acceptable distributed assessment of the betting commitment intervals without an unnecessary increase of the computational cost. Table 4.3 Some rules of the designed rule base of the proposed IGIB-RB system for local relative risk analysis Antecedent Consequent R1 ( 1 0.617? = ): IF HPC is {L}, TUR is {NF} and FRC is {L} THEN the relative microbial risk is  {( ,[0.7,0.9]),( ,[0.27,0.3]),( ,[0.01,0.05]),( ,[0.01,0.02]),( ,0)}VL LM H VH  R2 ( 2 0.633? = ): IF HPC is {L}, TUR is {NF} and FRC is {M} THEN the relative microbial risk is  {( ,[0.85,0.9]),( ,[0.1,0.3]),( ,[0.01,0.02]),( ,0),( ,0)}VL L MH VH  R3 ( 3 0.727? = ): IF HPC is {L}, TUR is {NF} and FRC is {H} THEN the relative microbial risk is  {( ,0.8),,( ,0.2),( ,0),( ,0)}VL L M H VH?  ??? ??? R13( 13 0.515? = ): IF HPC is {M}, TUR is {F} and FRC is {L} THEN the relative microbial risk is  {( ,[0.02,0.06]),( ,[0.04,0.07]),( ,[0.6,0.8]),( ,[0.3,0.5]),( ,0)}VL LM H VH R14( 14 0.523? = ): IF HPC is {M}, TUR is {F} and FRC is {M} THEN the relative microbial risk is  {( ,[0.2,0.3]),( ,[0.3,0.4]),( ,[0.4,0.7]),( ,0)}VL LM H VH?  R15( 15 0.523? = ): IF HPC is {M}, TUR is {F} and FRC is {H} THEN the relative microbial risk is  {( ,[0.25,0.4]),( ,[0.7,1]),( ,0),( ,0)}VL L M H VH?  ??? ??? R26( 26 0.507? = ): IF HPC is {H}, TUR is {HF} and FRC is {M} THEN the relative microbial risk is {( ,0),( ,0),( ,0),( ,[0.8,0.9])( ,[0.15,0.3])}VL L M H VH  R27( 27 0.502? = ): IF HPC is {H}, TUR is {HF} and FRC is {H} THEN the relative microbial risk is {( ,[0.15,0.3]),( ,[0.75,0.9]),( ,[0.05,0.06])}VL L MH VH??   137 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.900.51FRC (mg/L)Membership   0 20 40 60 80 100 120 14000.51HPC (Cfu/mL)Membership0 0.2 0.4 0.6 0.8 1 1.200.51TUR (NTU)MembershipVLHFFNFL MM H VHSP '12'HSP '34'SP '12' SP '34' LSP '12' SP '34' Figure 4.6 The fuzzy grades representing different levels of the antecedent parameters of the IGIB-RB system and the incoming fuzzy inputs for SPs ?QC117? and ?QC324? 1 2 3 4 5 600.20.40.60.81Number of combined rulesConflict measure  difBetPpm(Null)dm (a) 1 2 3 4 5 600.20.40.60.81Number of combined rulesConflict measure  difBetPpm(Null)dm (b) Figure 4.7 Different measures of conflict for the increasing number of aggregated rules for (a) SP ?QC117?, and (b) SP ?QC324? Table 4.4 Proportional rule contribution (PRC) measure for increasing number of aggregated rules Sampling point PRC1 PRC2 PRC3 PRC4 PRC5 PRC6 PRC7 SP?QC117? 0 33.619 13.261 7.112 5.183 3.505 1.712 SP?QC324? 0 4.102 2.736 1.419 0.860 0.577 0.425     138 Table 4.5 shows the distributed assessments of belief, plausibility, and betting commitment intervals for the set of input parameters where the first two activated rules are combined. The aggregated normalized betting commitment intervals of the hypotheses representing the pignistic probability associated with relative risk levels for SPs ?QC117? and ?QC324? are depicted in Figure 4.8 and Figure 4.9, respectively. The obtained results are compared with the distributed assessment of relative risk when the same experts expressed their judgments based on the RIMER belief rule structure. Table 4.5 The distributed interval assessment of belief, plausibility, and betting commitments of relative risk associated with drinking water at SPs ?QC117? and ?QC324? Sampling  Point Risk Grade Bel Interval Pls Interval BetP Interval Rank   ?QC117? VL [0.579, 0.590] [0.974, 0.985] [0.963, 0.978] 1 L [0.0137, 0.0240] [0.410, 0.421] [0.410, 0.417] 2 M 0 [0.004, 0.006] [0.001, 0.002] 3 H 0 0 0 4 or 5 VH 0 0 0 4 or 5 ________________________________________________________________________   ?QC324? VL [0.022, 0.037] [0.068, 0.098] [0.040, 0.066]  4 L [0.093, 0.115] [0.433, 0.472] [0.352, 0.391] 2 M [0.314, 0.335] [0.736, 0.769] [0.657, 0.695] 1 H [0.085, 0.100] [0.376, 0.413] [0.246, 0.279] 3 VH 0 [0.037, 0.045] [0.026, 0.031] 5  The small width of betting commitment intervals obtained for both SPs ?QC117? and ?QC324? demonstrates an advantage of the proposed approach in decision making. The small width of this interval helps us to make a more distinctive decision about the best hypothesis that represents the relative risk associated with drinking water.   139 VL L M H VH00.20.40.60.81  Minimum of Normalized BetPMaximum of Normalized BetP (a) VL L M H VH Null00.20.40.60.81  RIMER Approach (b) Figure 4.8 The distributed assessment of microbial risk of drinking water for SP ?QC117? using (a) the normalized betting commitment intervals, and (b) the belief distributed assessment based on the RIMER approach VL L M H VH00.20.40.60.81  Minimum of Normalized BetPMaximum of Normalized BetP (a) VL L M H VH Null00.20.40.60.81  RIMER Approach (b) Figure 4.9 The distributed assessment of the microbial risk of drinking water for SP ?QC324? using (a) the normalized betting commitment intervals, and (b) the belief distributed assessment based on the RIMER approach The comparison of the results obtained using the proposed BRB system for fifty-two SPs in the Quebec City?s main WDN (See Appendix A) versus the RIMER approach (Yang et al., 2006) is presented in Tables 4.6. The overall comparison of the results shows that both the extended BRB system and the RIMER approach often predict the relative risk associated with drinking water as ?low to medium {L-M}? or ?very low to low {VL-L}?. It is obvious that the proposed extended BRB system provides a greater support for individual grades representing the relative risk. This is mainly attributed to considering extra types of  140 uncertainty such as local ignorance. Meanwhile, the assumption of fuzzy evaluation grades for the relative risk in the consequent results in obtaining a large difference between the average belief and plausibility of each individual grade (See Table 4.5). However, if the preference ranking method proposed in Chapter 3 is employed, both methods provide the same assessment for most SPs. Our justification to obtain more intuitive results using the extended BRB system (specially in SPs that the results are significantly different from the RIMER approach) is taking into account the rule compatibility measure and incorporating the stopping criterion for rule combination (e.g., See SPs ?QC116?, ?QC127?, ?QC321?, ?QC322?, ?QC325?, ?QC326?, ?QC403?, and ?QC 407? in Table 4.6). More specifically, the relative risk of ?high {H}? obtained using the RIMER approach (e.g., SPs ?QC326?, ?Q327?, ?QC403?, and ?QC407?) is counterintuitive based on the consultation with the drinking water quality experts of the Quebec City?s distribution network. For instance, more detailed analysis of the results for two SPs ?QC117? and ?QC324? will follow. Table 4.6 Comparison of distributed assessment of average pignistic probability using the proposed extended BRB system and RIMER approach for fifty-two sampling points in Quebec City?s main water distribution network Comparison of Distributed Assessment of Average Pignistic Probability RIMER approach (Yang et al., 2006)                 Extended BRB (Proposed method) Sampling Point VL L M H VH VL L M H VH QC202 0.45 0.126 0.083 0.313 0.065 0.15 0.529 0.503 0.122 0.009 QC207 0.422 0.13 0.086 0.334 0.067 0.15 0.529 0.504 0.122 0.009 QC201 0.055 0.365 0.465 0.095 0.035 0.02 0.445 0.799 0.132 0.003 QC101 0.055 0.365 0.467 0.095 0.035 0.02 0.445 0.799 0.132 0.003 QC204 0.055 0.365 0.465 0.095 0.035 0.02 0.445 0.798 0.132 0.003 QC105 0.333 0.19 0.272 0.17 0.037 0.15 0.528 0.499 0.125 0.011 QC114 0.603 0.111 0.069 0.203 0.051 0.15 0.529 0.504 0.122 0.009 QC111 0.374 0.139 0.11 0.341 0.067 0.15 0.529 0.503 0.122 0.009 QC109 0.849 0.083 0.046 0.03 0.03 0.15 0.529 0.504 0.122 0.009  141 QC107 0.326 0.2 0.257 0.121 0.037 0.15 0.528 0.5 0.124 0.01 QC116 0.467 0.125 0.082 0.301 0.063 0.15 0.529 0.504 0.122 0.009 QC117 0.921 0.055 0.033 0.033 0.033 0.75 0.508 0.008 0.001 0.001 QC119 0.573 0.119 0.212 0.068 0.032 0.05 0.379 0.671 0.246 0.023 QC118 0.079 0.277 0.458 0.095 0.032 0.02 0.445 0.799 0.132 0.003 QC327 0.362 0.137 0.093 0.375 0.073 0.15 0.529 0.504 0.122 0.009 QC113 0.472 0.129 0.097 0.273 0.059 0.15 0.529 0.504 0.122 0.009 QC309 0.055 0.365 0.465 0.095 0.035 0.02 0.444 0.801 0.132 0.003 QC326 0.359 0.138 0.093 0.377 0.073 0.15 0.529 0.502 0.122 0.009 QC308 0.089 0.247 0.443 0.134 0.033 0.02 0.445 0.799 0.132 0.003 QC320 0.052 0.33 0.489 0.08 0.032 0.02 0.445 0.798 0.132 0.003 QC206 0.394 0.133 0.089 0.353 0.07 0.15 0.529 0.504 0.122 0.009 QC408 0.354 0.226 0.229 0.16 0.046 0.06 0.385 0.66 0.242 0.023 QC412 0.055 0.356 0.473 0.095 0.034 0.02 0.445 0.798 0.132 0.003 QC404 0.915 0.059 0.035 0.035 0.035 0.75 0.509 0.008 0.001 0.001 QC402 0.054 0.362 0.472 0.093 0.035 0.02 0.445 0.798 0.132 0.003 QC403 0.284 0.158 0.181 0.326 0.061 0.15 0.529 0.503 0.122 0.009 QC314 0.492 0.282 0.111 0.096 0.035 0.15 0.528 0.498 0.125 0.011 QC407 0.315 0.149 0.144 0.348 0.066 0.02 0.445 0.799 0.132 0.003 QC311 0.055 0.365 0.465 0.095 0.035 0.02 0.445 0.799 0.132 0.003 QC316 0.138 0.203 0.364 0.204 0.04 0.08 0.52 0.592 0.127 0.01 QC321 0.362 0.137 0.093 0.375 0.073 0.15 0.529 0.502 0.122 0.009 QC322 0.359 0.138 0.093 0.377 0.073 0.15 0.529 0.503 0.122 0.009 QC315 0.252 0.178 0.248 0.251 0.05 0.15 0.529 0.503 0.122 0.009 QC324 0.1 0.1 0.465 0.265 0.1 0.05 0.368 0.668 0.262 0.031 QC328 0.061 0.322 0.476 0.106 0.032 0.02 0.445 0.799 0.132 0.003 QC304 0.055 0.365 0.465 0.095 0.035 0.02 0.444 0.8 0.132 0.003 QC325 0.766 0.099 0.056 0.079 0.038 0.15 0.529 0.504 0.122 0.009 QC302 0.055 0.365 0.465 0.095 0.035 0.02 0.445 0.799 0.132 0.003 QC305 0.053 0.353 0.48 0.088 0.034 0.02 0.444 0.801 0.132 0.003 QC413 0.73 0.101 0.059 0.107 0.041 0.15 0.529 0.503 0.122 0.009 QC318 0.358 0.138 0.097 0.373 0.072 0.15 0.529 0.504 0.122 0.009 QC317 0.595 0.143 0.131 0.082 0.036 0.15 0.529 0.501 0.124 0.01 QC329 0.893 0.063 0.035 0.026 0.026 0.97 0.413 0.001 0.001 0.001 QC330 0.452 0.132 0.084 0.306 0.063 0.15 0.529 0.501 0.124 0.01 QC323 0.315 0.149 0.144 0.348 0.066 0.15 0.529 0.502 0.122 0.009 QC301 0.138 0.203 0.364 0.204 0.04 0.04 0.454 0.74 0.138 0.002 QC205 0.388 0.136 0.103 0.338 0.067 0.15 0.529 0.502 0.122 0.009 QC319 0.359 0.138 0.093 0.377 0.073 0.15 0.529 0.504 0.122 0.009 QC110 0.45 0.126 0.083 0.313 0.065 0.15 0.529 0.503 0.122 0.009 Qc115 0.1 0.245 0.43 0.121 0.033 0.02 0.445 0.798 0.132 0.003 QC102 0.264 0.173 0.23 0.268 0.053 0.15 0.529 0.504 0.122 0.009 QC203 0.06 0.327 0.477 0.1 0.032 0.02 0.445 0.798 0.132 0.003   142 By interpreting the assessment results at the pignistic level, using the hypothesis preference ranking method presented in Chapter 3, the relative risk of drinking water during the season at SP ?QC117? is estimated as ?very low {VL}? using both IGIB-RB system and the RIMER approach. However, at SP ?QC324?, the IGIB-RB system suggests the relative risk of ?low to medium {L-M}?, while the RIMER approach suggests the relative risk of ?medium to high {M-H}?, that is the RIMER approach overestimated the perceived risk levels. Meanwhile, the relative risk analysis using the IGIB-RB system compared to the assessment using the RIMER approach seemed more intuitive to the consulted drinking water experts. In fact, the condition in SP ?QC324? is typical of extremities of drinking water distribution systems in summer seasons when water temperature is relatively high. However, the estimated relative risk is not an absolute one and is based on various conservative hypotheses. That is, the relative risk associated with SP ?QC324? using IGIB-RB system is estimated as being relatively higher than the relative risk associated with other sampling points such as SP ?QC117? and other seasons (i.e., based on the experts? judgment the water quality complies with regulatory water quality microbiological standards). Hence, the estimation of risk using the proposed IGIB-RB system constitutes a flexible decision making tool for the managers and distribution network operators to compare water quality at different seasonal and spatial conditions. 4.6 Conclusions Interval grade and interval-valued belief structure rule-based (IGIB-RB) system for local information fusion was introduced in this chapter. The IGIB-RB system provides a generic inference engine for knowledge representation, which is superior to classic belief rule-based (BRB) systems. This system provides flexibility in representing uncertain knowledge and  143 reaches more intuitive results at the decision making level when cross validated with the assessments that directly are provided by experts. Besides, this chapter presented a computationally efficient manifestation of fuzzy evidential reasoning for rule aggregation in the proposed BRB system. This has become possible through presenting a set of definitions for conflict and consistency among IGIB structures, and a stopping criterion for rule aggregation. In the proposed IGIB-RB system, incommensurable and uncertain sources of information can be used to build the knowledge base. An uncertain rule expression matrix that includes ambiguity, vagueness, local ignorance, and interval-valued belief degrees is elicited from the experts. The consequent of IF-THEN rules are designed based on the IGIB structure to model this broad uncertainty manifestation. The rules are activated using fuzzy composition, and subsequently aggregated through the extended FDST presented in Chapter 3. The performance of the proposed IGIB-RB system was evaluated through a numerical example and a case study problem on the relative risk analysis of drinking water. The results obtained revealed that the proposed IGIB-RB system can reach intuitive assessments of the relative microbial risk of drinking water when various types of uncertain information are involved. The next chapter further extends the proposed IGIB-RB system to allow for spatiotemporal monitoring in complex systems along with additional feature extraction techniques and a fast implementation algorithm.  144  Chapter  5: Spatiotemporal monitoring: Distributed information fusion 5.1 Introduction Spatiotemporal monitoring for distributed complex systems requires specific signal processing, data interpretation, and event analysis platforms. Online sensors provide the hardware basis to collect spatiotemporal data. Water distribution networks (WDN) are considered as distributed complex systems with various dynamic variables that can influence the safety and quality of water. Water quality at the WDN level can be degraded due to various reasons such as a problem in the structural integrity of the system, pipe breakage, biofilm separation, failure of Chlorine boosters in reservoirs, or a disinfection breakthrough. According to Koch and McKenna (2011), accurate identification and detection of intentional or natural contaminants in drinking water pipes is critical to water supply security and management of health risks. Currently, there is a gap between real occurring contaminations in the WDN and the measured surrogate water quality parameters (WQP) used for online monitoring. This gap is filled with event detection algorithms (Kumar et al., 2007). There are a few contaminant-specific probes that can identify the target contaminants. However, the emerging sensor technology is more suitable in supporting the detection of contaminants with commonly online monitored WQPs (WERF, 2008). When surrogate WQPs are available, the detection of anomalous process behavior is largely dependent on computational methods and their discriminatory ability. The existing event detection systems are mainly based on the data collected from a single monitoring station to indicate the occurrence of a contamination event (Murray et al., 2010;  145 Yang et al., 2009). Byer and Carlson (2005) examined the response of several surrogate WQPs to a range of contaminants. Cook et al. (2006) also introduced an event detection algorithm based on case-based reasoning systems and the experiments performed earlier by Byer and Carlson (2005). Initial trials for distributed fusion from multiple monitoring stations have been reported (O?Halloran et al., 2006; Yang et al., 2008). McKenna et al. (2008) established statistically significant clusters of detections through considering the time and location of each detected event as the result of a random time-space point process. These methods are mainly focused on two monitoring locations to improve the aggregation results using one of the nodes as the reference to compensate for the calibration error of the other node, variable time delays and background noise. The emerging technologies for online monitoring of WQPs engender the employment of more advanced pattern recognition methods such as non-stationary control chart approach (Jarret et al., 2006) and baseline deviation and multivariate classification (Kroll & King, 2006). More specific information fusion methods were employed to combine various types of information including operational data (Hart et al., 2010), sampling location specific features (Raciti, 2012), and multiple sampling locations dataset (Koch & McKenna, 2011; Hou et al., 2013) to improve the event detection performance. A set of integrated algorithms for water quality event detection is employed in the CANARY software (Hart et al., 2009; Murray et al., 2010). The above methods typically use data processing and pattern recognition algorithms that only handle time series and objective data. However, several references argue that water quality monitoring in WDNs involves both subjective and numerical information that engenders the  146 use of expert or hybrid frameworks (Dawsey et al., 2006; Adam, 2009; Francisque at al., 2009a; Lu & Huang, 2009; Mounce et al., 2009). However, many of the currently employed expert systems are limited by their framework that requires experts to provide specific judgment about a certain number of water quality degradation scenarios. These frameworks barely allow for the incorporation of uncertain causal spatiotemporal relationships between WQPs and water quality estimates. Thus, a flexible knowledge representation framework that can capture spatiotemporal data and uncertain expert knowledge is required. The focus of this chapter is to extend the applicability of the BRB system to spatiotemporal monitoring problems through introducing a multi-level information fusion structure with dynamic fuzzy evidential reasoning and learning approaches. Several fast implementations of the Dempster?s combination rule for dynamic fusion have been proposed including the methods based on the fast M?bius transform (Kennes & Smets, 1991; Yaghlane et al., 2001; Den?ux & Yaghalne, 2002) and the structural analysis model (Yang et al., 1994; 2002a; 2002b; Huynh et al., 2006). Some other efficient inference engines for dynamic information fusion use several simplifying assumptions for fast implementation (Rubin, 1999; Herrera et al., 2000; Lawry, 2001; Guo et al., 2009). However, an efficient fuzzy evidential algorithm for dynamic spatiotemporal data aggregation is not readily available. This chapter introduces a multi-level information fusion framework that efficiently performs data aggregation in distributed systems such as the WDN at measurement, feature, and decision levels. The multi-level information fusion framework encompasses three sub-modules: i) spatiotemporal sensor data preprocessing, ii) primary and secondary (parametric) feature extraction, and iii) a flexible dynamic hierarchical BRB system. The hierarchical  147 BRB system is based on a networked fuzzy belief rule-based (NF-BRB) system and a high-level BRB system. The former is built based on expert knowledge and an adaptive threshold assignment method to analyze primary features, while the latter is specialized in water quality assessment based on secondary features. A hybrid learning method is employed for parameter and structure learning of the high-level BRB system. The schematic of the multi-level information fusion is presented in Figure 5.1.  Figure 5.1 The schematic of the multi-level information fusion framework for spatiotemporal monitoring   148 5.2 Problem statement In this section, the problem of spatiotemporal monitoring in a distributed system such as the WDN is introduced to explain the motivations for the proposed multi-level information fusion. Figure 5.2 represents a simple schematic diagram of a spanning tree  G = (V , E)  defined between two consequent nodes of the WDN. It is assumed that several surrogate WQPs at each node of the network are monitored with different sampling frequencies. It is assumed that events such as contaminations change the dynamic response of the water quality sensors. Assuming that the average retention time (RT) between the two target nodes is known, the problem is to provide online assessments of the water quality and generate early warning alarms when a contamination slug passes through one or several sensors. This chapter introduces methods that are suitable for generating early warning alarms based on the variations in the WQP sensor measurements. The current methods to detect anomalies in the WDN can be categorized in three groups including statistical (Allgeier et al., 2005; McKenna et al., 2008; Shang et al., 2007; Murray et al., 2010), empirical AI-based (Allgeier et al., 2005; Murray et al., 2010; Raciti et al., 2012), and data mining (McKenna et al., 2008; Yang et al., 2008; Yang et al., 2009; Murray et al., 2010) methods. Since the expert knowledge and spatiotemporal characteristics of the WDN are very important, flexible knowledge representation systems that can tackle different manifestation of uncertainty are required. Experts provide a set of temporal and spatiotemporal rules (associated with nodes and edges of the spanning tree  G ) to interpret the variations of sensor responses during specific time intervals at one or two nodes of the WDN considering the RT between the nodes. A large portion of the available information is based on incomplete data and uncertain information provided to the expert. Therefore, in this chapter extensions of fuzzy evidential  149 reasoning and belief rule- based systems will be employed for uncertainty manifestation and handling inconsistency in the inference process.  Figure 5.2 The schematic of two nodes of the WDN, the calculated average retention time, and the contamination slug in the water flow 5.3 Sensor data preprocessing In the proposed framework, prior to primary and secondary feature extraction from composite spatiotemporal WQP sensor measurements, a preliminary adaptive signal transformation is required. The adaptive signal transformation technique is briefly reviewed in the rest. 5.3.1 Adaptive transformation As the response of a chemical sensor such as residual Chlorine concentration, is formed by the combination of several factors, an adaptive signal transformation method can be applied before decomposition of the signal for means of feature extraction (Yang et al., 2009). The WQP signal  y(t)  can be considered as the composite of instrument response to a specific event ( ye(t) ), natural baseline wandering ( yb(t) ), instrument noise ( yn(t) ), drift ( yd(t) ), and  150 operation-related random variance (? ) as  y(t) = ye(t)+ yb(t)+ yn(t)+ yd(t)+?  (5.1) With the assumption of Gaussian distribution for instrument noise and operation related random error, and if the  yb(t)  and drift  yd(t)  are modeled using a polynomial function  ?QF(t) , the sensor response can be written as  y(t) =?Ei(t)+?QF(t)+?( yn(t),? )  (5.2) In order to separate the part of the signal attributed to the sensor response to the event,  ?QF(t)  and instrument noise  ?( yn(t),? )  should be found. If the mean value of the signal in the  ith  time window is  yi and the  ith  window is selected such that  ?QF(t)??QF ? yi, the relative change between the sensor output at time  t  and the mean value in the adaptive time window is obtained as  ?e(t) = 1?yi(t)yi= ?? (Ei(t)? Ei)yi??QF(t)??QFyi  ?e(t) ? ?Ei(t)? EiEi11+ k  k =?QF +?( yn,? )?Ei (5.3) (5.4) (5.5) Equations (5.4) and (5.5) provide an adaptive transformation of the sensor response for means of secondary feature extraction that will be presented in the next section. The  151 proposed adaptive transformation amplifies the variations in the sensor response amplitude due to anomalies and eliminates the effect of baseline variation and calibration issues on the secondary feature extraction. 5.4 Feature extraction for anomaly detection The early development EWS for water quality monitoring are functioning based on the direct measurements of WQPs. However, the emerging technologies that provide the possibility of online monitoring of WQPs engender the employment of more advanced pattern recognition methods applicable to time-series for anomaly detection. Currently, the most integrated algorithms for EWS in the WDN are provided in CANARY software (Hart et al., 2007; Hart et al., 2009; Murray et al., 2010) to detect water quality anomalies based on online monitored WQP time-series. In general, the features extracted from online WQPs can be classified into four categories: statistical features, temporal features, signal shape and composite features (Hou et al., 2013). Most of the time, a significant and persistent deviation from the regulatory ranges can be attributed to certain anomaly events. These deviations are quantified using the primary and secondary feature extraction methods summarized in the following. 5.4.1 Primary feature extraction To explain the primary temporal and spatiotemporal features extracted from sensor measurements in a networked system, assume that a spanning tree  G = (V , E)  represents the set of nodes  V  (i.e., sampling locations) and directional edges  E  (i.e., existing flow directions between sampling locations). Primary features represent spatiotemporal variations of WQPs between two nodes and temporal variations of WQPs at one node of the WDN.  152 Also, assume that  yq,vs and  yq,v?s represent the  qth  monitored WQP signal at the nodes  vs and  v?s, respectively. The spatiotemporal feature  evidqes ?s  that is associated with the edge  es ?s of the spanning tree  G  is defined as  evidqes ?s(n) =yq,v?s(n) ? yq,vs(n ? m)yq,vs(n ? m); n = m+1,...,nc (5.6) where  m  is the number of time sample lag between the nodes  vs and  v?swhich is a representative of approximate retention time (RT) between  vs and  v?s  nodes in the most recent time window. The temporal features  evidqvs  for the  qth  monitored WQP at node  vs  is defined as  evidqvs(n) =yq,vs(n) ? yq,vs(n ? ?m )yq,vs(n); n = ?m +1,...,nc (5.7) where  ?m  is the predefined time sample delay to calculate the percentage of change in the WQP measurement at node  vs. The NF-BRB system that will be presented in Section 5.5 is specialized in analyzing the primary features.  5.4.2 Secondary (parametric) feature extraction In some distributed systems that long-term historical anomaly data are available probabilistic models (e.g., Chu et al., 2006; Dereszynski & Dietterich, 2011) or empirical models (Mitchell, 1997) can be employed for event detection. However, in WDNs historical dataset are available but they cannot be easily attributed to water quality failure events (Yang e al., 2009). As a result, probabilistic and empirical models cannot be directly employed. Thus, in  153 the following an automated parametric feature extraction method is introduced to quantify the variations in the adaptive transformation of WQP sensor responses that are attributed to anomaly events. The variations in the adaptively transformed WQP signal  ?e  obtained using Equations (5.4) and (5.5) can be effectively modeled as the superposition of short-duration Gaussian functions. The following steps are repeated at each time-window  TK, K = 1,...,n  of the adaptively transformed WQP signal  ?e . Step 1: The three-parameter Gaussian function is selected as the basic elementary function to model the signal temporal variations as  gt, f ,? = g(n;t, f ,? ) = ( 2?? )?1/2 e?( n?t2? )2 +i f (n?t)  (5.8) where  t  and  f  are in R and ?  is in R+. The adaptively transformed WQP signal  ?e  in Equation (5.4) is approximated using a weighted some of elementary functions in white Gaussian noise  W (n)  as  ?e(n)= alei?lg(n;tl, fl,?l)l=1p?+W (n) (5.9) The main assumption behind this formulation is that the adaptively transformed signal  ?e  has a Gaussian distribution at each time-window  TK, K = 1,...,n . It should be noted that the addition of a short-duration anomaly event to the signal does not significantly change the distribution of the adaptively transformed signal  ?e . Step 2: A realization of Equation (5.9) as  154  ?e?0(n)= a0ei?0g(n;t0, f0,?0)+ w(n)  (5.10) is used.  ?0= [?02a0?0t0f0?0]T  denotes the parameter vector in which  ?02  is the noise variance,  a0  is the amplitude,  ?0  is the duration of the elementary function,  t0  is the time of located function,  ?0 is the phase, and  f0 is the frequency of the elementary function. Step 3: The Gaussian elementary function  gb is selected for modeling the relative change in the adaptively transformed signal  ?e(t) for  t ?TK. This selection must satisfy the minimum distance between  ?e(t)  and the orthogonal projection of it on  gb as  mingb||?em(t) ||2= mingb||?elo(t)? ?elo(t),gb(t) gb(t) ||2  (5.11) where ?elo(t) = ?e(t) . If  gb that is highly similar to  ?elo(t)  is found, we can find the solution to Equation (5.11), i.e.  maxgb| ?elo(t),glo(t) |2  (5.12) Step 4: Based on the estimated  gb the remainder of the relative change signal,  ?em, is found as  ?em(t)=?elo(t)? ?elo(t),gb(t) gb(t) = ?elo(t)? Alogb(t)  (5.13) Following this process,  ?e  is replaced with  ?em  and Steps 1 to 4 are repeated until we have  155  ?ep+1(t) = ?e(t)? All=1p?gk(t) ? ?0 (5.14) The above algorithm can extract significant variations in the adaptively transformed signal  ?e . If the above steps are repeated  p  times at time window  TK, the vector of secondary features is defined as  ? c(K ) = [?1(K ),...,?l(K ),...,?p(K )].  The remaining question is how to use the parameter vector  ? c  to differentiate certain anomaly events in the WDN. To provide a solution, the high-level BRB system with an augmented learning algorithm will be presented in Section 5.5.2. 5.5 Distributed multi-level information fusion: Hierarchical belief rule-based system This section presents a flexible distributed information fusion framework that can incorporate both WQP sensors measurements and imprecise and deficient subjective knowledge on the indication of each surrogate WQP. For a better illustration, the schematic of distributed multi-level information fusion for a WDN with six nodes and two defined strata is shown in Error! Reference source not found.. In the rest, structure of the hierarchical BRB system (including the NF-BRB system and the high-level BRB system), dynamic fuzzy evidential fusion algorithm, and the learning method to construct a consistent and locally optimum high-level BRB system are presented. 5.5.1 The structure of the NF-BRB system for spatiotemporal monitoring In this section, the NF-BRB system is proposed to model the uncertain knowledge about the WDN with spatiotemporally monitored surrogate WQPs. As a result, the NF-BRB system can model the expert knowledge about water quality events based on spatiotemporal variations of  156 WQPs and the relative RT between the monitoring locations in the WDN. Assume that the spanning tree  G = (V , E)  shows the connectivity of the nodes in the WDN.  Figure 5.3. The schematic of the multi-level information fusion for spatiotemporal monitoring  The value of input attributes can be represented by a set of  Lq fuzzy evaluation grades  {Fqf, f = 1,..., Lq}.  Assume that are  Msp spatiotemporal and  Mt temporal incoming pieces of evidence to the NF-BRB system at time instant  n,  are found using Equations (5.6) and (5.7). The set of  rt temporal rules  {Rk,k = 1,...,rt}  for the node  vsare defined with the general V2 V1 V3  V4 V5 V6 Bel4 Bel3  Stratum 1 AND AND AND Bel1 Bel2 AND Stratum 2 V1 V2 V1 V3 V5 V6 Secondary features Secondary features Secondary features Bel?1 Bel?2 Bel?4 Bel?3 V1 V4 Secondary features Dynamic Multi-level Fusion  Feature level Decision level Measurement level  157 form of the  kth  rule as   Rk: IF  evid1vs is  F1 f and ? and  evidMtvs  is  FMtf THEN the consequent of the rule (e.g., the temporal water quality at node  vs) is   (?A11,m11,k),(?A12,m12,k),...,(?Aij,mij,k),...,(?ANN,mNN ,k){ },  with the initial weight of rule  Rk as  ?kand the weight of attribute  evidqvs  as  ?q, and the consequent belief degrees as mij,k.  Similarly, The set of  rsp spatiotemporal rules  {Rl,l = 1,...,rsp} between the nodes  vs and  v?s are defined as  Rl:IF   evid1es ?s  is  F1 f and ?  and evidMspes ?s  is  FMspf,  THEN the consequent of the rule (e.g., the water quality based on WQPs at the nodes  vs and  v?s) is  (?A11,m11,l),(?A12,m12,l),...,(?Aij,mij,l),...,(?ANN,mNN ,l){ },  with the initial weight of rule  Rl as  ?land the weight of attribute  evidqes ?s  as ?q, and the consequent belief degrees as mij,l.  5.5.2 The structure of the high-level BRB system The high-level BRB system is specialized in analyzing secondary features extracted from adaptively transformed WQP signals using the algorithm explained in Section 3.2. This section presents the structure of the belief rules in the high-level BRB system. Fuzzy system modeling with hierarchical fuzzy partitioning is employed to construct the high-level BRB system. In order to make the computations independent from the units, all features are scaled  158 to the unit space. Based on a set of training datasets and univariate features at different time windows, the fuzzy clustering is performed using the hierarchical fuzzy partitioning algorithm (Guillaume & Charnomordic, 2004). Assume that the vector of extracted features from signal  yi,vs at time window  TK is found as  ? c(K ) = [?1(K ),...,?P(K )].  Assume that  ?s(K ) = [?s,1,...,?s,Lf]T , with fuzzy partitioning based on the elements  ?s,t(K ) , parameter Dt,sk represent the projection of the  kth  partitions on  ?s(K ), s = 1,..., P.  Thus, a set of  P ? Lfantecedent attributes are defined for the high-level BRB system. As a result, the rules  Rl, l = 1,...,rf of the high-level BRB system are defined as   Rl: IF  ?1,1(K )  is  D1,1k and ? and  ?P,1(K )  is  D1,Pk and ? and  ?1,Lf(K )  is  DLf,1k and ?  ?P,Lf(K )  is  DLf,Pk THEN the consequent of the rule (e.g., the drinking water quality at  vsin the time window  TK) is   (?A11,m11,l),(?A12,m12,l),...,(?Aij,mij,l),...,(?ANN,mNN ,l){ },  with the weight of the rule  Rl as  ?land the weight of features  ?s,t, s = 1,..., P, t = 1,..., Lf  as  ?st and belief degrees as mij,l.  All these parameters are adjusted through the learning algorithm that will be presented in Section 5.6. 5.5.3 Dynamic fuzzy evidential fusion for improved temporal risk analysis This section proposes the dynamic fuzzy evidential fusion method through extending the fuzzy Dempster-Shafer theory (FDST) presented in Chapter 3. This algorithm will be employed at different levels of the hierarchical BRB system to obtain a distributed  159 assessment of the relative risk associated with drinking water that is dynamically updated. The minimum and maximum values of pignistic probability of a continuous fuzzy hypothesis  ?H  as  (?H ,[BetP?(?H ), BetP+(?H )]) are found by solving the optimization problem  BetP(?H ) ?min/ max m(?Aij) h(? )d?01??Aij???????????.  (5.15) with the constraints  h?Aij,?H(? ) = inf(? ?H (tk ),? ?Aij (tk ))tk??Aij? ??Aij(tk)tk??A?ij? , tk ? ?A?ij i = 1,...N , j = i,..., N ,h?Aij,?H(? ) = 0, tk??A?ij , i = 1,...N , j = i,..., N ,aij? m(?Aij) ? bij, i = 1,...N , j = i,..., N ,m(?Aij) = 1.j?i?i?  (5.16)  (5.17) (5.18) (5.19) where the last two constraints are due to the interval-valued belief degrees. Initial belief rule combinations in both the NF-BRB system and the high-level BRB system are based on the method presented in Chapter 3. The initial rule combination provides a static betting commitment interval of the evaluation grades at each time step. The set of distributed betting commitment intervals at each time step  tk?TK is presented by  ?tk={(?Aii,[BetPii,k?, BetPii,k+]) | i = 1,2,..., N}, tk?TK. (5.20)   160 Definition 5.1 -(belief function transformation from distributed betting commitment intervals)- Prior to combining the betting commitment intervals at different time windows, the betting commitment intervals of individual fuzzy evaluation grades are mapped into the distributed belief degrees. The belief structure transformation  ? :?k??k is based on normalizing the betting commitment intervals as  ?i,k=(BetPii?+ BetPii+)(BetPii?+ BetPii+)i=1N?, i = 1,..., N , k = 1,...,nK. (5.21) Assume that the credibility factor of the assessment by the hierarchical BRB system is decaying over time. Thus, if a betting commitment distribution is obtained in the previous time window  TK, the discounted belief structure  (?K)? (tn?tk) for combination in the current time window  Tn is found by  ?i,K? (tn?tK)=? (tn? tk)BetPK(?Aii)BetPK(?Aii)i=1N?, i = 1,2,..., N ,   ??,K? (tn?tk)= 1?? (tn? tk)+ ? (tn? tk)??,K (5.22)  (5.23) where  tK is the center time of time window  TK, K = 2,...,n. The  ? (t)  is the decreasing function of the passed time with  ? (0) = 1  and  limt??? (t) = 0.  Since discounting rate from one time window to the next one is considerable, an exponential decaying factor with an accelerated rate as  ? (t) = dacc(t)e?? t  can be employed.  dacc is the accelerated discounting coefficient which is assigned such that a realistic discounting rates for previous time  161 windows are obtained. The iterated discounting of belief structures can be summarized as a discounting which has a total credibility factor equal to multiplication of all individual credibility factors as stated in Lemma 5.1. Lemma 5.1- Assume that there is a belief structure ?  and the credibility factors  ?1 and  ?2.  Then, the iterative discounting is as  (??1)?2= ??1?2.   Definition 5.2 - (Markovian requirement of the betting commitments combination with memory decaying) Assume that the belief structures  ?K, K = 1,...,n  are obtained through normalizing the betting commitment distributions of  n  time windows  TK, K = 1,2,...,n  with  ti> tj, ti?TK,tj?TK?1 for  K = 2,...,n. Let  fn(?1,?2,...,?n)  be the combination of all belief structures  ?1,?2,...,?nobtained through  fn(.) at current time window  Tn. If  dtK= tK? tK?1 where  tK is the center time of the time window  TK, K = 2,...,n,  there exists a function  g  that combines two belief structures such that  fn(?1,?2,...,?n) = g(g(...g(g((?1)? (dt2),?2)? (dt3),?3)? (dt4),...,?n?1)? (dtn),?n).  (5.24) Assume that the betting commitments of individual evaluation grades in the time window  TK are shown by  BetPK(?Aii), i = 1,..., N .  Let  ? (dt2) be the credibility factor of the assessment provided in time window  T1 when the assessment in time window  T2 has become available. The dynamic pignistic probability  DBetP2(.)  of individual evaluation grades is found by  162  DBetP2(?Aii) = ?s,1(dt2)?t,2?Au=?Ass??Att???h?Aii,?Au(? )d?01?h?Aii,?Au(? ) = inf(? ?Aii ,? ?Au )tk??Au? ??Au(tk)tk??A?ii? , ?Au = ?Ass ? ?Att  (5.25)  (5.26) If the two assessments at time windows  TK?1 and  TK were to be combined, the distance between the pignistic probability transformations at two consecutive time windows is found as  difBetPK?1,K= maxi?1,...,NDBetPK?1(?Aii)? DBetPK(?Aii)( ). (5.27) The uncommitted belief without the contribution from equal-pignistic-valued evaluation grades is defined by  pm?(?) = BetPl,K?1(?Aii)BetPl,K(?Aii)i=1Nl?? BetPl,K?1(?Aii)BetPl,K(?Aii)BetPl ,K?1=BetPl ,K?  (5.28) The credibility factor of the assessment in time window  TK?1 is defined based on  dtK= tK? tK?1and the betting commitment distributions as  ? (dtK) = e(??K+?K?cdifBetK?1,K)t  (5.29) where  ?c is assigned in a way to guarantee that  ? (dtK) is always smaller than one. ?K is assigned as  163  ?K=1, if ananomaly is found in thetime windowTK?1, if noanomaly is found in thetime windowTK????? (5.30) and the constant  ?K in Equation (5.29) is assigned as  ?K= ?K?1, difBetPK?1,K< ?1and pm?(?) < ?2?K= ?K?1+?add, difBetPK?1,K> ?1or pm?(?) > ?2?????  (5.31) where  ?add  is an additive forgetting factor that further reduces the credibility factor of the assessment in window  TK when the assessment is highly conflicting with its other adjacent time windows. To explain the dynamic fuzzy evidential fusion of  n  distributed betting commitment at time windows  TK, K = 1,...,n,  assume that the memberships of the elements of the intersection grade  ?A(i,i+1)  are defined by  ??A(i,i+1)(t) = min{??Ai(t),??Ai+1(t)}; t ??.  (5.32) The following formulation shows the recursive implementation for the function  g  in Equation (5.24) that satisfies the Markovian requirement with memory decaying. If no memory decaying is required all credibility factors  dtK, K = 2,...,n  should be equal to 1. Assume that the betting commitment distribution of fuzzy evaluation grades at time windows  TK, K = 2,...,n  are available (with the credibility factors  ? (dtK)  with  dtK= tK? tK?1). The set of discounted belief structures are found by  164  (?K?1)? (dtK)={(?Aii,?i,K?1? (dtK)) | ?i,K?1? (dtK)=? (dtK)BetPK?1(?Aii)BetPK?1(?Aii)i=1N?, i = 1,..., N} (5.33) where  ?i,K?1? (dtK )i=1N?= ? (dtK), K = 2,..., N . The remaining belief degree is assigned to the frame of discernment as global ignorance  ??,K?1= 1?? (dtK), K = 1,2,...,n?1. (5.34) Then, a recursive algorithm similar to Gou et al. (2009) can be formulated to obtain a dynamic distributed assessment of the betting commitments. The only difference is that the normalized pignistic probabilities are substituted with belief degrees. Then, the normalized assessments of individual and intersection evaluation grades are found as  ?n(?Ai) =?i,I (n)1? ??,I (n), i = 1,...N ,?n(?A(i,i+1)) =?(i,i+1),I (L)1? ?1N ,I (L), i = 1,...N ?1, (5.35) (5.36) The dynamic pignistic probabilities of individual fuzzy evaluation grades  ?Aii  are found by  DBetPn(?Aii) =?n(?Ai)+?( ?Ai , ?A(i,i+1))?n(?A(i,i+1))?( ?Ai , ?A(i,i+1))+?( ?Ai+1, ?A(i,i+1)) , i = 1,..., N ?1,  (5.37) where  ? :[0,1]2 ? [0,1]  is the similarity measure between two fuzzy subsets  ?A  and  ?B  on the list ?? as  165  ?( ?A, ?B) = | ?A? ?B |max(|?A |,|?B |).  (5.38) 5.6 Hybrid learning for the high-level BRB system Several algorithms for updating the BRB systems have been presented. The most famous learning algorithms for BRB systems are optimal learning (Yang et al., 2007a), sequential learning (Zhou et al. 2009; 2010; 2012) and expert intervention (Zhou et al., 2010; 2012). In this section, a hybrid learning algorithm for the high-level BRB system based on helpful features of previous methods is proposed. In the proposed method, the restricted rule firing and structure updating of the belief rules improve the accuracy of the identification through assessing the rule utility and thus, a less complex BRB system with a faster rule aggregation is obtained. The proposed learning algorithm is used to readjust the structure and parameters of the belief rule base. The set of input-output training samples for the high-level BRB system is formed as pairs  (??mc,??m)  where  ??mc  is the vector of secondary features in the time window  TK and  ??mis the result of dynamic fusion of the betting commitment distributions using the NF-BRB system for all  tk?TK. The expert intervention is incorporated through restricting rule firing in the NF-BRB system to the inputs for which the dynamic threshold   yvs ,nthr  for at least one WQP at time window  Tn  is passed. The dynamic threshold for  yvsis defined as  166  yvs,nthr=(1?? (n))nyvs,KK=1n?+? (n)yvs,exthr  (5.39) where  yvs,exthr  is the fixed threshold defined by the expert for the node vs and  ? = e??(n?1)  is assigned to guarantee that the dynamic threshold is closer to the threshold defined by the expert in the early stages of learning. Let  ?s(K ) for  s = 1,..., P be the  sth  vector of secondary features extracted from the time window  TK of the input signal  yvs.  Each element of the vector  ?s has a numerical value whose set of matching degrees with the antecedent evaluation grade  Dt,sk is found by  ? = (Dt,sk,?sk(?s,t(K ), Dt,sk), k = 1,..., Lt, s = 1,..., P, t = 1,..., Nf{ } (5.40) where  ?sk(.) is a matching function or a similarity measure between two fuzzy sets. The global degree of firing of the belief rule  Rl with the incoming feature vector  ?s(K )  is found by  ?l(?s(K )) =?k=1Lt(?l?kl(?s(K )))?l?km(?s(K ))m=1L??????????????? (5.41) where  ?kl(? c(K )) is calculated as  167  ?sl(? c(K )) = ?s=1P?t=1Lf ?st(?sk(?s,t(K )))l?stt=1Lf?s=1P???????????????????1??1??s=1P?t=1Lf ?st(?sk(?s,t(K )))l?stt=1Lf?s=1P????????????????????. (5.42) In both previous equations ?  symbols denote t-norm operators and ?  is the modification factor that compromises between the conjunctive and disjunctive operators. 5.6.1 Discussion of sensor fault tolerance In the proposed hierarchical BRB system, sensor fault tolerance can be considered. It has been shown that in complex networks as the ratio of faulty sensors to the fault free sensors increases, the dynamic fusion algorithm that are based on decision fusion at adjacent analysis windows are superior to static fusion algorithms as they assign higher weights to the nodes that more consistently report an anomaly. In the proposed framework both primary and secondary features are employed for final event detection at local strata of the WDN. In the proposed decision fusion algorithm the following steps were followed: 1) Obtain local decisions from every node based on temporal rule bases 2) Obtain an average local decision through evaluating temporal and spatiotemporal rules in each stratum using dynamic fuzzy evidential fusion in the NF-BRB system 3) Obtain average decision through dynamic fusion in the hierarchical BRB system In previous sections, analytical models for event detection through estimating the betting commitment distributions were provided. The failure of the EWS is defined when the acceptable threshold of the number of faulty sensors is passed.  It is easier to discuss the probability of false positive results at the pignistic level. When there is no faulty sensor,  (1? BetP( yvs,nthr))i  is the pignistic probability that  i  sensors have values that have passed the dynamic threshold  yvs,nthr  and  BetP( yvs,nthr))N?i  is the pignistic  168 probability that  N ? i  sensors have values below the dynamic threshold. Thus, pignistic probability of occurrence of a justifiable false positive result is  BetPfpr=Ni??????BetP( yvs,nthr)N?i(1? BetP( yvs,nthr))ii= ? N????N? (5.43) where ? is a threshold defined for the case that there is no fault-tolerant assumption in Equation (5.39). In the presence of  t  faulty sensors reporting a positive detection, only  ? N????? t  sensors out of  N ? t  sensors need to produce justifiable false positive results to get a final false positive result. The pignistic probability of a justifiable false positive result is  BetPfpr=Ni??????BetP( yvs,nthr)N?i?t(1? BetP( yvs,nthr))ii= ? N?????tN?t? (5.44) A similar procedure can be followed to define the pignistic probability of justifiable false negative results in both cases. This formulation can be used to readjust the receiver operating characteristic (ROC) curves when there is the assumption of faulty sensors. 5.6.2 Belief rule parameter tuning with optimal learning Since the training pairs  (??mc,??m)  obtained through the NF-BRB system are imperfect atypical rules my be obtained. For the  mth  pair of observed data, the BRB system is trained to minimize the distance between observed dynamic pignistic probability DBetPio(m)  and the pignistic probability  BetPi(m) . Such a requirement should be satisfied for all pairs of the observed data. This leads to an optimization problem on the set of all output evaluation grades as  169  minQ?i(Q) i = 1,..., N ,  (5.45) where  ?i(Q) =1MBetPi(m)? DBetPio(m)( )m=1M?2, i = 1,..., N , (5.46) and  BetPij(m)? DBetPijo(m)( ) is the residual at the  mth  time window and  Q = Q(mij,l,?st,?l)  is the parameter vector that needs to be readjusted. The multi-objective function that should be solved is  minQ{?1(Q),?2(Q),...,?i(Q),...,?N(Q)}  (5.47) with the constraints  0 ? mij,l?? mij,l+?1, i = 1,..., N , j = i,..., N ,l = 1,..., L,mij,l?j?i??1,i?l = 1,..., L,mij,l+j?i??1,i?l = 1,..., L,m(?Aij)j?i?= 1,i?l = 1,..., L,0 ??l?1, l = 1,..., L,0 ? ?st?1, s = 1,..., P, t = 1,..., Lf,?stt=1Lf?s=1P?= 1. (5.48) (5.49) (5.50) (5.51) (5.52) (5.53) (5.54) The above multi-objective optimization problem can be solved using the minimax method (Miettinen, 1999; Yang et al., 2000; Marler & Arora, 2004; Yang et al., 2007a) or other  170 solvers such as Matlab Optimization Toolbox. 5.6.3 Rule base structure updating with belief rule utility The average expected utility based on the previous feature vector observations  h(? c(1)),...,h(? c(n)) for the belief rule  Rl is  Ul= u(?Aij)mij,llimn?+?cl(? c(K )) / nK=1n?limn?+?cm(? c(K ))K=1n?m=1L?/ n,j=iN?i=1N?  (5.55) where  cl(? c(K )) =?l?kl(?s,t(K )). Thus, for a large number of passed time windows (i.e,  n?? ), one obtains the following approximation  El? cl(? c )? c? p(? c )d? c  (5.56) Assume that the  tth element of the vector  ?s(K ) = [e1(K ),...,et(K ),...,eLt(K )]T  is delimited to an interval of  [etL,etU]  and has a sampling density function of  pt(et).  The expectation of the rule  Rl due to the incoming feature vector  ? c(K )  is defined as  El??lIts(Dt,sk)t=1Lf?s=1P? (5.57) where  Its(Dt,sk) = ?tk(? c(K ), Dt,sk) pt(et)detetLetU? and  Dt,sk represents the  kth  input grade  171 of the  ? c  antecedent attribute in the rule  Rl. Then, the statistical utility of the rule  Rl is found by  limn??Uln?Elu(?Aij)mij,lj=iN?i=1N?Eli=1L?. (5.58) Thus, based on the updated values of the parameter vector  Q  in Section 5.5.1,  Uln  will be updated using Equations (5.57) and (5.58). Then, the rule  Rl is removed at the pruning stage if the following condition for a long enough sequence of time windows (i.e.,  m > mh) holds  Ul=UlKK=n?mn?m+1< ?U, for m > mh.   (5.59) 5.7 Problem setting for validation In order to show the utility of the proposed multi-level information fusion framework, a case study of online monitoring of six sampling locations in the Quebec City?s main WDN is considered. The problem for only one sampling location when online measurements of common WQPs are available was handled using the extended FDST for decision support (Chapter 3) and the IGIB-RB system for relative risk analysis (Chapter 4). To formulate the problem in the distributed setting for the Quebec City?s main WDN, a spanning tree  G ={V , E}  as presented in Figure 5.4 is formed. The set of temporal belief rules attributed to the nodes (i.e., monitoring locations)  V ={v1,v2,v3,v4,v5,v6}  and spatiotemporal belief rules attributed to the edges  E ={e12,e13,e14,e23,e56}  are elicited from the WDN experts. The  172 recorded hourly monitoring data for common WQPs including free residual Chlorine (FRC) concentration and turbidity (TUR) during January, July, and October 2011 are available. The consulted experts were provided with information about data collected at each monitoring location and the retention time (RT) between the nodes and the size of connecting pipes. The estimated range of RT from treatment plant ( v1) based on the size of the connecting pipes and measured pressures and flow rates are found through tuning an EPANET model of the Quebec City?s main WDN. The summary of these estimations are presented in Table 5.1.  Figure 5.3 The schematic of six sampling locations in the Quebec City's main WDN and their connectivity in the spanning tree G={V,E}   173 Due to the rechlorination at the reservoir, two strata  S1 and  S2 are defined as  S1={v1,v2,v3,v4} and  S2={v5,v6}. The antecedent attributes of the temporal rules of the NF-BRB system are defined as the percent change in FRC concentration ( Clf) and TUR measurements ( Tur ) as  evid1,vs(n) =Clf ,vs(n?1)?Clf ,vs(n)Clf ,vs(n?1),                                             (5.60)  evid2,vs(n) =Turvs(n)?Turvs(n?1)Turvs(n?1).                                             (5.61) Table 5.1 The information of the estimated RT from treatment plant (node v1) for different nodes of the Quebec City's main WDN Connecting pipe  Node  Pipe Size 30 inches 40 inches 42 inches Venturi ( v2) 0.45-1.23 hours 0.63-0.79 hours 0.64-0.93 hours Verdun ( v3) 3.75-7.40 hours 6.88-8.94 hours 0.64-0.93 hours Reservoir entry ( v4) 7.35-15.93 hours 11.5-15.35 hours N/A In the reservoir  170 hours St. Vallier ( v6) 170-171.84 hours N/A N/A   Five output fuzzy evaluation grades ?very low {VL}?, ?low {L}?, ?medium {M}?, ?good {G}? and ?very good {VG}? for the consequent of the hierarchical BRB system are defined to describe the water quality in the WDN.  174  Figure 5.4 The probability density function of residual FRC concentration at four nodes of the Quebec City's main WDN Based on the consultations with the Quebec City?s WDN experts, FRC concentration and TUR measurements at the treatment plant and other nodes of Quebec City?s main WDN are often in acceptable regulatory ranges. For instance, Figure 5.4 shows the probability density function of the FRC concentration at nodes  v1 to  v4 during January 2011. Both fixed threshold (Preleman et al., 2012) and dynamic threshold (Arad et al., 2013) methods for contamination event detection based on the FRC concentration signal in the WDN have been previously proposed. Due to considerable baseline variations of FRC signal at some nodes such as Venturi ( v2) node, the dynamic threshold scheme for FRC as in Equation (5.39) is employed to fire the temporal belief rules in the NF-BRB system. For illustrating the belief rule structures, the spatiotemporal rules for the edge  e12between treatment plant ( v1) and Venturi ( v2) nodes from the rule base of the first stratum are shown in Table 5.2.  175 Table 5.2 Initial spatiotemporal rules between water treatment plant (v1) and Venturi (v2) nodes (RT~ 1 hour) Spatiotemporal Rules between Treatment plant and Venturi sampling points If at treatment plant Cl2 <1.2 mg/L  cl2(t,1) : Chlorine concentration at treatment plant  cl2(t,2) : Chlorine concentration at Venturi node  With  ?cl2=cl2(t ?1,1)? cl2(t,2)   With  ?Tur =Tur(t,2)?Tur(t ?1,1)   If  ?cl2< 0.1cl2(t ?1,1) ,  ?Tur ? 0 , then WQ is G  If  ?cl2< 0.1cl2(t ?1,1) ,  ?Tur ? 0.1Tur(t ?1,1) then WQ is M-G  If ?cl2< 0.1cl2(t ?1,1)  WQ is L-M  If  ?cl2< 0.1cl2(t ?1,1) ,  ?Tur > 0.25Tur(t ?1,1) , WQ is ({L, 0.5}, {M, 0.5})  If  ?cl2> 0.1cl2(t ?1,1) ,  ?Tur ? 0 , WQ is M-G  If  ?cl2> 0.1cl2(t ?1,1) ,  ?Tur ? 0.1Tur(t ?1,1) , WQ is M  If  ?cl2> 0.1cl2(t ?1,1) ,  0.1Tur(t ?1,1) < ?Tur ? 0.25Tur(t ?1,1) , WQ is L-M  If  ?cl2> 0.1cl2(t ?1,1) ,  ?Tur > 0.25Tur(t ?1,1) , WQ is {(L, 0.75), (?, 0.25)}  In the next section, anomaly scenarios to validate the performance of the hierarchical BRB system are generated. Then, the four configurations of the multi-level information fusion (i.e., static vs. dynamic fusion, and no learning vs. hybrid learning) are constructed and their performance in anomaly event detection in WDN is evaluated.  176 5.8 Validation: Distributed online water quality monitoring 5.8.1 Distributed monitoring Often a variation from the baseline can be attributed to an anomaly if other evidence exist that support occurrence of an event. The secondary conditions that should be satisfied to warrant a justifiable alarm for an anomaly event are based on the predefined patterns in the online WQP signals. These patterns are quantified using the secondary feature extraction method presented in Section 5.3 and are used to train the high-level BRB system through the hybrid learning algorithm. The unprocessed sensor readings for FRC concentration and TUR signals for January, July and October 2011 at the treatment plant ( v1) and Venturi ( v2) nodes are shown in Figure 5.5. For illustration, the assessment results before adding any simulated anomaly patterns to the WQP signals of the month January using the initial static multi-level information without learning are shown in Figure 5.6. The results show that before adding the anomaly signals to the original signals the multi-level information fusion framework often estimates the water quality as ?medium {M}? or ?good {G}?. 5.8.1 Testing strategies with spatiotemporal data The central problem hindering full evaluation of monitoring and event detection frameworks at the WDN level is that actual water quality events are scarce. A practical alternative approach to test the monitoring and early warning systems is to project the signal patterns associated with water quality events to the measured WQPs in the actual WDN.  177  Figure 5.5 The unprocessed sensor measurements of residual FRC concentration and TUR at treatment plant and Venturi nodes during three months of 2011  Figure 5.6 The distributed assessment of betting commitments of water quality grades in January 2011 (normal condition without any added anomaly)  178 The tests to extract the anomaly patterns that appear in the WQP signals are conducted in a pilot facility. For instance, various water contamination tests in a pilot facility in the Sandia National Laboratories were performed (Yang et al., 2009), and various U-shaped, V-shaped, inverse U-shaped (Gaussian) and inverse V-shaped patterns in the WQP signals were observed.  FRC concentration signals with certain added anomalies based on the sample anomaly patterns in CANARY software (Murray et al., 2010; Yang et al., 2009) are simulated and added to the original FRC concentration signals at several time steps to test the performance of the multi-level information fusion framework for anomaly detection.  Figure 5.7 Anomaly patterns with positive and negative strength of 1 Figure 5.7 shows different patterns of anomaly of strength 1. These patterns can be scaled to model different contamination events (Hou et al., 2013). The inverse U-shaped and inverse V-shaped anomalies are added to the adaptively transformed TUR signal  ?eTur to model an increase in the TUR level, while, the U-shaped and V-shaped signals are added to adaptively  179 transformed residual chlorine signal  ?eClf to model a decrease in FRC concentration. Scenario 1: Simulated single anomaly events In this section, U-shaped pattern of strength 1 are added to the FRC signals at treatment plant ( v1) and Venturi ( v2) nodes to model a water quality failure event such as disinfection breakthrough in the treatment plant. It is assumed that a Chlorine booster after the Venturi ( v2) node has regulated the FRC concentration in other nodes of the WDN. The FRC concentration at six nodes of the network with the simulated anomaly events of strength 1 and the adaptively transformed FRC signals are shown in Figure 5.8.  Figure 5.8 The FRC concentration signals and their adaptive transformation at 6 nodes of the WDN during three months of the year 2011 In the rest, the results of the detection algorithm for the month January 2011 are presented. Figure 5.9 shows the FRC signals at treatment plant ( v1) and Venturi ( v2) nodes with the anomaly patterns added to the Venturi FRC measurements in January 2011.  180  Figure 5.9 The FRC concentration signals at the treatment plant (v1) and Ventri (v2) nodes with six U-shape anomaly patterns added to the Venturi measurements Figure 5.10 shows the estimated normalized betting commitment of water quality grades using the hierarchical BRB system with static fusion and without the learning algorithm. The aggregated assessments are obtained through defining a set of fuzzy utilities for output water quality grades. As it is observed, the aggregated assessments show a considerable number of false negative and positive results. Figure 5.11 shows the estimated normalized betting commitment of water quality grades using the hierarchical BRB system when dynamic fusion and hybrid learning algorithms are employed. It is observed that the detection rate of the dynamic fusion with learning is considerably higher than static fusion without learning. Particularly, the NF-BRB system that is only based on the subjective judgment and uses the static fusion of the spatiotemporal variations of the measured WQPs generates a high number of justifiable false positive results. On the other hand, the hierarchical BRB system with dynamic fusion obtains a locally optimum solution through finding the statistical rule utility and updating the structure and tunable parameters of the high-level BRB system.  181  Figure 5.10 The estimated betting commitment at each time window using the hierarchical BRB system with static fusion and without learning  Figure 5.11 The estimated betting commitment at each time window using the hierarchical BRB system with dynamic fusion and hybrid learning Scenario 2: Simulated composite contamination events In this scenario, composite anomaly events for addition to FRC concentration at Venturi ( v2) and St. Vallier ( v3) nodes are simulated through projecting the response of the FRC sensor to  182 eight contaminants injected at fixed time intervals in a pilot facility to actuall measurements in the WDN. The scaling of the event strength for each chemical at different concentrations has been adopted from the experimental results presented in (Yang et al., 2009). The adaptively transformed FRC concentration  ?eCl, for six experiments of consecutive injection of eight chemicals with fixed time intervals between the injections, measured at two sampling locations are shown in Table 5.3. Separate simulations of the FRC sensor response with anomaly patterns at Veturi ( v2) node are performed. The simulated adaptively transformed FRC signals and their reconstructions using the secondary feature extraction method in Section 5.3 are presented in Figure 5.13. The adequate reconstruction ability indicates that the selected features can fairly represent the important components of the composite FRC signals. The aggregated assessments of water quality for six experiments using the hierarchical BRB system with static fusion and without learning are presented in Figure 5.13. The time samples that are attributed to the detected anomaly events are annotated. Figure 5.14 depicts the aggregated assessments and the annotated time samples using the hierarchical BRB system with dynamic fusion and hybrid learning. Figure 5.14 indicates that dynamic fuzzy evidential fusion produces anomaly alarms that are proportional to the concentration of the contaminant. However, with incorporating the dynamic fusion with accelerated discounting rate, a contamination with low concentration can generate a high number of false negative results. This condition can be seen in the injection of terrific broth and tryptic soy broth at low concentrations of 0.47 and 0.12 mg/L after 500 hours in Experiments 3 to 6 as shown in Figure 5.14. However, in general the  183 hierarchical BRB system with dynamic fusion has a higher area under the receiver operation characteristic (ROC) curve due to a lower number of false positive results. Table 5.3 Relative changes of FRC signals as the composite response to injection of eight biological and chemical contaminants in a testing facility Contaminants Experiment Observed relative change of FRC concentration   Day Conc. (mg/L) 22.4 m from injection point 335.4 m from injection point Aldicarb MW 190.39 3 2.2 88.2?0.6 87.1?0.0 24 2.2 93.7?0.9 86.5?0.7 3 1.1 44.5?0.7 43.1?0.4 24 1.1 45.3?0.5 42.1?0.7 3 0.2 8.0?0.5 8.7?0.1 24 0.2 8.4?0.6 8.1?0.1 Lyphosate MW 169.08 22 3.0 96.6?0.0 95.1?0.0 23 3.0 97.3?0.6 95.2?0.0 23 1.5 62.5?2.1 58.7?0.8 22 0.4 31.7?0.7 31.0?0.9 23 0.4 36.2?0.5 33.4?0.3 Nicotine MW 162.23 25 3.8 39.2?0.4 79.7?0.5 28 3.8 48.0?0.5 88.5?0.6 25 1.9 24.9?0.3 44.9?0.4 28 1.9 29.3?0.6 51.7?0.6 25 0.4 7.9?0.2 13.0?0.4 28 0.4 9.6?0.1 15.9?0.0 Colchicine MW 399.44 3 3.6 88.2?0.6 87.1?0.0 24 3.6 91.2?0.7 86.4?0.7 3 1.8 44.5?0.7 43.2?0.5 24 1.8 45.3?0.5 42.2?0.3 3 0.4 8.3?0.7 8.2?0.1 E. coli in terrific broth 1 0.14 89.9?0.5 82.2?0.0 2 0.14 89.7?0.5 89.5?0.6 1 0.07 66.5?0.9 71.0?0.7 2 0.07 63.6?0.5 69.0?1.0 1 0.01 15.2?0.5 16.7?0.8 2 0.01 ND ND Nutrient broth 136 0.95 12.0?0.1 15.5?0.5 137 0.95 13.5?0.5 25.6?0.7 136 0.47 6.6?0.0 6.7?0.0 137 0.47 1?0.0 12.6?0.5 136 0.12 2.3?0.3 2.8?0.3 137 0.12 2.3?0.4 ND Terrific broth 102 2.2 21.1?0.3 26.0?0.3 112 2.2 21.4?0.1 25.4?0.5 102 1.1 10.6?0.0 12.9?0.5 112 1.1 11.7?0.0 14.2?0.1 112 0.2 3.4?0.0 4.0?0.1 Typtic soy broth 141 0.95 17.8?0.3 23.3?0.1 176 0.95 15.6?0.3 22.7?0.6 141 0.47 9.7?0.6 12.6?0.0 176 0.47 10.1?0.3 14.1?0.8 141 0.12 3.5?0.5 3.8?0.5 176 0.12 3.1?0.3 4.7?0.4  184  Figure 5.12 The simulated and estimated relative change of FRC concentration at Venturi (v2) node for six contamination experiments  Figure 5.13 The aggregated water quality assessment (solid line) and annotated water quality events (dotted time samples) using the hierarchical BRB system without learning (static assessment)  185  Figure 5.14 The aggregated water quality assessment (solid line) and annotated water quality events (dotted time samples) using the hierarchical BRB system with hybrid learning (dynamic assessment) The performance of four configurations of the multi-level information fusion for contamination event detection were evaluated. The ROC curves of static multi-level information fusion and dynamic multi-level information fusion with and without the hybrid learning algorithm demonstrate the detection performance of these configurations. The ROC curves of these detectors are depicted in Figure 5.15. Overall, the area under the ROC curves for the multi-level dynamic information fusion with hybrid learning has the highest value among all possible configurations. The results presented in Figures 5.11 to 5.13 show that in time samples that no anomaly is added often water quality is assessed as ?medium {M}? or ?good {G}? using both static and dynamic fusion frameworks. Also, in most time steps the aggregated betting commitment associated with water quality grade ?medium {M}? is greater than that of other water quality grades.  186  (a)  (b)  (c)  (d) Figure 5.15 The ROC curves of water quality event detection results using a) hierarchical BRB system (static fusion), b) hierarchical BRB system (dynamic fusion), c) hierarchical BRB system (static fusion with learning), and d) hierarchical BRB system (dynamic fusion with learning) The detection result of water quality degradation events in both Scenario 1 (single contamination events) and Scenario 2 (composite contamination events) indicates that dynamic multi-level information fusion with hybrid learning has a higher detection rate with fewer false negative and positive results compared to other configurations. It should be noted that the water quality estimate and alarm generations are highly dependent on the defined utility function, adaptive threshold used in the learning algorithm, and the length of analysis window. A twenty-four hour window was selected as the initial analysis window since most variations in the obtained signal without learning were persistent for less than twenty-four hours. Besides, employing dynamic thresholds for belief rule firing and the  187 expert interventions through tuning the certain parameters that change the performance of the learning algorithm has significantly improved the results. Based on consultations with the Quebec City?s main WDN experts, no major event or boiling notices had happened in the network for the period that the data were available. Thus, the proposed framework provided correct assessments of water quality when no anomaly was added to the WQP sensor measurements. Finally, the validation performed through the simulation of anomaly events strongly supported the efficiency of the multi-level information fusion with hybrid learning in the event detection. 5.9 Conclusions This chapter introduced several configurations of the multi-level information fusion for online water quality monitoring and early warning in the WDN. The proposed framework employs primary and secondary (parametric) feature extraction methods that provide the inputs to the hierarchical BRB system. The hierarchical BRB system comprises the NF-BRB system and the high-level BRB system, with or without the hybrid learning algorithm. The NF-BRB system provides a generic model for knowledge representation for distributed water quality monitoring in the WDN characterized by spatiotemporal variations of WQPs and RT between the nodes. The NF-BRB system was built based on the implications of relative change in the adaptively transformed WQP measurements. A hybrid learning algorithm adaptively readjusts the high-level BRB system that is specialized in interpreting the secondary features and constructs a locally optimum belief rule base. Besides, the assessments obtained using the hierarchical BRB system at each time window are aggregated through a dynamic fuzzy evidential fusion algorithm. The dynamic fusion substantially enhances the performance of the event detection through reducing the number of false psitive  188 results. The performance of the multi-level information fusion framework was validated through a set of simulated anomaly events. Several contamination events based on the results of the contamination experiments in a testing facility were simulated and an adaptive signal pattern projection technique was proposed to scale and add the anomaly patterns to the actual WQP measurements of the Quebec City?s main WDN. Four configurations of the multi-level information fusion framework (i.e., static vs. dynamic fusion, and no learning vs. hybrid learning) were tested for event detection and their performance was compared. The results obtained revealed that the proposed dynamic fusion algorithm with hybrid learning significantly improves the events detection performance.  189  Chapter  6: Conclusions 6.1 Summary Our Original research question asked how to perform multifaceted information fusion for spatiotemporal monitoring problems in complex systems, such as water distribution networks (WDN). In other words, how can we make an informed decision when both subjective and objective information with uncertainty are involved in a problem with sparsity in the spatial domain? Meanwhile, the specific complex problem of water quality monitoring in the WDN was investigated. Providing an effective decision support system (DSS) for complex distributed systems can offer a promising solution to different areas such as early warning systems (EWS), seismic and infrastructure monitoring, asset and resource management, and general recommender systems. In exploring a flexible DSS to address spatiotemporal monitoring and event detection in complex systems, this thesis has formulated spatiotemporal monitoring within several flexible knowledge representation frameworks based on extended fuzzy evidential reasoning and extended belief rule-based (BRB) systems. The proposed integrated framework allows incorporating incommensurate information, based on both expert knowledge and historical data, to the inference process when various patterns of information deficiency are involved. The followed procedure and proposed methods offer theoretically sound tools for information uncertainty management for condition assessment and risk analysis in complex systems. The necessity of assuring high levels of safety and reliability in the WDN motivated the investigation and validation of a novel multi-level information fusion framework for online  190 monitoring and relative risk analysis based on surrogate water quality parameters (WQP) and their spatiotemporal dynamics. The problem was broken down into multifaceted information fusion problems at the local and network levels in the WDN. At the local fusion level, two objective functions including the assessment of relative water quality for decision support and also the estimation of relative risk levels at specific sampling locations were considered. At the network level, the objective is to provide a distributed assessment of water quality and to effectively detect anomaly events in the network based on commonly monitored WQPs measured at different sampling locations. 6.1.1 Summary of the contributions The main contributions of this research are: 1) The interval grade and interval-valued belief (IGIB) structure was proposed to provide flexibility in knowledge representation. Specifically, two extra types of interval uncertainty in expert knowledge in addition to ambiguity and vagueness were modeled. The IGIB structure can be used for more flexible knowledge elicitation from experts, and can be employed to readjust the regulatory norms in online monitoring problems. 2) An extension of the FDST was proposed to aggregate the IGIB structures obtained from independent evidential sources. The proposed method was used to design a multifaceted information fusion module for water quality assessment in the WDN. 3) An expert local information fusion system based on extended BRB systems was developed to perform relative risk analysis using partial or deficient knowledge elicited from experts or the imperfect information collected in one monitoring location. The proposed method can handle inconsistency in the inference about the state of a complex system based on available heterogeneous sensor data and deficient subjective information.  191 4) A comprehensive multi-level information fusion module based on hierarchical BRB systems was proposed. The hierarchical BRB system detects the water quality degradation events based on primary and secondary (parametric) features extracted from WQP signals and obtains local and distributed assessments of water quality under uncertainty and inconsistency. 5) In order to evaluate the performance of each sub-module of the DSS, case study problems for quality assessment and relative risk analysis were investigated. Besides, an EWS for online monitoring in the WDN was designed and actual online monitoring data and simulated anomaly events were used to validate each sub-module of the proposed nontraditional DSS. 6.2 Main conclusions Main conclusions that are provided here are based on the followed design procedure and the obtained results in evaluating individual sub-modules of the proposed nontraditional DSS. The first objective of local information fusion for water quality monitoring was performed through incorporating the expert knowledge about the implications of monitored surrogate WQPs in one sampling location of the network. For this purpose, an extension of the FDST was proposed to model local ignorance and interval-valued belief degrees when the expert cannot make specific judgments about the relative water quality based on surrogate WQPs. The extended framework was used to elicit the expert knowledge through defining a vocabulary for water quality grades based on generalized fuzzy sets. Then, the deficient information represented on generalized fuzzy sets was mapped into interval grade and interval-valued belief degree (IGIB) structures. The obtained IGIB structures convey information in presence of interval uncertainty, vagueness, and epistemic uncertainty. The extended FDST was employed to aggregate the IGIB structures and to provide a distributed assessment of the local water quality in the WDN.  192 The inconsistency in the inference process for risk analysis in the WDN motivated proposing a more complex local information fusion method specialized in relative risk analysis of drinking water. The nonlinearity between the implications of online monitored WQPs and the relative risk is handled through an extended BRB system that can incorporate subjective epistemic uncertainties as well as nonlinear input-output relationships in its rule base. This capability was achieved through embedding the IGIB structure in the consequent of the BRB system. The previous developments set the basis to introduce a distributed multi-level information fusion framework that can handle deficient spatiotemporal information about a networked system. The WDN was modeled as a spanning tree with monitoring locations as its nodes and directional edges based on the flow direction between the nodes. The spatiotemporal dynamics related to the water flow in the pipes and other hydraulic characteristics of the system were factored in a digraph. Issues such as presence of water reservoir and specific RT between the graph nodes are considered in the hierarchical BRB system that is specialized in anomaly detection, and local and distributed relative risk analysis. Characteristics of the Quebec City?s main WDN such as approximated RT between sampling locations were determined using an EPANet model of the WDN. In general, the distributed water quality assessment and relative risk analysis problems modeled in this thesis are formulated as nonlinear function approximations of features extracted from surrogate WQPs, spatial variations of the monitored WQPs, and imprecise temporal relationships between monitored WQPs at different nodes. A networked fuzzy belief rule-based (NF-BRB) system was designed to incorporate this complex pattern of  193 interdependencies among system parameters. The initial assessments are provided using the NF-BRB system, which is based upon primary features of the surrogate WQPs at various sampling locations. Then, a high-level BRB system that is built based on objectively selected secondary features of surrogate WQPs is employed to improve the performance of the multi-level information fusion. A hybrid learning method based on rule parameter optimization and evaluation of belief rule utility was introduced to decrease the number of false positive and negative results and hence increase the overall detection performance. As an important contribution, several anomaly events that were measured during various contamination experiments in a pilot facility are simulated and projected to the monitored WQP signals of the Quebec City?s main WDN. The simulated events were added to the FRC concentration and TUR signals to evaluate the performance of the proposed framework. The initial anomaly patterns for eight biological and chemical contaminants were generated based on the experimental results. Then, CANARY open source software was used to simulate the extra anomaly patterns for different types of contaminants given the initially generated patterns. In generating the anomaly patterns, an important assumption on the scalability of the amplitude of the WQP sensor response to different contaminant concentrations have been made. The anomaly signals were scaled based on the adaptive transformation of FRC concentration during contamination experiments in the pilot facility. The FRC patterns during the contamination events were projected to the FRC measurements at different nodes of the Quebec City?s main WDN. These simulations extensively help in validating the performance of the event detectors in similar situations in the real WDN. The performance of four configurations of the multi-level information fusion frameworks in contamination event detection was evaluated. It was demonstrated that employing the hybrid learning and  194 dynamic fusion algorithms significantly improves the detection performance. This type of validation, which provides great advantages, is novel and not found in the literature of the field. To conclude, as the main contribution of this thesis, several knowledge representation and inference frameworks were presented through extending fuzzy evidential reasoning and BRB systems to provide more flexible, comprehensive, and multi-level knowledge representation frameworks that can be employed for early warning and detection of anomalies, and relative risk analysis within a complex environment. Finally, the utility and efficiency of the proposed frameworks were validated through extensive empirical dataset and additional anomaly event simulations. 6.3 Future work There are several possibilities that we see for exploiting the future research directions and improving the results obtained in this thesis. The current research has led to development of a comprehensive DSS that can be employed for online monitoring of complex and networked infrastructure systems. Specifically, this thesis focused on online water quality monitoring in medium-sized WDNs. Some good first steps are evaluating the proposed methods for larger WDNs. However, since the proposed DSS is quite general extra investigation to employ the proposed methods for other distributed systems such as power networks, pipelines, and computer networks is possible. Some opportunities for future research are outlined in the rest: ? There are several complexities in a real WDN that cannot be fully considered during analyzing the results from online measurement of surrogate WQPs. For instance, variations in the source water quality and treatment process at the treatment plant can  195 considerably change the way that the regulatory parameters of monitoring units are adjusted in practice. Besides, the regulations are changing with time and thus more fundamental changes in the rule bases may be required. One simple way to achieve this goal is to define separate rule bases for different times of the year. Also, a number of independent temporal rules for some sampling points can be generated. ? In this research, simulated anomaly patterns were added to the monitored WQP signals of a real WDN, thus it was possible to obtain a validation strategy. Other researchers have performed experimental tests for several contamination scenarios on pilot-scale water distribution facilities. The scalability of the results obtained under experimental and simulation conditions to the sensor responses in the real WDN in case of an actual event can be further studied. ? Several uncertainties may be involved in estimating the risk associated with an indicated anomaly event in the WDN. Previous studies that use the expert knowledge for risk analysis in the WDN are mainly based on the statistics of manual sampling results. In this thesis, a parametric feature extraction method and automated learning algorithm based on historical patterns were proposed to improve the anomaly event detection and relative risk analysis. However, it is helpful to investigate the possibility of direct knowledge elicitation from water quality experts and treatment plant technicians about the indications of the extracted parametric features. ? There are various condition assessment and risk quantization techniques that can be employed as part of a regular online monitoring program. This research introduced a flexible and comprehensive DSS used in online monitoring. However, the reliable and easy integration of data analysis platforms with the current data acquisition  196 technologies used by municipalities is a current challenge that requires further research. ? The procedure through which final assessments of a DSS are communicated with the water treatment plant operators is an important topic of research. This research has explored several ideas to enhance the interpretability of the assessments and uncertainties involved for better decision support. The trials to make a fully automated system for decision making and recommending the required actions (such as activating Chlorine boosters in the WDN) are the next anticipated steps. ? The proposed knowledge representation and DSS can be extended for other distributed decision making problems such as resource allocation, asset management, e-management, and general recommender systems. However, important questions in employing the proposed approach for applications such as control, and specific pattern recognition systems may arise, that necessitate proper use of the proposed methods. Some shortcomings in employing the proposed DSS for specific applications may stem from data structure and limiting requirements for knowledge elicitation and representation in the domain. In such cases, it is still possible to adopt some of the developed algorithms in this research within a hybrid inference framework. ? A major next step from an implementation point of view is improving the speed of the extended FDST algorithm presented in Chapter 3. This could lead to significant improvement that makes the algorithm more appropriate for larger networks and other distributed systems that require faster online data processing. 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The statistics of microbial and physicochemical water quality parameters (WQP) at fifty-two SPs of the Quebec City?s main WDN during the years 2003 to 2005 are presented in Table A.1 and Table A.2, respectively.  223  Figure A.1 Distribution of fifty-two SPs within the Quebec City?s main WDN  224 Table A.1 The statistics of microbial WQPs at fifty-two SPs of the Quebec City?s main WDN Sampling location Avg TUR Std TUR Avg HPC Std HPC Avg FRC Std FRC 1 QC202 0.36 0.21 23.02 75.78 0.19 0.19 2 QC207 0.34 0.09 15.76 21.01 0.18 0.13 3 QC201 0.28 0.14 0.27 0.86 0.82 0.15 4 QC101 0.29 0.15 5.86 24.42 0.70 0.19 5 QC204 0.27 0.12 0.26 0.67 0.88 0.22 6 QC105 0.36 0.29 49.07 173.86 0.40 0.19 7 QC114 0.42 0.48 13.64 21.40 0.14 0.13 8 QC111 0.31 0.12 26.94 81.75 0.32 0.19 9 QC109 0.53 0.50 9.69 36.11 0.24 0.31 10 QC107 0.44 0.51 43.50 193.05 0.44 0.31 11 QC116 0.37 0.21 16.73 30.38 0.05 0.08 12 QC117 1.11 0.92 39.65 33.74 0.02 0.04 13 QC119 0.53 0.69 70.11 190.65 0.16 0.15 14 QC118 0.37 0.15 4.50 8.95 0.58 0.11 15 QC327 0.29 0.12 8.58 15.85 0.14 0.11 16 QC113 0.38 0.19 16.78 91.26 0.31 0.20 17 QC309 0.28 0.26 22.11 74.78 0.71 0.20 18 QC326 0.19 0.07 0.17 0.39 0.27 0.09 19 QC308 0.32 0.15 3.28 15.44 0.54 0.20 20 QC320 0.36 0.13 0.21 0.43 0.66 0.14 21 QC206 0.31 0.19 7.26 13.62 0.09 0.10 22 QC408 0.32 0.14 62.67 211.34 0.34 0.29 23 QC412 0.26 0.15 0.11 0.31 0.64 0.21 24 QC404 0.77 0.39 16.70 42.07 0.13 0.16 25 QC402 0.30 0.41 0.67 1.56 0.76 0.21 26 QC403 0.28 0.19 0.57 1.34 0.39 0.14 27 QC314 0.33 0.21 53.86 178.41 0.12 0.11 28 QC407 0.25 0.24 0.42 1.48 0.96 0.28 29 QC311 0.26 0.14 3.63 27.08 0.75 0.15 30 QC316 0.22 0.11 1.64 8.74 0.48 0.17 31 QC321 0.29 0.10 0.29 0.47 0.25 0.12 32 QC322 0.21 0.08 27.25 74.31 0.08 0.09 33 QC315 0.33 0.26 5.24 18.73 0.43 0.13 34 QC324 0.26 0.14 107.00 132.51 0.13 0.09 35 QC328 0.24 0.08 2.36 2.58 0.60 0.13 36 QC304 0.24 0.13 29.49 86.91 0.79 0.18 37 QC325 0.47 0.59 13.17 11.29 0.05 0.09 38 QC302 0.26 0.17 2.68 19.65 0.68 0.11 39 QC305 0.33 0.48 21.76 135.29 0.86 0.23 40 QC413 0.46 0.35 3.52 7.87 0.20 0.15 41 QC318 0.29 0.17 7.65 58.38 0.29 0.13 42 QC317 0.49 0.26 39.29 157.36 0.37 0.17 43 QC329 0.59 0.85 3.27 4.15 0.25 0.20 44 QC330 0.35 0.19 39.18 71.90 0.03 0.05 45 QC323 0.26 0.15 0.14 0.53 0.36 0.13 46 QC301 0.24 0.12 19.26 140.71 0.48 0.13 47 QC205 0.33 0.13 0.26 0.56 0.31 0.16 48 QC319 0.26 0.23 17.87 52.50 0.26 0.19 49 QC110 0.36 0.23 3.38 8.28 0.25 0.21 50 QC115 0.36 0.14 0.22 0.49 0.53 0.14 51 QC102 0.31 0.13 7.03 45.83 0.42 0.18 52 QC203 0.30 0.14 0.79 2.31 0.61 0.20   225 Table A.2 The statistics of physicochemical WQPs at fifty-two SPs of the Quebec City?s main WDN Sampling location Avg Temp Std Temp Avg pH Std pH Avg TTHM Std TTHM 1 QC202 11.45 6.21 7.45 0.19 49.56 18.26 2 QC207 11.18 4.48 7.45 0.21 59.16 40.68 3 QC201 10.10 7.56 7.46 0.19 23.61 12.31 4 QC101 11.25 6.91 7.55 0.20 22.53 16.49 5 QC204 11.06 7.76 7.42 0.27 54.89 48.72 6 QC105 10.14 6.67 7.56 0.20 40.94 22.94 7 QC114 13.21 3.02 7.46 0.21 41.94 22.09 8 QC111 9.19 6.21 7.53 0.27 42.63 27.87 9 QC109 10.56 5.88 7.65 0.31 65.42 17.76 10 QC107 11.38 5.44 7.62 0.20 33.38 24.13 11 QC116 12.55 4.34 7.47 0.17 43.54 24.02 12 QC117 10.10 3.75 8.26 0.70 50.63 24.66 13 QC119 11.44 5.50 7.48 0.17 32.62 17.84 14 QC118 11.06 7.06 7.52 0.20 37.28 26.58 15 QC327 11.17 5.89 7.48 0.26 37.02 25.91 16 QC113 11.29 6.27 7.49 0.19 41.25 24.36 17 QC309 12.03 5.40 7.52 0.23 42.46 18.79 18 QC326 15.50 5.00 7.48 0.23 38.22 21.80 19 QC308 10.89 6.80 7.51 0.24 36.10 27.03 20 QC320 10.14 7.55 7.53 0.19 39.52 27.39 21 QC206 10.23 5.36 7.53 0.20 44.95 26.14 22 QC408 9.45 4.40 8.12 0.40 57.45 30.10 23 QC412 11.57 5.56 7.56 0.23 38.58 25.07 24 QC404 10.34 5.58 7.60 0.22 36.44 25.08 25 QC402 9.01 7.19 7.69 0.25 58.76 0.00 26 QC403 9.68 6.30 7.61 0.21 37.39 22.65 27 QC314 13.03 4.46 7.51 0.19 34.88 20.62 28 QC407 9.89 6.71 7.81 0.35 39.59 25.72 29 QC311 10.16 6.77 7.53 0.21 32.45 23.73 30 QC316 11.15 6.28 7.53 0.28 26.73 16.05 31 QC321 10.64 4.58 7.53 0.26 49.56 34.45 32 QC322 11.21 4.72 7.48 0.23 41.34 16.22 33 QC315 11.57 6.10 7.60 0.19 38.81 32.63 34 QC324 10.63 4.79 7.70 0.23 53.82 32.03 35 QC328 9.09 6.22 7.64 0.23 38.29 31.51 36 QC304 11.36 6.75 7.53 0.22 31.47 23.01 37 QC325 12.50 4.96 7.50 0.22 45.68 18.92 38 QC302 11.80 5.85 7.58 0.21 25.64 14.87 39 QC305 9.80 7.38 7.52 0.29 26.78 27.34 40 QC413 11.76 4.60 7.50 0.20 39.47 21.20 41 QC318 10.32 4.71 7.62 0.25 37.87 21.45 42 QC317 11.25 6.00 7.50 0.22 34.65 24.01 43 QC329 10.27 5.05 7.61 0.23 41.39 36.54 44 QC330 12.36 3.17 9.38 0.19 55.60 17.00 45 QC323 10.36 7.30 7.59 0.18 25.49 14.53 46 QC301 11.08 5.97 7.65 0.21 36.26 25.94 47 QC205 10.37 6.41 7.55 0.16 42.32 25.35 48 QC319 11.03 6.02 7.47 0.20 39.86 24.58 49 QC110 9.20 5.53 7.54 0.16 29.89 18.52 50 QC115 13.16 6.83 7.54 0.20 43.53 30.72 51 QC102 9.86 7.49 7.54 0.22 23.06 8.76 52 QC203 11.53 6.82 7.39 0.21 45.58 21.54  

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