International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

Sparse Bayesian learning with Gibbs sampling for structural health monitoring with noisy incomplete modal… Huang, Yong; Beck, James L. Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 1  Sparse Bayesian Learning with Gibbs Sampling for Structural Health Monitoring with Noisy Incomplete Modal Data  Yong Huang Associate Professor, School of Civil Engineering, Harbin Institute of Technology, Harbin, China James L. Beck  George W. Housner Professor, Division of Engineering and Applied Science, California Institute of Technology, Pasadena, United states  ABSTRACT: Most hidden damage that occurs in civil structures is in localized areas. In this paper, this information is exploited in a sparse Bayesian learning framework for inferring localized stiffness reductions as a proxy for structural damage that uses noisy incomplete modal data from before and after possible damage. The methodology proposes a new sparse Bayesian model that induces spatial sparseness in representations of the inferred stiffness reductions. To obtain not only the most plausible model of sparse stiffness reductions within a specified class of models, but also a quantification of its uncertainty, the method uses Gibbs sampling to generate samples from the posterior distribution for the structural stiffness parameters, system modal parameters and eigenequation-error precision parameter. The approach has five important benefits: (1) no matching of model and experimental modes is needed; (2) solving the eigenvalue problem of a structural model is not required; (3) all the uncertain parameters are sampled or estimated conditional on the modal data, and, therefore, no user-intervention is required; (4) the effective dimension for the Gibbs sampling only depends on the small number of parameter groups that are used for constructing the conditional PDFs for drawing samples; and (5) the inferred stiffness reductions are spatially sparse in a way that is consistent with a Bayesian Ockham's razor. A three-dimensional braced-frame model from the Phase II benchmark problem sponsored by the IASE-ASCE Task Group on Structural Health Monitoring is analyzed using the proposed method. The results show that the proposed approach reduces the occurrence of false and missed damage detections in the presence of modeling errors.  1. INTRODUCTION Inverse problems are essential in structural health monitoring (SHM), including localizing and assessing structural damage using measured vibration data (Ching and Beck, 2004; Ching et al., 2006; Farrar and Worden, 2013). However, real inverse problems in SHM are typically ill-posed, i.e., the uniqueness or existence of an inverse solution is not guaranteed when recovered from noisy and incomplete data. Furthermore, modeling errors should be explicitly acknowledged. The inherent uncertainty suggests that we should not just search for a single optimal model when solving inverse problems, but rather attempt to describe the family of all plausible models that are consistent with both the observations and our prior information. This leads us to consider inverse problems from a full Bayesian perspective, which provides a robust and rigorous framework due to its ability to account for model uncertainties.  To alleviate the ill-posedness of inverse problems, the prior distribution can be used to provide model regularization in the Bayesian framework, instead of the regularization term used in deterministic least-norm estimation methods. In this article, a novel method for the stiffness loss inverse problem is presented that exploits the prior knowledge that structural 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 2  damage typically occurs at a limited number of locations in a structure in the absence of its collapse. The method builds on developments in Sparse Bayesian learning (Tipping, 2001) and its application in Bayesian compressive sensing (Ji et al., 2008; Huang et al., 2014), where a Bayesian framework is presented for automatic model sparseness promotion, allowing higher-resolution for damage localization.  However, there are always computational challenges for solving Bayesian inverse problems, since they usually require the evaluation of multi-dimensional integrals that are often analytically intractable, especially for high-dimensional problems. Laplace’s method of asymptotic approximation (Beck and Katafygiotis, 1998) can be used, which is accurate when there is a sufficiently large amount of data available. Based on Laplace’s method, a new sparse Bayesian learning framework for stiffness loss inversion has been proposed recently (Huang and Beck, 2015). In this paper, we explore the use of Gibbs sampling (GS) (Geman and Geman, 1984; Gelfand et al., 1990; Ching et al., 2006), a special case of Markov chain Monte Carlo simulation, to efficiently sample the posterior PDF of the high-dimensional uncertain parameter vectors that arise in our stiffness loss inversion method. The basic idea here is to decompose the uncertain parameters into four groups and to iteratively sample the posterior distribution of one parameter group conditional on the other three groups and the available data.  2. BAYESIAN MODELING  Suppose ௦ܰ  sets of vibration time histories are measured in a structure and ܰ௠ dominant modes of the system are identified from each set of time histories. The MAP (maximum a posteriori)  estimates are taken from the modal identification as the “measured” natural frequencies 	૑ෝଶ ൌሾ ෝ߱ଵ,ଵଶ , … , ෝ߱ଵ,ே೘ଶ , ෝ߱ଶ,ଵଶ , … , ෝ߱ேೞ,ே೘ଶ ሿ் and mode shapes ૐ෡ ൌ ൣૐ෡ଵ,ଵ் , … ,ૐ෡ଵ,ே೘் ,ૐ෡ଶ,ଵ் , … ,ૐ෡ேೞ,ே೘் ൧் , where ૐ෡௥,௜ ∈ Թே೚	 gives the identified components of the system mode shape of the ݅௧௛ mode (݅ ൌ 1,… ,ܰ௠) at the ௢ܰ  measured DOFs (degrees of freedom) from the ݎ௧௛ data segment (ݎ ൌ 1,… , ௦ܰ).  2.1. Structural model class The formulation is based on modal parameter identification with low-amplitude vibration data where the structural behavior is well approximated by linear dynamics with classical normal modes for damage detection purposes. Under this hypothesis, a damping matrix need not be explicitly modeled since it does not affect the model mode shapes.  We take a class of linear structural models that has ௗܰ DOFs (degrees of freedom), a known mass matrix ۻ from structural drawings and an uncertain stiffness matrix ۹ that is represented as a linear combination of ሺ ఏܰ ൅ 1ሻ	 substructure stiffness matrices ۹௝,	݆ ൌ 1,2, … ఏܰ, as follows:                   ۹ሺીሻ=۹଴+∑ ߠ௝۹௝ேഇ௝ୀଵ             (1) where the nominal substructure stiffness matrices ۹௝ ∈ Թே೏ൈே೏ represent the contribution of the 		݆௜௛		 substructure to the overall stiffness matrix ۹  and ી ൌ ൣߠଵ, ߠଶ, … , ߠேഇ൧ ∈ Թேഇ  are corresponding stiffness scaling parameters. An inferred reduction of any ߠ௝, ݆ ൌ 1,… ఏܰ,	 is assumed to correspond to local damage in the ݆௜௛ substructure.  2.2. Prior for system mode shapes and structural stiffness scaling parameters  In the situations where measurement of only partial DOFs are available, it is advantageous to introduce system mode shapes ૖ ൌൣ૖ଵ் , … ,૖ே೘் ൧் ∈ Թே೏ே೘ൈଵ  (Ching and Beck, 2003; Ching and Beck,  2004; Ching et al., 2008), as well as system natural frequencies ૑ଶ ൌൣ߱ଵଶ, … , ߱ே೘ଶ ൧் ∈ Թே೘ൈଵ, to represent the actual underlying modal parameters of the linear dynamics of the structural system at all ௗܰDOFs corresponding to those of the structural models. These system mode shapes are not constrained to be exact eigenvectors corresponding to any structural model because there will always be modeling errors, so: 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 3                       ሺ۹ሺીሻ െ ߱௜ଶۻሻ૖௜ ൌ ܍௜	        (2) Independent zero-mean Gaussian vectors with covariance matrix ିߚଵ۷ே೏ are chosen to model the uncertain equation errors ܍௜ ∈ Թࢊࡺ, ݅ ൌ1,… ,ܰ௠, ݎ ൌ 1,… , ௦ܰ . This maximum entropy probability model (Jaynes, 2003) gives the largest uncertainty for ܍ subject to the first two moment constraints: ۳ሾሺ܍௜ሻ௞ሿ ൌ 0, ۳ሾሺ܍௜ሻ௞ଶሿ ൌିߚଵ, ݇ ൌ 1,… , ௗܰ.   With this choice in conjunction with Eq. (2), we construct the joint prior PDF on the system modal parameters ૖ and ૑ଶand stiffness scaling parameter ી as:       ݌ሺ૖,ωଶ, ી|ߚሻ ൌ ܿ଴ሺ2ିߚߨଵሻିே೘ே೏ ଶ⁄        ∙ exp ቄെఉଶ ∑ ฮሺ۹ሺીሻ െ ߱௜ଶۻሻ૖௜ฮே೘௜ୀଵଶቅ			 (3) where c଴	 is a normalizing constant and ‖∙‖ denotes the Euclidean norm. Note that mode shapes corresponding to the structural model are the most probable values a priori of the system mode shapes. The introduction of Gaussian errors in the eigenequations with variance ିߚଵ, provides a soft constraint instead of imposing the rigid constraint given by the eigenequations of the structural model. When ߚ → ∞,  the system mode shapes become tightly clustered around the mode shapes corresponding to the structural model specified by ી,		which are given by Eq. (2) with all ܍௜ ൌ ૙.  We can deduce the prior PDF for ી conditional on system modal parameters ૖ and ૑ଶ:   ݌ሺી|૖,૑ଶ, ߚሻ ൌ ݌ሺ૖,૑ଶ, ી|ߚሻ ݌ሺ૖,૑ଶ|ߚሻ⁄    ൌ ݌ሺ૖,૑ଶ, ી|ߚሻ ׬ ݌ሺ૖,૑ଶ, ી|ߚሻ݀ી⁄   	ൌ ࣨሺી|ሺ۶்۶ሻିଵ۶்܊, ሺ۶்۶ߚሻିଵሻ  (4) Similarly, the conditional prior PDFs for ૖ and ૑ଶ are obtained as (5) and (6), respectively:    ݌ሺ૖|૑ଶ, ી, ߚሻ ൌ ݌ሺ૖,૑ଶ, ી|ߚሻ ݌ሺ૑ଶ, ી|ߚሻ⁄    ൌ ࣨሺ૖|૙, ሺ۴்۴ߚሻିଵሻ                 (5)    ݌ሺ૑ଶ|૖, ી, ߚሻ ൌ ݌ሺ૖,૑ଶ, ી|ߚሻ ݌ሺ૖, ી|ߚሻ⁄    ൌ ࣨሺ૑ଶ|ሺ۵்۵ሻିଵ۵்܋, ሺ۵்۵ߚሻିଵሻ       (6)  where the definitions of ۶, ܊, ۴, ۵ and ܋ can be found in Huang and Beck (2015). For the error precision parameter	ߚ, we take the widely used Exponential distribution: ݌ሺߚ|ܾ଴ሻ ൌ Expሺߚ|ܾ଴ሻ ൌ ܾ଴ expሺെܾ଴ߚሻ  (7) which produces the maximum entropy prior for ߚ with the mean constraint of Eሺߚ|ܾ଴ሻ ൌ 1 ܾ଴⁄ . 2.3. Likelihood function for structural stiffness scaling parameters For the structure in service, any damage will typically occur at a limited number of substructures in the absence of structural collapse, i.e.,  ∆ી ൌ ી െ ી෡௨ can be considered as a sparse vector with relative few non-zero components, where ી  and ી௨	 are the stiffness scaling parameters for current (possibly damaged) and undamaged states, and ી෡௨  is the MAP (maximum a posteriori) estimate of ી௨ . During the calibration stage with the undamaged structure, we assume that the model is a globally identifiable (Beck and Katafygiotis, 1998) in ી௨ based on a large amount of time-domain vibration data, so that the posterior for ી௨	 is tightly clustered around a unique MAP estimate ી෡௨.  In the Bayesian probabilistic framework, the potential parameter change ∆ી is modeled as a zero-mean Gaussian vector, based on the Principle of Maximum Information Entropy (Jaynes, 2003), where we assign independent variances ߙ௝  for each component ߠ௝.  This is inspired by the idea of using the ARD Gaussian prior in sparse Bayesian learning to introduce sparseness (Tipping, 2001).  We use this probability model for ∆ી  to construct a likelihood function for ી where the MAP value ી෡௨  from the calibration stage is treated as pseudo-data:          ݌൫ી෡௨|ી, હ൯ ൌ ∏ ࣨ൫ߠ෠௨,௝|ߠ௝, ߙ௝൯ேഇ௝ୀଵ           ൌ ࣨ൫ી෡௨|ી, ۯ൯              (8) where ۯ ൌ diag൫ߙଵ, … , ߙேഇ൯. 2.4. Likelihood function for system modal parameters ࣘ and ࣓ଶ Using the Principle of Maximum Information Entropy again, the combined prediction errors and measurement errors for the system modal parameters ૑ଶ and ૖ are modeled independently as zero-mean Gaussian variables with unknown variances, so:       ݌൫ૐ෡ ,૑ෝଶ|૖,૑ଶ, ી൯ 	ൌ ݌൫ૐ෡|૖൯݌ሺ૑ෝଶ|૑ଶሻ     =ࣨ൫ૐ෡|ડ૖, ۷ߟ൯ࣨሺ૑ෝଶ|ۺ૑ଶ, ۷ߩሻ	    (9) 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 4  where ડ ∈ Թே೚ேೞே೘ൈே೏ே೘  with “1s” and “0s” picks the observed degrees of freedom in the “measured” mode shape data set from the full system mode shapes ૖ ; 	ۺ ൌ ൣ۷ே೘, … , ۷ே೘൧் ∈Թேೞே೘ൈே೘	is the transformation matrix between the vector of  ௦ܰ  sets of ܰ௠	 identified natural frequencies ૑ෝଶ  and the ܰ௠ system natural frequencies ૑ଶ.	 Parameters ߟ  and ߩ  are prescribed variances for the predictions of the identified mode shapes ૐ෡  and natural frequencies ૑ෝଶ from system mode shapes ૖ and system natural frequencies ૑ଶ, respectively. 3. GIBBS SAMPLING ALGORITHM To simplify the notation, we denote the pseudo-data (MAP estimates) used for structural model updating as ऎ෡ ൌ ቂૐ෡், ሺ૑ෝଶሻ், ൫ી෡௨൯்ቃ் for the operational stage, while for the calibration stage, ऎ෡ ൌ ൣૐ෡், ሺ૑ෝଶሻ், ሺી଴ሻ்൧்	where ી଴  is a chosen nominal vector. For damage detection, our goal is to compute 	݌൫ી|ऎ෡൯ , which is the marginal posterior PDF of the stiffness scaling parameters. It is calculated from the full posterior PDF over all unknown parameters ݌൫૖,૑ଶ, ી, ߚ|ऎ෡൯  by marginalizing over the parameters ૖, ૑ଶ	and ߚ. Bayes’ theorem gives this full posterior PDF as the normalized product of the likelihood functions in (8) and (9) and the prior PDFs in (3) and (7) but the resulting expression is intractable because the high-dimensional normalizing integral cannot be computed analytically. Instead, we implement GS to draw posterior samples from ݌൫૖,૑ଶ, ી, ߚ|ऎ෡൯ by decomposing the whole model parameter vector into the four groups ሼ૖,૑ଶ, ી, ߚሽ	 and sampling from one parameter group conditional on the other three groups and the available data. In the next section, the full conditional posterior PDFs for each parameter group are derived. 3.1. Full conditional posterior PDFs  3.1.1. Conditional posterior PDF for	૖ The conditional posterior PDF over parameter vector 	૖  is obtained by using Laplace’s asymptotic approximation (Beck and Katafygiotis, 1998) to integrate the “nuisance” hyper-parameters ߟ out:   ݌൫૖|ऎ෡,૑ഥଶ, ીഥ, ̅ߚ൯  ൌ ݌׬൫૖|ऎ෡,૑ഥଶ, ીഥ, ̅ߚ, ߟ൯ ݌൫ߟ|ऎ෡ ,૑ഥଶ, ીഥ, ̅ߚ൯݀ߟ     ൎ ݌൫૖|ऎ෡,૑ഥଶ, ીഥ, ̅ߚ, ߟ෤൯                            (10) where ߟ෤ ൌ argmax ݌൫ߟ|ऎ෡ ,૑ഥଶ, ીഥ, ̅ߚ൯. The above approximation is based on the assumption that the posterior ݌൫ߟ|ऎ෡ ,૑ഥଶ, ીഥ, ̅ߚ൯  has a unique maximum at	ߟ෤ (the MAP value of ߟ). The posterior PDF of ૖ given the specified ߟ෤ is computed from Bayes' theorem: ݌൫૖|ऎ෡ ,૑ഥଶ, ીഥ, ̅ߚ, ߟ෤൯ ∝ ݌൫ૐ෡|૖, ߟ෤൯݌൫૖|૑ഥଶ, ીഥ, ̅ߚ൯(11) Combining the Gaussian prior ݌൫૖|૑ഥଶ, ીഥ, ̅ߚ൯ in (5) and the Gaussian likelihood ݌൫ૐ෡|૖, ߟ෤൯ in (9) gives a Gaussian posterior PDF ࣨ൫૖|ૄ૖, ઱૖൯	for ݌൫૖|ऎ෡,૑ഥଶ, ીഥ, ̅ߚ, ߟ෤൯ where the mean and covariance matrix are:                        ૄ૖=ߟ෤ିଵ઱૖ડ்ૐ෡  (12)                 ઱૖=൫̅۴ߚത்۴ത ൅ ߟ෤ିଵડ்ડ൯ିଵ   (13) where ۴ത ൌ ۴ሺીഥሻ. This leaves the task of maximizing the PDF  ݌൫ߟ|ऎ෡ ,૑ഥଶ, ીഥ, ̅ߚ൯ to find the MAP value ߟ෤:     ݌൫ߟ|ऎ,૑ഥଶ, ીഥ, ̅ߚ൯ ∝ ݌൫ૐ෡|૑ഥଶ, ીഥ, ̅ߚ, ߟ൯	݌ሺߟሻ     ∝ ݌׬൫ૐ෡|૖, ߟ൯ ݌൫૖|૑ഥଶ, ીഥ, ̅ߚ൯݀૖ ∙ ݌ሺߟሻ     ∝ ࣨ൫ૐ෡|૙, ۷ߟ ൅ ̅ିߚଵડሺ۴ത்۴തሻିଵડ்൯ ∙ ݌ሺߟሻ  (14)We absorb the PDF ݌ሺߟሻ into the proportionality constant in (14) since it is chosen as a broad uniform prior. Maximizing the logarithm of (14) with respect to ߟ	leads to:                   ߟ෤ ൌ ቂ୲୰൫઱૖ડ೅ડ൯ାฮૐ෡ିડૄ૖ฮమቃଶே೚ேೞே೘  (15) where trሺ∙ሻdenote the trace of a matrix. In (15), both the posterior mean ૄ૖	 and covariance matrix ઱૖	 depend on ߟ,  and so an iterative method using (12), (13) and (15) is required.    3.1.2. Conditional posterior PDF for	૑ଶ Following the same strategy as in Section 3.1.1, the conditional posterior PDF for ૑ଶ	is given by:       ݌൫૑ଶ|ऎ෡ ,૖ഥ, ીഥ, ̅ߚ൯ ൎ 	݌൫૑ଶ|ऎ෡ ,૖ഥ, ીഥ, ̅ߚ, ߩ෤൯          ∝ ݌ሺ૑ෝଶ|૑ଶ, ߩ෤ሻ݌൫૑ଶ|૖ഥ, ીഥ, ̅ߚ൯                 ൌ ࣨሺ૑ଶ|ૄ૑మ, ઱૑మሻ             (16) 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 5  where the mean and covariance matrix are:             ૄ૑మ ൌ ઱૑మ൫̅۵ߚഥ்̅܋ ൅ ߩ෤ିଵ்ۺ૑ෝଶ൯         (17)               ઱૑మ ൌ ൫ߩ෤ିଵۺ்ۺ ൅ ̅۵ߚഥ்۵ഥ൯ିଵ       (18)                 ۵ഥ ൌ ۵ሺ૖ഥሻ		, ̅܋ ൌ ܋ሺ૖ഥ, ીഥሻ              ߩ෤ ൌ argmax ݌൫ߩ|ऎ෡ ,૖ഥ, ીഥ, ̅ߚ൯.  An iterative method is used for ߩ෤ based on              ߩ෤ ൌ ቂ୲୰൫઱૑మۺ೅ۺ൯ାฮ૑ෝమିૄۺ૑మฮమቃଶே೚ே೘       (19) in conjunction with (17) and (18). 3.1.3. Conditional posterior PDF for	ી Similarly, the conditional posterior PDF for ી  is derived as:    ݌൫ી|ऎ෡ ,૖ഥ,૑ഥଶ, ̅ߚ൯ ൎ 	݌൫ી|ऎ෡ ,૖ഥ,૑ഥଶ, ̅ߚ, હ෥൯       ∝ ݌൫ી෡௨|ી, હ෥൯݌൫ી|૖ഥ,૑ഥଶ, ̅ߚ൯                                 ൌ ࣨሺી|ૄી, ઱ીሻ             (20) where                   ૄી=઱ી൫̅۶ߚഥ்̅܊ ൅ ۯ෩ી෡௨൯                (21)                    ઱ી=൫̅۶ߚഥ்۶ഥ ൅ ۯ෩൯ିଵ      (22)       ۶ഥ ൌ ۶ሺ૖ഥሻ, ̅܊ ൌ ܊ሺ૖ഥሻ, ۯ෩ ൌ diagሺહ෥ሻ              હ෥ ൌ argmax ݌൫હ|ऎ෡ ,૖ഥ, ̅ߚ൯.  An iterative method is used for ߙ෤௝ based on               ߙ෤௝ ൌ ሺ઱ીሻ௝௝ ൅ ൫ી෡௨ െ ૄી൯௝ଶ              (23) in conjunction with (21) and (22). For the calibration stage, however, model sparseness is not expected and hence optimization of હ  is not required.  We fix all components ߙ௝	 with large values (e.g. ߙ෤௝ ൌ 10ଽ).   (48) 3.1.4 Conditional posterior PDF for ߚ We get the conditional posterior PDF for	ߚ as: ݌൫ߚ|ऎ෡ ,૖ഥ,૑ഥଶ, ીഥ൯ ൌ න݌൫ߚ|ऎ෡ ,૖ഥ,૑ഥଶ, ીഥ, ܽ଴, ܾ଴൯ ݌ሺܽ଴, ܾ଴|ऎ෡ ,૖ഥ,૑ഥଶ, ીഥሻ݀ܽ଴ܾ݀଴ ൎ ݌൫ߚ|ऎ෡ ,૖ഥ,૑ഥଶ, ીഥ, ෤ܽ଴, ෨ܾ଴൯ ∝ ݌ሺ૖ഥ,૑ഥଶ, ીഥ|ߚሻ	݌൫ߚ| ෤ܽ଴, ෨ܾ଴൯ ∝ Gamሺܽ଴ᇱ , ܾ଴ᇱ ሻ.                                           (24) where ൣ ෤ܽ଴, ෨ܾ଴൧ ൌ argmax ݌ሺܽ଴, ܾ଴|ऎ෡ ,૖ഥ,૑ഥଶ, ીഥሻ and the shape parameter ܽ଴ᇱ  and rate parameter ܾ଴ᇱ  are given by:                      	ܽ଴ᇱ =	 ෤ܽ଴ ൅ ܰ௠ ௗܰ 2⁄                    (25)     ܾ଴ᇱ ൌ ෨ܾ଴ ൅ ∑ ฮሺ۹ሺીഥሻ െ ഥ߱௜ଶۻሻ૖ഥ௜ฮே೘௜ୀଵଶ 2ൗ .  (26) By assuming uniform priors over ܽ଴ and ܾ଴	, the posterior PDF for them is derived as:  ݌൫ܽ଴, ܾ଴หऎ෡ ,૖ഥ,૑ഥଶ, ીഥ൯ ∝ ݌ሺ૖ഥ,૑ഥଶ, ીഥ|ܽ଴, ܾ଴ሻ ∝ න݌ሺ૖ഥ,૑ഥଶ, ીഥ|ߚሻ	݌ሺߚ|ܽ଴, ܾ଴ሻ ݀ߚ ൌ Γሺܽ଴ ൅ ܰ௠ ௗܰ 2⁄ ሻሺܽ଴ ܾ଴⁄ ሻே೘ே೏/ଶΓሺܽ଴ሻሺ2ܽߨ଴ሻே೘ே೏/ଶ  ቄ1 ൅ ଵଶ௕బ ቀ∑ ฮ൫۹ሺીഥሻ െ ഥ߱௜ଶۻ൯૖ഥ௜ฮே೘௜ୀଵଶቁቅିே೘ே೏/ଶି௔బ (27) By direct differentiation of the logarithm of (27) with respect to ܾ଴ , the MAP estimate of ܾ଴  is given by:        ෨ܾ଴ ൌ ௔బே೘ே೏ ∑ ฮሺ۹ሺીഥሻ െ ഥ߱௜ଶۻሻ૖ഥ௜ฮே೘௜ୀଵଶ    (28) 3.2. Gibbs sampling algorithm The GS algorithm is summarized as follows. GS Algorithm for generating ܯ	samples 1.Initialize the samples with ̅ߚሺ଴ሻ ൌ 100,ሺ૑ഥଶሻሺ଴ሻ ൌ ∑ ૑ෝݎݏ2ܰݎൌ1 ܰݏ⁄ and ીഥሺ଴ሻ ൌ ી଴,  a chosen nominal vector, if in the calibration stage, while if in the monitoring stage, ીഥሺ଴ሻ ൌ ી෡௨. Let	݊ ൌ 1.2. For ݊ ൌ 1	to ܯ  3. Sample the system mode shapes     ૖ഥሺ௡ሻ ∼ ݌൫૖ഥ|ऎ෡ , ሺ૑ഥଶሻሺ௡ିଵሻ, ીഥሺ௡ିଵሻ, ̅ߚሺ௡ିଵሻ൯; 4. Sample the system natural frequencies     ሺ૑ഥଶሻሺ௡ሻ ∼ ݌൫૑ഥଶ|ऎ෡ ,૖ഥሺ௡ሻ, ીഥሺ௡ିଵሻ, ̅ߚሺ௡ିଵሻ൯; 5. Sample the precision parameter   ̅ߚሺ௡ሻ ∼ ݌൫ߚ|ऎ෡ ,૖ഥሺ௡ሻ, ሺ૑ഥଶሻሺ௡ሻ, ીഥሺ௡ିଵሻ൯; 6. Sample the model structural parameters               ીഥሺ௡ሻ ∼ ݌൫ી|ऎ෡ ,૖ഥሺ௡ሻ, ሺ૑ഥଶሻሺ௡ሻ, 	̅ߚሺ௡ሻ൯; 7. Let ݊ ൌ 	݊ ൅ 1 8. End for  Samples ቄ૖ഥሺ௡ሻ, ሺ૑ഥଶሻሺ௡ሻ, ߚതሺ௡ሻ, ીഥሺ௡ሻ: ݊ ൌ1,… ,ܯቅ are obtained. Assuming that the Markov chain created by the GS is ergodic (Gelman et al., 1995), the GS samples are distributed as the target joint PDF ݌൫૖,૑ଶ, ી, ߚ|ऎ෡൯  when ݊  is sufficiently large that the Markov chain reaches its stationary state. Samples from the marginal distribution ݌൫ી|ऎ෡൯ are then obtained by simply examining the GS samples ીഥሺ௡ሻ. According to our experience, GS is 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 6  ergodic when the regions of high values of ݌൫૖,૑ଶ, ી, ߚ|ऎ෡൯ are effectively connected. In the implementation of the GS algorithm, it is common to discard the samples before the Markov chain reaches its stationary state. The time period required for the Markov chain to reach its stationary state is called the burn-in period, but this is not easy to ascertain. In the algorithm, we determine the burn-in period simply by visual inspection of the Markov chain samples through time. To determine how many Markov chain samples to generate after burn-in is achieved, we check the convergence of the samples of interest, i.e. the stiffness scaling parameters	ી. 3.3. Evaluation of damage probability  The evaluation of possible damage location and severity is achieved by computing the probability that any stiffness parameter of a substructure has decreased more than a prescribed fraction ݂ . Using the samples of ી  obtained from the monitoring stage (possibly damaged) and the calibration stage (undamaged), we estimate the probability of damage using the following approximation:      ௝ܲௗ௔௠ሺ݂ሻ ൌ ܲ൫ߠ௝,௡ௗ ൏ ሺ1 െ ݂ሻߠ௝,௡௨ |ऎ෡ௗ,ऎ෡௨൯      ൎ ଵெ∑ ۷ൣ̅ߠ௝,௡ௗ ൏ ሺ1 െ ݂ሻ̅ߠ௝,௡௨ ൧ெ௡ୀଵ  (29)                                                    where ۷ሾ∙ሿ  is the indicator function, which is unity if the condition is satisfied, otherwise it is zero;  ̅ߠ௝,௡ௗ 	and ̅ߠ௝,௡௨  denote the ݄݊ݐ samples of the stiffness parameter of the ݆݄ݐ substructure for the monitoring and calibration stages, respectively, updated from the data ऎ෡ௗ ൌ  ൣ૑ෝ௨ଶ ,ૐ෡௨, ી෡௨൧  and ऎ෡௨ ൌ ൣ૑ෝ௨ଶ,ૐ෡௨, ી଴൧, respectively.  4. ILLUSTRATIVE EXAMPLE We apply the proposed approach to the brace-damage cases in the IASC-ASCE Phase II Simulated Structural Health Monitoring Benchmark structure (Bernal et al., 2002), which is a finite element model of a four-story, two-bay by two-bay steel braced-frame with 120 DOFs. Four damage patterns are considered which are simulated by reducing the elastic moduli of certain braces in the structural model. The reader is referred to Ching and Beck (2003, 2004) for detailed information. Both the full and the partial-sensor scenarios are considered in the experiment. For the full-sensor scenario (suffix .fs later), measurements are available at the center of each side at each floor with the directions parallel to the side in either the positive ݔ  direction or ݕ	 direction. While for the partial-sensor scenario (suffix .ps later), only measurements at the third floor and the roof are available. For the simulated monitoring data, ten segments of simulated discrete-time histories with equal length of 20s are generated for each measured DOF using the finite element model. Simulated noise is added with variance equal to 10% of the mean square of the time histories. Using the MODE-ID modal identification procedure, ten sets of modal data ( ௦ܰ ൌ 10) are identified and are shown in Ching and Beck (2003). For locating the faces sustaining brace damage, a 3-D 12-DOF shear-building numerical model is employed that has rigid floors and three DOFs per floor: translations parallel to the ݔ and ݕ axes and rotation about the z axis. The stiffness matrix ۹ is then parameterized as:               ۹ሺીሻ ൌ ۹଴ ൅ ∑ ∑ ߠ௦௙۹ഥ௦௙௙௦              (30)where 4,…,1=ݏ refers to the story number and f=	′ ൅ ݔᇱ, ′ െ ݔᇱ,ᇱ൅ ݕᇱ, ′ െ ݕᇱ indicates the face of the respective floor. Four stiffness parameters are used for each story to give a stiffness scaling parameter vector ી with 16 components.   The sub-structure stiffness reductions for the above four brace damage patterns are summarized as follows: 1) DP1B: 11.3% reduction in ߠଵ,ା௬  and ߠଵ,ି௬ ; 2) DP2B: 5.7% reduction in ߠଵ,ା௬  and ߠଵ,ି௬ ; 3) DP3B: 11.3% reduction in ߠଵ,ା௬  and ߠଵ,ି௬;  5.7% reduction in ߠଷ,ା௬  and ߠଷ,ି௬; 4) DP3Bu: 11.3% reduction in ߠଵ,ି௬ and 5.7% reduction in ߠଷ,ି௬.  Applying the GS procedure in both the calibration (undamaged) and monitoring (damaged) stages, Markov chain samples of all stiffness scaling parameters ી , system modal parameters ૖  and ૑ଶ,  and equation error 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 7  precision ߚ	are obtained. We focus here on the results of the monitoring stage. The Markov chain samples of the four stiffness scaling parameters at the “൅ݕ"  face in each story are shown in Figure 1, for the DP3B.fs and DP3B.ps cases. The Markov chain seems to reach its stationary state after a burn-in period of roughly 1000 samples. In Figure 2, all 3000 samples after burn-in are plotted in the ൛ߠଵ,ା௬,ߠଶ,ା௬ൟ space and ൛ߠଷ,ା௬,ߠସ,ା௬ൟ  space. It is clearly seen that the reduction of the sub-structure stiffnesses at the “ ൅ݕ"  face in the first and third stories are correctly detected for this case. Moreover, the amount of the identified stiffness loss is roughly correct, i.e., using the mean of the sample values, there is about 12% loss in ߠଵ,ା௬, and 6% loss in ߠଷ,ା௬. Comparing the results of the partial-sensor and full-sensor scenarios in terms of their posterior uncertainty in ી , it is clear that the samples spread more out in the ી space for the DP3B.ps case, because there is less modal data to constrain inference of the model parameters.  The probability of damage estimated using (30) for the sixteen stiffness scaling parameters in the four damage cases are shown in Figures 3 and 4, for the full and partial sensor scenarios, respectively. For all stiffness scaling parameters in the full-sensor scenarios (Figure 3), the damage patterns are reliably detected in both qualitative and quantitative ways. All the identified stiffness ratios are close to their actual values. Furthermore, no occurrence of false damage detection is observed in the full-sensor scenario. This is a benefit of the proposed sparse Bayesian formulation which reduces the uncertainty of the unchanged components. It produces sparse models by learning the hyper-parameter હ, where ߙ෤௝ → 0	implies that ሺ઱ીሻ௝௝ →0  and ીഥ௝ → ൫ી෡௨൯௝. From the results of the partial-sensor scenario in Figure 4, it is seen that the effectiveness of the proposed approach degrades somewhat because of the fewer measured DOFs and a false damage detection occurs in this case: the mean damage fraction for  ߠଶ,ା௫ is 2% for the DP3Bs.ps case, where no actual damage occurs in this substructure. In addition, the mean damage fraction in ߠଵ,ି௬	 is under-estimated as 8%, compared to the actual damage of 11.3%.   Figure 1: Posterior Markov chain samples for the stiffness scaling parameters for the DP3B case.   Figure 2: 3000 posterior samples of structural stiffness scaling parameters plotted in the ൛ߠଵ,ା௬,ߠଶ,ା௬ൟ space and 	൛ߠଷ,ା௬,ߠସ,ା௬ൟ  space, for (a), (b): the DP3BU.fs and (c),(d):	the DP3BU.ps case  Figure 3: Estimated damage probability curves for each substructure: (a) DP1B.fs; (b) DP2B.fs; (c) DP3B.fs; (d) DP3Bu.fs.  0 2000 40000.80.91 1,+yDP3B.fs0 2000 40000.80.91 DP3B.ps0 2000 40000.911.1 2,+y0 2000 40000.911.10 2000 40000.911.1 3,+y0 2000 40000.911.10 2000 40000.911.1n 4,+y0 2000 40000.911.1n-0.05 0 0.05 0.1 0.15 0.200.51-0.05 0 0.05 0.1 0.15 0.200.51-0.05 0 0.05 0.1 0.15 0.200.51probability of Exceedance, P jdam-0.05 0 0.05 0.1 0.15 0.200.51Damage extent, f(a)(b)(c)(d)1,-y1,-y1,+y3,-y1,-y1,+y1,+y3,+y3,-y 1,-y12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 8   Figure 4: Estimated damage probability curves for each substructure: (a) DP1B.ps; (b) DP2B.ps; (c) DP3B.ps; (d) DP3Bu.ps.  5. CONCLUSION We have presented a sparse Bayesian learning approach with Gibbs sampling for inferring substructural stiffness loss from noisy incomplete modal data. The effective dimension of the parameter updating with this algorithm only depends on the four parameter groups that are used for constructing the conditional PDFs to draw samples; this is very important for real applications that involves estimation of high-dimensional model parameter vectors. Moreover, the new approach is capable of estimating the uncertainty the system modal parameters as well as the eigenequation-error precision parameters from the Gibbs samples. The damage detection results show that the new approach is effective in identifying local structural damage. The occurrence of missed and false damage detections is effectively suppressed.  6. ACKNOWLEDGMENTS This research was done when the first author was a Postdoctoral Scholar at Caltech. This work was supported by the NSF (Award No. EAR-0941374) and the NSFC (Grant No. 51308161). 7. REFERENCES Beck, J.L., and Katafygiotis, L.S. (1998). “Updating models and their uncertainties. I: Bayesian statistical framework.” Journal of Engineering Mechanics, 124 (4) (1998), 455–461. Bernal, D., Dyke, S.J., Lam, H.-F., and Beck, J.L. (2002). “Phase II of the ASCE benchmark study on SHM.” In: Proceedings of the 15th Engineering Mechanics Division Conference of the American Society of Civil Engineering, 125(9), 1048–1055 Ching, J., and Beck, J.L. (2003) “Two-step Bayesian structure health monitoring approach for IASC-ASCE phase II simulated and experimental benchmark studies.” Technical Report EERL 2003-02, Caltech, Pasadena, CA. Ching, J., and Beck, J.L. (2004). “New Bayesian model updating algorithm applied to a structural health monitoring benchmark.” Structural Health Monitoring, 3 (4) (2004), 313–332. Ching, J., Muto, M., and Beck, J.L. (2006). “Structural model updating and health monitoring with incomplete modal data using Gibbs Sampler." Computer-Aided Civil and Infrastructure Engineering, 21 (4) (2006), 242–257.  Farrar, C.R., and Worden, K. (2013). “Structural Health Monitoring: a Machine Learning Perspective.” John Wiley& Sons. Gelfand, A.E., Hills, S.E., Racine-Poon, A., and Smith, A.F.M. (1990). “Illustration of Bayesian inference in normal data models using Gibbs sampling.” Journal of American Statistical Association, 85 (1990), 972–85. Geman, S., and Geman, D. (1984). “Stochastic relaxation, Gibbs distribution and the Bayesian restoration of images.” IEEE Transactions on Pattern Analysis and Machine Intelligence, 6: 721–41 Gelman, A., Carlin, J. S., Stern, H. S. and Rubin, D. B. (1995). “Bayesian Data Analysis.”  Chapman & Hall. Huang, Y., Beck, J.L., Wu, S., and Li, H. (2014) “Robust Bayesian compressive sensing for signals in structural health monitoring.” Computer-Aided Civil and Infrastructure Engineering, 29(3), 160–179 Huang, Y., and Beck, J.L. (2015). “Hierarchical sparse Bayesian learning for structural health monitoring with incomplete modal data”, International Journal for Uncertainty Quantification, (In press), arXiv: 1408.3685. Jaynes, E.T. (2003). “Probability Theory: The Logic of Science.” Cambridge University Press. Ji, S., Xue, Y., and Carin, L. (2008). “Bayesian compressive sensing." IEEE Transactions on Signal Processing, 56(6), 2346 –2356. Tipping, M.E. (2001). “Sparse Bayesian learning and the relevance vector machine.” Journal of Machine Learning Research, 1, 211–244. -0.05 0 0.05 0.1 0.15 0.200.51-0.05 0 0.05 0.1 0.15 0.200.51-0.05 0 0.05 0.1 0.15 0.200.51probability of Exceedance, P jdam-0.05 0 0.05 0.1 0.15 0.200.51Damage extent, f(b)(c)(d)1,-y1,-y(a)1,+y1,+y1,-y3,-y2,+x3,+y1,+y1,-y3,-y

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