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Diameter increment models for the mixed swamp forests of Sarawak Chai, Francis 1991

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DIAMETER INCREMENT MODELS FOR THE MIXED SWAMP FORESTS OF SARAWAK By Fr.«?,ncis Cliai Yan Chiew B. Sc. (Forestry) Agriculture University of Malaysia A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF T H E REQUIREMENTS FOR T H E DEGREE OF MA S T E R OF FORESTRY in T H E FACULTY OF GRADUATE STUDIES DEPARTMENT OF FORESTRY We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA August 1991 © Francis Chai Yan Chiew, 1991 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of F O R E S T R Y r The University of British Columbia Vancouver, Canada Date A d & K S r / o , n°t\ DE-6 (2/88) Abstract Information on the growth and yield of Mixed Swamp Forests in Sarawak is inadequate. Tree diameter increment models were built as an initial effort towards developing a growth and yield modelling system for forests in Sarawak. First, the level of modelling for better predictions of diameter increment of commer-cial trees was investigated. The three levels of modelling were: (1) developing a single equation for all commercial species, (2) a separate equation for each of two groups, (3) a separate equation for each individual species. Second, linear and nonlinear models for seven important commercial species were developed. Diameter increment was modelled as a function of both tree and stand predictor variables. Only one nonlinear function, based on theoretical concepts, was attempted. Third, the usefulness of crown position and previous diameter growth for predicting current diameter growth was examined. Several objective measures such as mean bias, mean squared bias, and mean absolute deviation were calculated from validation data to indicate accurracy of predictions. Modelling at the individual species level resulted in better predictions than at the single group or two group level. The linear weighted models tested as better predictors than the nonlineax model. Previous diameter growth, in the absence of better predictor variables, was found to be more useful than crown position for predicting current diameter growth. n Table of Contents Abstract ii List of Tables vii List of Figures x Acknowledgement xii 1 Introduction 1 2 Description of the Sarawak Forests 5 2.1 Peat Swamp Forests 6 2.2 Management and Silviculture in Mixed Swamp Forests 9 3 Literature Review 11 3.1 Growth and Yield Modelling in General 11 3.1.1 Whole Stand Models 11 3.1.2 Stand Class Models 13 3.1.3 Individual Tree Models 16 3.2 Growth and Yield in the Tropics 18 3.2.1 Modelling Approaches 18 3.2.2 Diameter Increment 20 3.2.3 Ingrowth 23 3.2.4 Site Quality/Competition 23 iii 4 Materials and Methods 25 4.1 Study Data Base 25 4.1.1 Location of the Data 25 4.1.2 Measurement Procedures 27 4.2 Data Preparation 31 4.3 Data Analysis 34 4.3.1 Selection of Modelling Level 34 4.3.2 Model-Building 35 4.3.3 Model Evaluation 42 4.3.4 Importance of Crown Position and Previous Diameter Growth For Prediction 45 5 Results 47 5.1 Level of Modelling for Better Prediction 47 5.1.1 Pseudo-R2 50 5.1.2 Mean Bias and Mean Squared Bias 50 5.1.3 Mean Absolute Deviations 55 5.2 Model-Building 56 5.2.1 Linear Models 56 5.2.2 Nonlinear Models 64 5.2.3 Validation of Models 67 5.3 Importance of Crown Position and Previous Diameter Growth For Prediction 71 5.3.1 Crown Position (CP) 71 5.3.2 Previous Diameter Growth (LDG) 72 6 Discussion 74 6.1 Level of Modelling for Better Prediction 74 iv 6.2 Linear Models 75 6.3 Linear Versus Nonlinear Models 77 G.3.1 Accuracy of Prediction 77 6.3.2 Other Considerations 78 6.4 Importance of Crown Position (CP) and Previous Diameter Growth (LDG) For Prediction 80 7 Conclusions and Recommendations 82 7.1 Conclusions 82 7.2 Recommendations 83 References Cited 85 Appendices 95 A List of Desirable and Acceptable Species in the Mixed Swamp Forest 95 A . l Desirable Species (List 1) 95 A. 2 Acceptable Species (List 2) 96 B List of Species by Growth and Shade Tolerance 98 B. l Desirable Species: 98 B . l . l Moderately Fast or Fast Growing, mainly Light Demanding: . . . 98 B.1.2 Mainly Slow Growing and Shade Tolerant: 99 B.2 Acceptable Species 99 B.2.1 Fast Growing, mainly Colonizing Species 99 B.2.2 Slow Growing, Shade Tolerant 100 C Model-Building for Species Groups 101 v C . l 'Best Models' from All Possible Regressions 101 C . l . l ' A l l Trees' Data 101 C.1.2 'Best Trees' Data 101 C. 2 Nonlinear Models 101 C.2.1 'AU Trees' Data 101 C.2.2 'Best Trees' Data 101 D Distribution of Bias by Predicted Diameter Increment and Diameter 105 D. l Plots of Bias for ' A l l Trees' Validation Data 105 D.2 Plots of Bias for 'Best Trees' Validation Data 105 vi List of Tables 2.1 Distribution of Peat Swamp Forest (xlOOO hectares) 8 4.1 List of Yield Plots in Mixed Swamp Forests 27 4.2 Recoding of Crown Position 39 5.1 Some Fit Statistics for the Common Model, ' A l l Trees' Model-Building Data 48 5.2 Some Fit Statistics for the Common Model, 'Best Trees' Model-Building Data 49 5.3 Pseudo-R2 for the Common Equation for Level of Modelling, Validation Data 50 5.4 Mean Bias and Mean Squared Bias for the Common Equation, ' A l l Trees' Validation Data 51 5.5 Mean Bias and Mean Squared Bias for the Common Equation, 'Best Trees' Validation Data 51 5.6 Mean Bias (cm) per tree by Diameter Class for ' A l l Trees' Validation Data 52 5.7 Mean Bias (cm) per tree by Diameter Class for 'Best Trees' Validation Data 53 5.8 Calculated Standard Error of Estimates and Percentage Error Prediction, ' A l l Trees' Validation Data 54 5.9 Calculated Standard Error of Estimates and Percentage Error Prediction, 'Best Trees' Validation Data 55 5.10 Mean Absolute Deviations (cm) from Validation Data Sets . . . . . . . . 56 5.11 Mean Absolute Deviation (cm) for ' A l l Trees' Validation Data 57 5.12 Mean Absolute Deviation (cm) for 'Best Trees' Validation Data 58 vii 5.13 Results of'Best Model' from Al l Possible Regressions for ' A l l Trees' Data 60 5.14 Results of 'Best Model' from Al l Possible Regressions for 'Best Trees' Data 60 5.15 Results of Adding Interaction Terms to 'Best' Model for ' A l l Trees' Data 61 5.16 Summary of Goldfeld-Quandt Test on the 'Final' Model, ' A l l Trees' Data 61 5.17 Summary of Goldfeld-Quandt Test on the 'Final' Model, 'Best Trees' Data 62 5.18 Summary of Goldfeld-Quandt Test on the Weighted Equation, ' A l l Trees' Data 62 5.19 Summary of Goldfeld-Quandt Test on the Weighted Regression, 'Best Trees' Data 63 5.20 Statistics of Unweighted Linear Regression for Ramin, 'Best Trees' Data 63 5.21 Parameter Estimates of Weighted Linear Regression for Shorea spp, ' A l l Trees' Data 64 5.22 Parameter Estimates of Weighted Linear Regression for R A M I , JONG, SEPP and K A P A , ' A l l Trees' Data 65 5.23 Parameter Estimates of Weighted Linear Regression for Shorea spp, 'Best Trees' Data 66 5.24 Parameter Estimates of Weighted Linear Regression for JONG, SEPP and K A P A , 'Best Trees' Data 66 5.25 Parameter Estimates for Nonlinear Model, ' A l l Trees' Model-Building Data 67 5.26 Parameter Estimates for Nonlinear Model, 'Best Trees' Model-Building Data 67 5.27 Pseudo-R2, Mean Bias, Mean Squared Bias (MSB) and Mean Absolute Deviation (MAD) by Model, ' A l l Trees' Validation Data 68 5.28 Pseudo-R2, Mean Bias, Mean Squared Bias and Mean Absolute Deviation by Model, 'Best Trees' Validation Data 69 5.29 Pseudo-R2, Mean Bias, Mean Squared Bias and Mean Absolute Deviation by Model and Data Set : 70 viii 5.30 Mean Bias, Mean Squared Bias and Mean Absolute Deviation for the Models by Diameter Class 70 5.31 Mean Squared Bias, Mean Absolute Deviation, Percentage Error Predic-tion for Model With and Without Crown Position 71 5.32 Mean Squared Bias, Mean Absolute Deviation and Percentage Error Pre-diction for Model With and Without Competitive Index 72 5.33 Mean Squared Bias, Mean Absolute Deviation and Percentage Error Pre-diction for Model With and Without L D G 73 C . l Results of 'Best Model' for Species Groups from A l l Possible Regressions for ' A l l Trees' Data 102 C.2 Parameter Estimates of Unweighted Linear Regression for Species Groups, ' A l l Trees' Data 103 C.3 Results of 'Best Model' for Species Groups from Al l Possible Regressions for 'Best Trees' Data 103 C.4 Parameter Estimates of Unweighted Linear Regression for Species Groups, 'Best Trees' Data 104 C.5 Parameter Estimates for Species Group Nonlinear Model, ' A l l Trees' Data 104 C.6 Parameter Estimates for Species Group Nonlinear Model, 'Best Trees' Datal04 IX List of Figures 2.1 Peat Swamp Forests of Sarawak 7 4.1 Layout of Plot and Quadrat of Sarawak Yield Plot 26 4.2 Silvicultural Scoring of Crown Position in Yield Plots 29 4.3 Silvicultural Scoring of Crown Form in Yield Plots 30 5.1 Residuals Plot for Unweighted Linear Model for Ramin, 'Best Trees' Data 59 D . l Bias, BI1 against Predicted Diameter Increment, DIP1, ' A l l Trees', One Group Modelling 106 D.2 Bias, BI2 against Predicted Diameter Increment, DIP2, ' A l l Trees', Two Group Modelling 107 D.3 Bias, BI3 against Predicted Diameter Increment, DIP3, ' A l l Trees', Species Modelling 108 D.4 Bias, BI1 against Diameter, D, ' A l l Trees', One Group Modelling . . . . 109 D.5 Bias, BI2 against Diameter, D, ' A l l Trees', Two Group Modelling . . . . 110 D.6 Bias, BI3 against Diameter, D, ' A l l Trees', Species Modelling I l l D.7 Bias, BI1 against Predicted Diameter Increment, D I P l , 'Best Trees', One Group Modelling 112 D.8 Bias, BI2 against Predicted Diameter Increment, DIP2, 'Best Trees', Two Group Modelling 113 D.9 Bias, BI3 against Predicted Diameter Increment, DIP3, 'Best Trees', Species Modelling 114 x D.10 Bias, B I l against Diameter, D, 'Best Trees', One Group Modelling . . . . 115 D . l l Bias, BI2 against Diameter, D, 'Best Trees', Two Group Modelling . . . 116 D.l2 Bias, BI3 against Diameter, D, 'Best Trees', Species Modelling 117 XI Acknowledgement I would like to thank the following people and organizations for making this study a reality. My appreciation and thanks go to the State Government of Sarawak for granting me study leave. Many thanks go to the Director of Forests and the other members of the the directorate in the Sarawak Forest Department for their support. Thanks are due to the A S E A N Institute of Forest Management for financial assistance. Thanks also go to the staff members of the Forest Research Branch for collecting the data used in my study. The Faculty of Forestry, U B C also generously provided me computer funding for the extensive statistical analyses. I would like to thank my supervisor, Dr. V . LeMay particularly, and the other members of my thesis committee, namely Drs. P. Marshall, A. Kozak and M . Bonnor for their direction and guidance. Also, thanks are due to the other Forestry Biometrics graduate students for their advice and help when needed. I would also like to thank my wife, Lily, for her patience, constant support and encouragement. xii Chapter 1 Introduction The growth and yield of the Mixed Swamp Forests of Sarawak, is a subject of importance and interest for many reasons. First, the Mixed Swamp Forest is the most extensive forest type in the Peat Swamp Forests constituting 80 % of its area. Second, due to the easy access and the occurrence of the highly valuable Ramin (Gonystylus bancanus (Miq.) Kurz), these were the first forests to be logged commercially. These forests have contributed significantly to the economic growth of Sarawak particularly from 1950 to 1970. Ramin, initially, and other mixed swamp species form the basis of a major log export industry. Log production from the interior hill forests, started in the late 1960's, has now overtaken swamp log production. Third, since 1970, the Sarawak Forest Depart-ment has an effective control over the management of the Peat Swamp Forests. It is the Forest Department's objective to manage the Mixed Swamp Forests on a sustained yield basis. This includes the imposition of minimum diameter cutting limits for commercial tree species harvested and an empirical cutting cycle of 45 years. Fourth, since 1961, the logged-over Mixed Swamp Forests have been silviculturally treated and by the end of 1988, approximately 220,000 ha were treated within a year after they were harvested. Fifth, a series of 62 permanent sample plots have been established since 1972 in the regenerating forests after harvesting. These plots, called Yield Plots, were periodically measured since establishment to study the development of the logged-over forests and to predict the next harvest. Sixth, very few studies on the growth and yield of Peat Swamp Forests have been done in the past. One study by FAO (1974a) was based on the growth 1 Chapter 1. Introduction 2 rates and mortalities of dominant and codominant trees from six research plots. Projec-tions were made for the length of optimum cutting cycle by species/species group. This study suggested 40 to 60 years to obtain the greatest mean annual volume increment. Seventh, as the area of old-growth Peat Swamp Forests continues to decrease, the pres-sures for cutting the second-growth are mounting. Finally, the number of known species and species that are commercially important in the Peat Swamp Forests is much less than in the Hill Mixed Dipterocarp Forests, and therefore growth of individual species or groups is easier to model. A large gap exists in the knowledge of growth and yield modelling between tropical and temperate forests. In temperate countries, models range from relative^ simple variable density whole stand models to more complex diameter distribution models, whereas in the tropics such models are greatly lacking. While no complete or comprehensive growth modelling system exists in Peninusular Malaysia (Wan Razali, 1986), a computer model based on the classical stand projection technique was developed in Sarawak in 1982 with the assistance of FAO (Kofod, 1982). This stand projection model was based on mean rates of growth; the accuracy of predicted values depends largely on the input data. Research is urgently needed in developing models for important growth components. One such component is diameter growth. For this study, the development of diameter growth models was restricted to the Mixed Swamp Forests of Sarawak for reasons already explained. These growth models were fitted based on two-thirds of the available data and validated by using the remaining one-third of the data. The emphasis was on developing a model for each of the following species groups and/or individual species: 1. Non-commercial trees; 2. Al l commercial trees; Chapter 1. Introduction 3 3. Commercial trees, fast growing light demandersj 4. Commercial trees, slow growing shade tolerants; 5. The following individual commercial species: (a) Meranti species group (Shorea spp) (b) Ramin (Gonystylus bancanus (Miq.) Kurz.) (c) Jongkong (Dactylocladus stenostachys Oliv.) (d) Sepetir (Copaifera palustris (Sym.) De Wit) (e) Kapur (Dryobalanops rappa Becc). Both linear and nonlinear regression models were developed for predicting diameter increment from a list of independent variables. Specifically, this study sought to answer the following questions on diameter growth of trees in the Mixed Swamp Forests: Question 1 Within linear models, modelling can be done by developing a single equation for all commercial species, separately for two groups (light demanders and shade toler-ants), or based on individual commercial species. Which level of modelling is better in terms of prediction? Question 2 Are linear models better predictors than nonlinear models at the individual species level of modelling? Question 3 Does including crown position in the model add significantly to predictions? Chapter 1. Introduction 4 Question 4 Does including the previous diameter growth in the model improve the prediction of diameter growth? This thesis has been organized in the following format. Chapter 2 gives a description of the types of forests in Sarawak, particularly the Mixed Swamp Forest. An account of the silviculture and management of the Mixed Swamp Forest is also given. Chapter 3 is a literature review of growth and yield modelling in general and in the tropics, the latter emphasizing tree diameter increment models. Chapter 4 details the materials and methods used in modelling diameter increment while Chapter 5 presents the results of the modelling efforts. The results are discussed in Chapter 6 while some recommendations and conclusions are drawn in Chapter 7. Chapter 2 Description of the Sarawak Forests Sarawak, on Borneo Island, is the biggest of fourteen states of the Federation of Malaysia. The total land area is 12.4 million ha of which 9.4 million ha are forested. The climate of Sarawak is hot and humid with an average daily temperature of 27° C at sea level. The mean annual rainfall is 3000 to 4000 mm. There is no distinct dry season and the North East Monsoon brings heavy rainfall to the State from November to February. The natural vegetation is largely an evergreen mixed tropical rainforest dominated by the family Dipterocarpaceae. Forest management in Sarawak is under the jurisdiction of the State Government; the Sarawak Forest Department is the agency responsible for managing the forests. The forests of Sarawak can be classified as: (1) Beach Forest, (2) Mangrove Forest, (3) Peat Swamp Forest, (4) Kerangas Forest, (5) Hill Mixed Dipterocarp Forest. The Beach Forest and the Kerangas Forest are non-commercial. The Beach Forest is restricted to sandy coastal soils and is small in area. The Kerangas Forest is located on white sandy soils, on beach terraces, or on weathered steep ridges. The Mangrove Forests are found along the more sheltered areas of the coast and in river estuaries. The main commercial species within this type are of the genera Rhi-zophora and Bruguiera. The major commercial use is production of wood chips for export for the manufacture of rayon. The Mangrove Forests have been traditionally used by coastal inhabitants as sources of poles, fuelwood, charcoal, housing thatch, and sugar. The Peat Swamp Forests cover about 14,700 km 2 and have made a major contribution to the state economy since 1946 when Ramin was first accepted in the international 5 Chapter 2. Description of the SarawaJc Forests 6 market. Currently, about 50 different tree species are harvested but the main ones are: Ramin : Gonystylus bancanus (Miq.) Kurz Jongkong : Dactylocladus stenostachys Oliv. Swamp merantis : Shorea spp. Sepetir : Copaifera palutris (Sym.) De Wit Kapur paya : Dryobalanops rappa Becc. Alan : Shorea albida Sym. The Hill Mixed Dipterocarp Forest, which comprises 63 % of the total land area, occupies the area from the inland limit of the fresh water swamp forests to the lower limit of the montane forests. This forest is dominated by the family Dipterocarpaceae accounting for 65 to 80 % of the net industrial stemwood volume of trees greater than 30 cm at reference height (FAO, 1974b). 2.1 Peat Swamp Forests As this study focuses on the Peat Swamp Forests, a further description follows. The Peat Swamp Forests are located behind the coastline and extend inland along the lower reaches of the main river systems. They extend over approximately 1.5 million ha and are concentrated in the vicinity of the towns of Sri Aman (Simanggang), Sibu and Miri (Figure 2.1). The distribution of the Peat Swamp Forest is given in Table 2.1 (Source: Lee, 1979). While it is to be noted that the entire area of Peat Swamp Forest is public land, only 47 % (Forest Reserves and Protected Forests) is under the control of the Forest Department. The peat swamps are dome-shaped in structure, which is more pronounced further from the sea. The depth of peat ranges from 7 to 15 m. The peat is a mass of semi-decomposed and undecomposed woody materials containing roots, branches and tree Chapter 2. Description of the Sarawak Forests 7 Source: FAO, 1974a Figure 2.1: Peat Swamp Forests of Sarawak Chapter 2. Description of the Sarawak Forests 8 Table 2.1: Distribution of Peat Swamp Forest (xlOOO hectares) Forest Type Forest Reserves and Stateland Total Protected Forests Forests Mixed Swamp 435 739 1174 Alan Batu 114 13 127 Alan Bunga 68 8 76 Padang Alan 37 4 41 Padang Paya 35 2 37 Total 689 766 1455 Source: Lee, 1979 trunks. The subsoil of the peat swamp is a stiff clay and white or yellow in colour. Plant nutrients are very low. The peat samples on ignition lose more than 75 % in weight and have pH usually less than 4.0. Anderson (1961) described the Peat Swamp Forests of Sarawak in detail and divided them into six types or phasic communities based on structure, physiognomy and flora. For practical planning, the Forest Department has grouped the various phasic communities into five distinct forest types (FAO, 1974a). These generally occur in the sequence: 1. Mixed Swamp Forest on the outer rim 2. Alan Batu Forest 3. Alan Bunga Forest 4. Padang Alan Forest 5. Padang Paya Forest at the apex of the dome The Mixed Swamp Forest is typically uneven-aged, multi-storied, with many com-mercial species. Dominant trees may have heights of 30 to 45 m and diameters of 56 to well over 81 cm (Lee, 1979). Adjacent to it is a similar forest type called the Alan Batu Chapter 2. Description of the Sarawak Forests 9 Forest which is dominated by large scattered alan trees. The Alan Bunga Forest is even canopied and the alan is almost the sole dominant tree. The Padang Alan Forest is dense and polelike, composed of almost pure alan with diameters seldom exceeding 40 cm. In the centre, the Padang Paya Forest is actually a group of several minor types. This may be characterized by dense stands of pole size trees, not dominated by alan, or by an open savanna-like forest comprised of stunted trees and patches of shrubs. 2.2 Management and Silviculture in Mixed Swamp Forests The Mixed Swamp Forests are the most extensive and important of all the Peat Swamp Forest types and cover 80 % of the total peat swamp area. The Peat Swamp Forest areas under governmental control are divided into a series of Regional Management Plan areas. Each Regional Management Plan area is a group of self-sustaining permanent forest areas. Prior to logging, an inventory is done to estimate the minimum timber volume above a desired cutting limit. The annual harvest and the corresponding area to cut is then prescribed based on this estimate. The minimum cutting diameter outside bark at breast height (dbhob) limits are usually 30 cm for ramin, no limit for alan, and 45 cm for other commercial tree species. Approximately 8000 to 10000 ha have been logged annually and the average commercial yield is 85 m 3 per ha (Lee, 1979). In the harvested forests, within a year after logging, a silvicultural treatment is per-formed. This involves the poison-girdling of all unsound, damaged, or badly shaped trees of all undesirable species over 20 cm diameter at breast height (Lee and Lai, 1977). In these treated forests, usually ten or more years after logging, permanent sample plots called yield plots were established. These one ha sized plots were established at an inten-sity of 0.25 % or one in 400 ha to collect information on growth rates of potential crop trees, on recruitment, and mortality (Lee and Lai, 1977). Chapter 2. Description of the Sarawak Forests 10 Each yield plot is further subdivided into one hundred 10 m X 10 m subplots or quadrats. In each quadrat, the best potential crop tree greater or equal to 5 cm dbhob with a good crown and stem was recorded for its size, crown position, crown form, and for its impeder or competitor. The impeder or competitor may be a vigorous woody climber, an undesirable species tree, or even a defective tree of commercial species outside the quadrat, which is shading the best potential crop tree. This assessment was done more frequently the first five years after establishment, and at least once in every five years, thereafter. Between 1972 and 1977, a total of 42 yield plots were established and measured as just described. Starting in 1983, all trees > 10 cm diameter at breast height over bark (dbhob) were identified by species and their crown positions and sizes recorded. Another 19 plots were established in 1987 for which more detailed tree measurements were recorded such as tree lean, stability, decay and presence of climbers. This detailed measurement procedure has since been adopted for the older established plots in their latest assessments. Chapter 3 Literature Review 3.1 Growth and Yield Modelling in General Growth and yield information is vital to forest management planning. The uses of growth and yield information are many. The two main applications for growth and yield infor-mation are in production planning and in silvicultural research (Alder, 1980). Growth and yield information can be used for (1) updating and projecting forest inventories, (2) determining harvest levels/allowable cut, (3) harvest scheduling, and (4) analysing alternative stand or silvicultural treatments (Wan Razali, 1986). Three broad approaches (Vanclay, 1988a) to growth and yield modelling can be clas-sified as: 1. whole stand models; 2. stand class or size class distribution models; and 3. individual tree models. 3.1.1 Whole Stand Models Whole stand models use stand statistics such as stocking, age, basal area, and site in-dex as inputs. This category ranges from simple yield tables or growth percentages to complicated mathematical equations or functions. Their appeal to forest managers rest in their ability to utilize conventional forest inventory information, the fast computation 11 Chapter 3. Literature Review 12 time, and the simplicity (Munro, 1974). The disadvantages are that individual tree infor-mation is totally lacking (Munro, 1974) and stand structure information .may sometimes be lacking (Wan Razali, 1986). The development of diameter distribution models seems to overcome this difficulty. Hann and Bare (1981) indicated that the lack of individual tree information was not critical as this was not needed to answer most of the uneven-aged management questions. Vanclay (1988a) stated that "whole stand models provide forecasts which are too general to be of much use in mixed forests". The earliest models in this category were the yield tables, namely the normal, empir-ical or variable density yield tables. Normal tables are constructed from data of stands judged to be fully stocked or normal, while empirical tables use plots of average stocking. Variable density yield tables tabulate yields for a range of densities for each age and site combination. Since the 1960's, differential or difference equations have been used to express the rate of change of various components of stands (Ek and Monserud, 1975). Numerical integration of the rate equations over time provide yield predictions (Clutter, 1963). Moser and Hall (1969) derived time independent nonlinear yield functions from the rate equations for uneven-aged hardwood stands. Moser (1972) quantified the three basic growth components namely ingrowth, mortality and survivor growth and then (Moser, 1974) developed equations for predicting these by size classes. Ek's (1974) mixed species model described ingrowth, mortality, and survivor growth in terms of the number of stems per two inch diameter classes with nonlinear difference models. Ek's system of equations is recursive in nature, while those of Moser employs the concept of simultaneous equations. A recursive system has functions which depend on only initial conditions. An advantage of the recursive system is that it facilitates the application of mathematical programming to develop optimal tree size class distribution (Adams and Ek, 1974). Lynch and Moser (1986) described a technique for predicting stand tables for each Chapter 3. Literature Review 13 of two species groups in a mixed stand by a system of differential equations relating the rates of change in per acre values of basal area, sum of diameters, and number of trees. Yandle et al. (1987) described a model for mixed hardwood stands in which the assumption that future diameter growth is related to past diameter growth is used for projecting tree diameter distributions over time. This model incorporated a measure of shade tolerance to partially account for species growth differences. Bowling et al. (1989) described a model for thinned stands of mixed Appalachian hardwoods which allowed predictions of diameter distributions by species group. Both Lynch and Moser (1986) and Bowling et al. (1989) made use of the parameter recovery method and the Weibull probability density function to predict diameter distributions. 3.1.2 Stand Class Models Stand class models or diameter class models (Davis and Johnson, 1987) simulate each diameter class based on the average tree. Stand characteristics are obtained by aggre-gating over all classes. -The stand classes may be comprised of diameter classes or more flexible groups. The models may be implemented using stand table projection, transition matrices or cohorts. This approach provides the greatest utility for forecasting mixed forests (Vanclay, 1988a). (1) Stand Table Projection A stand table is the frequency distribution of trees per unit area by diameter class. A stand table projection is the adjustment of this distribution over time by considering diameter growth, removals, mortality, and ingrowth. This method has a long history of use and is readily adapted to mixed forests and computers (Korsgaard, 1984 and Vanclay, 1988a). The differences among the three variations of stand table projection Chapter 3. Literature Review 14 centre on how stems are distributed within a diameter class and how the diameter growth information is applied to estimate class growth (Davis and Johnson, 1987). The first assumes that all trees in a diameter class fall at the class midpoint and that all trees grow at the same rate. This method essentially projects the class boundaries so that future classes contain the same trees. Depending on the structure of the stand in relation to the average increment, a considerable overestimate or underestimate of growth may result. Mixed stands, which usually have several stand fractions or species groups each with different growth rates, would render this approach unsuitable. The second method assumes that the trees in each diameter class are uniformly dis-tributed and that average diameter increment is applied to each tree. The proportion of trees moving into higher diameter classes is calculated by a movement ratio determined from the class width and increment. Bias is introduced if stem distribution is not uni-form but may be minimized by using a small class width. This method was used by Kofod (1982) in a computer model for Hill Mixed Dipterocarp Forests in Sarawak. The same model was used by Korsgaard (1988) for a mangrove forest in Bangladesh. Vanclay (1988a) reported that this method, though extensively used in Queensland, Australia, has been abandoned due to its restriction on the distribution of diameters. The third method applies variable diameter increment to actual diameters. The percentages of trees moving up diameter classes from each class is based on the actual distribution of growth rather than assuming a uniform distribution. This method is more accurate, but is more data demanding and can be more efficiently modelled using matrices (Vanclay, 1988a). Clutter et al. (1983), in a detailed critical review, found much potential for misuse of the stand table projection method. However, when other models are not available or not appropriate, a stand table projection may be the only practical way to estimate growth as it is highly flexible and easily implemented. In the stand projection system developed Chapter 3. Literature Review 15 for Sarawak, the stand is divided into a matrix and individual growth rates and mortality rates are applied to each element of the matrix (Korsgaard, 1984). The disadvantages of stand table projection relate to the many parameters required and the difficulty in adjusting growth as affected by site and competition (Vanclay, 1988a). (2) Transition M a t r i x Models A transition matrix model predicts the final number of trees by a matrix multiplication of a state vector, containing the initial number of trees per unit area by diameter classes, by a Markov matrix, containing the probabilities of movement. Transition matrix models are based on two assumptions, namely the Markov and stationary assumptions. The first requires that the probability of any event must depend only on the initial state, and the second requires that these probabilities do not change over time. The weaknesses of the model as listed by Vanclay (1988a) are that competition, regeneration and mortality cannot be varied, and predictions must be an integer multiple of the remeasurement period. A considerable number of such models have been developed in forestry despite the restrictive assumptions. Bruner and Moser (1973), and Moser (1978) discussed a model using 25 states: 23 for trees 8 inches above, and one state each for mortality and harvesting. The Queensland Department of Forestry in Australia has used this approach for some years (Vanclay, 1988a). Buongiorno and Michie (1980) modified the concept of the Markov process to develop a matrix model for a hardwood forest which enables the analysis of management alternatives. Harrison and Michie (1985) indicated it is possible by a* matrix factorization approach to overcome the restriction that projection intervals must be integer multiples of the remeasurement period. Solomon et al. (1986) developed a two-stage matrix model (FIBER) for predicting growth and yield of multiple species forest stands in New England, United States. The annual transition probabilities were expressed as functions of tree size and stand density. Bonnor and Magnussen (1988) Chapter 3. Literature Review 16 predicted stem distributions for unevenaged forest stands in Eastern Ontario, Canada by using empirical growth probability matrices. Values for the transition matrices were calculated by major species groups from growth probability equations based on constants or first- or second-degree polynomials. The advantages of this method are that it is easy to use, computationally efficient, and useful where ample data are available. (3) Cohort Models Cohorts are groups of trees with similar characteristics such as size or species, and within each group, the growth of the mean tree is modelled (Vanclay, 1988a). The cohorts may be deciles, or percentiles whose boundaries are determined by the distribution of trees in the stand (Vanclay, 1983). For each cohort, the group identity, the mean size and number of stems within the cohort is monitored. Clutter and Allison (1974) used 25 cohorts each with an equal frequency of four percent to model radiata pine (Pinus radiata D. Don) plantations. Monserud and Ek (1977) used height classes to predict the development of uneven-aged hardwoods. The cohort approach was used by Vanclay (1987, 1989) for developing a growth model for North Queensland rainforests in Australia. His rainforest growth model admits a maximum of 200 cohorts for each stand. Stems from the same species group whose diameters at breast height differ by less than five mm were grouped by him into a single cohort. The efficiency of this approach lies on the initial formation and management of cohorts. With appropriate growth functions, it can offer great reliability and flexibility for yield forecasting in mixed forests (Vanclay, 1988a). 3.1.3 Individual Tree Models Individual tree models are the most complex of the three categories. Growth is simulated by growing each individual tree and aggregating the trees to give estimates of stand Chapter 3. Literature Review 17 growth and yield. Models are generally divided into two classes: (i) distance dependent models, and (ii) distance independent models. Distance dependent models require inter-tree distance (Munro, 1974), while the spa-tial positions of trees are not required in distance independent models. Their common inputs include diameter, tree height or crown class of trees (Vanclay, 1988a). Most indi-vidual tree models calculate a competition index and use it to determine which trees live or die, and their level of growth (Davis and Johnson, 1987). While individual tree models are capable of producing detailed tree and stand information, they have not proven to predict the development of individual trees reliably (Munro, 1974). Also, in the case of distance dependent models, expensive and sparsely available tree coordinates are re-quired, large computer storage is needed for tree information, and excessive computer time required to calculate and assess a meaningful biological measure of competition (Munro, 1974 and Wan Razali, 1986). Most of the individual tree models are concerned with even-aged forests or planta-tions (Dudek and Ek, 1980). Models developed for mixed species using the distance independent approach are typified by the PROGNOSIS model (Stage, 1973; Wykoff et al, 1982) and the STEMS (Stand and Tree Evaluation and Modelling System) model (Belcher et al., 1982). An example of a distance dependent model for mixed species has been demonstrated by Ek and Monserud (1974) in their FOREST model. Ek and Dudek (1980) recommended distance dependent models to solve species composition problems. Meldahl et al. (1987) selected a distance independent tree model for modelling mixed forests in Southern United States due to its inherent flexibility. Adlard et al. (1988) stated that the models of widest application in modelling tropical mixed forests "are probably the representative tree models and the individual tree distance independent models". Chapter 3. Literature Review 18 3.2 Growth and Yield in the Tropics Growth and yield literature is rather limited for mixed tropical forests. Revilla (1981), in a review for the Asia region, admitted that his review was hampered by the 'relative scarcity' of growth and yield studies. He concluded that (or Malaysia and the Philippines, the reported studies in these countries do not reflect the abundancy of the data. The studies done were on tree growth and mainly girth or diameter increment studies by species and girth/diameter class. Stand density and site quality were absent in these studies. Most of the published studies, as cited in FAO (1983), were concerned with diameter, basal area distributions, diameter growth, and stand mortality after logging. Most growth information from mixed tropical forests has been analysed at the tree level. The main reasons were that tree diameter limits are used for cutting control, and diameter distribution is of more importance than whole stand increment. The diameter distribution in uneven-aged forests normally follows a reverse-J shape. Examples can be found in FAO (1983). The distribution can be described mathematically by the exponential or Weibull distributions. The distribution of basal area is different from that of diameter and is more useful for studying stands. While the diameter distri-bution is strongly influenced by the very large number of small trees present, these small trees contribute little to stand basal area. The distribution of volume by size classes follows closely that of basal area. 3.2.1 Modelling Approaches Modelling approaches for mixed tropical forests were either based on the analysis of in-dividual tree increment or mean tree increment. Simple models based on tree increments are (i) time of passage, and (ii) stand table projection. Time of passage is defined as the years required for a crop to recover after felling Chapter 3. Literature Review 19 under selection management. As this method requires no mortality rate, it is suitable when only increment data are available. Trees are grouped into diameter classes and the mean increment for each class is computed. The time of passage is calculated for each class. The total time taken for trees to grow from the regeneration stage to maturity or any desired diameter cutting limit is the sum of all times of passage through all diameter classes up to that cutting limit. The use of regression in the calculation is possible (Singh, 1981 as cited in FAO, 1983). The mean annual volume increment can be derived from inventory data. One disadvantage of this method is that the cutting cycle may be overestimated, because of high variation in increment within a diameter class and between species or species groups (Wan Razali, 1986). However, if the mean increment of the final crop tree can be estimated, then a realistic and useful result may be obtained. The FAO (1974a) study on production potential in the peat swamp forests of Sarawak made use of the time of passage method to predict the next harvest and the cutting diameter to be used. The basic concepts behind the stand projection model have been described earlier. The model developed in Sarawak by Kofod (1982) uses a combination of the concept of time of passage and de Liocourt's quotients or q-values to project future numbers of trees, de Liocourt's quotient expresses the rate at which the number of trees decline with increasing diameter. Low values of q, mean a relatively higher proportion of trees in the larger diameter classes compared to higher values of q. For the Mixed Dipterocarp Forests of Sarawak, the q-value was around 1.65 for the diameter range 10 to 30 cm and 1.25 for the diameter range 30 to 60 cm when based on five cm diameter classes (Kofod, 1982). The model developed by Kofod is available in both a manual and computerized version. Due to the simplicity of the model, Kofod recommended it to be used only for comparing the gross effects of various rates of growth and mortality on treated stands, and not for yield forecasts. Jonkers (1982) used the Kofod model to project five basic Chapter 3. Literature Review 20 stands simulating five intensities of harvesting and under three treatments 40 years into the future. Chai and Sia (1988) made use of the same computer model to simulate the development of basal area and volume for silvicuiturally treated forests in the mixed swamp forests. Korsgaard (1988) suggested that height and height increments as well as form factors be added to the model to simulate the volume development. The whole stand yield model approach is simple in concept as the yield can be as-sessed at any point in time by conventional inventory techniques. No information on mortality, ingrowth or increment is required as yield is the net volume of products such as merchantable timber which can be extracted from a stand. However, a large number of permanent growth plots spanning at least two measurement periods of not less than 10 years are necessary for determining the coefficients of the static yield functions (FAO, 1983). The yield function for a mixed forest can be expressed in the form of: Y = f(YEAL, SITE, B A L ) (3.1) where Y E A L represents age or 'years after logging', SITE represents site index or 'mean height of dominant trees' and B A L represents stocking or 'residual basal area after logging'. Multiple regression techniques can then be applied to obtain the relationship between yield and the independent variables. This approach has been used in the Philippines (Revilla, 1981). 3.2.2 Diameter Increment Tree diameter increment variation in tropical forests is large even by diameter class, being of the order of 60 % (coefficient of variation), and correlation coefficents between diameter increment and diameter for various species lie between 0.4 and 0.7 (FAO, 1983). Chapter 3. Literature Review 21 These variations may be attributable to climate, site variation, and competitive status of the trees. It is observed that for most species, the diameter increments are less than one cm per year. In Malaysia, the dipterocarp species have diameter increments of 0.6 to 0.8 cm per year (Bryan, 1980 and Tang and Wan Razali, 1981). While the shape of the regression curves for mean increment against diameter for the normal situation is a weakly ascending curve (FAO, 1983), other shapes are possible and are influenced by age of the trees, microsite or genetic factors, and competitive status. A study by Bryan (1980) for the Mixed Dipterocarp Forests of Sarawak showed the effect of crown position, independently of diameter class, on increment. The dominant and emergent trees have increments around 50 % greater than the subdominant and the understorey trees. The growth of trees can be expressed as either diameter increment or basal area increment. There is no conclusive evidence that diameter increment is a better dependent variable to model than basal area increment. Either can be used if account is taken of the error distribution (Vanclay, 1983, 1988a). Wan Razali (1986) concluded from his preliminary data analysis however that diameter growth was a better dependent variable based on a more satisfactory residuals and better fit than basal area growth. Equations used for modelling diameter increment could be empirical or theoretical; many empirical equations have been developed using stepwise regression analysis. Wan Razali (1986) modelled diameter increment using unweighted linear, weighted linear, and nonlinear regression. He favoured weighted linear models over the other models based on better residuals. Vanclay (1988a) stated that although theoretically it is preferable to use generalised least squares regression, ordinary least squares generally can be used for remeasured plot data if the number of plots is large relative to the number of measure-ments. A larger number of plots might result in lower correlations between variables or most observations being independent. He further observed that diameter functions pro-viding sensible predictions should be constrained to pass through the origin or having a Chapter 3. Literature Review 22 small positive value, rise to abroad plateau, decrease and asymptotically approach zero or reach a reasonable maximum diameter. As the shape of diameter increment distribution will influence the regression approach used, and also can be the basis for species grouping, this must be analysed first before selecting a regression model. The predictor variables to be used in the diameter increment function can be individual tree variables or stand variables. Revilla (1981) and Wan Razali (1986) suggested the following independent variables: o Initial tree diameter at breast height - D B H (cm). • Total basal area of all species per plot - B A T (m 2/ha). • Species group basal area per plot - G B A (m 2/ha). • Individual tree diameter growth in the last growth period - LDG(cm/year). • Site indicator - SI. • Competition index such as microstand density, crown size or crown class - CI. • Age or years after logging - A . • Numerous combinations of predictor variables or their interactions and logarithmic terms such as: 1/A, 1/DBH, InA, InDBH, ln(diameter increment), SI/A, S I / A 2 , S I /DBH, S I / D B H 2 , SI*CI etc. Wan Razali's (1986) studies indicated that the growth predictors vary with species group, but initial tree diameter (DBH) and diameter increment of the preceding growth period (LDG) are important. Chapter 3. Literature Review 23 3.2.3 Ingrowth Ingrowth is defined by Davis and Johnson (1987) as the volume of new trees growing into measurable size during the measurement period. Alder (in FAO, 1983) referred to ingrowth or recruitment as the process by which trees grow into measurable size during a growth period and may include regeneration or those seedlings yet to reach measurable size. Ingrowth is used here to refer to the number of trees growing into measurable size. Regeneration is often a sporadic process and the establishment of ,seedlings on the forest floor depends on seed dormancy factors and light conditions. A distinction must be made between true ingrowth and ingrowth associated with plot design. The "leading desirable" system of measurement proposed by Dawkins (1952), and practised in central and West Africa and also in Sarawak, often gives rise to false ingrowth as a new leading desirable is preferred over the old one. Thus, no published information on ingrowth is available for many plots measured in this manner. Nicholson (1979) considered ingrowth as unlikely to affect yield estimates in the current cutting cycle and hence could be neglected in stand projections. Bryan (1980) showed that ingrowth trees have higher increment than stems already present. Tang and Wan Razali (1981) reported that the mean ingrowth varies with plots and time after logging. As the data on ingrowth trees are limited, more studies are needed to make any definite conclusions. 3.2.4 Site Quality/Competition Revilla (1981) and Wan Razali (1986) stressed the importance of site productivity or quality to improve growth and yield predictions. Wan Razali (1986) suggested using site indicator species, soil types and nutrient availability for expressing site potential in mixed tropical forests. Vanclay (1988b) reported that a study done in Queensland used species composition of stands, and remotely sensed data with soil type information to predict Chapter 3. Literature Review 24 site productivity. A better measure of competition needs to be developed to facilitate its usage in individual tree models. Crown position or crown illumination is commonly collected on tropical forest permanent plots and is suitable as a competition index (FAO, 1983). Alder (in FAO, 1983) suggested that an index suitable for microstands in mixed forests could be defined as the ratio of the subject tree diameter to the mean diameter of the 100 largest trees per ha. For a 20 m X 20 m square or subplot, the index would be the ratio of the subject tree diameter to the mean diameter of the four largest trees. He recommended this index over crown illumination index as it eliminates subjectivity. Lorimer (1983) recommended for general use the index i = l where Dj is the diameter of competitor j and D{ is the diameter of subject tree i. He defined a competitor tree as one of an equal or higher crown class than the subject tree. More studies are required to determine suitable competitive indices for mixed species stands in the tropics. Chapter 4 Materials and Methods 4.1 Study Data Base 4.1.1 Location of the Data The data used in this study come from yield plots established in ten Mixed Swamp Forest localities. These plots were laid out randomly at an intensity of 0.25 % or one in every 400 ha, in forests which vary in the years elapsed after logging. The oldest established plots were set up in 1972 in the Simunjan and Sedilu permanent forest areas1 , while the newest established plots were set up in 1987 in Bawan Forest Reserve (FR) and Batang Lassa Protected Forest(PF) (Table 4.1). The yield plots in seven localities, Yield Plots 07 to 29, have been measured as often as seven times. The plots in three other localities, Yield Plots 30 to 48,2 were measured as often as six times. Plots established in 1987, Yield Plots 65 to 83, have been measured twice. The plots were measured at regular intervals every 3 to 5 years. The detailed layout of the one ha plot is shown in Figure 4.1. The basic recording unit in each plot is a subplot or quadrat of 10 x 10 m. Each tree in a quadrat is identified by a quadrat number, tree number and species code (4-letter code, Appendix A). 1 either a protected forest or a forest reserve 2 Yield Plots 49-64 were established in Mangrove forests 25 Chapter 4. Materials and Methods 26 On O i l 021 022 " O i l 0)2 041 0*2 OSI 052 -01- -02 -03 -04- -05 014 101 01) • • 024 102 0*1 0 2 > ' - . 0 )4 0»2 O H o)) 061 044 on 04) 072 034 061 _033^ 062 •09 08 07- 06 104 10) . , 064 H I 121 0 » 3 , 122 Q64 hTjT 061 132 074 141 073 142 064 I 3 I 063 , 132 -I I - 12 14 201 113 202 124 123 162 134 161 1)3 <62 144 ITl 143 172 134 161 13) - 2 0 - 19- 18 17- 16 204 20) • • 164 211 2 i t 221 21 i 214 2 D a > 224 2 2 | 184 •8) | 2)1 2)2 23-22) 2)4 17) 11 164 242 231 24" 23) . . 244 16) 2 32 25 24) ^ 234 233 P Scale 20 /77 20 m 3 cm 3cm N A \..:Belian peg x.Red 1lagging Source: Lee and Lai, 1977 Figure 4.1: Layout of Plot and Quadrat of Sarawak Yield Plot Chapter 4. Materials and Methods 27 Table 4.1: List of Yield Plots in Mixed Swamp Forests Locality Yield Plot Established Measurements Years Since ( F R ° / P F b ) Number (Year) Freq. c (latest) Felling 1. Simunjan 07-10(4) 1972 6(1988) 28-31 2. Sedilu 11-12(2) 1972 7(1988) 30-31 3. Triso 13-16(4) 1973 7(1989) 28-35 4. Sebuyau 17-18(2) 1973 7(1989) 25-32 5. Saribas 19-22(4) 1973 7(1989) 33-34 6. Daro 23-26(4) 1973 7(1989) 27-28 7. Tatau 27-29(3) 1973 7(1989) 19-20 8. Btg . d Lassa 30-36(7) 1976 4(1985) 22-24 9. Loba Kabang 37-42(6) 1976 6(1985) 30-33 10. Bawan 43-48(6) 1977 4(1985) 25-29 11. Bawan 65-72(8) 1987 2(1988) 15-22 12. Btg. Lassa 73-83(11) 1987 2(1988) 22-27 a F R = Forest Reserve h PF=Protected Forest ' ; Freq.=Frequency dBtg.=river 4.1.2 Measurement Procedures In all plots established and measured before 1983, the following measurement procedure was followed: (a) for the 'best' potential crop trees3 or 'Leading Desirable' trees, the species, diameter at breast height outside bark (dbhob), crown position, crown form and 'impeders' were recorded; (b) a stand table of all trees greater than 10 cm dbhob by 10 cm diameter classes for three groups of trees, namely 'Desirable', 'Undesirable' and dead trees, was recorded. The Desirable species are those tree species given silvicultural preference as potential crop trees by The Sarawak Forest Department, and are also those commercially extracted. A list of Desirable species used for silvicultural operations is given in Appendix A. Those 3one tree per quadrat Chapter 4. Materials and Methods 28 tree species not included under the Desirable species group are classified in the Undesir-able or noncommercial group. A further subdivision of Desirable species into fast growing light demanders and slow growing shade tolerant trees is possible as shown in Appendix B. The Leading Desirable is the best tree belonging to the Desirable species group with a good complete living crown and good stem form in a favourable position within a subplot or quadrat (Lee and Lai, 1977). The diameter of this tree must be between 5 cm and 150 cm dbhob. The crown position recorded indicates the relative positions of the crown and the amount of illumination received (Figure 4.2). The crown form recorded for the leading desirable reflects the relative size and development of the crown (Figure 4.3). The impeders or competitors recorded may be vigorous woody climbers, Undesirable species, or defective trees shading the leading desirables. An impeder can also be the Leading Desirable of the next quadrat. Climbers are recorded as ' C and tree impeders are recorded by their 4-letter codes. Starting in 1983, i.e., for plots established between 1972 to 1977, dbhob, species, and crown position were recorded for all trees greater than 10 cm dbhob. For plots estab-lished in 1987, a very detailed individual tree measurement procedure was implemented. This involved recording, for each living tree, the species, dbhob, crown position, crown form, tree lean and stability, tree decay, and the presence of woody climbers. A trunk height measurement for the best potential crop trees greater than 30 cm dbhob has been measured on all plots since 1987. This detailed measurement procedure was followed for all assessments of.the yield plots performed in 1988 and 1989. Chapter 4. Materials and Methods 29 Source: Lee and Lai, 1977 Figure 4.2: Silvicultural Scoring of Crown Position in Yield Plots Chapter 4. Materials and Methods 30 Complete cir cl« Buloton penuh * ' f ^ ^ ( o ) 1 5 \ J Perfect ' /( Terboik trregulor circle J /—>. ° ° f 4 f Good 1 Boik A Holt - crown ^ (^ *^  Junjong J \ seporoh Vf"} / ^^^v 1 bulol J ) ft ^ 3 Toleroble Sedong Le»* thon holf-crown Junjong \ 3 / ^ \ to' jompoi 1Q S. ^ seporoh V I ^—> If ^ y f Poor Burok Ont or few bronchet j-^^> Sotu otou X- Z sedikil ^(hS* dohon - dohon / ? ] Very poor Terton burok Source: Lee and Lai, 1977 Figure 4.3: Silvicultural Scoring of Crown Form in Yield Plots Chapter 4. Materials and Methods 31 4.2 Data Preparation A substantial amount of checking for data entry errors and/or wrong diameter measure-ments in the data set was done by personnel of the Sarawak Forest Department. At each measurement occasion, the currently collected data was checked in the field against the old measurements and errors rectified as much as possible. At the office, illogical diameter increments were screened using the U P D A T E 4 program. The basal area per plot figures for all tree species, and for commercial tree species only for each of the 42 yield plots was calculated previously for measurements made up to 1983 by a F O R T R A N program in Sarawak. The whole data set can be considered to consist of two subsets of measurements. One subset is comprised of those data collected before 1983 in which diameter was only measured for the 'best' commercial or 'Leading Desirable' trees. This subset also contains the frequency of trees by 10 cm diameter classes of the commercial, non-commercial and dead trees. The second data subset has individual tree diameter measurements for all trees, and other tree information, and was collected from 1983 onwards. The first data subset is referred to as the 'best trees' data while the second data subset is called 'all trees' data. As the individual tree approach to modelling diameter growth was used in this study, a list of individual trees containing three categories of information was prepared. 1. plot, quadrat, tree and species identification number; 2. diameter increment in cm/year (the dependent variable); 3. plot and tree attributes and other information which are to be used as the inde-pendent variables. These are for trees > 10 cm dbhob and are: 4 A computer program developed at the Commonwealth Forestry Institute, University of Oxford, U K . Chapter 4. Materials and Methods 32 (a) dbhob in cm, diameter breast height5 over bark; 00 initial crown position or competitive index as recommended by Alder (FAO, 1983); initial total basal area per ha; (d) initial basal area per ha of the commercial species group; initial basal area per ha of the non-commercial species group; (0 years since harvest; (g) stems per ha; quadratic mean diameter. "Initial" measurements are those recorded at the beginning of the growth period. This list was prepared for each tree from the data set of yield plots 7 to 29 for one measurement period, namely 1983-88 for plots 7 to 12 and 1984-89 for plots 13 to 29. Another list of individual trees was prepared from the data set collected before 1983. This list was made up of the 'best' or 'Leading Desirable' trees measured and is available for several measurement periods. This list contains the same information as the first list but with an additional independent variable for some measurements, namely the diameter increment for the preceding measurement interval. The diameter increments were calculated for the same individual 'best trees' over time. In this study, the 'all trees' and 'best trees' data sets were each split into two, the model-building data and the validation or prediction data. One third of the total number of trees for each species was randomly selected as the validation data for each data set. The data from the yield plots in the mixed swamp forests poses a few problems. 51.3m above ground level Chapter 4. Materials and Methods 33 (1) Diameter and Height Measurements A problem with diameter measurement is associated with buttressed or fluted trees. The point of measurement of diameter at breast height is 1.3 m vertically above ground for unbuttressed trees or 3 m for trees which are buttressed at breast height. The point of measurement for buttressed tree may have been moved without recording the new position leading to negative diameter increments. As this study takes a single tree approach to diameter increment modelling, all tree with negative increments were excluded from subsequent analysis. The trees were used in calculating stand basal area, however. Also, the lack of height data precluded its use in the development of diameter increment models in this study. (2) Change of Species The species were identified and recorded by 4-letter codes. Due to uncertainty over the correct species, changes in species may happen over two or more successive measurements. In this study, this problem was dealt with by assuming that, the correct species is that identification which was more frequently recorded. If this was not possible, the data associated with the 'unidentified' tree were excluded from those to be analysed. This was only done for the individual tree parameters. However, the trees were still included for calculating the stand parameters. (3) Disappearance or Reappearance of Trees Though cases of trees disappearing could be considered as mortality, errors do occur resulting from failure to measure or record data for a tree. The latter situation is rare in this data base and hence all trees disappearing were assumed dead. Trees reappearing may indicate confused identity or an earlier missed measurement or ingrowth. In this study, trees appearing which are greater than 10 cm dbhob and not over a reasonable upper limit for the given growth period were considered ingrowth trees. Chapter 4. Materials and Methods 34 (4) Site Indicator The concept of site index, which is important to growth and yield modelling for even-aged forests, is difficult to apply for uneven-aged forests. Age is a nebulous definition for tropical mixed forests though 'years after logging' has been used (Revilla, 1981) to imply age. Environmental factors and the presence of indicator species which may be used as substitutes for site index were not measured in the existing data. Thus, this study did not explore other means of expressing the potential productivity of a site. 4.3 Data Analysis 4.3.1 Selection of Modelling Level A prediction equation was developed for each of the three levels of models denned as: 1. all commercial species (first level); 2. two groups, one consisting of all the light demanding trees and the other, the shade tolerant trees (second level); 3. each individual species (third level). A common model for each analysis data set ('best trees' and 'all trees' data sets) was selected based on 'common' important variables. The important variables are those retained in the model using the all possible regression approach, based on increase in R 2 , decrease in residual mean square (RMS), lowest Mallow's C p and the lowest prediction sum of squares (PRESS) statistic. Important independent variables which occur more frequently in the equations fitted at the third level were used to form the common model. A common model was used so that any gain in prediction could be attributed to the level of modelling and not due to the number of independent variables present in the model. Chapter 4. Materials and Methods 35 For each analysis data set, the common model was fitted at the three levels using the data reserved for model-building. The common model was fitted at the third level for only those species with sufficient number of observations for model-building. The validation data for those species for which model-building was possible were then combined to form a common validation data set. This common validation set was used to calculate mean bias, mean squared bias and mean absolute deviation (see page 43-45 for definitions) for the three levels of modelling separately. The distribution of bias by predicted diameter increment and by dbhob were graphically plotted and examined for any trend. The level of modelling which resulted in the least mean bias, mean squared bias, and mean absolute deviation was deemed better at prediction. 4.3.2 Model-Building Using the research by Revilla (1981) and Wan Razali (1986) as a guide, I modelled diameter increment for the selected level using the following approach. Both linear and nonlinear regression models were fitted to the data sets. Linear Models A linear model for each species or species group and for the 'all trees' (post-1983) and for the 'best trees' or Leading Desirables (pre-1983) data set was developed. The linear model may be expressed as Y = XP + E (4.2) where Y X = P = diameter growth rate in cm/year matrix of predictor variables matrix of coefficients to be estimated and Chapter 4. Materials and Methods 36 E — random error. This model states that diameter increment is a linear and additive function of the pre-dictor variables. The multiple linear regression procedures available with SAS (SAS Institute, 1985) were used to estimate the parameters. The independent variables to be tried in the regression equation were based in part on those used by Revilla (1981) and Wan Razali (1986). These were: 1. tree diameter outside bark at breast height - D (cm), 2. square of diameter breast height - D 2 , 3. total basal area of all species per plot - B T (m 2/ha), 4. commercial species group basal area per plot - B C , 5. commercial species to all species group basal area per plot - B C / B T or RS 6. noncommercial species group basal area per plot - B N , 7. total number of stems per ha, all species - ST, 8. total number of stems per ha, all commercial species - SC, 9. total number of stems per ha, all noncommercial species - SN, 10. quadratic or average mean diameter, all species - QDT or A D T 11. quadratic or average mean diameter, all commercial species - QDC or A D C , 12. quadratic or average mean diameter, all noncommercial species - QDN or A D N , 13. individual tree diameter growth in the preceding growth period 6 - L D G , 6length of 3-5 years Chapter 4. Materials and Methods 37 14. competition index 7 as suggested by Alder (FAO, 1983) - CI, 15. crown position - CP, 16. age or years since logging - A G E , 17. interaction of crown position, a qualitative variable and the other independent variables. The chosen independent variables have been either measured in the yield plots or could be derived from those already measured. The diameter increment of a tree is influenced by its size (DBH), its competitive status (CP or CI), and by its stand density. The stand density variables were expressed in terms of basal area per ha, stems per ha and quadratic or average mean diameters and depends on time elapsed since logging (AGE). The relative density of commercial trees to all trees (RS) may influence the diameter growth rate of a tree as the commercial trees are the relatively faster growers in the stand. The previous diameter growth rate (LDG) may reflect the impact of site and inter-tree competition (Wan Razali, 1986) and is correlated to the current diameter increment. The 'all possible subsets regression' procedure in SAS called P R O C R S Q U A R E was used as this procedure can guarantee finding the 'best' model. This procedure considered all possible regression models a.nd identified a few 'best' subsets according to criteria such as R 2 , MSE and the C p statistic. A good model to be chosen is one in which all independent variables are significant at an alpha level of 0.05, the residual mean square (MSE or RMS) is small and the R 2 value is high. Two other criteria which were used to determine the best equation are Mallow's C p statistic and the PRESS statistic. With C p , the best model is one in which this value is either equal to or closest to p, (Draper 7ratio of diameter of the subject tree to mean diameter of the four largest trees in a 20 m x 20 m subplot Chapter 4. Materials and Methods 38 and Smith, 1981) the number of parameters estimated in the model. With the PRESS statistic, models with smaller values are considered better models as they have smaller prediction errors (Neter et al. , 1990). Selecting the final regression model with C p statistic requires identifying the model that offer the best compromise between minimizing C r (smaller total mean squared error), and having a C p value that is approximately equal to p (smallest bias). The list of independent variables also include some multicollinear ones, namely B T with BC and B N and ST with SN and SC. If these are simultaneously entered in the model, singularity of the variance-covarince matrix may occur, resulting in failure of the regression. The residual plots from fitting the models were also used for selecting the 'best' model. Models with satisfactory residuals are preferred over those with residuals indicating het-eroscedasticity, and lack of fit. Lastly, the cost of measuring the variables required in a particular model was considered in selecting one model over another. First, the R S Q U A R E procedure was executed using all the potential independent variables including crown position to determine the 'best' subset model. This was done in order to reduce the number of variables in the model to some good subsets based o n R 2 , RMS, PRESS and the C p statistic rather than on significance of variables because of probable heteroscedasticity. Also, as the data used was observational, lack of orthogo-nality among the independent variables is expected. As such, the least square result for each independent variable depends on which other variables are in the model (Rawlings, 1988). For any model subset size, the all possible regressions technique can ensure find-ing the 'best' model. Stepwise regression methods may identify a good subset, but not necessarily the 'best' model, and may lead to different 'best' subset models. Thus, it was not used in this study. Based on the selected subset model, a standard multiple regression using PROC R E G in SAS was carried out to consider incorporating some important interaction terms. The Chapter 4. Materials and Methods 39 Table 4.2: Recoding of Crown Position Old New Competitive Status 1 0 suppressed and 2 0 intermediate trees 3 0 4 1 codominant and 5 1 dominant trees interaction terms attempted were restricted to those formed by combining the other independent variables with crown position. This was done because crown position is a qualitative variable. Other interaction terms were not considered so as to limit the size of the 'final' model. Crown position was recoded from the original five categories to two categories as in Table 4.2. The recoding was done for two reasons. Firstly, crown position scoring in the field has a subjective element and is relatively more accurate for scores obtained for codominant and dominant trees. Also, the recoding to two categories avoided the fitting of a large equation made up of numerous interaction terms. As the residuals were expected to be heteroscedastie, the probability values as obtained from the SAS PROC R E G run were used to indicate the important interaction terms to be retained. Before labeling this model (with important variables and interaction terms) as the 'final' model, the signifi-cance of each variable had to be checked. In order to check significance, the regression assumptions must be checked. Residual analysis was performed to check violation of the regression assumptions and to assess the 'lack of fit'. An F-test for lack of fit was not done since repeated observations of the independent variables were not available and there was no evidence of lack of fit in any of the residual plots. Chapter 4. Materials and Methods 40 The homogeneity of variance of the error terms was checked using the Goldfeld-Quandt test (Goldfeld and Quandt, 1965). This test was carried out as follows. The observations were ordered in increasing variance of residuals by the value of the predicted dependent variable. The k central observations were removed and a linear regression each was done on the first and last ((n-k)/2) observations. Next, R was calculated as the ratio of M S E 2 to M S E j , where M S E X and M S E 2 are the residual sums of squares from the first and second regression. Under hornoscedasticity, R has the F distribution with [(n-k-2m-2)/2,(n-k-2m-2)/2] degrees of freedom where m is the number of independent variables. If R was less than the critical F values for the degrees of freedom specified, then hornoscedasticity was concluded. A weighted regression procedure was used if it was discovered that the residuals did not have uniform variance with respect to the error terms. Weighting was done by using either the reciprocal of initial dbh (1/D) or the reciprocal of a power of the predicted value. Wan Razali (1986) weighted his linear regressions by 1/D as he observed that variance of the residuals varied proportionally with initial dbh. Weighting with a power of the predicted value was one among many recommended by Judge et al. (1985) to correct for heteroscedasticity. The significance of the interaction terms and of the other independent variables was checked in the weighted regressions using a significance level of 0.05. If significant, these variables are retained to form the 'final' models. The weighted models were used in the validation. Since the 'all trees' data set is comprised of only one measurement interval for all individual trees, the linear model did not include L D G . Also, the second data set was measured for the 'best' potential crop trees, only. It was envisaged that the models fitted from the 'best trees' data set would display better fit than models fitted from the first data set because of the inclusion of L D G . The increments for the 'best trees' data set would be expected to be correlated to its previous diameter growth. Chapter 4. Materials and Methods 41 Nonlinear Models A nonlinear model in which the parameters are additive and nonlinear of the following general form was used. Y = f[0,X] + E (4.3) where Y = diameter growth rate A' = array of predictor variables 9 — array of coefficients to be estimated / [ j ] = a nonlinear function and E -— random error. Nonlinear models examined were those that incorporated the concept of catabolism and anabolism. This approach was used by Hahn and Leary (1979) in STEMS and tried by Wan Razali (1986). The equation may be represented in the following form: Y = ex{xx) + e2{x2)6> + e4(xa) + E (4.4) where Y = diameter increment, 8i(Xi) = the "intercept"term, e2(X2)63 = the "catabolic" term, ^ 4 ( ^ 3 ) = the "anabolic" term, E = the error term, Xi = CI, competitive index for the 'all trees' data', or LDG, the previous diameter growth, Chapter 4. Materials and Methods 42 for the 'best trees' data set, X2 = DBH, diameter breast height and X3 = BT, the total basal area of all species per plot The model was fitted to diameter growth using the Levenberg-Marquardt algorithm of the Nonlinear Regression (NLR) procedure of SPSS Version 4.0 using both model-building data sets, by species. Only one nonlinear model form was attempted. This model form was adopted from that given by Hahn and Leary (1979) and Wan Razali (1986). It was observed that in fitting the linear regressions, both competitive index and the previous diameter growth accounted for a large portion of the variation in the dependent variable. Thus, these two independent variables were incorporated as first terms in the nonlinear model. The nonlinear models were then evaluated against the linear models by using the validation data. 4.3.3 Model Evaluation Model validation was defined by Vanclay (1983) and by Bruce and Wensel (1987) as determining the quality of model predictions. Ideally, this means using a data set in-dependent of that used to construct the model. Snee (1977) and Neter et al. (1990) suggested various ways of validating a model. These can be summarized as follows: 1. Examining the model's performance on the model-building data set or self-validation. 2. Comparison of the model predictions and coefficients with theoretical expectations, earlier empirical results, and simulation results. 3. Collection of new data to check the model and its predictive ability. 4. Using an independent data set to check the model's prediction or the technique of cross-validation. Chapter 4. Materials and Methods 43 Validation techniques used by various growth and yield modellers involved splitting the data set in various proportions. Hann (1980) and Holdaway and Brand (1983) used one-fifth of their data for validation. West (1981) and Meldahl et al. (1987) used one-fourth, while Hann (1980) and Wan Razali (1986) used one-third of their data for validating their models. The data for validation could be systematically selected by plot from the data base (Holdaway and Brand, 1983), or randomly selected by plot (Wan Razali, 1986) or by tree. In this study, the technique of cross-validation which splits the whole data set into two, namely the model-building set and the validation or prediction set, was used. The model-building data was randomly selected by tree, from two-thirds of the available data while the remaining one-third formed the validation data. A number of objective measures were obtained to assess the validity of the models as follows: 1. Pseudo-R2 or I 2 coefficient. 2. Mean bias or mean squared bias. 3. Mean absolute deviation. (1) Pseudo-R 2 coefficient This coefficient is calculated in the same way as the coefficient of multiple deter-mination or the R 2 value as follows: Pscud°-R, = 1-Wrlf (4-5) or PWO-*2=1-SST^ (4'6) where Chapter 4. Materials and Methods 44 Y\ = actual zth Y value Y'i = predicted zth Y value Y mean Y value e; = bias or 'residual' n = number of cases in the validation data set. The calculation of R 2 requires predicted values from the model-building data set. The predicted diameter increemnt values used in deriving the pseudo-R2 are the values predicted by a model based on the model-building data set, but calculated for the validation data set. The value of the pseudo-R2, like that of R 2 . indicate the degree of goodness of fit as if the model had been fitted to the validation data set. A higher pseudo-R2 value for a model or level of modelling relative to another model or level would indicate that the former model or level fits or predicts better than the latter model or level. (2) Mean bias or mean squared bias The mean bias is calculated as: Tn (e) M B = ^ l=lK l} (4.7) n and the mean squared bias as: M S B = ^ l=lK l} (4.8) n If the mean bias is small, this indicates that the model constructed from the model-building data set predicts well for the cases in the validation data set. If the bias is large and positive for some cases, and large and negative for others, this could result in their cancelling out each other, resulting in a very small mean bias. A Chapter 4. Materials and Methods 45 better measure is the mean squared bias. Neter et al. (1990) states that if the mean squared bias (mean squared prediction error) "is fairly close to M S E based on the regression fit to the model-building data set, then the error mean square MSE for the selected regression model is not seriously biased and gives an appropriate indication of the predictive ability of the model." They stated that if MSB is much larger than MSE, then MSB should be relied on to indicate the predictive ability of the selected regression model. (3)Mean absolute deviation The mean absolute deviation or M A D can be expressed as: M A D = ^ ' = 1 ^ . (4.9) n This overcomes the problem of positive values of bias cancelling out the negative ones. Small values of M A D would indicate good predictive ability of the selected model for the validation data set. Each linear and nonlinear model was tested against its validation data for each species or species group. The bias, mean squared bias, and mean absolute deviation were computed. The calculation of the objective measures was also done by 10 cm diameter classes. A comparison of the objective measures for the linear and nonlinear models was then made to determine which was the better predictor. 4.3.4 Importance of Crown Position and Previous Diameter Growth For Prediction Crown Position (CP) The importance of crown position was also assessed in terms of mean squared bias and mean absolute deviation using the validation data sets. Weighted linear models Chapter 4. Materials and Methods 46 were already fitted for both model-building data sets when crown position was in the model. The weighted models were fitted again to the data sets, without crown position as one of the independent variables. This was only done for equations in which the crown position was found to be significant in the model. The mean squared bias and mean absolute deviation were then computed from the validation data for the model with and without crown position. The change or decrease in MSBs and MADs as obtained from the validation data was used to indicate the relative importance of crown position. Previous Diameter Growth (LDG) The importance of previous diameter growth was assessed in the same manner as crown position. The weighted linear model developed from the 'best trees' data set was used as it contains L D G as a predictor variable. The mean squared bias and mean absolute deviation were computed from the validation data when L D G was in and not in the model. The decrease in MSBs and MADs indicated the magnitude of improvement in prediction when L D G was included in the model. Chapter 5 Results 5.1 Level of Modelling for Better Prediction A common model was fitted for the three levels of modeling for both the 'all trees' model-building and 'best trees' model-building data subsets. The two common models predicting diameter increment are linear equations of the following inde-pendent variables: A l l Trees: DI = f(D, D 2 , CI, A G E , RS, BT, ST, QDT, CP) Best Trees: DI = f(D, D 2 , L D G , RS, SC, A D N , CP) The fit statistics by level for the common model for 'all trees' model-building data subset and for the 'best trees' model-building data subset are given in Ta-bles 5.1 and 5.2. The pseudo-R2, mean bias, and mean squared bias for the common models for the three levels of modelling were calculated using the validation data for each data subset. The distributions of average bias by predicted diameter increment and by dbhob were plotted and examined for any trend. One other measure, the mean absolute deviation, was computed to gauge how well the different level models predict. 47 Chapter 5. Results 48 Table 5.1: Some Fit Statistics for the Common Model, ' A l l Trees' Model-Building Data Light Number Species/Sp Group Requirement of S E E a PRESS SSE b Coding Cases (cm) Al l Com. c Spp 4344 0.172 0.319 442.99 440.62 Light Dem. L T d 2977 0.201 0.329 324.71 321.97 Shade Tol. ST e 1379 0.194 0.222 68.76 67.56 J O N G ' LT 152 0.339 0.199 6.74 5.56 M B U A LT 227 0.320 0.309 23.36 20.76 MLIL LT 496 0.202 0.294 43.99 41.93. M P A Y LT 178 0.194 0.272 14.22 12.42 K A P A LT 217 0.405 0.187 8.34 7.21 GERO LT 149 0.260 0.319 16.47 14.14 M D P D LT 89 0.338 0.295 10.33 6.86 A K A U LT 115 0.207 0.396 20.88 16.50 K E L A LT 126 0.259 0.337 16.22 13.20 K P P Y LT 226 0.156 0.242 13.81 12.67 T E R E LT 205 0.484 0.270 16.12 14.25 R A M I ST 175 0.121 0.169 5.71 4.73 SEPP ST 267 0.171 0.202 11.59 10.51 K E B A ST 86 0.093 0.268 7.16 5.47 K T P Y ST 118 0.079 0.176 4.09 3.35 N Y B A ST 92 0.238 0.309 10.27 7.82 N Y J A ST 274 0.404 0.194 11.49 9.98 SIMP ST 189 0.454 0.130 3.50 3.03 "Standard Error of Estimate h Residual or Error Sum of Squares cCommercial d l ight demanders c shade tolerants •^4-letter code for species as in Appendix A Chapter 5. Results 49 Table 5.2: Some Fit Statistics for the Common Model. ;Best Trees' Model-Building Data Light Number Species/Sp Group Requirement of Kdj SEE PRESS SSE Coding Cases (cm) A l l Com. Spp 1615 0.582 0.269 117.64 116.37 Light Dem. LT 1060 0.625 0.273 79.79 78.49 Shade Tol. ST 554 0.498 0.220 27.47 26.53 JONG LT 81 0.623 0.187 3.23 2.56 M B U A LT 106 0.486 0.264 8.33 6.81 MLIL LT 201 0.562 0.240 12.26 11.08 M P A Y LT 66 0.451 0.228 3.98 3.02 K A P A LT 115 0.562 0.200 5.36 4.27 GERO LT 62 0.343 0.374 10.07 7.55 A K A U LT 51 0.332 0.370 9.32 5.88 K E L A LT 57 0.633 0.254 4.12 3.15 T E R E LT 98 0.665 0.254 7.50 5.81 R A M I ST 156 0.510 0.137 3.12 2.79 SEPP ST 168 0.430 0.176 5.49 4.97 N Y J A ST 52 0.679 0.202 3.04 1.79 Chapter 5. Results 50 Table 5.3: Pseudo-R2 for the Common Equation for Level of Modelling. Validation Data Data Set Number Level of Modelling of Cases One Group Two Group Species 'AH Trees' 1666 0.180 0.237 0.376 'Best Trees' 596 0.627 0.617 0.617 5.1.1 Pseudo-R2 Table 5.3 indicates that there seems to be no difference in the value of pseudo-R2 for the three levels of modelling based on the 'best trees' validation data subset. •However, based on the 'all trees' validation data set, modelling at the individual species level resulted in a pseudo-R2 value twice that for modelling at the one broad group level. 5.1.2 Mean Bias and Mean Squared Bias The mean actual and predicted diameter increments, mean bias and mean squared bias for each of the three levels of modelling for the 'all trees' validation data set are given in Table 5.4. Table 5.5 gives the same information for the 'best trees' validation data set. For both data sets, the mean bias is smallest at the species level of modelling. There are underpredictions (positive mean bias) and overpredictions (negative mean bias) for the 'all trees' and 'best trees' validation data respectively. The 'SEE ' is also shown in Tables 5.4 and 5.5. 'SEE ' was calculated as the square root of the mean squared bias. The percentage error prediction was computed as the ratio of the SEE to the mean diameter increment (actual) and expressed as a percentage (Rawlings, 1988, p.188). Both 'SEE ' and percentage error prediction in Tables 5.4 show for the 'all trees' data that there is, in general, an improvement Chapter 5. Results 51 Table 5.4: Mean Bias and Mean Squared Bias for the Common Equation, ' A l l Trees' Validation Data Level of Modeling One Group Two Groups Species Actual DI (era)" 0.3760 0.3760 0.3760 Predicted DI (cm) 0.3619 0.3541 0.3764 Mean Bias (cm) 0.0141 0.0219 0.0086 Mean Squared Bias (cm2) 0.1096 0.1020 0.0834 'SEE ' (cm) 6 0.331 0.319 0.289 Percent Error Pred. c 88.0 85.0 76.8 "Diameter Increment bPseudo Standard Error of Estimate cPrediction Table 5.5: Mean Bias and Mean Squared Bias for the Common Equation, 'Best Trees' Validation Data Level of Modeling One Group Two Groups Species Actual DI (cm) 0.5376 0.5376 0.5376 Predicted DI (cm) 0.5453 0.5485 0.5422 Mean Bias (cm) -0.0077 -0.0109 -0.0046 Mean Squared Bias (cm2) 0.0554 0.0553 0.0570 'SEE ' (cm) 0.235 0.235 0.239 Percent Error Pred. 43.8 43.7 44.4 in prediction in terms of smaller mean squared bias when models are developed at the individual species level. The magnitude of the improvement in prediction (in reduced mean squared bias) depends on the species for which the common model was developed. Tables 5.6 and 5.7 give the mean bias by dbh classes. Both underpredictions and overpredictions occur across the range of diameters. It can also be seen that the species level of modelling produces the smallest mean bias in predicting diameter Chapter 5. Results 52 Table 5.6: Mean Bias (cm) per tree by Diameter Class for ' A l l Trees' Validation Data D B H Class(cm) Number of Cases Level of Modelling One Group Two Groups Species 10-15 698 -0.015(0.243) -0.002(0.241) -0.001(0.234)a 15-20 338 0.057(0.328) 0.066(0.316) 0.026(0.291) 20-25 225 0.046(0.376) 0.049(0.355) 0.003(0.321) 25-30 127 0.094(0.464) 0.095(0.444) 0.013(0.395) 30-35 90 0.060(0.409) 0.065(0.390) 0.030(0.318) 35-40 56 •0.167(0.349) -0.118(0.354) -0.003(0.283) 40-45 44 -0.103(0.350) -0.072(0.316) 0.050(0.293) > 45 88 0.010(0.404) -0.043(0.389) -0.010(0.368) "Figures enclosed within brackets are standard deviation of bias increment for dbh classes < 45 cm for the 'all trees' validation data. Modelling at the individual species level for the 'all trees' data can reduce the percentage error prediction by as much as half (Table 5.8). Separate models for Ramin, Sepetir, Jongkong and Kapur lead to reduction in percentage error pre-diction by 31, 12, 44 and 18%. However, the mean squared biases are the same for the three levels of modelling for the 'best trees' data (Table 5.9). There was no improvement for Ramin 1 and Sepetir, but slight improvement for Jongkong, Kapur, Meranti lilin, Meranti paya and Terentang. It also indicates that models developed at the species level for Meranti buaya and a few others (Nyatoh jangkar, Geronggang, Akau and Kelampu), were poorer predictors than the models for the two groups or a single model for all species. The distribution of bias by predicted diameter and bias by diameter for the 'all trees' validation data set are shown in Figures D . l to D.3 and D.4 to D.6 respectively (Appendix D. l ) . Figures D.7 to D.9 and D.10 to D.12 show the plot of bias by predicted diameter and by diameter for the 'best trees' validation data Refer to Appendix A for 4-letter species codes Chapter 5. Results 53 Table 5.7: Mean Bias (cm) per tree by Diameter Class for 'Best Trees' Validation Data D B H Class(cm) Number of Cases Level of Modelling One Group Two Groups Species 10-15 126 -0.032(0.206) -0.026(0.203) -0.0020(0.198) 15-20 127 0.007(0.229) -0.002(0.226) 0.009(0.213) 20-25 99 -0.003(0.236) -0.018(0.235) -0.003(0.245) 25-30 78 -0.032(0.251) -0.048(0.250) -0.042(0.251) 30-35 58 0.016(0.248) 0.011(0.249) 0.023(0.260) 35-40 48 -0.022(0.267) -0.017(0.264) -0.011(0.263) 40-45 45 0.031(0.230) 0.044(0.234) 0.008(0.279) > 45 15 0.009(0.306) 0.052(0.317) -0.085(0.354) set (Appendix D.2). In general, from Figures D . l to D.6, both positive biases (underprediction) and negative biases (overprediction) occur across the range of predicted diameters and diameters. The spread of bias against predicted diameter increment increases with predicted diameter increment. Also, the range of predicted diameters for the model developed at the species level is larger. The plots for the 'best trees' validation data set reveal that both positive biases (underprediction) and negative biases (overprediction) occur across the range of predicted diameter increment and diameter. Figures D.7 to D.12 however, do not indicate the trend of bias increasing with predicted diameter increment as in Figures D . l to D.6. vThe plots did not indicate which level of modelling is better in terms of better spread of bias or magnitude of bias. The biases were also plotted by species against predicted diameter increment and actual diameter. In general, these plots reveal the same pattern as that already presented. Chapter 5. Results 54 Table 5.3: Calculated Standard Error of Estimates and Percentage Error Prediction, ' A l l Trees' Validation Data  Mean Standard Error Percentage Error Species N Q Actual Estimate (cm) Prediction Di Incfc One Gp Two Gp Species One Gp Two Gp c Species R A M I 84 0.286 0.273 0.208 0.185 96.0 73.3 65.0 JONG 75 0.296 0.318 0.344 0.188 107.3 116.2 63.7 SEPP 131 0.260 0.243 0.211 0.211 93.6 81.3 81.4 M B U A 112 0.471 0.420 0.403 0.411 89.3 85.7 87.3 MLIL 246 0.350 0.303 0.300 0.294 86.6 85.7 83.9 M P A Y 88 0.288 0.268 0.276 0.269 93.3 96.1 93.7 K A P A 107 0.200 0.244 0.254 0.208 122.0 126.8 103.8 GERQ 74 0.572 0.414 . 0.393 0.375 72.4 68.7 65.5 M D P D 45 0.510 0.387 0.381 0.359 75.9 74.8 70.4 A K A U 57 0.651 0.476 0.461 0.407 73.1 70.9 62.6 K E L A 62 0.683 0.461 0.442 0.378 67.6 64.7 55.4 K E B A 42 0.378 0.220 0.241 0.231 58.3 63.7 61.1 K T P Y 58 0.266 0.287 0.276 0.267 107.9 104.0 100.5 K P P Y 112 0.350 0.288 0.296 0.277 82.3 84.4 76.2 N Y B A 44 0.524 0.437 0.454 0.431 83.4 86.6 82.4 N Y J A 136 0.246 0.211 0.202 0.192 85.6 81.9 78.1 SIMP 92 0.118 0.232 0.175 0.128 196.5 148.0 108.4 T E R E 101 0.774 0.496 0.461 0.348 64.1 59.6 44.9 "Number of Cases b Diameter Increment (cm) 0 Group Chapter 5. Results 55 Table 5.9: Calculated Standard Error of Estimates and Percentage Error Prediction, 'Best Trees' Validation Data Species N Mean Actual Di Inc Standard Error Estimate (cm) Percentage Error Prediction One Gp Two Gp Species One Gp Two Gp Species R A M I 77 0.360 0.121 0.121 0.120 33.5 33.7 33.5 J O N G 40 0.414 0.209 0.218 0.202 50.6 52.7 48.9 SEPP 83 0.389 0.217 0.215 0.217 55.9 55.4 55.8 M B U A 52 0.537 0.276 0.279 0.293 51.3 52.0 54.6 MLIL 99 0.492 0.247 0.246 0.241 50.2 50.0 49.0 M P A Y 33 0.406 0.193 0.191 0.187 47.6 47.1 45.9 K A P A 56 0.388 0.241 0.240 0.237 62.1 61.9 61.2 GERO 30 0.950 0.290 0.286 0.318 30.6 30.1 33.5 A K A U 25 0.757 0.270 0.277 0.287 35.7 36.6 37.9 K E L A 28 0.894 0.253 0.268 0.272 28.3 30.0 30.4 N Y J A 26 0.418 0.318 0.298 0.341 76.1 71.1 81.5 T E R E 48 1.024 0.249 0.246 0.232 24.3 24.0 22.7 5.1.3 Mean Absolute Deviations The overall mean absolute deviations for the two validation data sets are summa-rized in Table 5.10. This indicates that for the 'all trees' data set, prediction using an equation at the species level resulted in less deviation than using two equations or a single equation. This result is the same as that indicated for the mean squared bias calculations. The results from the two validation data sets were further analysed at the species level to examine if the better prediction also holds true. Better predictions are possible using each species equation for Ramin, Jongkong, Sepetir, Meranti buaya, Meranti paya and Kapur for the 'all trees' data set (Table 5.11). Modelling at the species level also resulted in smaller MAD's for the other commercial species (e.g., Akau, Kelampu, Ketiau paya, Kumpang paya, Nyatoh jangkar, Simpoh and Chapter 5. Results 56 Table 5.10: Mean Absolute Deviations (cm) from Validation Data Sets Data Set Number of Cases Level of Modelling One Group Two Groups Species ' A l l Trees' 1666 0.249 (0.218) 0.237 (0.215) 0.208 (0.201)° 'Best Trees' 596 0.177 (0.155) 0.177 (0.155) 0.176 (0.162) "standard deviation of bias Terentang). Modelling at the species level for the 'best trees' data resulted in smaller MAD's for Jongkong, Meranti lilin, Meranti paya, Kapur and Terentang but larger MAD's for Meranti buaya, Geronggang, Akau, Kelampu and Nyatoh jangkar (Table 5.12). 5.2 Model-Building 5.2.1 Linear Models Based on the results of testing the three model-building levels, the individual species » level was chosen. Linear models for diameter increment were fitted to both model-building data subsets for the seven individual species using PROC R S Q U A R E . This first stage of model building gave the 'best' model, and results are summarized in Tables 5.13 and 5.14 for the 'all trees' and 'best trees' data respectively. Appendices C . l to C.4 give the results of all possible regressions for the four species groups. The results of testing for important interaction terms to be added to the 'best' model are summarized in Table 5.15. This shows that only the 'best' models for Ramin, Jongkong, Sepetir and Kapur developed from the 'all trees' data set were affected. The 'best' models for the seven species for the 'best' trees data have no Chapter 5. Results 57 Table 5.11: Mean Absolute Deviation (cm) for ' A i l Trees' Validation Data Species Name N Level of Modelling One Group Two Groups Species R A M I 84 0.229 0.153 0.119 J O N G 75 0.274 0.303 0.138 SEPP 131 0.190 0.154 0.153 M B U A 112 0.319 0.309 0.277 MLIL 246 0.238 0.238 0.234 M P A Y 88 0.212 0.223 0.197 K A P A 107 0.192 0.199 0.157 GERO 74 0.309 0.291 0.290 M D P D 45 0.269 0.266 0.286 A K A U 57 0.368 0.359 0.302 K E L A 62 0.360 0.341 0,293 K E B A 42 0.185 0.205 0.199 K T P Y 58 0.180 0.163 0.164 K P P Y 112 0.237 0.244 0.210 N Y B A 44 0.286 0.294 0.309 N Y J A 136 0.168 0.146 0.150 SIMP 92 0.197 0.152 0.089 T E R E 101 0.397 0.367 0.268 Chapter 5. Results 58 Table 5.12: Mean Absolute Deviation (cm) for 'Best Trees' Validation Data Species Name N Level of Modelling One Group Two Groups Species R A M I 77 0.088 0.091 0.087 JONG 40 0.159 0.166 0.144 SEPP 83 0.149 0.146 0.147 M B U A 52 0.218 0.220 0.228 MLIL 99 0.197 0.196 0.193 M P A Y 33 0.148 0.148 0.143 K A P A 56 0.201 0.196 0.177 GERO 30 0.237 0.236 0.252 A K A U 25 0.201 0.203 0.248 K E L A 28 0.197 0.208 0.217 N Y J A 26 0.234 0.227 0.237 T E R E 48 0.200 0.193 0.179 interaction terms added. These models, and others not affected by the addition of interaction terms, were regarded as the 'final' models. The results of testing for homogeneity of variance using the Goldfeld-Quandt test for the 'final' models are summarized in Table 5.16 and 5.17 for the two data sets. The results indicate that heteroscedasticity was present except for the Ramin 'best trees' data. The plot of residuals against predicted values for the unweighted linear model of Ramin is given in Figure 5.1. The results of testing for homogeneity of variance on the weighted equations are summarized in Table 5.18 and 5.19 for the 'all trees' and 'best trees' data respectively. The results of the linear (unweighted) regression for Ramin developed from the 'best trees' data subset are given in Table 5.20. The parameter estimates for the weighted regressions are summarized in Tables 5.21 to 5.22 and 5.23 to 5.24 for the 'all trees' and the 'best trees' data respectively. Chapter 5. Results 59 PLOT OF ZRESID WITH PRED +•----•----•----•----+----+----+----•----•----+----+-.--3* 1 1 1 1 2+ 1 1 1 I 1 1 1 1 2 2 1 1 1+ 1 11 1 1 1 I I 12 1 111 13 1 11112 1 1 1 2 1 11 131 1 11 1 1 11 1211 11 O* 1 1 21121 2 1 1 1 11 1 1 11 2 1 1 1 2 31 21 11 1 112 131 1 1 1 1 1 2 1 13 1 1* 11 121 1 11 2 1 1 1 2 1 1 -2+ 1 11 1 •3* •4* • 5+ • +..+..+ •.... + .—• ... + ....•...-•..--•-.-.• + -.4 - .2 0 .2 .4 .6 .8 1 1 2 Pred ic ted Value Figure 5.1: Residuals Plot for Unweighted Linear Model for Ramin, 'Best Trees' Data Chapter 5. Results 60 Table 5.13: Results of 'Best Model' from A l l Possible Regressions for ' A l l Trees' Pat a Species/ No. of Variables T>2 S E E P 2 c P SSE P P R E S S P Sp Group Cases in Model (cm) R A M I 176 CI,AGE,RS 0.105 0.186 -2.4 5.97 6.26 JONG 152 SN,D,QDT,D 2 BT,ST,RS,BN 0.410 0.204 10.4 5.86 7.65 QDN,QDC SEPP 267 D,CP,SC,QDC 0.272 0.189 4.4 9.26 10.01 RS,ST,QDN,BN M B U A 227 D,D 2 ,QDN,QDT 0.327 0.308 8.5 20.76 22.73 QDC,ST,BT MLIL 496 D , D 2 , B N , C P 0.223 0.290 4.7 41.00 42.62 AGE,SN,QDN M P A Y 178 CI,SN,BC,RS 0.210 0.269 -1.1 12.54 13.41 K A P A 219 CP,CI ,QDC,D 2 D,BC,BN,SN 0.449 0.200 8.8 8.34 10.15 Q D T . A G E Table 5.14: Results of 'Best Model' from A l l Possible Regressions for 'Best Trees' Data Species/ No. of Variables Kdj S E E P c P SSE P P R E S S P Sp Group Cases in Model (cm) R A M I 156 L D G , D , D 2 0.515 0.136 5.1 2.79 3.02 BC,ST J O N G 81 L D G , S T , C P , D 2 0.694 0.169 4.7 2.07 2.63 D , B T , A D T SEPP 168 L D G , C P , S T , B N 0.473 0.169 4.9 4.59 5.11 R S , A D C , A D N M B U A 106 L D G , D 2 , A D N 0.532 0.252 4.0 6.27 7.56 BN,SN,RS MLIL 201 L D G , D 0.567 0,238 -0.7 11.23 11.66 M P A Y 66 D 2 , A D T , L D G 0.528 0.212 4.7 2.68 3.37 S C , A G E K A P A 115 LDG,SC,CP 0.576 0.196 -3.6 4.29 4.68 Chapter 5. Results 61 Table 5.15: Results of Adding Interaction Terms to 'Best' Model for ' A l l Trees' Data Species/ No. of Variables S E E P SSE P PRESSp Sp Group Cases in Model (cm) R A M I 156 CI,AGE,RS CP,CI*CP 0.138 0.183 5.69 6.08 JONG 81 SN.D,QDT,D 2 ,BT ST,RS,BN,QDN 0.418 0.203 5.71 7.63 QDC,CP,D*CP SEPP 168 D,SC,QDC,RS,ST,QDN BN,CP,D*CP,SC*CP 0.311 0.184 8.59 9.81 ST*CP,QDN*CP,BN*CP K A P A 115 CI ,QDC,D 2 ,D BC,BN,SN,QDT 0.474 0.196 7.96 9.29 CP,D*CP Table 5.16: Summary of Goldfeld-Quandt Test on the 'Final' Model, ' A l l Trees' Data Species N MSEi MSE 2 R df F SigQ RAMI 176 0.02264 0.05200 2.30 54,54 1.58 * J O N G 152 0.01208 0.07924 6.97 38,38 1.71 * SEPP 267 0.01233 0.06273 5.09 76,76 1.46 MBUA 227 0.03794 0.16801 4.43 72,72 1.46 * MLIL 496 0.05832 0.12297 2.11 157,157 1.30 * MPAY 178 0.03285 0.12133 3.69 55,55 1.58 * KAPA 219 0.01868 0.06814 3.09 58,58 1.55 ° * = significant at an a level of 0.05 ns = not significant at an a level of 0.05 Chapter 5. Results 62 Table 5.17: Summary of Goldfeld-Quandt Test on the 'Final' Model, 'Best Trees' Data Species N MSEj. M S E 2 R df F Sig a R A M I 156 0.01252 0.02012 1.61 44,44 1.65 ns JONG 81 0.00759 0.05123 6.75 19,19 2.15 * SEPP 168 0.01503 0.04722 3.14 48,48 1.60 * M B U A 106 0.02290 0.08788 3.84 28,28 1.87 * MLIL 201 0.03689 0.07157 1.94 67,67 1.52 * M P A Y 66 0.00655 0.08249 12.59 16,16 2.33 K A P A 115 0.00850 0.07123 8.38 36,36 1.76 * ° * = significant at an a level of 0.05 ns — not significant at an a level of 0.05 Table 5.18: Summary of Goldfeld-Quandt Test on the Weighted Equation, ' A l l Trees' Data Species N M S E i M S E 2 R df F SigQ Weight R A M I 176 0.00100 0.00188 1.88 55,55 1.57 D b J O N G 152 0.00033 0.00419 12.70 39,39 1.69 * D SEPP 267 1.9567 2.3613 6.6321 2.7811 3.39 1.18 43,39 77,79 1.68 1.46 * ris pc P M B U A 227 2.7930 3.7447 1.34 69,73 1.46 ns P MLIL 496 3.0862 3.6493 1.18 157,157 1.32 ns D M P A Y 178 2.3425 3.3578 1.43 57,57 1.54 ns P K A P A 219 3.9546 6.3011 1.59 65,64 1.52 * P " * = significant at an a level of 0.05 ns = not significant at an a level of 0.05 6 reciprocal of dbh °power of predicted diameter increment -Chapter 5. Results 63 Table 5.19: Summary of Goldfeld-Quandt Test on the Weighted Regression, 'Best Trees' Data Species N M S E i M S E 2 R df F Sig a Weight JONG 81 0.00021 0.00271 12.90 20,20 2.12 D 2.1404 6.0794 2.84 18,21 2.12 P SEPP 168 3.2989 4.6173 1.40 52,51 1.60 ns P M B U A 106 0.00243 0.00395 1.63 29,29 1.84 ns D MLIL 201 0.00299 0.00242 0.81 64,64 1.51 ns D M P A Y 66 4.0059 2.0140 0.50 12,16 2.42 ns P K A P A 115 2.5130 4.5337 1.80 36,36 1.75 * P " * = significant at an a level of 0.05 ns — not significant at an a level of 0.05 Table 5.20: Statistics of Unweighted Linear Regression for Ramin, 'Best Trees' Data Regression Standard Contribution" Variable Coefficient Error t-Stat to R 2 Intercept -0.220628 0.116809 -1.89 L D G 0.542990 0.0625356 8.68 0.2357 D 0.0220558 0.00679113 3.25 0.0330 D2 -0.000313362 0.000112771 -2.78 0.0241 BC -0.0108004 0.00354262 -3.05 0.0291 ST 0.000181986 0.000062557 2.91 0.0265 R-Squared Adjusted R-Squared Residual Mean Square Standard Error of Estimate Residual Sum of Squares (RSS) Degrees of Freedom for RSS PRESS 0.5311 0.5155 0.0186254 0.136 2.794 . 150 3.021 "Contribution to R 2 or squared semipartial correlation is the amount by which R 2 would be reduced if that variable were to be removed from the regression model. Chapter 5. Results 64 Table 5.21: Parameter Estimates of Weighted Linear Regression for Shorea spp, ' A l l Trees' Data Variable M B U A MLIL M P A Y Intercept -2.677440 -3.082202 0.285641 D 0.044783 0.025863 I? 2 -0.000470 -0.000225 CI 0.378335 A G E 0.008078 BT -0.046933 B N -0.166250 ST 0.001589 SN 0.003747 -0.000369 QDT 0.157948 QDN -0.043192 0.183311 CP 0.125147 5.2.2 Nonlinear Models Only one nonlinear model was attempted in fitting to the two mo del-building data sets. The Levenberg-Marquardt algorithm in SPSS Version 4.0 was used with the derivatives supplied in the run statement. Convergence of the solution was faster with derivatives than if the derivatives were not supplied. The solutions from SPSS Version 4.0 were checked against those obtained by the Gauss-Newton algorithm of SAS. Exactly the same solutions were obtained by both algorithms. The starting values for the nonlinear regression were estimated in the following manner. The nonlinear regression for the 'all trees' data, for example, was: DI = 9XCI + 62De= + 8 4 B T Linear regression (without intercept) was applied to DI — diCI + 0 4 B T to estimate starting values for 9t and 64. Then a logarithmic transformation was applied to DI = Q 2 D 6 z resulting in ln(DI) = ln62 + 63ln(D). A regression of Chapter 5. Results 65 Table 5.22: Parameter Estimates of Weighted Linear Regression for R A M I , JONG, SEPP and K A P A , ' A l l Trees' Data Variable R A M I J O N G SEPP K A P A Intercept 0.192199 -11.038678 -3.718799 -0.503197 D 0.018832 0.007668 0.019495 D 2 -0.000183 -0.000435 CI 0.421195 A G E -0.006047 BT 0.498339 B N -1.319996 -0.178709 RS -7.120668 2.216462 ST -0.007717 0.005097 SN 0.024212 0.000312 SC -0.006709 QDT -1.359253 0.015968 QDN 1.887388 0.215604 QDC 0.304483 -0.032819 CP 0.257637 D*CP -0.006448 0.017419 CI*CP -0.394020 SC*CP -0.000518 QDN*CP 0.023850 Chapter 5. Results 66 Table 5.23: Parameter Estimates of Weighted Linear Regression for Shorea spp, 'Best Trees' Data Variable M B U A MLIL M P A Y Intercept 2.091587 -0.098276 -2.402001 D 0.008215 D 2 0.000237 L D G 0.775182 0.768692 0.540306 A G E 0.011860 B N 0.061633 RS 1.697208 SC 0.000593 SN -0.001169 A D T 0.113127 A D N -0.188120 Table 5.24: Parameter Estimates of Weighted Linear Regression for JONG, SEPP and K A P A , 'Best Trees' Data Variable J O N G SEPP K A P A Intercept 4.662580 -1.1466721 0.254596 L D G 0.498428 0.608893 0.530122 B T 0.053887 B N -0.049452 RS -1.260629 ST -0.000934 0.000767 SC -0.000536 A D T -0.302141 A D C 0.027260 A D N 0.076866 CP 0.080664 0.080519 0.195569 Chapter 5. Results Table 5.25: Parameter Estimates for Nonlinear Model, ' A l l Trees' Model-Building Species N 0i 02 03 04 M S E R A M I 176 0.115234 0.162570 0.097107 -0.000540 0.03635 JONG 152 0.257014 1.636235 -0.552524 -0.005672 0.06319 SEPP 267 -0.059743 0.094637 0.490805 -0.003947 0.04099 K A P A 219 0.193671 0.056275 0.597157 -0.005264 0.05448 M B U A 227 -0.095997 0.206551 0.449991 -0.011087 0.10931 MLIL 496 0.037587 0.133325 0.510618 -0.007613 0.09037 M P A Y 178 0.232501 0.094591 0.380819 -0.003830 0.07550 Table 5.26: Parameter Estimates for Nonlinear Model, 'Best Trees' Model-Building Data Species N 0i 02 03 04 M S E R A M I 156 0.636843 0.039105 0.369333 -0.000715 0.02040 J O N G 81 0.711419 -0.040100 0.298945 0.003447 0.04007 SEPP 168 0.624576 0.034074 0.502989 -0.001275 0.03183 K A P A 115 0.494740 0.087698 0.522035 -0.011043 0.04852 M B U A 106 0.809725 -1.337327 -0.929833 0.004072 0.07149 MLIL 201 0.739493 0.005284 1.095985 -0.002111 0.05709 M P A Y 66 0.360242 0.000013 2.787626 0.004105 0.05806 ln(DI) upon ln(D) yielded starting values for 62 and 03-The models for the seven individual species for the 'all trees' data subset and the 'best trees' data model-building data sets are summarized in Tables 5.25 and 5.26, respectively. The results of modelling for the four species groups are given in Appendix C.2. 5.2.3 Validation of Models The results of the validation in terms of pseudo-R2, mean bias, mean squared bias and mean absolute deviation using the 'all trees' and 'best trees' validation data 67 Data Chapter 5. Results 68 Table 5.27: Pseudo-R2, Mean Bias, Mean Squared Bias (MSB) and Mean Absolute Deviation (MAD) by Model, ' A l l Trees' Validation Data Species N a Pseu< io -R 2 Mean Bias (cm) MSB (cm2) M A D (cm) Linear Nonlin Linear Nonlin Linear Nonlin Linear Nonlin b R A M I 84 0.134 0.092 -0.011 -0.012 0.035 0.037 0.124 0.129 JONG 76 0.240 0.050 0.010 0.066 0.176 0.221 0.199 0.224 SEPP 132 0.131 0.097 0.046 0.032 0.107 0.111 0.173 0.180 K A P A 109 0.171 0.234 0.007 -0.001 0.104 0.096 0.184 0.186 M B U A 112 0.107 0.019 0.094 0.056 0.176 0.193 0.313 0.329 MLIL 246 0.207 0.178 -0.018 -0.021 0.086 0.089 0.232 0.240 M P A Y 88 0.287 0.298 -0.002 -0.004 0.069 0.068 0.190 0.190 "Number of observations b Nonlinear subsets are summarized in Table 5.27 and 5.28 respectively. The linear model predicted better than nonlinear model for the 'all trees' valida-tion data (Table 5.27). Except for Kapur and Meranti paya, the pseudo-R2 values are larger for the linear model than for the nonlinear model. The mean squared bi-ases are also smaller for the linear model for all the individual species except Kapur and Meranti paya. Similarly, the mean absolute deviations are slightly smaller for the linear model than for the nonlinear model. The linear model predicted better than nonlinear model for Ramin, Jongkong, Sepetir and Meranti paya using the 'best trees' validation data (Table 5.28). For Kapur, the MSB and M A D are larger for the linear model than for the nonlinear model. The figures for pseudo-R2, MSB, and M A D did not indicate which model predicts better for Meranti lilin. An overall validation was carried out by combining the validation data for the seven individual species for each of the two data sets. The relevant individual species equation was applied to each case of the validation data to compute the objective measures. The results are summarized in Table 5.29. The pseudo-R2 Chapter 5. Results 69 Table 5.28: Pseudo-R2, Mean Bias, Mean Squared Bias and Mean Absolute Deviation by Model, 'Best Trees' Validation Data Species N Pseudo-R2 Mean Bias (cm) MSB (cm2) M A D (cm) Linear Nonlin Linear Nonlin Linear Nonlin Linear Nonlin R A M I 77 0.452 0.364 0.023 0.021 0.014 0.017 0.089 0.094 JONG 40 0.567 0.584 0.053 0.039 0.044 0.043 0.148 0.156 SEPP 83 0.396 0.399 -0.044 0.012 0.048 0.047 0.146 0.151 K A P A 56 0.548 0.597 0.022 0.012 0.055 0.049 0.178 0.173 M B U A 52 0.536 0.491 0.035 0.012 0.073 0.081 0.209 0.226 MLIL 99 0.509 0.508 -0.003 -0.006 0.060 0.060 0.198 0.198 M P A Y 33 0.530 0.502 0.034 -0.036 0.030 0.132 0.135 0.184 value was larger for the linear model, while the MSB and M A D values were smaller relative to their values for the nonlinear model in the case of the 'all trees' data. However, the pseudo-R2 value was smaller for the linear model than the nonlinear model for the 'best trees! data. Also, the MSB value was larger and the M A D value was smaller for the linear model than the nonlinear model for the 'best trees' data. This indicates that the linear models were slightly better predictors than the nonlinear models for the 'all trees' data set. The overall validation result for the 'best trees' data did not confirm this. An analysis of MSB and M A D by 10 cm diameter class indicated that the linear models predicted slightly better than the nonlinear model for some diameters (Table 5.30), and slightly worse for others. Chapter 5. Results 70 Table 5.29: Pseudo-R2, Mean Bias, Mean Squared Bias and Mean Absolute Deviation by Model and Data Set Data Set Model Pseudo-R2 Mean Bias (cm) MSB (cm2) M A D (cm) No. of Cases A l l Trees Linear 0.2086 0.0148 (0.324) 0.1049 (0.468) 0.2093 (0.247)° 847 Nonlin 0.1569 0.0106 (0.334) 0.1117 (0.555) 0.2178 (0.254) Best Trees Linear 0.3588 0.0107 (0.252) 0.0633 (0.356) 0.1643 (0.191) 440 Nonlin 0.4443 0.0095 (0.234) 0.0549 (0.181) 0.1661 (0.165) "standard deviation of bias Table 5.30: Mean Bias, Mean Squared Bias and Mean Absolute Deviation for the Models by Diameter Class Data Set Model Criterion Diameter Class (cm) 10 < 19 20 < 29 30 < 39 40 < 49 > 50 A l l Trees Linear M B (cm) 0.0199 -0.0152 0.0762 -0.0121 0.0622 MSB (cm2) 0.0734 0.0851 0.2607 0.1436 0.2070 M A D (cm) 0.1871 0.2183 0.2580 0.2426 0.3240 Nonlinear M B (cm) -0.0009 0.0060 0.1209 0.0155 -0.0726 MSB (cm2) 0.0728 0.0964 0.3319 0.0930 0.2292 M A D (cm) 0.1904 0.2372 0.2934 0.2178 0.3464 Best Trees Linear M B (cm) 0.0003 0.0194 0.0559 -0.0373 MSB (cm2) 0.0354 0.0526 0.0598 0.2028 M A D (cm) 0.1411 0.1785 0.1643 0.2293 Nonlinear M B (cm) -0.0054 0.0255 0.0563 -0.0376 MSB (cm2) 0.0355 0.0524 0.0593 0.1309 M A D (cm) 0.1461 0.1819 0.1665 0.2145 Chapter 5. Results 71 5.3 Importance of Crown Position and Previous Diameter Growth For Prediction 5.3.1 Crown Position (CP) The validation results when crown position was present in the weighted linear model are shown in Table 5.31. The last column in this table gives the percentage error prediction. Table 5.31: Mean Squared Bias, Mean Absolute Deviation, Percentage Error Prediction for Model With and Without Crown Position Data Set C P a in Species N DI b MSB 'SEE ' C M A D P E P d Model (cm) (cm2) (cm) (cm) A l l Trees Yes R A M I 84 0.2843 0.0353 0.1878 0.1243 66.1 No 0.0349 0.1869 0.1277 65.7 Yes MLIL 246 0.3498 0.0865 0.2941 0.2325 84.1 No 0.0875 0.2959 0.2339 84.6 Yes A l l Comm.6 2171 0.3764 0.1218 0.3490 0.2475 92.7 No 0.1256 0.3544 0.2512 94.2 Best Trees Yes J O N G 40 0.4140 0.0444 0.2108 0.1482 50.9 . No 0.0464 0.2155 0.1514 52.0 Yes SEPP 83 0.3888 0.0477 0.2184 0.1456 56.2 No 0.0496 0.2228 0.1468 57.3 Yes K A P A 56 0.3875 0.0546 0.2337 0.1779 60.3 No 0.0571 0.2391 0.1835 61.7 Yes A l l Comm 797 0.5770 0.0755 0.2748 0.1941 47.6 No 0.0765 0.2766 0.1950 47.9 "significant at 0.05 probability level for species/species group as listed in this table 6 Actual Diameter Increment cPseudo Standard Error of Estimate ^Percentage Error Prediction C A U commercial species Including crown position in the model improves the prediction slightly (Ta-ble 5.31). The percentage error prediction was reduced about one per cent. The Chapter 5. Results 72 MSBs (except for Ramin) and MADs decreased slightly when CP was in the model. The contribution of competitive index (CI) for predicting diameter increment was assessed in the same manner as crown position (Table 5.32). Its presence had more effect in reducing the percentage error prediction for Ramin, but practically no effect in reducing percentage error prediction for the model developed for all the commercial species as a single group. Table 5.32: Mean Squared Bias, Mean Absolute Deviation and Percentage Error Predic-tion for Model With and Without Competitive Index Data Set CI in Species N DI MSB 'SEE ' M A D P E P Model (cm) (cm2) (cm) (cm) ' A l l Trees' Yes R A M I 84 0.2843 0.0353 0.1878 0.1243 66.1 No 0.03995 0.1999 0.1249 70.3 Yes A l l Comm. 2171 0.3764 0.1218 0.3490 0.2475 92.7 No 0.1224 0.3498 0.2481 92.9 5.3.2 Previous Diameter Growth (LDG) The results of validation when L D G was in and out of the model are summarized in Table 5.33. It can be seen that including L D G in the model improves the prediction in terms of percentage error prediction by as much as 20 % for Jongkong, Meranti buaya, and Meranti paya. The MSBs and MADs are considerably lower when L D G was in the model than when it was not included. Chapter 5. Results 73 Table 5.33: Mean Squared Bias, Mean Absolute Deviation and Percentage Error Predic-tion for Model With and Without L D G L D G in Species N DI MSB 'SEE' M A D P E P Model (cm) (cm2) (cm) (cm) Yes R A M I 77 0.3596 0.0144 0.1200 0.0891 33.4 No 0.0179 0.1338 0.1068 37.2 Yes JONG 40 0.4140 0.0444 0.2108 0.1482 50.9 No 0.0901 0.3001 0.2239 72.5 Yes SEPP 83 0.3888 0.0477 0.2184 0.1456 56.2 No 0.0729 0.2699 0.2041 69.4 Yes K A P A 56 0.3875 0.0546 0.2337 0.1779 60.3 No 0.0906 0.3010 0.2201 77.7 Yes M B U A 52 0.5371 0.0734 0.2710 0.2090 50.5 No 0.1438 0.3792 0.2827 70.6 Yes M P A Y 32 0.4063 0.0299 0.1730 0.1351 42.6 No 0.0712 0.2668 0.2226 65.7 Yes A l l Comm 797 0.5770 0.0755 0.2748 0.1941 47.6 No 0.1159 0.3405 0.2545 59.0 Chapter 6 Discussion 6.1 Level of Modelling for Better Prediction Modelling at the individual species level resulted in better predictions of diameter growth when tested against the validation data. This was especially so for the 'all trees' data set. The Mixed Swamp Forests in Sarawak have 45 commercially known species and this consists of many genera. Each tree species is silviculturally different in terms of light requirements, diameter growth rate, and maximum attainable dbh. There is a large amount of variation in diameter growth between species. The light demanders are generally faster growers while the shade tolerant trees are relatively slow growing. By modelling diameter growth at the individual species level, these differences between species were in part accounted for, leading to better predictions. The results are not obvious when the models were tested against the 'best trees' validation data set. A possible explanation is that at the individual species level of model building, some models for the 'best trees' data were built from very small number of observations and as such do not 'generalize' to other situations. In fact, the SEEs for the model built from the 'best tree' data for Kapur, Geronggang and Nyatoh jangkar (0.200, 0.374 and 0.202 respectively) were larger than their corresponding values (0.187, 0.319 and 0.194 respectively) for the 'all trees' data set. The SEEs for models built from the 'best trees' data were generally smaller than those for models built from the 'all trees' data set. Another explanation is 74 Chapter 6. Discussion 75 that validation for the 'best trees' was done for a smaller range of species and the trees selected for measurments were more uniform. 6.2 Linear Models Weighting was done for all the models except that based on the Ramin 'best trees' data as the error terms were heteroscedastic. Without weighting, there would be a loss in precison of the estimates of the regression coefficients and the usual ordinary least square estimates of the variances of the coefficients would be biased. Weight-ing was done mainly by two methods, one that assumes that variance increases with initial dbh and the other that assumes it is proportional to a power of the predicted value. The first method is simpler to carry out and was effective as a weight for Meranti lilin (both data sets) and Meranti buaya ('best trees' data set). This method was attempted by Wan Razali (1986) for correcting heteroscedasticity in his linear models of diameter growth of trees of Peninsula Malaysia. The second method, requires values of the predicted dependent variable from an initial regres-sion run. It was more effective in weighting those models that were not amenable to the first method. However, the weighting by the second technique failed for models developed for Jongkong and Kapur from both data sets (Tables 5.17 and 5.18). The Goldfeld-Quandt test indicated heteroscedasticity for both weighting methods for these two species; however, this could also be due to some type of misspecification such as omitted variables or an incorrect functional form (Judge et al, 1985). Nevertheless, there was some improvement in the distibution of residuals as reflected in the lower F-value for the Goldfeld-Quandt test with the weighted model than with the unweighted model. The Goldfeld-Quandt test results are in error if serial correlation is present (Epps Chapter 6. Discussion 76 and Epps, 1977) or if there is a lack of fit (Goldfeld and Quandt, 1965). The F-test for lack of fit was not done as repeated observations were not available with the independent variables and there was no evidence of lack of fit on the residual plots. Heteroscedasticitjr for the models mentioned above could be due to nonnormality of some of the variables concerned (Rawlings, 1988; Tabachnick and Fidell, 1989). This was indeed the case with the linear model for Ramin developed from the 'all trees' data set. The dependent variable is highly positively skewed while one of its independent variables, A G E , is highly negatively skewed. The skewness value for the two variables is 1.33 and -0.92 respectively. Logarithmic transformation could be applied to both variables before fitting, but was not attempted as the weighting method was preferred to preserve the original scale of the dependent variable. Also, logarithmic transformation of the dependent variable results in a biased estimate of it. In this study, the plots of residuals against predicted diameter increment for the unweighted linear models and also for some nonlinear models show an interesting pattern. An abrupt edge is formed in the lower half of the residuals plot. This line was also observed by Wan Razali (1986) in the residual plot for his models. He explained that this line corresponds to the observed diameter increment equal to zero and the residual equals the negative value of the predicted increment. He stated that the pattern was caused by the relatively large number of points having zero increment. Wan Razali (1986) determined that L D G , B C , RS and D*BC were common predictor variables in his weighted linear models for four tropical tree species groups. For this study using data from the Mixed Swamp Forests, the significant variables in the linear models varied with the species and the data sets. The significant Chapter 6. Discussion 77 variables for the models developed from the 'all trees' data set were a combination of D, D 2 , and various other stand variables. Crown position (CP) was significant in the 'all trees' model for R A M I and MLIL. The linear models for the 'best trees' have L D G in combination with other stand variables. Crown position appears as a significant variable for the 'best trees' model for J O N G , SEPP and K A P A . The variable, RS, expressing the relative density of commercial to non-commercial trees is important for J O N G and SEPP models for the 'all trees' data and for M B U A and SEPP models for the 'best trees' data. CI, considered as an alternative for CP w-as significant for only two models, R A M I and M P A Y , with the 'all trees' data. Some of the significant variables are being routinely measured while the rest could be derived from the measured ones. The algebraic sign for the coefficients of D, L D G and CP are all positive, reflecting their positive correlations with diameter growth. However, this holds only if the independent variables are not collinear or highly correlated. 6.3 Linear Versus Nonlinear Models 6.3.1 Accuracy of Prediction In general, the results of validation in terms of mean squared bias and mean absolute deviation indicate that linear models were better predictors than the nonlinear models. For the 'all trees' data the nonlinear model for Kapur and Meranti paya predicted better than the linear model as assessed by pseudo-R2 and MSB (Table 5.26). In the case of Kapur, the overall poorer prediction of the linear model was due to relatively poorer predictions for the dbh classes > 20.0 cm. The MSBs for 20-30, 30-40 and 40+ cm dbh classes for the linear model were 0.080, 0.170 and Chapter 6. Discussion 78 0.615 respectively, as compared to 0.051, 0.127 and 0.536 for the nonlinear model. The relatively poorer prediction of the linear model for Meranti paya was due to its poorer prediction for the dbh class > 40 cm (0.122 as compared with 0.108 for MSB). Table 5.27 indicated that the nonlinear model for Kapur for the 'best trees' data predicted better than the nonlinear model. Analysis of the mean squared biases by dbh class for the predictions for this model indicated smaller values than the nonlinear model across the range of diameters. The mean bias figures in Tables 5.26 and 5.27 indicated that the linear as well as the nonlinear model overpredicted (negative bias) and underpredicted (positive bias) depending on the species for which the model was built. The magnitudes of mean biases were small, however, the largest being 0.094 cm in the case of the linear model for Meranti buaya. Wan Razali (1986) stated the magnitudes of overpredictions and underpredictions for the four species group models developed by him were small and mostly less than 0.05 cm. This was also the case for the biases for the 'best trees' models in this study. 6.3.2 Other Considerations The plots of residuals against the predicted value for the weighted linear models were visually compared against those for the nonlinear models. The former exhib-ited a more satisfactory pattern, with the points spread quite uniformly about the zero line. The plots for the nonlinear models were more wedgeshaped. The linear models were relatively easier to fit than the nonlinear models. The linear models were fitted using the 'all possible regressions' procedure of SAS (PROC RSQUARE) . Even 15 independent variables, using PROC R S Q U A R E , which makes use of the leaps-and-bounds algorithm of Furnival and Wilson Chapter 6. Discussion 79 (1974), did not pose a problem in computing. The nonlinear models were harder to fit. First, due to cost considerations, the, nonlinear models fitted were all re-stricted to only four parameters. A larger model fitted on a large data set would be quite expensive to fit. Second, initial starting values for the parameters are nec-essary. These can be difficult to derive. Even though SAS PROC NLIN provides a grid search for the best starting values, this would entail much computing, and hence was not utilised to provide reasonable starting values. Third, an appropriate algorithm, which often is restricted to what is available with the popular statisti-cal packages, must be selected. Gallant (1987) stated that the more widely used methods of computing nonlinear least squares estimators are the modified Gauss-Newton method and the Levenberg-Marquardt algorithm. Seber and Wild (1989) observed that the former is not as robust as good implementations of the latter. The Levenberg-Marquardt algorithm available with SPSS Version 4 was used. This algorithm is also capable of checking whether the supplied derivatives were correct. Fourth, a global rather than a local solution is desired in fitting a nonlinear model. For checking the solution, the same algorithm can be tried with different starting points or a different algorithm can be attempted. The latter method using the Gauss-Newton algorithm (PROC NLIN in SAS) was chosen in this study to ensure a global solution. In this study, only one nonlinear form was attempted. Wan Razali (1986) considered five nonlinear models, and based on residual plots, normal probability plots and size of residual mean squares, concluded that only one of the nonlinear models performed better than the linear models. The linear models in this study predicted better than the nonlinear model. However, as the linear models must be viewed as purely approximations of the system and are meant for predictions, Chapter 6. Discussion 80 care must be taken to ensure they are not used to make large extrapolations. An appropriately formulated nonlinear model may be more realistic and would be expected to give better predictions, either for interpolation or extrapolation. More studies need to be done on the potential of the nonlinear models for prediction. It is to be noted that the nonlinear models made use of only three variables (four parameters), whereas the linear models had as many as ten variables. Biases and loss of predictability are introduced if relevant variables are eliminated while estima-tion and prediction may be improved by eliminating irrelevant variables (Rawlings, 1988). The nonlinear models could be improved further by incorporating any other relevant variables. A comparson of the 'best'1 three variables linear models with the nonlinear models in terms of mean square error indicates that the former have smaller values, implying better fits and perhaps better prediction than the latter. 6.4 Importance of Crown Position (CP) and Previous Diameter Growth (LDG) For Prediction It is rather surprising that crown position, regarded as having an important influ-ence on the diameter increment, contributed very little to accuracy in predicting di-ameter increment for commercial trees of the Mixed Swamp Forests. Bryan (1980) demonstrated for Dipterocarp Forests of Sarawak, that emergent and dominant trees have 50 % greater increments than the subdominant and understorey trees. Chai (1988) in developing regression models for four Shorea spp groups in three Hill Mixed Dipterocarp Forest sites in Sarawak stated that the R 2 value increased when crown position was incorporated in the models. He used two dummy or indicator variables to classify the crown positions. The canopy conditions in mixed forests 1as determined by P R O C R S Q U A R E in SAS Chapter 6. Discussion 81 make classifications of crown status difficult and not free of personal bias. Thus, competitive index (CI) as suggested by Alder (in FAO, 1983) appeared attractive. However, based on the results as given in Table 5.31, competitive index was not superior to crown position in contributing towards accuracy in predictions. The influence of L D G on prediction could be assessed from the unique contribu-tion of the independent variables shown (Table 5.19). The unique contribution to R 2 is the amount by which R 2 would be reduced if that variable were removed from the regression. For example, the total unique contribution of the five independent variables add up to 0.3484 of which LDG is 0.2357 (approximately 68 %) while the remaining (0.5311-0.3484) or 0.1827 is due to the joint contribution of the five variables in the case of the linear model for Ramin ('best trees'). This indicates that L D G is the most important of the predictor variables considered for predicting diameter growth of Ramin. Table 5.31 indicates that L D G was also an important predictor variable for the other individual species and for all the species as a single group. Wan Razali (1986) in developing weighted linear models for four species groups also observed that L D G was common to all models and that it contributed the most to explaining diameter growth of each species group. Chapter 7 Conclusions and Recommendations 7.1 Conclusions Diameter growth models for three Shorea spp and four other commercially impor-tant species in Mixed Swamp Forests of Sarawak were developed and validated. The following conclusions can be drawn from this study: 1. It is better, in terms of accuracy of predictions, to model diameter increment based on individual tree species rather than on a group or groups. The species level of modelling resulted in as much as 10 % reduction in percentage error prediction. 2. The weighted linear model predicted better than the selected nonlinear model for Ramin, Jongkong, Sepetir, Meranti buaya and Meranti lilin when validated on the 'all trees' data set. Upon validation on the 'best trees' data set, the weighted linear model predicted better than the nonlinear model in the case of Ramin, Meranti buaya and Meranti paya. 3. Heteroscedasticity was a problem when fitting the models. This was corrected by a weighted regression in which the weight was either the reciprocal of initial dbh or the reciprocal of a power of the predicted independent variable. 4. Crown position and its alternative, competitive index, were found to con-tribute little to the accuracy of predictions, while the previous diameter growth 82 Chapter 7. Conclusions and Recommendations 83 rate was shown to be very important in explaining current diameter growth. 7.2 Recommendations The following suggestions are made with regards to further studies in growth and yield of Mixed Swamp Forests in Sarawak: 1. No site productivity measure was available for the present study. This could be an important omitted variable which may have improved the prediction of diameter growth. The average total height of dominant trees remaining after logging has been used as an indicator of site productivity in the Philippines (Canonizado, 1978). The mean total height of either Kapur and/or the Shorea spp 45+ cm dbh could be used as a site indicator for Mixed Swamp Forests in Sarawak. 2. Trunk height measurements for potential crop trees > 30 cm have been taken in the latest assessments of the mixed swamp Yield Plots. In order to cover a larger range of diameters, and at the same time restricting the number of trees which need to be measured, the trunk height measurement can be done for one 'Leading Desirable' tree per quadrat. The trunk height measurements are required for computing stemwood volume of trees. 3. Detailed measurements for all individual trees, as was done since 1983, should be continued for all the mixed swamp Yield Plots. Since previous diameter increment has been shown to be important in predicting current diameter growth, another detailed or third measurement needs to be carried out, at the minimum, to obtain more accurate prediction models using data from these plots. Chapter 7. Conclusions and Recommendations 84 4. In this study, the best linear model was determined using the all possible regression approach. This approach fits every possible combination of all the specified variables. The possibility exists that a better model could be built using different transformations of, or interactions between, the independent variables. Inclusion of transformed variables was not considered seriously in this study. Future diameter increment models for forests in Sarawak could be developed, which incorporate various tranformed variables and/or their interaction terms. 5. Due to the increasing availability of powerful computers and optimization al-gorithms, the nonlinear models (i.e., the more theoretical models) for studying diameter growth merit serious attention. References Cited Adams, D. M . and Ek, A. R. 1974. Optimizing the management of uneven-aged forest stands. Can. J . For. Res. 4: 274-287. Adlard, P. J . , Spilsbury, M . J . and Whitmore, T. C. 1988. Current thinking on modelling the tropical moist forest. Paper presented at IUFRO Growth and Yield in Tropical Mixed/Moist Forest Conference, June 20-24, 1988, Kuala Lumpur. Alder, D. 1980. Forest volume estimation and yield prediction Vol 2- yield predic-tion. FAO Forestry Paper 22/2, FAO, Rome. 194p. Anderson, J . A. R. 1961. Peat swamp forests of Sarawak and Brunei in relation to their silviculture, Vol I. Univ., of Edinburgh, unpublished Ph.D. thesis. Belcher, D. M . , Holdaway, M . R. and Brand, G. J . 1982. A description of STEMS: The Stand and Tree Evaluation and Modeling System. USDA.For. Serv. Gen. Tech. Rep. NC-79. 18p. Bonnor, G. M . and Magnussen, S. 1988. Predicting stem diameter distributions from growth probability equations. USDA For. Serv. Gen. Tech. Rep. NC-120. pp. 192-200. 85 References Cited 86 Bowling, E. H. , Burkhart, H. E. , Burk, T. E. and Beck, D. E. 1989. A stand-level multispecies growth model for Appalachian hardwoods. Can. J . For. Res. 19: 405-412. Bruce, D. and Wensel, L. C. 1987. Modelling forest growth: Approaches, defini-tions and problems. Paper at IUFRO Forest Growth Modelling and Prediction Conference, Minneapolis, M N , Aug 24-28, 1987. pp. 1-8. Bruner, H. D. and Moser, J . W. 1973. A Markov chain approach to the prediction of diameter distribution in uneven-aged forest stands. Can. J . For. Res. 3: 409-417. Bryan, M . B. 1980. Studies of timber growth and mortality in the mixed dipterocarp forests of Sarawak. Consultancy report to FAO Project MAL/76/008. Buongiorno, J . and Michie, B. R. 1980. A matrix of uneven-aged forest manage-ment. Forest Science 26(4): 609-625. Canonizado, J . A. 1978. Simulation of selective forest management regimes. Malay. Forester 41(2): 128-142. Chai, E. 0. K. 1988. Regression Models for Shorea species in three primary hill mixed dipterocarp forests in Sarawak. Forest Deparment, Sarawak. 31p. Chai, F. Y . C. and Sia, P. C. 1988. Stand table projections for a mixed swamp forest of Sarawak. Paper presented at IUFRO Growth and Yield in Tropical Mixed/Moist Forests Conference, 20-24 June, 1988, Kuala lumpur. References Cited 87 Clutter, J . L. 1963. Compatible growth and yield models. Forest Science 9(3): 354-371. — : and Allison, B. J . 1974. A growth and yield model for Pinus radiata in New Zealand. In: Growth Models for Tree and Stand Simulation (Fries, J . ed.). Royal Coll. For. Stockholm. Res. Note 30. Fortson, J . C , Pienaer, L. V . , Brister, G. and Bailey, R. L. 1983. Timber Management: A Quantitative Approach. New York: Wiley. 333p. Davis, L. S. and Johnson, K. N . 1987. Forest Management, Third Edition. Mcgraw-Hill Book Inc. 790p. Dawkins, H. C. 1952. Experiments in low percentage enumerations of tropical high forest. Empire Forestry Rev., U.K. 31(2). Draper, N . and Smith, H. 1981. Applied Regression Analysis, 2nd edition. New York: Wiley. 709p. Dudek, A. and Ek, A. R. 1980. A bibliography of worldwide literature on individual tree based stand growth models. Department of Forest Resources, University of Minnesota, Staff Paper Series No. 12. 33p. Ek, A. R. 1974. Nonlinear models for stand table projection in northern hardwood stands. Can. J . For. Res. 4: 23-27. References Cited 88 and Monserud, R. A . 1974. FOREST: A computer model for the growth and reproduction of mixed species stands. Univ. Wisconsin, College of Agric. and Life Sciences Research Rep. R2635. 13p. and Monserud, R. A. 1975. Methodolgy for modeling forest stand dynamics. Staff Paper Series No. 2. 30p. and Dudek, A . 1980. Development of individual tree based stand growth simulators: Progress and Applications. Department of Forest Resources, Univ. of Minnesota, St. Paul, Staff Paper Series No. 20. 25p. Epps, T. W. and Epps, M . L. 1977. The robustness of some standard tests for auto-correlation and heteroskedasticity when both problems are present. Econometrica 45: 745-753. FAO. 1974a. A study of stand conditions and production potential in the peat swamp forests of Sarawak. FO:DP/MAL/72/009 Working Paper 27, FIDP, FAO, Kuala Lumpur. 1974b. An inventory of the mixed dipterocarp forest of Sarawak, 1969-1972. Volume I: Methodology, Technique and Results, Technical Report no. 2. 209p. 1983. Growth and yield of mixed tropical forests: Part I-Current Knowledge; Part II-Forecasting Techniques. Unpublished consultancy report for FAO by Denis Adler, Oxford, 1983. References Cited 89 Furnival, G. M . and Wilson. R. B. 1974. Regressions by leaps and bounds. Tech-nometrics 16: 499-511. . Gallant, A. R. 1987. Nonlinear Statistical Models. New York: Wiley. 610p. Goldfeld, S. M . and Quandt, R. E. 1965. Some tests for hornoscedasticity. J . Amer. Stat. Assn. 60: 539-547. Hahn, J . T. and Leary. R. A. 1979. Potential diameter growth function. In: A Generalized Growth Projection System. USDA For. Serv. Gen. Tech. Pap. NC-49. pp. 61-79. Hann D. W. 1980. Development and evaluation of an even- and uneven-aged pon-derosa pine/Arizona fescue stand simulator. USDA For. Serv. Res. Pap. INT-267. 95p. and Bare. B. B. 1981. Uneven-aged forest management: State of the art (or science). USDA For. Serv. Gen. Tech. Rep. INT-50. 18p. Harrison T. P. and Michie, B. R. 1985. A generalized approach to the use of matrix growth models. Forest Science 31(4): 850-856. Holdaway, M . and Brand, G. J . 1983. An evaluation of the STEMS tree growth projection system. USDA For. Serv. Res. Pap. NC-234. 20p. Jonkers, W. B. J . 1982. Options for silviculture and management of the mixed dipterocarp forest of Sarawak. FAO Working Paper No. 11, FO: MAL/76/008, Forest Department, Kuching. 117p. References Cited 90 Judge, G. G., Griffiths, W. E . } Hill , R. C , Liitkepohl, H. and Lee, T. C. 1985. The Theory and Practice of Econometrics. Second edition. John Wiley and Sons, New York. 1019p. Kofod, E. 0 . 1982. Standtable projections for the Mixed Dipterocarp Forest of Sarawak. Field Document No. 9, FAO/MAL/76/008, Forest Department, Kuching. 107p. Korsgaard, S. 1984. A manual for the standtable projection simulation model. The Research Council for Developmental Research, Copenhagen. 70p. 1988. The standtable projection simulation model. Paper presented at IUFRO Growth and Yield in Tropical Mixed/Moist Forests Conference. 20-24 June, 1988. Kuala Lurnpur. Lee, H. S. 1977. Manipulation and regeneration of the Mixed Swamp Forests in Sarawak. The Malayan Nature Journal 31(1): 1-9. 1979. Natural regeneration and reforestation in the peat swamp forests of Sarawak. Ministry of Agric, For., and Fisheries, Ibaraki, Japan. Trop. Agric. Res. Series No. 12. pp. 51-60. and Lai, K. K. 1977. A manual of silviculture for the permanent forest estate of Sarawak. Silviculture Pamphlet No. S.R. 1/77 Forest Department, Sarawak. 57p. Lorimer C. G. 1983. Tests of age independent competition indices for individual trees in natural hardwood stands. For. Ecol. Manage., 6: 343-360. References Cited 91 Lynch, T. B. and Moser, J. W. 1986. A Growth model for mixed species stands. Forest Science 32(3): 697-706. Meldahl, R. S., Bolton, R. K. and Marian Eriksson. 1987. Development of a mixed species projection system for southern forests. In: Proceedings, IUFRO Growth Modelling and Prediction Conference, Minneapolis, Minn. pp. 102-109. Monserud, R. A. and Ek, R. A . 1977. Prediction of understorey tree height growth in northern hardwood stands. Forest Science 23(3): 391-400 Moser, J . W. 1972. Dynamics of an uneven-aged forest stand. Forest Science 18(3): 184-191. 1974. A system of equations for the components of forest growth. In: Growth Models for Tree and Stand Simulation (Fries, J . ed.). Royal College of Forestry, Stockholm, Res. Notes No. 30. pp. 260-287. 1978. Discrete time Markov processes. Proc. 1977 Midwest Forest Mensurationists Meeting. USDA For. Serv. Gen. Tech. Rep. NC-46. pp. 21-27. Moser, J . W. and Hall, 0. F. 1969. Deriving growth and yield functions for uneven-aged forest stands. Forest Science 15(2): 183-188. Munro, D. D. 1974. Forest growth models- a Prognosis. In: Growth Models for Tree and Stand Simulation (Fries, J . ed.). Royal College of Forestry, Stockholm, Res. Notes No. 30. pp. 7-21. References Cited 92 Neter, J., Wasserman, W. and Kutner, M . H. 1990. Applied Linear Statistical Models: Regression, Analysis of Variance, and Experimental Designs. 3rd edition. Irwin Inc. 1181p. Nicholson, D. I. 1979. The effects of logging and treatment on the mixed Dipter-rocarp forests of southeast Asia. Consultancy Rep. FO: MISC/79/8, FAO, Rome. Rawlings, J . O. 1988. Applied Regression Analysis: A Research Tool. Wadsworth, Inc.. Belmont, California. 553p. Revilla, A. V . 1981. A review of growth and yield studies on the tropical mixed forests in the Asia region. Paper presented at joint FAO/IUFRO meeting on growth and yield studies in mixed indigenous forests, Jan 26-30, 1981, Los Banos, Philipp-pines. SAS Institute, Inc. 1985. SAS User's Guide: Statistics, Version 5 edition. Cary, North Carolina: SAS Institute, Inc. Seber, G. A . F. and Wild, C. J . 1989. Nonlinear Regression. New York:Wiley. 768p. Stage, A. R. 1973. Prognosis model for stand development. USDA For. Serv. Res. Pap. INT-137. 32p. Snee, R. D. 1977. Validation of regression models: methods and examples. Tech-nometrics 19(4): 415-428. References Cited 93 Solomon. D. A. . Hosmer, R. A. and Hayslett, H . T., Jr. 1986. A two-stage matrix modelfor predicting growth of forest stands in the Northeast. Can. J . For. Res. 16: 521-528. Tabachnick, B. G. and Fidell, L. S. 1989. Using Multivariate Statistics. Second Edition. Harper & Row, Publishers Inc., New York. 746p. Tang, H. T. and Wan Razali Mohd. 1981. Report on growth and yield in in-land mixed indigenous forests in Peninsular Malaysia, Forestry Dept., Peninsular Malaysia. Vanclay, J. K . 1983. Techniques for modelling timber yield from indigenous forests with special reference to Queensland. M.Sc. thesis, University of Oxford. 194p. 1987. A stand growth model for yield regulation in North Queensland rainforests. Paper presented at IUFRO Forest Growth modeling and Prediction Conference, Minneapolis, M N , August 23-27, 1987. 8p. 1988a. A stand growth model for yield predictions in rainforests: Design, Implementation and Enhancements. Paper presented at IUFRO Growth and Yield in Tropical Mixed/Moist Forests Conference, June 20-24, 1988, Kuala Lumpur. 1988b. Site productivity assessment in rain forests: an objective ap-proach using indicator species. Paper presented at IUFRO Growth and Yield in Tropical Mixed/Moist Forests Conference, June 20-24, 1988,-Kuala Lumpur. References Cited 9 4 1989. A growth model for North Queensland Rainforests. For. Ecol. Manage., 27: 245-271. Wan Razali Mohd. 1986. Developement of a generalized forest growth and yield modelling system for mixed tropical forests of Peninsular Malaysia. Ph.D. thesis, Univ. of Wash., Seattle. 177p. West, P. W. 1981. Simulation of diameter growth and mortality in regrowth euca-lypt forest of southern Tasmania. Forest Science 27(3): 603-616. Wykoff, W. R., Crookston, N . L. and Stage, A. R. 1982. User's Guide to the Stand Prognosis Model. USDA For. Serv. Gen. Tech. Rep. INT-133. Yandle, D. 0., Myers, J . R., Mayo J . H. and Lamson, N . I. 1987. Development and application of a stand-based growth model for mixed hardwood stands. Paper pre-sented at IUFRO Forest Growth Modeling and Prediction Conference, Minneapolis, M N , August 24-28, 1987. Appendix A List of Desirable and Acceptable Species in the Mixed Swamp Forest A . l Desirable Species (List 1) Vernacular name Botanical name Code Durian burong Durio carinatus Mast. D U R B Engkabang bungkus Shorea macrantha (Brandis) Sym. E N G B Jongkong Dactylocladus stenostachys Oliv. J O N G Kapur paya Dryobalanops rappa Becc. K A P A Keruing paya Dipterocarpus coriaceous V. SI. K E R P Kerukup Shorea pachyphylla Ridl. ex Sym. K E R K Meranti buaya Shorea uliginosa Foxw. M B U A Meranti lilin Shorea teysmanniana Dyer ex Brandis MLIL Meranti lop Shorea scabrida Sym. M L O P Meranti paya Shorea platycarpa Heim. M P A Y Mersawa paya Anisoptera marginata Korth. MSWA Ramin Gonystylus bancanus (Miq.) Kurz. R A M I Semayur Shorea inaequilateralis Sym. S E M A Sepetir paya Copaifera palustris (Sym.) De Wit SEPP 95 Appendix A. List of Desirable and Acceptable Species in the Mixed Swamp Forest 96 A.2 Acceptable Species (List 2) Ako Xylopia coriifolia Ridley A K A U Ako tembaga Xylopia fusca Maingay ex Hk. f. et Th. A K A T Benuah padang Macaranga pruinosa (Miq.) Muell. -Arg. B E N U Bintangors Calophyllum Linn. BINT Entuyut Tetramerista glabra Miq. E N T U Geronggang paya Cratoxylum arborescens (Vahl) B l . GERO Geronggang padang Cratoxylum glaucum Korth. G E R P Kelampu Sandoricum emarginatum Hiern. K E L A Kepayang babi Mezzettia leptopoda (Kk. f. et Th.) Oliv. K E B A Keruntum Combretocarpus rotundatus (Miq.) Dans. K R U N Ketiau paya Ganua motley ana (de Vr.) Pierre ex Dubard K T P Y Kumpang paya Horsfieldia crassifolia (Hk. f. et Th.) Warb. K P P Y Kumpang pendarahan Myristica lowiana King K P D H Medang bulu Litsea paludosa Kosterm. M D B U Medang engkala Litsea resinosa B l . M D E N Medang keli Litsea gracilipes Hook F. M D K E Medang padang Litsea crassifolia (Bl.) Boerl. M D P D Medang pasir Litsea turfosa Kosterm. M D P A Medang puteh Litsea nidularis Gamble M D P U Menggris Koompassia malaccensis Benth. MGRS Minggi Paratocarpus venenosus (Zoll. & Mor.) Becc. • MIGI endix A. List of Desirable and Acceptable Species in the Mixed Swamp Forest 97 Nyatoh babi Nyatoli jangkar Pelai paya Pitoh Rengas bulu Rengas kasar Rengas paya Serabah Simpor Terentang Source: Lee and Lai, 1977 ssp. foruesii (King) Jarrett Palaquium pseudorostratum H . J. Lam N Y B A Palaquium walsurifolium Pierre N Y J A Alstonia pneumatophora Backer ex L. G. P E L A den Berger. Swintonia glauca Engl. P T O H Melanorrhoea specioso Ridley R E B U Melanorrhoea tricolor Ridley R E K A Melanorrhoea beccarii Engl. R E P Y Goniothalamus andersonii J . Sinclair SERB Dillenia pulchella (Jack) Gilg. SIMP Campnosperma coriaceum (Jack) Hall. f. ex T E R E v. Steenis Appendix B List of Species by Growth and Shade Tolerance B . l Desirable Species: B . l . l Moderately Fast or Fast Growing, mainly Light Demanding: Vernacular name Botanical name Engkabang bungkus Shorea macrantha Geronggang paya Cratoxylum arborescens Geronggang padang Cratoxylum glaucum Jongkong^ Dactylocladus stenostachys Kapur paya Dryobalanops rappa Kelampu Sandoricum emarginatum Meranti buaya Shorea uliginosa Meranti lilin Shorea teysmanniana Meranti lop Shorea scabrida Meranti paya Shorea platycarpa Pelai paya Alstonia pneumatophora Semayur Shorea inaequilateralis 98 Appendix B. List of Species by Grovsth and Shade Tolerance 99 B.1.2 Mainly Slow Growing and Shade Tolerant: Ramin Gonystylus bancanus Sepetir paya Copaifera palustris Bintangor dudok Calophyllum rhizophorum Bintangor paya Calophyllum hosei Durian burong Durio carinatus Kepayang babi Mezzetia leptopoda Keruing paya Dipterocarpus coriaceus Ketiau paya Ganua motleyana Mersawa paya Anisoptera marginata Minggi Paratocarpus venenosus Nyatoh babi Palaquium pseudorostratum Nyatoh jangkar Palaquium walsurifolium B.2 Acceptable Species B.2.1 Fast Growing, mainly Colonizing Species Ako Ako tembaga Xylopia corrifolia Xylopia fusca endix B. List of Species by Growth and Shade Tolerance Benuah padang Kumpang paya Kumpang pendarahan Medang bulu Medang engkala Medang padang Medang pasir Medang puteh Serabah Terentang B.2.2 Slow Growing, Entuyut Kerukup Keruntum Menggris Pitoh Rengas bulu Rengas kasar Rengas paya Simpor Source: Lee, 1977 Macaranga maingayi Horsfieldia crassifolia Myristica lowiana Litsea grandis Litsea resinosa Litsea crassifolia Litsea turfosa Litsea nidularis Goniothalamus andcrsonii Campnosperma coriaceum Shade Tolerant Tetramerista glabra Shorea pachyphylla Combretocarpus rotundatus Koompassia malaccensis Swintonia glauca Melanorrhoea speciosa Melanorrhoea tricolor Melanorrhoea beccarii Dillenia pulchella Appendix C Model-Building for Species Groups C . l 'Best Models' from All Possible Regressions C . l . l 'All Trees' Data C . l .2 'Best Trees' Data C.2 Nonlinear Models C.2.1 'All Trees' Data C.2.2 'Best Trees' Data 101 Appendix C. Model-Building for Species Groups 102 Table C . l : Results of 'Best Model' for Species Groups from A l l Possible Regressions for ' A l l Trees' Data Species/ No. of Variables Kdj S E E P c P SSE P PRESS P Sp Group Cases in Model Non-comm J 6059 CP,ST,CI,QDN D , B T , Q D T , D 2 0.119 0.222 12.2 299.1 300.6 AGE,BN,SC,QDC AGE,BN,SC,QDC A l l commb 4356 CP,BN,QDN,ST SC,RS,QDT 0.192 0.306 12.1 408.0 410.8 C I , D 2 , D , A G E Light demc 2977 BC,CP,SC,QDN QDT,BT,ST 0.218 0.326 10.7 314.6 317.6 CI,SN,AGE,D Shade tol d 1379 D 2 , B T , Q D T BC,QDN,CP 0.200 0.221 9.0 66.9 68.3 aNon-commercial specis ' 'Al l commercial specis c Light demanding species dShade tolerant species Appendix C. Model-Building for Species Groups 103 Table C.2: Parameter Estimates of Unweighted Linear Regression for Species Groups, ' A l l Trees' Data Variable Non-comm All comm Light dem Shade tol Intercept 0.810766 -1.480589 -0.992312 -1.659500 D 0.008187 0.023763 0.031570 0.010269 D 2 -0.000079 -0.000320 -0.000373 -0.000105 CI 0.062334 0.201683 0.153071 0.111110 A G E -0.001563 -0.008408 -0.007220 -0.005865 BT 0.023226 -0.116622 -0.093571 B N 0.056879 -0.161299 BC 0.159594 0.084869 RS -0.914197 ST -0.001071 0.002972 0.002706 SC 0.001587 -0.003472 -0.004261 SN 0.002290 QDT -0.038346 -0.125233 0.024672 QDC 0.006231 QDN -0.073697 0.171899 0.204746 0.091020 CP -0.089028 0.076709 0.098341 0.042574 Table C.3: Results of 'Best Model' for Species Groups from A l l Possible Regressions for ' Best Trees' D at a Species/ Sp Group No. of Cases Variables in Model S E E P c p SSE P P R E S S P A l l comm 1614 LDG,CP,RS A D C , D , D 2 0.583 0.269 4.9 116.2 117.3 Light dem 1060 L D G , C P , A D N D , D 2 , A D C 0.627 0.272 3.1 78.2 79.3 ' Shade tol 554 L D G , C P , R S , D 2 D , A D N , A G E , B N SC,ADC,SN,ADT 0.510 0.218 12.5 25.7 27.1 Appendix C. Model-Building for Species Groups 104 Table C.4: Parameter Estimates of Unweighted Linear Regression for Species Groups, 'Best Trees' Data Variable All comm Light dem Shade tol Intercept -0.085460 -0.600240 -0.227190 D 0.016567 0.019868 0.011512 D 2 -0.000307 -0.000421 -0.000244 L D G 0.707597 0.771646 0.570463 A G E 0.006314 RS -0.226640 -1.333810 B N -0.043455 SC 0.000872 SN 0.000480 A D T -0.047759 A D C 0.004890 0.006910 0.037109 A D N 0.018171 0.049941 CP 0.042164 0.060499 0.107936 Table C.5: Parameter Estimates or Species Group Nonlinear Model, ' A l l Trees' Data Species 0i 02 03 0 4 M S E Non-comm A l l comm Light dem Shade tol 0.036758 0.232770 0.257837 0.097188 0.194240 0.177024 0.134742 0.127067 0.308835 0.276218 0.369054 0.354169 -0.007729 -0.005842 -0.005176 -0.004793 0.06164 0.10261 0.11795 0.05058 Table C.6: Parameter Estimates 'or Species Group Nonlinear Model, 'Best Trees' Data Species 0i 02 03 0A MSE A l l comm Light dem Shade tol 0.726801 0.793388 0.591955 0.039819 0.003081 0.232641 0.374448 0.706183 0.096261 -0.001448 0.000393 -0.005723 0.07362 0.07618 0.05127 Appendix D Distribution of Bias by Predicted Diameter Increment and Diameter D . l Plots of Bias for 'AH Trees' Validation Data D.2 Plots of Bias for 'Best Trees' Validation Data 105 Appendix D. Distribution of Bias by Predicted Diameter Increment and Diameter 106 BI1 1 .2 + 1 .0 + 0 . 8 + 0 . 6 + 0.4 + 0 . 2 + 0 . 0 + -• 0 . 2 l •0.4 + • 0 . 6 + • 0 . 8 i * * * * ** * * ****** * * * * * * * *** ****** ** ***** * ********* ****** **** ********** *_********* ********** ******** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ****** ****** ****** ****** ****** ****** ****** ****** ****** ****** ****** ****** * * * * ** * * * * * * ** ***** * * * * * * * ** * * * * * * * * * * * * * * * ** ***** ******** ***** * ****** ******* ************ ****** ** ********_* *_. ******* ** ****** *** * ******** ** *********** ************ ********** ****** * ** * * * * * * * * * * * * * * * * * * * * * * * * * * - 1 . 0 + - 1 . 2 + I -- + + + + + + + +-0 . 0 0 . 2 0.4 0 . 6 0 . 8 1 .0 1 .2 1.4 DIP1 Figure D.l: Bias, BI1 against Predicted Diameter Increment, DIP1, 'All Trees', One Group Modelling Appendix D. Distribution of Bias by Predicted Diameter Increment and Diameter 107 BI2 1.2 + 1.0 + 0.8 + 0.6 + 0.4 + 0.2 i 0.0 +--0.2 l -0.4 + -0.6 + -0.8 l -1.0 + * * * * * * * * * ** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ** * * ** *** * * * * * * * * * ******* * * * * * * * * ***** * ***** * * * * ** * ******* * **** * * * * * * * * * * **** ***** * * * * * * * * * * * ******* ***** * * * * * * * * * * * * * * * ***** *** ** * * * * * * * * * * * **** ** * * * * ** * * * ** * * * * * ******* ****** * * ** ****************** *** * * * **********************—** *_. ******************** *** * ** ********************** ** * ****************** ***** ************* *** * * * ************* ** * * *********** ** ** ****** **** ** * * * * * * * * * * * * * * * *** -1.2 + • - + -• 1.4 0.0 0.2 0.4 0.6 0.8 DIP2 1.0 1.2 Figure D.2: Bias, BI2 against Predicted Diameter Increment, DIP2, 'All Trees', Two Group Modelling Appendix D. Distribution of Bias by Predicted Diameter Increment and Diameter 108 B I 3 1 .2 + 1 .0 + 0 . 8 + 0 . 6 + 0.4 + 0 . 2 + 0 . 0 +-- 0 . 2 + -0.4 + - 0 . 6 + - 0 . 8 + - 1 . 0 + ** * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * ** * * * * * *** * * * * * * * * *** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ** ** * * * * * * * * ** * * * **** **** *** * * **** ***** * ****** **************** **** * * * * * ** * ** * * * * * * * ************* ****** **** * ************ ******* * *** **** *********** **** **** ** ** A * * * * * * * * * * * * * * — * * * — * — * * * — * * _ — * *************** ** * * * *************** ** ***** * **** ************ ***** ** * * ***************** ****** * ** *************** * * ** * * ******* ** *** ** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * - 1 . 2 + • - + - • 0 . 0 • - + - -0 . 2 • - + - -1.2 •-+— 1.4 0.4 0 . 6 0 . 8 DIP3 1 .0 Figure D.3: Bias, BI3 against Predicted Diameter Increment, DIP3, 'All Trees', Species Modelling Appendix D. Distribution of Bins by Predicted Diameter Increment and Diameter 109 BI1 1.6 + 1 .4 + 1 .2 + 1 . 0 + 0 . 8 + 0 . 6 + 0 . 4 + 0 . 2 + 0 . 0 + — - 0 . 2 + - 0 . 4 + - 0 . 6 + - 0 . 8 + * * * * * *** * *** * ** * * ** ***** ***** * ** * * * * * * * * * * ** * * * * * ***** * * ** * ********* * * * * ********** * * * *** ***** * * * ** * ** ********** ** * * * * ********* * * * * * ******** * * * * * ** * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * *** * * * * * * * * * * * * * * * * * * * * * * .************* *-** ***_. ************ ** **** * * * ********** *** ** *** ** ** ****************** *** * ********************* * * ****************** * * * ***************** * * * ***** **** * ** * *** *** * * * ** * * * - 1 . 0 + + + + + + + + + + + + -10 20 30 40 50 60 70 80 90 100 110 D Figure D.4: Bias, BI1 against Diameter, D, 'All Trees', One Group Modelling Appendix D. Distribution of Bias by Predicted Diameter Increment and Diameter 110 B I 2 1 . 6 + 1 .4 + 1 .2 + 1 . 0 + 0 . 8 + 0 . 6 i 0 . 4 + 0 . 2 i 0 . 0 i— - 0 . 2 + - 0 . 4 + - 0 . 6 + - 0 . 8 + * * * * * * * * * ******* * ****** ** * * * ******* * * * * * * * * ***** ****** * ****** * ** * * * * ********* * * * * * ********** * *********** ** ** * * ************* * ************** * ** * * ********* **** * ************ *** **** ** ************ * * * * .******************** * * ****************** * * ******************* ** ** * * ****************** * * * ***************** * ** **** * *********** **** *> * * * ******** ******** * * * * *** **** * * **** * * * * * * * * * ** * * * * * * * * - 1 . 0 + + + + + + + + + + + + -10 20 30 40 50 60 70 80 90 100 110 D Figure D.5: Bias, BI2 against Diameter, D, 'All Trees', Two Group Modelling Appendix D. Distribution of Bias by Predicted Diameter Increment and Diameter 111 B I 3 1 . 6 + 1 . 4 + 1 .2 + 1 . 0 + 0 . 8 + 0 . 6 i 0 . 4 + 0 . 2 i 0 . 0 + -• 0 . 2 + • 0 . 4 + • 0 . 6 + • 0 . 8 + • 1 . 0 + * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * .**** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * i * * * it* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * i ************** *** ** ** **************** *_** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ** * * ************* ******** * * **** ***** * ****** ** * ******* * * * * * ** * *** * ** * * * * ** * * * * * * • - + -20 • - + -90 —+ +--100 110 10 30 40 50 60 D 70 80 Figure D.6: Bias, B I 3 against Diameter, D, 'All Trees', Species Modelling Appendix D. Distribution of Bias by Predicted Diameter Increment and Diameter 112 B I 1 1 . 0 + 0 . 8 + 0 . 6 + 0.4 + 0 . 2 + 0 . 0 +-- 0 . 2 + -0.4 + - 0 . 6 + - 0 . 8 i ** * * * * * * * * * * * * * * * * * * * * * ** ** * ** ** * *** ** * * * * * * * * * * * * * * * * * * * * ** ** *********** * * * * ** ** ****** ** ******** *** * ************************* * * ** . A * * * * * * * * * * * * * * * * * * * * * * * * * — — — * _ * **************** * **** **** * ************** ***** * * ** ********* * ***** ****** * * *** *** * ** * * * * *** *** **** * * ** ** * * * * * * * * * * * * * * * * ** * * ** * * * * - 1 . 0 + • - + + + + + +--0 . 0 0.4 0 . 8 1.2 1 . 6 2 . 0 DIP1 Figure D.7: Bias, BI1 against Predicted Diameter Increment, DIP1, 'Best Trees', One Group Modelling Appendix D. Distribution of Bias by Predicted Diameter Increment and Diameter 113 BI2 1 .0 + 0 . 8 + 0 . 6 + 0.4 + 0 . 2 + 0 . 0 +« • 0 . 2 + •0.4 + • 0 . 6 + 0 . 8 + • 1 . 0 + * * * * * * * * * * * * * * * * * * 4c * * * * * * ** * * * * ***** * * * ** * * * * * * * * ****** * * * * * *** ** ***** * * * * * * * * * * * *** * * * *** ** * * * * * * * * * * * * * * * ********* ** * ** * * * * * ** * * * ** * ********* ********** * * * * *** * ******** ** *** * * * * * * *** * ********* ** ** ** * * * * * * ** * * * * * ***** * * ** * ** * * * * * * * * * * ** * * * *** ** ** * * * * * * * * * ** * * * ** * ** * * * * * * .*. * * * * ** * ** -- + + + + + + --0 . 0 0.4 0 . 8 1 .2 1 . 6 2 . 0 DIP2 Figure D.8: Bias, B I 2 against Predicted Diameter Increment, D I P 2 , 'Best Trees', Two Group Modelling Appendix D. Distribution of Bias by Predicted Diameter Increment and Diameter 114 BI3 1 .0 + 0 . 8 + 0 . 6 + 0.4 + 0 . 2 + 0 . 0 +--- 0 . 2 i -0.4 + - 0 . 6 + - 0 . 8 + - 1 . 0 + * * * * * * * * * * * * ** * * * * *  * * * * * * * t * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * — A * * * * * * * * * * * * * * * * * * * * * * - * - * * * _ * *. ************* *** ** * *** ** * * ******************* * *** * * * * **** **** * ** **** ** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 1 ** ** ** * * ._+ — 0 . 0 - - + - • 0.4 •-+--1.2 ._ + _ . 2 . 0 0 .8 1 . 6 DIP3 Figure D.9: Bias, BI3 against Predicted Diameter Increment, DIP3, 'Best Trees', Species Modelling Appendix D. Distribution of Bias by Predicted Diameter Increment and Diameter 115 BI1 1 .0 + 0 . 8 + 0 . 6 + 0 . 4 + 0 . 2 + 0 . 0 + -•0 .2 + •0 .4 + 0 . 6 + 0 . 8 + • 1 . 0 + * * * * * ** * * * * * * * * * * ** * * * * * ._* * *** * * * * * ****** ** * * * * * * * * * ** * * *** * ** ***** ***** * *** * * ** *** * * * * * * * **** ********* *** *** * * *** * * ****************** ***** ***** **** * ***********-************* **** *****_*. * * * * * * * * * * * * * * * * * * **** * * * * * ** **** * * * * * * * * * * * * * *** ** * * * * * ** ** * ****** ***** * ** *** * ** **** ****** * * *** * ** * * * * * * * * * ** * * *** * * * * * * * * * ** * * * * * ** * * * * * * - + + + + + + 10 20 30 40 50 60 D Figure D.10: Bias, BI1 against Diameter, D, 'Best Trees', One Group Modelling Appendix D. Distribution of Bias by Predicted Diameter Increment and Diameter 116 BI2 1 . 0 + 0 . 8 + 0 . 6 + 0 . 4 + 0 . 2 i 0 . 0 + -0 . 2 + 0 . 4 + 0 . 6 + 0 . 8 + 1 . 0 + * * * * * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * ** ****** * *** ** * * * * * * * * ** * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * * ***** ** * **** * * * * * * * * * * * ********** * * * * ***** * * * * *** **** * * * *** ********** * * * * **_*** * *_ * ** * * ********** * * ***** **** * * * ** * * * * * * ***** ** * * * * * * * * * * * **** * * * *** * * ****** * * ** * * * * ** * * * ******** * * * * * * * * * * * *** *** * * *** * ** * * * * * * * * * * * *** * * * * * * * * * * * * * * * * — + • 10 • - + • 60 20 30 40 50 Figure D.ll: Bias, BI2 against Diameter, D, 'Best Trees', Two Group Modelling Appendix D. Distribution of Bias by Predicted Diameter Increment and Diameter 117 BI3 1 .0 + 0 . 8 + 0 . 6 + 0 . 4 + 0 . 2 + 0 . 0 + — • 0 . 2 + • 0 . 4 + • 0 . 6 i • 0 . 8 + • 1 . 0 + ** * * ** * ** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * i * * * * * ** * ********* ** * * ********* * * * * *** *********** * *** ** *********** * * * * * *********** * ** * ****** *** * *** * * *** **** ** ** * * * * * * * * * * ** * * * * * * * * * *** * * * * * * * * * * * * * * * **** ** * * * * ** * * * * * * * * * * * * * * * * * * * * * * * * * ** ** * * * * * * * * * * * * * * * * ** ** * * * * * * * * * * * * ** * * ** * ** * * * * * * * * * * 10 - - + -20 • -+• 30 • - + • 40 --+-50 • -+-60 Figure D.12: Bias, BI3 against Diameter, D, 'Best Trees', Species Modelling 

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