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0e5ffcac6e6fe3b297a9c5ad1d640fe4fb56f679 ASSESSING UNCERTAINTY IN FOREST DYNAMIC MODELS: A CASE STUDY USING SYMFOR By DENIS RIBEIRO DO VALLE A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2008 2008 Denis Ribeiro do Valle To my family, in particular to my wife Natercia Moura do Valle. ACKNOWLEDGMENTS I thank my advisor Christina Staudhammer and the committee members who have given me the freedom to pursue my research ideas and have greatly supported my search for the best methods to address these ideas. TABLE OF CONTENTS page A C K N O W L E D G M E N T S ..............................................................................................................4 LIST OF TABLES ......... ..... .... ....................................................6 LIST OF FIGURES .................................. .. ..... ..... ................. .7 ABSTRAC T .......................................................................................... CHAPTER 1 INTRODUCTION ............... ................. ........... .............................. 10 O v erv iew ................... ...................1...................0.......... O bje ctiv e s ................... ...................1...................1.......... 2 M E T H O D S .......................................................................................................13 M odel Description .............................. ............ ...... ........... 13 T ap ajo s D ata set ................................................................................................13 M modeling A ssum options A nalyzed...................................................................... ...............15 Dynamic Equilibrium Assumption (DEA) ....................................... ...............15 M aximum Size A ssum ption (M SA)...................................................... ..... .......... 17 M modifications to SIM FLO R A .............................................. ..................... ...............18 O ne H hundred Y ear Sim ulations.............................................................................. ............20 D ata A n a ly sis ................. ................................................................................. ............... 2 1 Variance Component Analysis .......................................................... ............... 22 Probability of Each Model Given the Data (r) ......................................................23 3 R E SU L T S ...........................................................................................32 Description of the Baseline Simulation Set for the Unlogged Forest............... ................ 32 M ean M modeling R results ............... ........ ...................... ... ................ .... ..32 Probability of Each Model and Comparison of Uncertainty Sources.................................33 4 D ISC U S SIO N ..............................................................................................42 Types of A ssum options ................................ .. ........ .. ........................ ................. 42 C om prison of U uncertainty Sources.......................................................................... ... ... 43 5 C O N C L U SIO N S ................. ......................................... ........ ........ ..... .... ...... .. 49 L IST O F R E F E R E N C E S .............................................................................. ...........................50 B IO G R A PH IC A L SK E T C H .............................................................................. .....................57 LIST OF TABLES Table page 21. Examples of assumptions contained in a sample of forest dynamic models ...................26 22. Original growth, recruitment" and mortalitytt submodel parameters...........................27 23. Approaches and modified parameters used to implement assumptions ..........................28 24. Species group characteristics (Phillips et al., 2004) and parameters used in DEA Fine T uning m odel. .............................................................................. 29 25. Analysis of Variance used to determine uncertainty due to plots and model sto c h a stic ity ................................................... ..................... ................ 3 0 26. Analysis of Variance used to determine parameter uncertainty for the baseline m odel ........... ................................................. .................. ........ ...... 30 31. Percentage change in the unlogged forest...................................... ........................ 36 32. Estimated posterior probability of each model, given the recruitment data, the m ortality data, and both datasets com bined ............................................ ............... 36 LIST OF FIGURES Figure pe 21. Nested design used for simulations....................... ..... ............................. 31 31. Comparison of observed versus simulated stand level data .............................................37 32. Mean simulation results from different models (baseline + 4 assumptions) over a tim e w window of 100 years ................................................................................. ..... ..38 33. Overall variance of the mean when data are not taken into account in estimating model probabilities (i.e., assumptions are not evaluated in light of the data), partitioned between parameter uncertainty, starting conditions effect, model stochasticity, and assum options effect ........................................ ........................... 39 34. Overall variance of the mean when data are taken into account in estimating the probability of each model (i.e., assumptions are evaluated in light of the data), partitioned between parameter uncertainty, starting conditions effect, model stochasticity, and assum options effect ........................................ ........................... 40 35. Multimodel average (continuous line) and 95% confidence interval (dashed line), shown when data are not used (grey) and when data are used (black) to estimate the probability of each m odel. ......................................... ........................... 41 41. Combinations of competition index and DBH contained in the data used to calibrate the growth submodel for speciesgroup 10............................................ ............... 47 42. Overall variance of the mean when data are taken into account in estimating the probability of each model (i.e., assumptions are evaluated in light of the data), including only the simulation sets from the baseline model and the model with the modified growth submodel, partitioned between parameter uncertainty, starting conditions effect, model stochasticity, and assumptions effect......................................48 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science ASSESSING UNCERTAINTY IN FOREST DYNAMIC MODELS: A CASE STUDY USING SYMFOR By Denis Ribeiro do Valle August, 2008 Chair: Christina Staudhammer Major: Forest Resources and Conservation Despite its local, regional and global importance, the Amazon forest faces multiple threats. Forest dynamic models have been widely used to evaluate the impact of a number of anthropogenic impacts on the forest, such as timber logging and climate change. I hypothesize that these models, however, have usually failed to report the full uncertainty associated with their projections. I analyzed two commonly used assumptions in forest modeling: dynamic equilibrium assumption and maximum size assumption. I then quantified four sources of model uncertainty using the tropical forest simulation model SIMFLORA: model stochasticity, parameter uncertainty, starting condition effect, and modeling assumptions. My results suggest that modeling assumptions, a commonly neglected source of uncertainty, can have a greater effect than other sources of uncertainty that are more commonly taken into account, such as parameter uncertainty, particularly when assumptions are used to deal with submodel extrapolations. Also, to reduce assumption uncertainty in particular, and overall model uncertainty in general, it is of fundamental importance to use the available data to determine the probability of each model (i.e., data are used to evaluate the different assumptions adopted in the modeling process). Furthermore, targeted experimental studies are crucial to generate data that can be used to avoid the use of some of these assumptions. Using SIMFLORA as a case study, my results indicate that the overall modeling uncertainty is likely to be underestimated if all four sources listed above are not simultaneously considered. Finally, the method developed in this thesis to partition overall variance of the mean into different uncertainty sources can be applied to quantify the uncertainty of other models, not restricted to forest dynamic models. CHAPTER 1 INTRODUCTION Overview The Brazilian Amazon contains about 40% of the world's remaining tropical rainforest and its vital role in global biodiversity, terrestrial carbon storage, regional hydrology and climate has been widely recognized (Nepstad et al. 1999, Laurance et al. 2001, Malhi et al. 2002, Nepstad et al. 2002). At the same time the use of its natural resources is essential for millions of rural Amazonians' health and livelihoods. For instance, many medicinal plant species found in the forest are the sole health care option for many rural poor (Shanley and Luz 2003), game is frequently the most important source of protein and fat to these people (Redford 1992) and, on a regional scale, timber logging is one of the most important rural activities for the economy (Lentini et al. 2003). Despite the local, regional and global importance of Amazonia's natural resources, the Brazilian Amazon faces multiple threats. In 2003, the total deforested area was estimated at 15% of the original area (SoaresFilho et al. 2006), and this area is increasing at one of the highest rates in the world, with a mean annual increase of 18,100 km2 yr1 (Malhi et al. 2008). Furthermore, the remaining forest is not intact. Generally uncontrolled, selective logging affects an annual area ranging from 10,000 20,000 km2 (Nepstad et al. 1999, Asner et al. 2005), which can greatly increase fire risk, one of the greatest threats to the forests of Amaz6nia (Nepstad et al. 2001, Cochrane 2003). By 2010 approximately a tenth of the Brazilian Amazon is planned to be designated as forest concession area, where selective logging is to take place (Verissimo et al. 2002). In this context, sustainable use of natural resources and the balance between satisfying immediate human needs and maintaining other ecosystem functions will require quantitative knowledge about the ecosystem's present and future responses (Clark et al. 2001, DeFries et al. 2004). Numerous forest dynamic models have been developed to try to make reliable longterm and largescale prediction using available shortterm and smallscale empirical data (Pacala et al. 1996, Kammesheidt et al. 2001). There has also been a growing awareness of the importance of quantifying modeling uncertainties, with some leading science journals (e.g., Ecology, Ecological Modeling, Global Environmental Change) devoting issues solely to this theme (Clark 2003, Dessai et al. 2007, Lek 2007). Nevertheless, when uncertainties from forest dynamic model projections are presented, they either refer to model stochasticity (GourletFleury et al. 2005, Degen et al. 2006), effect of starting conditions (Phillips et al. 2004, van Gardingen et al. 2006), or parameter uncertainty (Pacala et al. 1996). A fourth source of uncertainty refers to the assumptions used when designing the model (e.g., the choice of equations to represent ecological processes); this can be a key source of uncertainty (Varis and Kuikka 1999, Qian et al. 2003, Brugnach 2005, der Lee et al. 2006). I do not know of any article that reports model uncertainties due to these assumptions and that analyzes all these sources of uncertainty jointly. As a consequence, I expect that the uncertainty in model forecasts has, in general, been under estimated. Objectives The objectives of this thesis, based on a commonly used forest dynamic model, are to: a) quantify model uncertainty derived from model stochasticity, parameter estimation, starting conditions and modeling assumptions; and, b) compare these sources of uncertainty in order to evaluate which sources contribute the most to the overall model uncertainty. Using SIMFLORA as a case study, the main hypotheses I will test are: a) forest dynamic model uncertainty has been underestimated by not simultaneously including the sources of uncertainty identified above; and b) model assumptions are the greatest source of overall model uncertainty. As a result, I intend to propose a new method to determine overall model uncertainty. CHAPTER 2 METHODS Model Description The Silviculture and Yield Management for Tropical Forests (SYMFOR) is a modeling framework that combines a management model, which allows the user to specify silvicultural activities in mixed tropical forest (e.g., harvest, thinning, poisoning, enrichment planting), with an empirical spatially explicit individual treebased ecological model, which simulates the natural processes of recruitment, growth and mortality. The SYMFOR model, originally designed for use in Indonesia (Phillips et al., 2003), was adapted for use in Guyana (Phillips et al. 2002b) and then for the Brazilian Amazon (Phillips et al. 2004). In Brazil, the model was further adapted by incorporating new management options and translating the user interface into Portuguese to become the model SIMFLORA. To date, three studies within the Brazilian Amazon have been published using SIMFLORA, two of them based on the Tapaj6s dataset (Phillips et al. 2004, van Gardingen et al. 2006) and one of them based on the Paragominas dataset (Valle et al. 2007). Other SYMFOR articles and reports can be found at www.symfor.org. All results presented in this manuscript are assessed in relation to SIMFLORA's overall (all trees with diameter at breast height, DBH > 5 cm) and commercial basal area projections based on the Tapaj6s dataset. Commercial basal area is defined throughout this thesis as the basal area of trees from commercial species with DBH greater than the legal minimum logging diameter for the Brazilian Amazon region (i.e., 45 cm). Tapajos Dataset The series of plots at Tapaj6s km 114 comprise 60 Permanent Sample Plots (PSPs), each of 0.25 ha, initially measured (all trees with DBH > 5 cm) in 1981 in unlogged primary forest. Twelve of these plots were left unlogged (region 14) while a silvicultural experiment with a randomized block design was installed in the remaining 48 plots (region 11). In this silvicultural experiment, all treatments were logged for timber in 1982 with similar logging intensities across treatments but with different thinning intensities applied in 19931994. The series of plots at Tapaj6s km 67 (region 12) comprise 36 Permanent Sample Plots (PSPs), each of 0.25 ha. This stand was logged in 1979 but permanent plots were installed only in 1981. Detailed description of the forest and these experiments can be found elsewhere (Silva et al. 1995, Silva et al. 1996, Alder and Silva 2000, Phillips et al. 2004, Oliveira 2005). To initialize the model for the 100yr simulations, I needed census data (xycoordinates, species group, and diameter from each tree within the plot) from unlogged plots in order to be able to simulate stand dynamics in two scenarios: an unlogged scenario and a simulated logging scenario. Therefore, I only used the 1981 prelogging data from region 11 and 14. Because the model requires 1 ha plots, it was necessary to join 4 plots to create a composite 1 ha plot, generating a total of 15 plots. Only plots from the same experimental block (prior to the experimental logging treatments) were joined together, in an effort to avoid withinplot variability and to increase betweenplot variability. Another set of simulations were conducted to compare the observed with the simulated data. To compare how well SYMFOR simulates stand dynamics without having to worry about how well it simulates the logging and thinning treatments effectively applied in the field, simulations were initialized with the 1981 unlogged forest data for those plots that were not logged (region 14) and with the 1981 and 1983 logged forest data from region 12 and region 11, respectively. For this comparison, the observed data consisted of the time series from regions 11, 12, and 14 prior to the thinning treatment. The same procedure to create composite 1 ha plots was used and, although this procedure mixed different treatments into a single composite plot, it does not affect the simulations as experimental treatments only differed in terms of thinning treatments applied in 19931994 and only prethinning data were used. Modeling Assumptions Analyzed Based on a literature review of forest stand dynamic models applied in the Amazon Basin, I identified several major modeling assumptions, from which two of the most common assumptions were chosen for the present analysis. A brief description of these assumptions is given below. Dynamic Equilibrium Assumption (DEA) The dynamic equilibrium assumption is a common forest dynamics modeling assumption (Kammesheidt et al. 2001, Porte and Bartelink 2002), generally being interpreted as a stable basal area and/or tree density on species group and/or stand level for an undisturbed forest. This assumption is implemented by adjusting the mortality and/or the recruitment submodels in such a way as to force the model to exhibit this equilibrium. The pragmatic justification for this procedure is that recruitment and mortality data are notoriously noisy and therefore empirical parameters are likely to be poorly estimated and need to be adjusted. Indeed, longterm forest monitoring studies in general are not well suited to collect extensive data on tree mortality: large sample sizes are required, mortality causes are not easily determined and errors on plot measurement (e.g., trees that were missed during measurement, lost their numbers or were harvested without record) can have a significant impact on mortality estimates (Alder and Synnott 1992, Alder and Silva 2000, Alder 2002). Also, recruitment data are plagued by the difficulty of species identification of seedlings and small trees and are generally highly stochastic (Vanclay 1994). As a consequence, in a recent review of mixed forest models, recruitment was often found to be poorly modeled (Porte and Bartelink 2002). To force the model to exhibit a dynamic equilibrium, two approaches have usually been adopted (Table 21): * Approach la: In the first approach (DEA Recruitment) every tree that dies, either due to natural mortality or logging, is replaced by a newly recruited tree with the minimum diameter of measurement. * Approach lb: The second approach (DEA Fine Tuning) uses an iterative method to fine tune (modify) parameters derived from the data so that the model exhibits the desired behavior (i.e., the dynamic equilibrium for an undisturbed forest). My approach was to modify these parameters within their confidence intervals. Tuning of the model is commonly done in many existing multicomponent forest growth models, both empirical and mechanistic (Gertner et al. 1995). The dynamic equilibrium assumption has a long tradition in fisheries, forestry and ecology. For instance, the concept of maximum sustainable yield, both in fisheries and forestry, is based on the idea that, in the long term, the number of individuals tends to remain constant when it is equal to the carrying capacity of the ecosystem. In ecology, the assumption that in the longterm an undisturbed forest exhibits a dynamic equilibrium is widespread. For instance, the assumption that the forest is in the steady state has been used for the metabolic theory of ecology (Brown et al. 2004), to generate corrections of recruitment rate (Sheil and May 1996) and net primary productivity (Malhi et al. 2004), to determine instantaneous decomposition rates (Palace et al. 2008), and to derive the expected steadystate diameter distribution based on demographic rates (Coomes et al. 2003, Kohyama et al. 2003, MullerLandau et al. 2006). Despite its wide use, the dynamic equilibrium assumption has been recently contested based on empirical findings that reveal that undisturbed tropical forests have been accumulating biomass and have shown increased turnover rates (Phillips and Gentry 1994, Phillips et al. 1998, Baker et al. 2004, Lewis et al. 2004, Phillips et al. 2008). Maximum Size Assumption (MSA) Although large trees comprise a major fraction of aboveground forest biomass (Clark and Clark 1996, Chambers et al. 1998, Chambers et al. 2001, Keller et al. 2001, Chave et al. 2003), the relative scarcity of these individuals and difficulty of measurement limits data collection for maximum tree size, large tree growth and mortality rate. As a consequence, simulation models often create trees considered to be unrealistically large or old (Porte and Bartelink 2002). Two approaches have generally been adopted to prevent trees from growing to unrealistic sizes (Table 21): * Approach 2a: The first approach (MSA Mortality) is to arbitrarily enhance mortality probability over a given diameter threshold. My approach was to increase the mortality rate to 100% for trees that reached the maximum diameter (based on the Tapajos dataset) of its species group. * Approach 2b: The second approach (MSA Growth) is to fit a diameter increment function in which increment tends to zero as tree diameter tends towards the species maximum size or simply to assume (as I have done) that increment drops to zero after the tree reaches the species maximum size. There is mixed empirical evidence for these assumptions. For instance, a low mortality and a continued diameter growth was observed for large individuals (>70 cm DBH) in a tropical forest in Costa Rica (Clark and Clark 1996, Clark and Clark 1999). In the Brazilian Amazon, the largest trees were observed to have the highest growth rates (Vieira et al. 2004). A final approach would be to estimate the mortality rate based on species maximum size (or age) and mean diameter increment or diameter distribution (Chave 1999, Mailly et al. 2000, Alder et al. 2002, Kohler et al. 2003, Degen et al. 2006). This approach was not analyzed because it is generally adopted only when no data on mortality are available. I limited my analyses to these assumptions in order to keep my simulations, results and discussions more concise. However, I do acknowledge that there are numerous other assumptions within forest dynamic models, such as: a) assuming independent annual diameter growth and/or recruitment rate (i.e., no serial correlation); b) using coarse estimates of gap sizes and gap formation frequency, sometimes based on "best guesses" (Phillips et al. 2004); c) assuming that all trees above the minimum felling diameter from mediumsized and large, mid and late successional species are harvestable, instead of identifying which trees are from commercial species (Kammesheidt et al. 2001); and d) assuming no harvest loss or a fixed harvest loss (e.g., due to hollowed trees, logs not found, stumps cut too high), which, depending on the model, can vary from 30 to 60% (Huth and Ditzer 2001, Kammesheidt et al. 2001, Phillips et al. 2004). Modifications to SIMFLORA Two of the uncertainty sources (model stochasticity and effect of starting conditions) are already simulated by SIMFLORA, and hence could be assessed directly. In contrast, the uncertainty derived from modeling assumptions had to be assessed by either changing initial parameter values (see original parameter estimates in Table 22) or by modifying SIMFLORA's source code. Modifications due to modeling assumptions are briefly described in Table 23 and the modified parameters are shown in Table 24. Uncertainty due to parameter estimation was assessed by allowing parameters of the main simulated ecological processes (i.e., growth, recruitment and mortality submodels) to vary. This uncertainty arises due to unmeasured covariates (termed "process error") or errors in measurement (termed "observation error" or "measurement error") (Ellner and Fieberg 2003, Clark and Bjornstad 2004). There are many subcategories of process error and each implies different strategies to simulate uncertainty in parameter estimates. For instance, variation among individuals might be simulated by randomly drawing parameter estimates for each tree at the beginning of the simulation or when it is recruited and keeping them fixed throughout the simulation, while variation due to climate yeartoyear differences might be simulated by randomly drawing parameter estimates for each year and keeping them fixed for all trees during that given year. Although it would be interesting to assess the differences among these alternative strategies, I chose to simulate uncertainty in parameter estimates as if it were solely due to measurement error. Therefore, I randomly drew parameter estimates in the beginning of each run, keeping them constant throughout the run, similar to the error analyses conducted by others (Pacala et al. 1996, Wisdom et al. 2000). The data required to assess parameter estimation uncertainty were the parameter estimates and their covariance matrices within a given submodel. Estimation of the original nonlinear regression parameters (using PROC NLIN; SAS Institute Inc. 2000) for the recruitment model (F = rle r + rI + r4, in which parameter rs was set to zero) failed to converge for two species groups in the original SIMFLORA calibration, yielding high standard errors. Therefore, for these two species groups, parameters rl and r2 were set to zero and parameters r3 and r4 and their respective covariance matrix were reestimated after the model successfully converged (Table 2 2). The original growth model was calibrated without taking into account the repeated measures nature of the diameter increment data. The justification for this was that parameter estimates were unbiased and biased standard errors of the parameters would not be used. I modified the covariance matrices provided by PROC NLIN by reestimating the variances of the mean parameters, assuming that the number of independent data was equal to the number of trees, arguably biasing high the parameter variances. Overall parameter uncertainty is expected to be lower than what will be reported throughout this manuscript. Once the covariance matrix for each equation was obtained, a vector of n random multivariate normal parameter estimates (X) was generated by using a vector of n (the number of correlated parameters) independent standard normal numbers (Z), a vector of the mean parameter estimates (u) and the lower triangular Cholesky matrix (A), given by eqn. 21: X = u + AZ [21] Given that each submodel for each species group was calibrated separately, I assumed that parameters between submodels or between species groups were not correlated. To avoid biologically unrealistic growth and recruitment, diameter increment and recruitment rates were constrained by imposing an upper limit equal to the observed (species group specific) 99th percentile of the diameter increment and recruitment rate, respectively. One Hundred Year Simulations I simulated stand dynamics for two extreme scenarios: i) an undisturbed forest; and ii) a heavily loggedover forest (where a simulated logging extracted all trees >45 cm DBH from commercial species resulting in a mean logging intensity of 75 6 m3 ha1 [mean 95% confidence interval]). The logging was simulated in the beginning of the run and was exactly the same for all simulations in order to ensure an identical starting point for all subsequent stand projections. These two extreme scenarios were chosen so that the potential range of the assumption effects on projected overall and commercial basal area could be assessed. Five sets of simulations (baseline + one set for each assumption) were generated, each consisting of twenty 100yr simulations for each plot and scenario (undisturbed and heavily loggedover forest; Fig. 21A). The baseline simulation for my study used the parameter set estimated directly from the data (Table 22; P. Phillips unpublished manuscript) and, despite not being completely free from assumptions, it is the simulation that most accurately reflects the data used for calibration. One extra set of simulations was run solely to determine uncertainty due to parameter estimation, consisting of 500 100yr simulations for each plot and scenario. Parameters were drawn randomly every two repetitions, resulting in a nested experimental design (individual runs nested within parameter sets nested within plots; Fig. 21B) which allowed my statistical analysis to disentangle uncertainty due to model stochasticity, parameter estimates and the effect of starting conditions (details are given in the Data Analysis section). I chose to determine parameter estimation uncertainty with a separate set of simulations because mean model projections when parameters were allowed to vary differed markedly from the baseline simulation (with parameters fixed at their mean values). This is a consequence of Jensen's inequality acting on the nonlinear submodels used by SIMFLORA (examples and discussion regarding Jensen's inequality effect can be found in Pacala et al. 1996, Ruel and Ayres 1999). Data Analysis Let L = {S,..., S, } be a finite set of model alternatives, x be the data, y be the response variable, A, and Co be the expected value and the variance, respectively, of the response variable given the data and the ith model alternative (i.e., E(y I x, S) = ,U; Var(y I x, S) = o2 ), and the probability of the ith model given the data be z, (i.e., P(S, I x) = z, ). Draper (1995) showed that m m Var(yI x,L) = ,oC2 + (j, ( /)2 In other words, the variance of the response variable is 1 1 the sum of the within model variance and the between model variance, both weighted by the probability of each model given the data. m m Using similar arguments, it can be shown that Var(  x, L)= Co + (/, /)2 , 1=1 1Ti where C 2; is the variance of the mean of the ith model alternative. This equation can be further expanded by decomposing the within model variance of the mean into the variances of the mean due to model stochasticity, due to plots (i.e., different starting conditions), and due to parameter uncertainty (o o p,, and o,,, respectively). The uncertainty due to modeling m assumptions is defined here as the variance between models, given by az (/, /)2 1=1 Therefore, the key equation in our study that allowed me to partition the overall variance of the mean into the different uncertainty sources is given by eqn. 22: ~Var~ \(, L) / C 2" + C2 + C,) (' U)2 [22] 1=1 1=1 The elements of the within model variance of the mean (o o2p and C2 ) in eqn. y,ms,l y,p,l y, pu, 22 were estimated by dividing the corresponding variance component (determined by the variance component analysis detailed below) by the appropriate number of observations (n). The estimation the probability of each model given the data (;ri) is also detailed below. Variance Component Analysis The variance due to starting conditions and due to model stochasticity (oC2 m, and C , respectively) were determined using the expected means squares from an Analysis of Variance (ANOVA) with one random effect (Table 25) estimated for every 10yr time step. Using the simulation set in which parameters were allowed to vary, I determined the uncertainty due to parameter estimation ( 2,,, ) by using the variance components analysis summarized in Table 26. This table was also estimated separately for every 10yr time step. The variance due to parameter uncertainty was the only result used from this set of simulations. Due to the prohibitively large amount of computer time necessary to determine o,2 for all models and since I was interested in the magnitude and not the exact value of this parameter, I determined o2, only for the baseline model and assumed it to be the same for all of the analyzed models. Probability of Each Model Given the Data (z,) Bayes rule was used to combine the information from the mortality, growth, and recruitment datasets into a single number, namely the posterior probability of each model (onwards referred to simply as probability of each model). The use of Bayes rule for model 1 (Mi), for example, is given by eqn. 23: L(D Mi) x L(D, I MI) x L(Dg I ,) x Zr(M,) z(M I, Dr, ZL(D IM,)x L(D, IM,)x L(Dg M,) x (M,) [23] where L is the likelihood, zi(M,) and zi(M, I D,,D,, Dg) are the prior and posterior probabilities, respectively, of model M,; and Di, D,, and Dg denote the mortality, recruitment, and growth datasets, respectively. Equal priors were assigned to each model (i.e., i(M,) = 1/5). The likelihood of each dataset given each model (L(D M,),L(D, M,), and L(Dg I M,)) was not easily estimated since the data within each dataset were not independent. To adequately determine the likelihood of each dataset given each model, I modeled each dataset as a function of SYMFOR's equations and inserted random effects to circumvent the lack of independence of individual observations, as described below: * Recruitment of trees: Let RJsk" be the number of recruited trees from species group s, in 10 x 10 m subplot j, within plot k, in time interval n (with duration of tn years). I assumed that Rskn Poisson(t, *, where F is the annual recruitment rate. The annual recruitment rate F is a function of the annual recruitment rate estimated by SYMFOR's F recruitment equation ( SYMFOR, recruitment equation shown in Table 22), and subplot and plot random effects (Esubplot and EPot, respectively), given by F FSYMFOR exp(Esplot + EP1) This equation was built so that F would always be greater or equal to zero and, when the random effects were equal to zero, F would be equal to SYMFOR S Mortality of trees: Let Yhkn denote if tree h, from species group s, subplot, and plot k, died (1) or stayed alive (0) in time interval n (with duration of tn years). I assumed that Yjkn ~ Bernoulli(p), where p is the probability of dying. This probability was estimated as = 1 (1 M)'", where AM is the annual mortality rate given by SMSYMFOR/MSYMFOR) Xp(Esubplot ) M = exp(EPl This equation can be summarized as 1 + [SYMFOR /(1 MSYIFOR )] exp(Ebplot ) logit( ) = logit(MSYOR) + Esubpot Similar to the notation used before, MSYMoR is the annual mortality rate as estimated by SYMFOR's mortality submodel (Table 22), and Esubplot is the random subplot effect. This equation was built so that AM would always be between zero and one and, when the random effect was equal to zero, AM would be equal to M SYMFOR ' Growth data were not used to estimate the probability of each model because all models shared the same growth submodel (except for the MSA Growth model, where the growth data do not contribute to discerning this model from the others). All random effects were assumed to come from a normal distribution with mean zero and variance to be estimated. The decision of which random effects to include was based on the number of levels of each random effect (a small number of levels would not allow a good estimation of the variance associated with it; Bolker et al., in prep.) and a preliminary analysis of the correlation structure of the residuals. Flat priors (uniform between 0 and 10,000) were given to the precisions of the normal distributions. Gibbs sampling through WinBUGS 1.4 (Bayesian inference Using Gibbs Sampling for Windows; Spiegelhalter et al. 1996) was used twice: first to estimate the modes of the variances of the random effects, second to estimate the mean likelihood of each model (with variances fixed at their estimated modes). The mean likelihood of each model is an approximation to the marginal likelihood, averaged over all possible values of the random effects. For the first step, 3 chains (with different initial values) were run to check for convergence (using Gelman and Rubin's convergence statistic, as modified by Brooks and Gelman (1998), a value lower than 1.2 was taken as indicative of convergence), each with 5,000 iterations. To estimate the mean likelihood of each model, given the data, a single chain was used with 5,000 iterations. Table 21. Examples of assumptions contained in a sample of forest dynamic models (see main text for a description of these assumptions) Assumptions Models DEA DEA MSA MSA References Recruit. F. Tuning Mortal. Growth SELVA GEMFORM MYRLIN SYMFOR SIMFLORA TROLL FORMIND 2.0 CAFOGROM S (GourletFleury and Houllier 2000, GourletFleury et al. 2005) S(Chambers et al. 2001, Chambers et al. 2004) (Alder unpublished manuscript, Alder 2002, Marshall et al. 2002) (Alder 2002, Alder et al. 2002, Nicol et al. 2002) i (Phillips et al. 2003, Phillips et al. 2004, van Gardingen et al. 2006, Valle et al. 2007) S (Chave 1999) S (Kammesheidt et al. 2001, Kohler et al. 2003) S (Alder and Silva 2000) (Favrichon 1998) Table 22. Original growth, recruitmentft and mortalityftt submodel parameters Species Growth model Recruitment model Mortality model groups ao al a2 as a4 rl r2 r3 r4 bd mo mi 1 0.003 0.008 0.045 0.001 0.117 0.029 3.047 0.000 0.006 7.5 2.6 1.8 2 0.005 0.011 0.050 0.000 0.114 0.045 0.000 0.000 0.000 3.8 2.9 3.0 3 0.745 0.739 0.000 0.001 0.195 0.000 0.000 0.044 0.014 15.0 2.0 1.1 4 0.004 0.021 0.159 0.001 0.103 0.301 0.825 0.000 0.249 5.0 3.0 2.4 5 0.001 0.022 0.029 0.000 0.002 0.701 0.020 0.000 0.696 7.5 2.3 1.2 6 0.051 0.081 0.005 0.001 0.152 0.008 2.806 0.000 0.001 7.5 2.5 1.6 7 2.263 2.246 0.000 0.002 0.368 0.074 2.293 0.000 0.004 7.5 4.5 3.3 8 0.003 0.175 0.163 0.001 0.600 0.000 6.906 0.000 0.003 15.0 2.5 0.9 9 0.009 0.333 0.078 0.005 0.428 0.059 0.788 0.000 0.051 5.0 8.9 3.6 10 0.007 0.081 0.029 0.001 0.135 0.000 0.000 0.002 0.004 15.0 4.3 2.8 Parameters were estimated by P. Phillips (unpublished manuscript) except for bolded parameters (see main text for detailed S explanation). tGrowth submodel: I =D(ao + ae "')+ ac+aa4 ttRecruitment submodel:F = rle' + l +'1. tttMortality submodel: mo mifD 10 x 10 m square (trees yr1), Iis the diameter increment (cm yr1), Mis the mortality rate (%), bdis the upper limit of the first diameter class (cm), and ao, al, a2, a3, a4, rl, r2, r3, r4, mo, and mi are the estimated parameters. For further details regarding variable definitions and equation forms, refer to Phillips et al. (2002a, 2004). Table 23. Approaches and modified parameters used to implement assumptions Assumption Approaches Description Modified parameters Recruitment model modified in order to replace any killed or DEA naturally dead tree by a new recruited tree. This new tree has Only recruitment model structure Recruitment the same tree attributes from the dead tree except for the DBH, modified in source code. Undisturbed forests which was set to 5 cm. are in a dynamic Simulation of an undisturbed forest was run for 100 years and equilibrium DEA model parameters were finetuned (modified iteratively within Fe T g their 95% confidence interval) to exhibit a constant species Table 24 Fine Tuning composition (constant basal area and tree density per species group) during this time window. MSA T Only mortality model structure Tree species have a MSA Mortality was increased to 100% for trees with DBH > Dmax. Only mortality model structure maximum attainable Mortality modified in source code m axim um attainable   MSA A Only growth model structure size MSA Growth was reduced to zero for trees with DBH > Dmax. n ro mode srucur Growth modified in source code. DEA and MSA stand for Dynamic Equilibrium Assumption and Maximum Size Assumption, respectively. Dmax is the maximum diameter (cm) of a species group, based on the available data. Table 24. Species group characteristics (Phillips et al., 2004) and parameters used in DEA Fine Tuning model (modified parameters in bold) Species groups characteristics DEA Fine Tuning Group name (reference) D95 G ri r4 mo m1 1 Slow growing midcanopy 41.8 0.21 0.029 0.006 2.3 2.0 2 Slow growing understory 15.9 0.09 0.045 0.000 2.3 2.9 3 Medium growing midcanopy 57.2 0.29 0.000 0.013 1.9 1.4 4 Slow growing lower canopy 27.7 0.18 0.301 0.238 2.9 2.7 5 Medium growing upper canopy 72.5 0.26 0.701 0.696 1.9 1.2 6 Fast growing upper canopy 76.0 0.54 0.001 0.008 3.3 2.0 7 Fast growing pioneers 35.8 0.54 0.040 0.100 5.0 3.8 8 Emergents climax 104.0 0.37 0.000 0.005 3.0 0.9 9 Very fast growing pioneers 38.7 1.26 0.030 0.100 10.4 4.9 10 Very fast growing upper canopy 78.2 0.94 0.000 0.004 4.3 3.9 D95 is the 95th percentile of the cumulative diameter frequency distribution (cm), G is the average growth rate (cm yr1), rl and r4 are parameters from the recruitment submodel, and mo and mi are parameters from the mortality submodel. Table 25. Analysis of Variance used to determine uncertainty due to plots and model stochasticityt Source of variation DF E(MS) Plot npiotl 2 +n 2 y,ms, rep y,p, Error nplot *((nep) a2 y,ms,z Total nplot'"n .,1 t These parameters were estimated separately for each model (baseline model plus one model for each of the four assumptions), for every logging scenario (logged and unlogged forest), and for every 10yr time step. This analysis corresponds to the nested design shown in Fig. 21A. O2,. and 2 are the variances due to model stochasticity and due to plots (i.e., due to different starting conditions), respectively, for the ith model; nplot and nep are the number of plots (i.e., 15) and number of repetitions per plot (i.e., 20), respectively. Table 26. Analysis of Variance used to determine parameter uncertainty for the baseline model Source of variation DF E(MS) Plot npo1 ,ms,, + np ypu, + nrps lps y,p,l Par. uncertainty lplot*( n 1) 2, + nl, S 2ps M. stochast. nplot* npn*(nrps1) 2 Total nplot* 11 '"I 1 tThis parameter was estimated separately for each logging scenario (logged and unlogged forest), and for every 10yr time step. This analysis corresponds to the nested design shown in Fig. 21B. 2 ,,,, 2, ,,, and ,2 are the variances due to plots (i.e., due to different starting conditions), due to parameter uncertainty, and due to model stochasticity, respectively, for the ith model; nEiot, n and n' are the number of plots (i.e., 15), number of repetitions per parameter set (i.e., 2), and the number of randomly drawn parameter sets per plot (i.e., 250), respectively. Figure 21. Nested design used for simulations. R and PS stand for repetitions and parameter sets, respectively. A) Nested design used to determine uncertainty due to assumptions, to starting conditions effect and to model stochasticity. This design was used for the 5 models evaluated (baseline + 4 assumptions); B) Nested design used to determine parameter uncertainty. Only the baseline model was used for these simulations. CHAPTER 3 RESULTS Description of the Baseline Simulation Set for the Unlogged Forest Simulations of the unlogged forest from region 11 and region 14, without any of the dynamic equilibrium or maximum size assumptions (i.e., baseline model), predicted drastic changes over a period of 100 years in species group composition, both in basal area and tree density, particularly for the pioneer species groups (species groups 7 and 9). Also, tree density and basal area of big trees (DBH > D95 [the 95th percentile of the diameter distribution in the beginning of the simulation]) increased for nearly all species groups (Table 31). These somewhat counterintuitive results for an undisturbed forest are shown to indicate why a forest modeler might want to adoptpost hoc assumptions that force the model to behave as expected. The effect of these assumptions is analyzed below. Mean Modeling Results Models with different assumptions did not predict drastically different stand dynamics for short simulation lengths, within the time span of available field data (Fig. 31). Furthermore, a visual comparison of the observed versus the simulated stand level results does not allow a clear cut determination of which assumptions best fitted the observed data, which might mislead one to conclude that uncertainty due to assumptions is negligible. Nevertheless, the average longterm (i.e., 100 years) simulation result changed drastically according to the adopted assumptions, particularly for the dynamic equilibrium assumptions, suggesting that assumptions are indeed a major source of uncertainty in model projections (Fig. 32). Unfortunately, empirical data covering this time span do not generally exist and therefore it is not possible to discern which of the simulation results best reflects reality. Again, as shown previously, the baseline scenario clearly indicates a mean trend of overall basal area that is outside the natural variability currently found within the area (black circles in Fig. 32), both for the unlogged and logged scenarios. One might assume that the basal area of the unlogged simulations after 100 years should be within the range of the recently measured basal area (as often assumed in the forest modeling literature), in which case one would judge the model with the DEA Tuning assumption to be more biologically sound when compared to other models (i.e., baseline, MSA Growth, MSA Mortality, and DEA Recruitment). Another important aspect is that the effect of adding a given assumption to the baseline model may depend on the management scenario being simulated (see Fig. 32). For instance, when the projections of the DEA Recruitment model are compared to the projections from the baseline model, it is clear that the difference in their projections is larger in the logged scenario then in the unlogged scenario from year 10 to year 60. As a consequence, even if all simulations were carried out with a single set of assumptions (e.g., DEA Recruitment), it would not necessarily follow that the comparison of logging scenarios (e.g., different cutting cycles, logging intensities, and/or harvesting systems) would remain unchanged since this assumption might have different effects depending on the specific logging scenario being simulated. Probability of Each Model and Comparison of Uncertainty Sources The dynamic equilibrium assumption (DEA) and the maximum size assumption (MSA) are often adopted after the model has been calibrated and some of its outputs assessed, typically as a way to avoid what is judged to be unrealistic model behavior. Also, field data are generally not used to evaluate these assumptions, particularly because model modifications due to assumptions are often seen as "minor" changes in parameters. Thus, to assess the uncertainty of adopting these assumptions without using field data to evaluate them, equal weights were set for all models (;, = 1/5). The effects of all sources of uncertainty tended to increase with time except for the effect of starting conditions. Also, the joint effect of all of the chosen assumptions (i.e., MSA Growth, MSA Mortality, DEA Recruitment, DEA Fine Tuning) was the greatest source of uncertainty (except in relation to commercial basal area in the unlogged scenario), probably due to the fact that, in contrast to the other sources of uncertainty, between model variance does not decrease with an increased number of simulations or plots measured. In general, the starting condition effect was the second greatest source of uncertainty, possibly due to the small number (i.e., 15) of 1 ha plots being simulated. Parameter uncertainty and model stochasticity were of similar magnitude and considerably smaller than assumption and starting conditions effects. Clearly, overall model uncertainty is often dominated by the uncertainty due to assumption effects (Fig. 33). When the data are used to estimate the probability of each model (which implies that the assumptions are evaluated based on the field data), these results drastically changed, substantially reducing the uncertainty due to assumptions. As expected, the submodel data equally supported the baseline, MSA Growth and MSA Mortality models (i.e., these three models had the same posterior model probability; Table 32). Although commercial basal area was best predicted by the DEA Fine Tuning model (data not shown) and this assumption was implemented by carefully changing parameter values within their confidence intervals, the sub model data clearly conflicted with the DEA models, yielding an approximately zero probability for this model. With equal weights for the MSA Growth, MSA Mortality and baseline models, the greatest source of uncertainty became the starting conditions effect while all the remaining sources of uncertainty were of similar magnitudes (Fig. 34). It is important to note that, over the 100yr simulation, using data to estimate the probability of each model resulted in a 13fold decrease in overall variance of the mean for the stand basal area, both for logged and unlogged simulations, and a 5fold and 2fold decrease in overall variance for the commercial basal area in the logged and unlogged simulations, respectively (note the different scales used in Fig. 33 when compared to Fig. 34). Furthermore, the multimodel average (average of the projections from all models, weighted by the probability of each model) also substantially changed when data were used to estimate model probabilities (Fig. 35). The reduction in overall variance of the mean and the change in the multimodel average have practical relevance from a forest management perspective. For instance, when data are taken into account in estimating the probability of each model, the overall basal area in the logged forest is expected to return to its prelogging levels within 20 to 30 years, instead of within 20 to > 100 years, and the postlogging recovery of commercial basal area after 100 years is 4659%, instead of 3767%. Table 31. Percentage change in the unlogged forest Species All trees Large trees (DBH > D95) groups Basal area Tree density Basal area Tree density 1 10 22 46 30 2 33 25 25 40 3 18 7 200 182 4 20 23 69 18 5 10 30 53 33 6 173 89 407 324 7 444 202 1092 836 8 7 45 18 25 9 1015 987 2209 2015 10 70 57 415 175 Total 29 2 141 82 Percentage change was determined as (100*[YlooYo]/Yo), where Y, is the plot characteristic in year i and D95 is the 95th percentile of the diameter distribution in the beginning of the simulation. Table 32. Posterior probabilityT of each model, given the recruitment data, the mortality data, and both datasets combined Recruitment Mortality Both datasets Model . data data combined Baseline 0.333 0.25 0.333 MSA mortality 0.333 0.25 0.333 MSA growth 0.333 0.25 0.333 DEA recruitment 0.000 0.25 0.000 DEA tuning 0.000 0.00 0.000 SThe posterior probability was estimated with Bayes theorem by combining the prior probability (each model had an equal prior probability, 1/5) and the likelihood (estimated using WinBUGS). Recruitment and mortality data came from logged and unlogged forests (regions 11, 12, and 14). o DoEA Recrlt ACalendar ye, I C "pA 8 M oAG . 1980 1985 1990 1995 Calendar year 1982 1984 1986 1988 1990 1992 Calendar year . .  9 0 1982 I I I 1982 1984 1986 1988 1990 Calendar year , 8 I I I I S0) z D 1982 1984 1986 1988 1990 1992 Calendar year oo 0 ,a I_ (0 0 1980 1985 1990 1995 Calendar year F Figure 31. Comparison of observed versus simulated stand level data. Observed stand level results are shown in black (empty black circles are 1ha plots and black line is their average) and average simulation results from models with different assumptions, over the time window of the observed data, are shown by grey lines. Overall basal area is shown in left panels (A,C,E) and commercial basal area is shown in right panels (B,D,F). Top panels (A and B) show region 11 (logged forest), middle panels (C and D) show region 12 (logged forest), and lower panels (E and F) show region 14 (unlogged forest). A B L0 E o 0 0 g 0 Baseline SDEA Recruit. o DEA Tuning Co4 MSA Mortal. MSA Growth oC 0D 0 S O 0 0 (0 E  E CO 0 C D) 0 20 40 60 80 100 0 20 40 60 80 100 Time (years) Time (years) Figure 32. Mean simulation results from different models (baseline + 4 assumptions) over a time window of 100 years. Simulations were initiated with 15 onehectare plots from regions 11 and 14, and logging (in logged scenario) occurred in year 0. Initial observed overall and commercial basal area values for each of the 15 plots are represented by black circles placed in year 100 to illustrate how mean simulation results, except from the DEA Tuning model, can extrapolate beyond the original data range. Left panels (A and C) show results from logged forest and right panels (B and D) show results from unlogged forest; upper panels (A and B) show overall basal area and lower panels (C and D) show commercial basal area. SA 0 B (D C > .. . Cl lI o .. I 0 10 20 30 40 50 D8 70 80 9 100 0 10 20 30 40 50 S0 70 80 90 100 O' C Param. uncertainty D 5 [ Starting cond. effect  E lo 0 Model Stochasticity  cC [ Assump. effect ; ,; a l l l l l l l l li l I : : 'l /"{ N I n I I IE I^a I I I I I o 0 1 30 3 40 0 0 0 70 80 90 100 0 10 20 30 40 50 0S 70 80 90 100 Time (years) Time (years) Figure 33. Overall variance of the mean when data are not taken into account in estimating model probabilities (i.e., assumptions are not evaluated in light of the data), partitioned between parameter uncertainty, starting conditions effect, model stochasticity, and assumptions effect. Results from logged forest are shown in left panels (A and C) and results from unlogged forest are shown in right panels (B and D); upper panels (A and B) show results regarding overall basal area and lower panels (C and D) show results regarding commercial basal area. Ca M I IO p e Sctarting condi effect mod (C m a D ashow rescts arears) Co CDVww v m o i  i    W3 .3 3 partitioned between parameter uncertainty, starting conditions effect, model stochasticity, and assumptions effect. Results from logged forest are shown in left panels (A and C) and results from unlogged forest are shown in right panels (B and 0 0 0 10 20 30 40 60 60 70 80 90 100 0 10 20 30 40 50 00 70 80 90 100 Cu * Param uncertainty m D S Starting cond. effect Model Stochasticity ii W Cj Assump. effect CD D); upper panels (A and B) show results regarding overall basal area and lower panels (C and D) show results regarding commercial basal area. Unlogged A / Without data Ij With data C .  20 40 60 20 40 60 B r z D 100 100 0 Time (years) Time (years) Figure 35. Multimodel average (continuous line) and 95% confidence interval (dashed line), shown when data are not used (grey) and when data are used (black) to estimate the probability of each model. Results for unlogged forest are shown in left panels (A and C) and results for logged forest are shown in right panels (B and D). Upper panels (A and B) show overall basal area and lower panels (C and D) show commercial basal area. Logged CHAPTER 4 DISCUSSION Types of Assumptions Many alternative assumptions can be considered biologically reasonable. Model projections, however, are greatly influenced by the specific set of assumptions that are chosen, particularly when these assumptions are not evaluated in light of the data (i.e., data are not used to estimate the probabilities of each model). For instance, data are not generally used to compare the baseline model to the fine tuned model. Furthermore, the same assumption can often be implemented in many different ways, potentially resulting in very different longterm projections, as shown for the DEA models. As a consequence, it is crucial that the uncertainty associated with these assumptions be adequately reported. The use of the data to estimate the probability of each model revealed a clear distinction between the analyzed assumptions. The Dynamic Equilibrium Assumption (DEA, as implemented in this thesis) forced the model towards the desired model behavior, either by completely ignoring the recruitment data (as in the DEA Recruitment model), or by slightly changing the fitted parameters from the recruitment and mortality submodels (as in the DEA Fine Tuning model). Both DEA approaches directly conflicted with the data. Similarly, the Maximum Size Assumption (MSA) was originally implemented by fine tuning the mortality parameter of large trees so that the undisturbed forest would keep a somewhat constant large tree density per species group after 100 years of simulation. Despite the highly variable and small dataset for large trees, this assumption (as previously implemented) nevertheless strongly conflicted with the mortality data. In contrast to the Dynamic Equilibrium Assumption (DEA), the Maximum Size Assumption (MSA, as implemented in this thesis) was used to fill a knowledge gap that exists in the calibration dataset, therefore not conflicting with the calibration data. Indeed, a multi component iterative model can easily start extrapolating outside the data range without an obvious indication to the user that this extrapolation is occurring. Trees with DBH > Dmax are one of the most obvious model extrapolations and assumptions regarding the dynamics of large trees are needed. Other extrapolations are far more subtle and frequently go unrecognized. Fig. 41, for instance, depicts the combinations of diameter and competition index for which the growth submodel of species group 10 would extrapolate outside the range of field observations. Knowledge gaps within the calibration dataset are indeed numerous and therefore assumptions must be used to decide what the model should do in these situations. When not acknowledged, the implicit assumption is that the statistical relationship between the dependent variable and the covariates can extrapolate correctly outside the data range. Thus, two types of assumptions are recognized in my study: the first type of assumption changes the model despite its conflict with the empirical data (as in the dynamic equilibrium assumption), while the second type does not conflict with the data since the changes to the original submodel (calibrated with the data) only result in different submodel extrapolations (as the maximum size assumption). I believe that only the second type is justified. Nevertheless, if the first type of assumption is to be used, then the uncertainty associated with it should be properly reported. For instance, if the data are ignored in estimating the probability of each model, uncertainty due to this assumption can be estimated as the squared difference between the average result from the projections with and without the assumption (i.e., this assumes both models have the same probability). Comparison of Uncertainty Sources The field of statistics has traditionally acknowledged parametric uncertainty once a particular model form has been chosen. The exclusion of model construction and selection uncertainty has been shown, however, to result in overoptimistic predictive or inferential uncertainty, which can have serious implications and result in overconfident decision making (Draper 1995, Hoeting et al. 1999). Likewise, probably the most studied source of uncertainty in the ecological modeling literature has been parameter estimate uncertainty, either with a local or global sensitivity analysis (Saltelli et al. 2000, Ellner and Fieberg 2003). The problem of ignoring model uncertainty is likely to be exacerbated in situations where model extrapolations from available data are needed for decision making given that models that are very different mathematically can have similar fits to the data but wildly different predictions outside the data range (Chatfield 1995, Draper 1995). I have shown here that assumptions used when submodels are extrapolated (i.e., MSA) can have an effect of similar magnitude to other more traditional sources of uncertainty, such as parameter uncertainty. The uncertainty associated with the maximum size assumption, however, was relatively small. This observation requires further investigation, in that it begs another question: Could the effect of assumptions, when related to submodel extrapolation, be the largest source of uncertainty? To compare the effect of adopting different strategies when the growth submodel extrapolated, I modified the growth submodel so that the speciesgroup mean diameter increment was used whenever the combination of covariates (diameter and competition index) extrapolated the data range (as depicted in Fig. 41). Simulations with this modified growth submodel were then compared to simulations with the baseline model (which assumes that diameter increment is correctly estimated by the growth submodel when it is extrapolated). This comparison revealed that assumption effect, even when used only to deal with submodel extrapolations, can indeed be the largest source of uncertainty (Fig. 42). Given that models are, by definition, simplifications of reality containing numerous assumptions, how can all these assumptions be taken into account? One might argue that even if the simulations are biased, the comparison of different logging scenarios (e.g., different cutting cycles and harvest intensity combinations) simulated with the same set of assumptions would generally be unbiased (Phillips et al. 2003). Unfortunately, my results revealed that this might not always be true given that the assumption effect may greatly depend on the scenario being simulated (e.g., logged vs. unlogged forest; Fig. 32 and Fig. 42). To take these assumptions into account, I believe that models should be built with many redundant components (e.g., different equations to represent a given submodel or alternative assumptions when submodels are extrapolated) and simulations performed using different subsets of these components. Logically, this approach is only possible for models built in a modular fashion. Then, different data sources could be used to assign updated probabilities to each model structure. Also, model averaging has been increasingly suggested as a method that can incorporate uncertainty due to model construction and selection (Wintle et al. 2003, Ellison 2004). However, to my knowledge, model averaging has rarely been extended to more complex iterative models, such as forest dynamic models. Uncertainty due to assumptions can only be reduced with carefully planned experiments or targeted collection of observational data. For instance, thinning experiments might help to determine growth patterns for combinations of diameter and competition index outside the range of the data currently available. Likewise, the usual data collection for tropical forest dynamics, through the monitoring of one hectare permanent plots (Alder and Synnott 1992), are unlikely to generate sufficient data on large tree dynamics, due to their low density. In the absence of such a data, model projections that integrate the results from models with different assumptions are likely to be much more robust than projections based on a single model (and consequently a single set of assumptions). 0 CA 0 oo 20 40 60 80 DBH (cm) Figure 41. Combinations of competition index and DBH contained in the data used to calibrate the growth submodel for speciesgroup 10. This figure reveals the regions (indicated by the question marks), in independent variable space, that will require an extrapolation of the growth submodel of speciesgroup 10. L0 W CO CD SA A sm B o c I I I I I I I I I I o I I I I I I I I I I I S 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 moe C I Param. uncertainty o D ] Starting cond. effect t Model Stochasticity m ,t c are Assump. effect ) 701 E o > co 1 11 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 Time (years) Time (years) Figure 42. Overall variance of the mean when data are taken into account in estimating the probability of each model (i.e., assumptions are evaluated in light of the data), including only the simulation sets from the baseline model and the model with the modified growth submodel, partitioned between parameter uncertainty, starting conditions effect, model stochasticity, and assumptions effect. Results from logged forest are shown in left panels (A and C) and results from unlogged forest are shown in right panels (B and D); upper panels (A and B) show results regarding overall basal area and lower panels (C and D) show results regarding commercial basal area. CHAPTER 5 CONCLUSIONS * It is crucial that available data be used to determine the probability of each model (i.e., data are used to evaluate the different assumptions adopted in the modeling process), to reduce assumption uncertainty in particular, and overall model uncertainty in general. * Uncertainty due to modeling assumptions can be of greater or similar magnitude when compared to the other sources of uncertainty that are more commonly assessed. * Targeted experimental studies are needed to reduce the need to extrapolate submodel results and to reduce uncertainty due to modeling assumptions. * Overall modeling uncertainty is likely to be underestimated when these four sources of uncertainty (i.e., model stochasticity, parameter estimation, starting conditions and modeling assumptions) are not jointly taken into account. 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Thomas, N. Best, and W. Gilks. 1996. BUGS 0.5: Bayesian inference using Gibbs sampling. Manual (version ii). Medical Research Council Biostatistics Unit, Cambridge, UK. Valle, D., P. Phillips, E. Vidal, M. Schulze, J. Grogan, M. Sales, and P. van Gardingen. 2007. Adaptation of a spatially explicit individual treebased growth and yield model and long term comparison between reducedimpact and conventional logging in eastern Amazonia, Brazil. Forest Ecology and Management 243:187198. van Gardingen, P. R., D. R. Valle, and I. S. Thompson. 2006. Evaluation of yield regulation options for primary forest in Tapajos National Forest, Brazil. Forest Ecology and Management 231:184195. Vanclay, J. K. 1994. Modelling forest growth and yield: applications to mixed tropical forests. CAB International, Oxford, UK. Varis, 0., and S. Kuikka. 1999. Learning Bayesian decision analysis by doing: lessons from environmental and natural resources management. Ecological Modelling 119:177195. Verissimo, A., M. A. Cochrane, and C. Souza Jr. 2002. National forests in the Amazon. Science 297:1478. Vieira, S., P. B. Camargo, D. Selhorst, R. Silva, L. Hutyra, J. Q. Chambers, I. F. Brown, N. Higuchi, J. Santos, S. C. Wofsy, S. E. Trumbore, and L. A. Martinelli. 2004. Forest structure and carbon dynamics in Amazonian tropical rain forests. Oecologia 140:468 479. Wintle, B. A., M. A. McCarthy, C. T. Volinsky, and R. P. Kavanagh. 2003. The use of Bayesian model averaging to better represent uncertainty in ecological models. Conservation Biology 17:15791590. Wisdom, M. J., L. S. Mills, and D. F. Doak. 2000. Life stage simulation analysis: estimating vitalrate effects on population growth for conservation. Ecology 81:628641. BIOGRAPHICAL SKETCH Denis Ribeiro do Valle was born in Sao Paulo, Brazil, having completed his undergraduate studies in forestry at ESALQ, the Agriculture School of the University of Sao Paulo. During his undergraduate studies, his experience was mostly with evenaged forestry, with internships in cellulose companies such as International Paper and Arauco Forestal. Before finishing his undergraduate studies, however, he participated on an experimental logging of mahogany in the state of Acre, becoming fascinated with the challenges of unevenaged forestry. In August 2003, Denis started working as an assistant researcher at a nongovernmental organization (NGO) called Amazon Institute for the People and the Environment (IMAZON), within the Dendrogene Project (a cooperation between IMAZON, Embrapa Amazonia Oriental, Brazil, and the University of Edinburgh, Scotland). Most of his work was related to forest dynamic modeling, focusing on questions regarding sustainable timber logging. In August 2006, Denis started his master's studies at the University of Florida. PAGE 1 1 ASSESSING UNCERTAINTY IN FOREST DYNAMIC MODELS: A CASE STUDY USING SYMFOR By DENIS RIBEIRO DO VALLE A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2008 PAGE 2 2 2008 Denis Ribeiro do Valle PAGE 3 3 To my family, in particular to my wife Natercia Moura do Valle. PAGE 4 4 ACKNOWLEDGMENTS I thank m y advisor Christina Staudhammer and the committee members who have given me the freedom to pursue my research ideas and have greatly supported my search for the best methods to address these ideas. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4LIST OF TABLES................................................................................................................. ..........6LIST OF FIGURES.........................................................................................................................7ABSTRACT.....................................................................................................................................8CHAPTER 1 INTRODUCTION..................................................................................................................10Overview....................................................................................................................... ..........10Objectives...............................................................................................................................112 METHODS.............................................................................................................................13Model Description..................................................................................................................13Tapajos Dataset.......................................................................................................................13Modeling Assumptions Analyzed........................................................................................... 15Dynamic Equilibrium Assumption (DEA)...................................................................... 15Maximum Size Assumption (MSA)................................................................................ 17Modifications to SIMFLORA.................................................................................................18One Hundred Year Simulations.............................................................................................. 20Data Analysis..........................................................................................................................21Variance Component Analysis........................................................................................ 22Probability of Each Model Given the Data (i ).............................................................233 RESULTS...............................................................................................................................32Description of the Baseline Simu lation Set for the Unlogged Forest..................................... 32Mean Modeling Results..........................................................................................................32Probability of Each Model and Comparison of Uncertainty Sources..................................... 334 DISCUSSION.........................................................................................................................42Types of Assumptions............................................................................................................42Comparison of Uncertainty Sources....................................................................................... 435 CONCLUSIONS.................................................................................................................... 49LIST OF REFERENCES...............................................................................................................50BIOGRAPHICAL SKETCH.........................................................................................................57 PAGE 6 6 LIST OF TABLES Table page 21. Examples of assumptions containe d in a sam ple of forest dynamic models..................... 2622. Original growth, recruitment and mortality submodel parameters........................... 2723. Approaches and modified parame ters used to implement assumptions............................2824. Species group characteristics (Phillips et al., 2004) and parameters used in DEA Fine Tuning model................................................................................................................... ..2925. Analysis of Variance used to determine uncertainty due to plots and model stochasticity........................................................................................................................3026. Analysis of Variance used to determine parameter uncertainty for the baseline model..................................................................................................................................3031. Percentage change in the unlogged forest..........................................................................3632. Estimated posterior probability of e ach model, given the recruitment data, the mortality data, and both datasets combined....................................................................... 36 PAGE 7 7 LIST OF FIGURES Figure page 21. Nested design used for simulations.................................................................................... 3131. Comparison of observed vers us simulated stand level data............................................... 3732. Mean simulation results from differ ent models (baseline + 4 assumptions) over a time window of 100 years..................................................................................................3833. Overall variance of the mean when da ta are not taken into account in estimating model probabilities (i.e., assumptions are not evaluated in light of the data), partitioned between parameter uncertain ty, starting conditions effect, model stochasticity, and assumptions effect.................................................................................3934. Overall variance of the mean when da ta are taken into account in estimating the probability of each model (i.e., assumptions are evaluated in light of the data), partitioned between parameter uncertain ty, starting conditions effect, model stochasticity, and assumptions effect.................................................................................4035. Multimodel average (continuous line) a nd 95% confidence interval (dashed line), shown when data are not used (grey) and wh en data are used (black) to estimate the probability of each model.................................................................................................. 4141. Combinations of compe tition index and DBH contained in the data used to calibrate the growth submodel for speciesgroup 10....................................................................... 4742. Overall variance of the mean when da ta are taken into account in estimating the probability of each model (i.e., assumptions are evaluated in light of the data), including only the simulation sets from th e baseline model and the model with the modified growth submodel, partitione d between parameter uncertainty, starting conditions effect, model stochas ticity, and assumptions effect.........................................48 PAGE 8 8 Abstract of Thesis Presen ted to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science ASSESSING UNCERTAINTY IN FOREST DYNAMIC MODELS: A CASE STUDY USING SYMFOR By Denis Ribeiro do Valle August, 2008 Chair: Christina Staudhammer Major: Forest Resources and Conservation Despite its local, regional and global importanc e, the Amazon forest faces multiple threats. Forest dynamic models have been widely used to evaluate the impact of a number of anthropogenic impacts on the forest, such as ti mber logging and climate change. I hypothesize that these models, however, have usually failed to report the full uncertainty associated with their projections. I analyzed two commonly used assumptions in forest modeling: dynamic equilibrium assumption and maximum size assumption. I then qua ntified four sources of model uncertainty using the tropical forest simulation mode l SIMFLORA: model stochasticity, parameter uncertainty, starting condition effect, and mode ling assumptions. My results suggest that modeling assumptions, a commonly neglected source of uncertainty, can have a greater effect than other sources of uncertainty that are more commonly taken into account, such as parameter uncertainty, particularly when a ssumptions are used to deal with submodel extrapolations. Also, to reduce assumption uncertainty in particular, and overall model uncer tainty in general, it is of fundamental importance to use the available data to determine the probability of each model (i.e., data are used to evaluate the different a ssumptions adopted in the modeling process). Furthermore, targeted experimental studies are crucial to generate data that can be used to avoid PAGE 9 9 the use of some of these assumptions. Using SIMF LORA as a case study, my results indicate that the overall modeling uncertainty is li kely to be underestimated if all four sources listed above are not simultaneously considered. Finally, the method developed in this thesis to partition overall variance of the mean into different uncertainty s ources can be applied to quantify the uncertainty of other models, not restricted to forest dynamic models. PAGE 10 10 CHAPTER 1 INTRODUCTION Overview The Brazilian Amazon contains about 40% of the world's remaining tropical rainforest and its vital role in global biodiversity, terres trial carbon storage, regional hydrology and climate has been widely recognized (Nepstad et al. 1999, Laurance et al 2001, Malhi et al. 2002, Nepstad et al. 2002). At the same time the use of it s natural resources is essential for millions of rural Amazonians' health and livelihoods. For instance, many medicinal plant species found in the forest are the sole health care option for many rural poor (Shanley and Luz 2003), game is frequently the most important source of protei n and fat to these people (Redford 1992) and, on a regional scale, timber logging is one of the mo st important rural activities for the economy (Lentini et al. 2003). Despite the local, regional and global impor tance of Amazonias na tural resources, the Brazilian Amazon faces multiple threats. In 2003, the total deforested area was estimated at 15% of the original area (SoaresFil ho et al. 2006), and this area is in creasing at one of the highest rates in the world, with a m ean annual increase of 18,100 km2 yr1 (Malhi et al. 2008). Furthermore, the remaining forest is not intact Generally uncontrolled, se lective logging affects an annual area ranging from 10,000 20,000 km2 (Nepstad et al. 1999, As ner et al. 2005), which can greatly increase fire risk, one of the greatest threats to the forests of Amaznia (Nepstad et al. 2001, Cochrane 2003). By 2010 approximately a tenth of the Brazilian Amazon is planned to be designated as forest concession area, where select ive logging is to take place (Verissimo et al. 2002). In this context, sustaina ble use of natural resources an d the balance between satisfying immediate human needs and maintaining other ecosystem functions wi ll require quantitative PAGE 11 11 knowledge about the ecosystems present and future responses (Clark et al. 2001, DeFries et al. 2004). Numerous forest dynamic models have been de veloped to try to make reliable longterm and largescale prediction using av ailable shortterm and smallscale empirical data (Pacala et al. 1996, Kammesheidt et al. 2001). There has also been a growing awareness of the importance of quantifying modeling uncertainti es, with some leading scie nce journals (e.g., Ecology, Ecological Modeling, Global Environm ental Change) devoting issues so lely to this theme (Clark 2003, Dessai et al. 2007, Lek 2007). Nevertheless, when uncertainties from forest dynamic model projections are presented, th ey either refer to model stochasticity (GourletFleury et al. 2005, Degen et al. 2006), effect of starting conditions (Phillips et al. 2004, van Gardingen et al. 2006), or parameter uncertainty (Pac ala et al. 1996). A fourth source of uncertainty refers to the assumptions used when designing the model (e.g., th e choice of equations to represent ecological processes); this can be a key source of uncer tainty (Varis and Kuikka 1999, Qian et al. 2003, Brugnach 2005, der Lee et al. 2006). I do not know of any article that reports model uncertainties due to these assumptions and that analyzes a ll these sources of uncertainty jointly. As a consequence, I expect that the uncertainty in model forecasts has, in general, been underestimated. Objectives The objectives of this thesis, based on a commonly used fore st dynam ic model, are to: a) quantify model uncertainty derived from mode l stochasticity, parameter estimation, starting conditions and modeling assumptions; and, b) compar e these sources of uncertainty in order to evaluate which sources cont ribute the most to the overall model uncertainty. Using SIMFLORA as a case study, the main hypot heses I will test ar e: a) forest dynamic model uncertainty has been underestimated by not simultaneously including the sources of PAGE 12 12 uncertainty identified above; and b) model assumptions are the gr eatest source of overall model uncertainty. As a result, I intend to propos e a new method to determine overall model uncertainty. PAGE 13 13 CHAPTER 2 METHODS Model Description The Silviculture and Yield Management for Tropical Forests (SYMFOR) is a modeling framework that combines a management model, wh ich allows the user to specify silvicultural activities in mixed tropical forest (e.g., harvest, thinning, pois oning, enrichment planting), with an empirical spatially explicit individual treebased ecological model, which simulates the natural processes of recruitment, growth a nd mortality. The SYMFOR model, originally designed for use in Indonesia (Phillips et al., 2003), was adapted for use in Guyana (Phillips et al. 2002b) and then for the Brazilian Amazon (Phill ips et al. 2004). In Brazil, the model was further adapted by incorporating new management options and translating the user interface into Portuguese to become the model SIMFLORA. To date, three studies within the Brazilian Amazon have been published using SIMFLORA, two of them based on the Tapajs dataset (Phillips et al. 2004, van Gardingen et al. 2006) and one of them based on the Paragominas dataset (Valle et al. 2007). Other SYMFOR articles and reports can be found at www.symfor.org All results presented in this manuscript are assessed in relation to SIMFLORAs overall (all trees with diameter at breast height, DBH 5 cm) and commercial basal area projections based on the Tapajs data set. Commercial basal area is defined throughout this thesis as the basal area of trees from comm ercial species with DBH greater than the legal minimum logging diameter for the Br azilian Amazon region (i.e., 45 cm). Tapajos Dataset The series of plots at Tapajs km 114 comprise 60 Permanent Sample Plots (PSPs), each of 0.25 ha, initially measur ed (all trees with DBH 5 cm) in 1981 in unlogged primary forest. Twelve of these plots were left unlogged (regi on 14) while a silvicultu ral experiment with a PAGE 14 14 randomized block design was installed in the remain ing 48 plots (region 11). In this silvicultural experiment, all treatments were logged for timber in 1982 with similar logging intensities across treatments but with different thinning intensit ies applied in 19931994. Th e series of plots at Tapajs km 67 (region 12) comprise 36 Permanent Sample Plots (PSPs), each of 0.25 ha. This stand was logged in 1979 but permanent plots were installed only in 1981. Detailed description of the forest and these experi ments can be found elsewhere (Sil va et al. 1995, Silva et al. 1996, Alder and Silva 2000, Phillips et al. 2004, Oliveira 2005). To initialize the model for the 100yr simulati ons, I needed census data (xycoordinates, species group, and diameter from each tree with in the plot) from unlogged plots in order to be able to simulate stand dynamics in two scenar ios: an unlogged scenario and a simulated logging scenario. Therefore, I only used the 1981 prelogging data from region 11 and 14. Because the model requires 1 ha plots, it was necessary to jo in 4 plots to create a composite 1 ha plot, generating a total of 15 plots. Only plots from the same experimental block (prior to the experimental logging treatments) were joined together, in an effort to avoid withinplot variability and to increase betweenplot variability. Another set of simulations were conducted to compare the observed with the simulated data. To compare how well SYMFOR simulates stand dynamics without having to worry about how well it simulates the logging and thinning treatments effec tively applied in the field, simulations were initialized w ith the 1981 unlogged forest data fo r those plots that were not logged (region 14) and with the 1981 and 1983 l ogged forest data from region 12 and region 11, respectively. For this comparison, the observed data consisted of the time series from regions 11, 12, and 14 prior to the thinning treatment. The sa me procedure to create composite 1 ha plots was used and, although this procedure mixed differen t treatments into a single composite plot, it PAGE 15 15 does not affect the simulations as experimental treatments only differed in terms of thinning treatments applied in 19931994 and only prethinning data were used. Modeling Assumptions Analyzed Based on a literature review of forest sta nd dynam ic models applie d in the Amazon Basin, I identified several major modeling assumpti ons, from which two of the most common assumptions were chosen for the present analysis A brief description of these assumptions is given below. Dynamic Equilibrium Assumption (DEA) The dynam ic equilibrium assumption is a co mmon forest dynamics modeling assumption (Kammesheidt et al. 2001, Porte and Bartelink 2002), generally be ing interpreted as a stable basal area and/or tree density on species group and/or stand level for an undisturbed forest. This assumption is implemented by adjusting the mortality and/or the recruitment submodels in such a way as to force the model to exhibit this e quilibrium. The pragmatic justification for this procedure is that recruitment and mortality data are notoriously noisy and therefore empirical parameters are likely to be poorly estimated and need to be adjusted. Indeed, longterm forest monitoring studies in general are not well suited to collect extens ive data on tree mortality: large sample sizes are required, mortality causes are not easily determined and errors on plot measurement (e.g., trees that were missed during measurement, lost their numbers or were harvested without record) can have a signifi cant impact on mortality estimates (Alder and Synnott 1992, Alder and Silva 2000, Alder 2002). Als o, recruitment data are plagued by the difficulty of species identification of seedlings and small trees and are generally highly stochastic (Vanclay 1994). As a consequence, in a recent re view of mixed forest models, recruitment was often found to be poorly modele d (Porte and Bartelink 2002). PAGE 16 16 To force the model to exhibit a dynamic equ ilibrium, two approaches have usually been adopted (Table 21): Approach 1a : In the first approach (DEA Recruitmen t) every tree that dies, either due to natural mortality or logging, is replaced by a newly recruited tree with the minimum diameter of measurement. Approach 1b: The second approach (DEA Fine Tuning) uses an iterative method to finetune (modify) parameters derived from the da ta so that the model exhibits the desired behavior (i.e., the dynamic equilibrium for an undisturbed forest). My approach was to modify these parameters within their conf idence intervals. Tuni ng of the model is commonly done in many existing multicomponent forest growth models, both empirical and mechanistic (Ger tner et al. 1995). The dynamic equilibrium assumption has a long tr adition in fisheries, forestry and ecology. For instance, the concept of maximum sustainable yield, both in fisheries and forestry, is based on the idea that, in the long term, the number of individuals tends to remain constant when it is equal to the carrying capacity of the ecosystem. In ecology, the a ssumption that in the longterm an undisturbed forest exhibits a dynamic equ ilibrium is widespread. For instance, the assumption that the forest is in the steady state has been used for the metabolic theory of ecology (Brown et al. 2004), to generate corrections of recruitment rate (Sheil and May 1996) and net primary productivity (Malhi et al. 2004), to dete rmine instantaneous decomposition rates (Palace et al. 2008), and to derive the expected steadystate diameter distribut ion based on demographic rates (Coomes et al. 2003, Kohyama et al. 2003, MullerLandau et al. 2006). Despite its wide use, the dynamic equilibrium assumption has been recently contested based on empirical findings that reveal that undi sturbed tropical forests have been accumulating biomass and have shown increased turnover rate s (Phillips and Gentry 1994, Phillips et al. 1998, Baker et al. 2004, Lewis et al. 2004, Phillips et al. 2008). PAGE 17 17 Maximum Size Assumption (MSA) Although large trees comprise a m ajor fraction of aboveground forest biomass (Clark and Clark 1996, Chambers et al. 1998, Chambers et al. 2001, Keller et al. 200 1, Chave et al. 2003), the relative scarcity of these in dividuals and difficulty of measurement limits data collection for maximum tree size, large tree growth and morta lity rate. As a consequence, simulation models often create trees considered to be unrealistically large or old (Porte and Bartelink 2002). Two approaches have generally been adopted to prevent trees from gr owing to unrealistic sizes (Table 21): Approach 2a : The first approach (MSA Mortality) is to arbitrarily enhance mortality probability over a given diameter threshold. My a pproach was to increase the mortality rate to 100% for trees that reached the maximum diameter (based on the Ta pajos dataset) of its species group. Approach 2b: The second approach (MSA Growth) is to fit a diameter increment function in which increment tends to zero as tree diam eter tends towards the species maximum size or simply to assume (as I have done) that incr ement drops to zero af ter the tree reaches the species maximum size. There is mixed empirical evidence for these assumptions. For instance, a low mortality and a continued diameter growth was observed for large individuals (>70 cm DBH) in a tropical forest in Costa Rica (Clark a nd Clark 1996, Clark and Clark 1999) In the Brazilian Amazon, the largest trees were observed to have the highest growth rates (Vieira et al. 2004). A final approach would be to estimate the mo rtality rate based on species maximum size (or age) and mean diameter increment or diam eter distribution (Chave 1999, Mailly et al. 2000, Alder et al. 2002, Kohler et al 2003, Degen et al. 2006). This approach was not analyzed because it is generally a dopted only when no data on mortality are available. I limited my analyses to these assumptions in order to keep my simulations, results and discussions more concise. However, I do acknowledge that there are numerous other assumptions within forest dynamic models, such as: a) assuming indepe ndent annual diameter PAGE 18 18 growth and/or recruitment rate (i .e., no serial correlation); b) usi ng coarse estimates of gap sizes and gap formation frequency, sometimes based on best guesses (Phillips et al. 2004); c) assuming that all trees above the minimum felling diameter from mediumsized and large, midand late successional species are harvestable, instead of identifying which trees are from commercial species (Kammesheidt et al. 2001); and d) assuming no harvest loss or a fixed harvest loss (e.g., due to hollowed trees, logs not found, stumps cut too high), which, depending on the model, can vary from 30 to 60% (Huth and Ditzer 2001, Kammesheidt et al. 2001, Phillips et al. 2004). Modifications to SIMFLORA Two of the uncertainty sources (m odel stocha sticity and effect of starting conditions) are already simulated by SIMFLORA, and hence coul d be assessed directly. In contrast, the uncertainty derived from modeling assumptions ha d to be assessed by either changing initial parameter values (see original parameter estima tes in Table 22) or by modifying SIMFLORAs source code. Modifications due to modeling assump tions are briefly descri bed in Table 23 and the modified parameters are shown in Table 24. Uncertainty due to parameter estimation was assessed by allowing parameters of the main simulated ecological processes (i.e ., growth, recruitment and mortal ity submodels) to vary. This uncertainty arises due to unmeasured covari ates (termed "process error") or errors in measurement (termed "observation error" or "measurement error") (Ellner and Fieberg 2003, Clark and Bjornstad 2004). There are many subcategories of process error and each implies different strategies to simulate uncertainty in parameter estimates. For instance, variation among individuals might be simulated by randomly draw ing parameter estimates for each tree at the beginning of the simulation or when it is recruited and keeping them fixed throughout the simulation, while variation due to climate yeartoyear differences might be simulated by PAGE 19 19 randomly drawing parameter estimat es for each year and keeping them fixed for all trees during that given year. Although it would be intere sting to assess the differences among these alternative strategies, I chose to simulate uncertainty in parameter estimates as if it were solely due to measurement error. Therefore, I randomly drew parameter estimates in the beginning of each run, keeping them constant throughout the r un, similar to the error analyses conducted by others (Pacala et al. 199 6, Wisdom et al. 2000). The data required to assess parameter estima tion uncertainty were the parameter estimates and their covariance matrices within a given su bmodel. Estimation of the original nonlinear regression parameters (using PR OC NLIN; SAS Institute Inc. 2000) for the recruitment model (43 12rIrerFIr, in which parameter r3 was set to zero) failed to converge for two species groups in the original SIMFLORA calibration, yielding high standard errors. Therefore, for these two species groups, parameters r1 and r2 were set to zero and parameters r3 and r4 and their respective covariance matrix were reestimated after the m odel successfully converged (Table 22). The original growth model was calibrated without taking into account the repeated measures nature of the diameter increment data. The justification for this was that parameter estimates were unbiased and biased standard e rrors of the parameters would not be used. I modified the covariance matrices provided by PROC NLIN by reestimating the variances of the mean parameters, assuming that the number of independent data was eq ual to the number of trees, arguably biasing high the parameter variances. Overall parame ter uncertainty is expected to be lower than what will be reported throughout this manuscript. Once the covariance matrix for each equation was obtained, a vector of n random multivariate normal parameter estimates (X) was generated by using a vector of n (the number of PAGE 20 20 correlated parameters) independent standard normal numbers (Z), a vector of the mean parameter estimates () and the lower triangular Cholesky matrix (A), given by eqn. 21: AZX [21] Given that each submodel for each species gr oup was calibrated separately, I assumed that parameters between submodels or between sp ecies groups were not correlated. To avoid biologically unrealistic growth a nd recruitment, diameter increment and recruitment rates were constrained by imposing an upper limit equal to the observed (species group specific) 99th percentile of the diameter increment and recruitment rate, respectively. One Hundred Year Simulations I sim ulated stand dynamics for two extreme scen arios: i) an undisturbed forest; and ii) a heavily loggedover forest (where a simulated logging extracted all trees 45 cm DBH from commercial species resulting in a mean logging intensity of 75 6 m3 ha1 [mean 95% confidence interval]). The logging was simulated in the beginning of the run and was exactly the same for all simulations in order to ensure an identical starting point for all subsequent stand projections. These two extreme scenarios were chosen so that the potential range of the assumption effects on projected overall and comm ercial basal area could be assessed. Five sets of simulations (baseline + one se t for each assumption) were generated, each consisting of twenty 100yr simulations for each plot and scenario (undisturbed and heavily loggedover forest; Fig. 21A). The baseline si mulation for my study used the parameter set estimated directly from the data (Table 22; P. Phillips unpublished manuscript) and, despite not being completely free from assumptions, it is the simulation that most accurately reflects the data used for calibration. PAGE 21 21 One extra set of simulations was run solely to determine uncertainty due to parameter estimation, consisting of 500 100yr simulations fo r each plot and scenario. Parameters were drawn randomly every two repetitions, resulting in a nested experi mental design (individual runs nested within parameter sets nested within plots; Fig. 21B) which allowed my statistical analysis to disentangle uncertainty due to model stocha sticity, parameter estimates and the effect of starting conditions (details are given in the Da ta Analysis section). I chose to determine parameter estimation uncertainty with a separa te set of simulations because mean model projections when parameters were allowed to vary differed markedly from the baseline simulation (with parameters fixed at their mean values). This is a consequence of Jensens inequality acting on the nonlinear submodels used by SIMFLORA (examples and discussion regarding Jensen's inequality effect can be found in Pacala et al. 1996, Ruel and Ayres 1999). Data Analysis Let } ,...,{1mSSL be a finite set of model alternatives, x be the data, y be the response variable, i and 2ibe the expected value and the variance, respectively, of the response variable given the data and the ith model alternative (i.e.,iiSxyE ), (; 2),(iiSxyVar), and the probability of the ith model given the data be i (i.e., i ixSP ) (). Draper (1995) showed that m i ii m i iiLxyVar1 2 1 2)( ),(. In other words, the variance of the response variable is the sum of the within model variance and the between model variance, both weighted by the probability of each model given the data. Using similar arguments, it can be shown that m i ii m i iyiLxyVar1 2 1 2 ,)( ),(, where 2 iyis the variance of the mean of the i th model alternative. This equation can be further PAGE 22 22 expanded by decomposing the within model variance of the mean in to the variances of the mean due to model stochasticity, due to plots (i.e., di fferent starting conditions), and due to parameter uncertainty ( 2 ,, imsy, 2 ,, ipy, and 2 ,, ipuy, respectively). The uncer tainty due to modeling assumptions is defined here as th e variance between models, given by m i ii 1 2)(. Therefore, the key equation in our study that allo wed me to partition the overall variance of the mean into the different uncertainty sources is given by eqn. 22: m i ii m i ipuyipyimsyiLxyVar1 2 1 2 ,, 2 ,, 2 ,,)() (),( [22] The elements of the within model variance of the mean ( 2 ,, imsy, 2 ,, ipy, and 2 ,, ipuy) in eqn. 22 were estimated by dividing the correspond ing variance component (determined by the variance component analysis de tailed below) by the appropriate number of observations ( n). The estimation the probability of each model given the data (i ) is also detailed below. Variance Component Analysis The variance due to starting conditions and due to model stochasticity (2 ,, imsyand 2 ,, ipy, respectively) were determined using the expected means squares from an Analysis of Variance (ANOVA) with one random effect (Table 25) estimated for every 10yr time step. Using the simulation set in which parameters were allowed to vary, I determined the uncertainty due to parameter estimation (2 ,, ipuy) by using the variance components analysis summarized in Table 26. This table was also estimated separately for every 10yr time step. The variance due to parameter uncertainty was the only result used from this set of simulations. Due to the prohibitively large amount of comp uter time necessary to determine 2 ,, ipuy for all models and PAGE 23 23 since I was interested in the ma gnitude and not the exact value of this parameter, I determined 2 ,, ipuyonly for the baseline model and assumed it to be the same for all of the analyzed models. Probability of Each Model Given the Data (i ) Bayes rule was used to combine the information from the mortality, growth, and recruitment datasets into a single number, na mely the posterior probability of each model (onwards referred to simply as probability of each model). The use of Bayes rule for model 1 ( M1), for example, is given by eqn. 23: )()()()( )()()()( ),,(1 1 1 1 1 1 i g ir im g r m grmMMDLMDLMDL MMDLMDLMDL DDDM [23] where L is the likelihood, ) (iM and ),,(grmiDDDM are the prior and posterior probabilities, respectively, of model Mi; and Dm, Dr, and Dg denote the mortality, recruitment, and growth datasets, respectively. Equal priors were assigned to each model (i.e., 5 /1)( iM ). The likelihood of each dataset given each model (), (),(ir imMDLMDL and ) (igMDL ) was not easily estimated since the data within each data set were not independent. To ad equately determine the likelihood of each dataset given each model, I modeled eac h dataset as a function of SYMFORs equations and inserted random effects to circumvent the lack of indepe ndence of individual observations, as described below: Recruitment of trees: Let sjknR be the number of recruited trees from species group s, in 10 x 10 m subplot j, within plot k, in time inte rval n (with duration of tn years). I assumed that ) *(~ FtPoisson Rn sjkn, where F is the annual recruitment rate. The annual recruitment rate F is a function of the annual recru itment rate estimated by SYMFORs recruitment equation (SYMFORF recruitment equation shown in Table 22), and subplot and plot random effects (subplotE andplotE respectively), given by PAGE 24 24 ) exp(* plot subplot SYMFOREE FF This equation was built so that F would always be greater or equal to zero and, when the random effects were equal to zero, F would be equal to SYMFORF Mortality of trees: Let hsjknY denote if tree h, from species group s, subplot j, and plot k, died (1) or stayed alive (0) in time interval n (with duration of tn years). I assumed that ) (~pBernoulli Yhsjkn, where p is the probability of dying. This probability was estimated asntM p) 1(1 where M is the annual mortality rate given by )exp(*)] 1/( [1 )exp(*)] 1/( [ subplot SYMFOR SYMFOR subplot SYMFOR SYMFORE M M E M M M This equation can be summarized as logit( M ) = logit(SYMFORM ) + subplotE. Similar to the notation used before, SYMFORM is the annual mortality rate as estimated by SYMFORs mortality submodel (Table 22), and subplotE is the random subplot effect. This equation was built so that M would always be between zero and one and, when the random effect was equal to zero, M would be equal to SYMFORM Growth data were not used to estimate the probability of each mode l because all models shared the same growth submodel (except for the MSA Growth model, where the growth data do not contribute to discerning this model from the others). All random effects were assumed to come from a normal distribution with mean zero and variance to be estimated. The decision of which random effects to include was based on the number of leve ls of each random effect (a small number of levels would not allow a good es timation of the variance associated with it; Bolker et al., in prep.) and a preliminary analysis of the correlation structur e of the residuals. Flat priors (uniform between 0 and 10,000) were given to the precisions of the normal distributions. Gibbs sampling through WinBUGS 1.4 (Bayesian inference Using Gibbs Sampling for Windows; Spiegelhalter et al. 1996) was used twice: first to estim ate the modes of the variances of the random effects, second to estimate the mean likelihood of each model (with variances fixed at their estimated modes). The mean like lihood of each model is an approximation to the marginal likelihood, averaged over all possible values of the random effects. For the first step, 3 PAGE 25 25 chains (with different initial values) were run to check for convergence (using Gelman and Rubins convergence statistic, as modified by Brooks and Gelman (1998), a value lower than 1.2 was taken as indicative of convergence), each with 5,000 itera tions. To estimate the mean likelihood of each model, given the data, a si ngle chain was used with 5,000 iterations. PAGE 26 26Table 21. Examples of assumptions contai ned in a sample of forest dynamic models (see main text for a description of these assumptions) Assumptions Models DEA Recruit. DEA F. Tuning MSA Mortal. MSA Growth References SELVA (GourletFleury and Houllier 2000, GourletFleury et al. 2005) (Chambers et al. 2001, Chambers et al. 2004) GEMFORM (Alder unpublished manuscript, Alder 2002, Marshall et al. 2002) MYRLIN (Alder 2002, Alder et al. 2002, Nicol et al. 2002) SYMFORSIMFLORA (Phillips et al. 2003, Phillips et al. 2004, van Gardingen et al. 2006, Valle et al. 2007) TROLL (Chave 1999) FORMIND 2.0 (Kammesheidt et al. 200 1, Kohler et al. 2003) CAFOGROM (Alder and Silva 2000) (Favrichon 1998) PAGE 27 27Table 22. Original growt h, recruitment and mortal ity submodel parameters Growth model Recruitment model Mortality model Species groups a0 a1 a2 a3 a4 r1 r2 r3 r4 bd m0 m1 1 0.003 0.008 0.045 0.001 0.117 0.029 3.047 0.000 0.006 7.5 2.6 1.8 2 0.005 0.011 0.050 0.000 0.114 0.045 0.000 0.000 0.000 3.8 2.9 3.0 3 0.745 0.739 0.000 0.001 0.195 0.000 0.000 0.044 0.014 15.0 2.0 1.1 4 0.004 0.021 0.159 0.001 0.103 0.301 0.825 0.000 0.249 5.0 3.0 2.4 5 0.001 0.022 0.029 0.000 0.002 0.701 0.020 0.000 0.696 7.5 2.3 1.2 6 0.051 0.081 0.005 0.001 0.152 0.008 2.806 0.000 0.001 7.5 2.5 1.6 7 2.263 2.246 0.000 0.002 0.368 0.074 2.293 0.000 0.004 7.5 4.5 3.3 8 0.003 0.175 0.163 0.001 0.600 0.000 6.906 0.000 0.003 15.0 2.5 0.9 9 0.009 0.333 0.078 0.005 0.428 0.059 0.788 0.000 0.051 5.0 8.9 3.6 10 0.007 0.081 0.029 0.001 0.135 0.000 0.000 0.002 0.004 15.0 4.3 2.8 Parameters were estimated by P. Phillips (unpublished manuscrip t) except for bolded parameters (see main text for detailed explanation). Growth submodel: 43 10) (2aCaeaaDIDa. Recruitment submodel:43 12e rIrrFIr. Mortality submodel: Db bD if if m m M 5 5d d 1 0. D is the diameter at breast height (cm), C is the diameterindependent competition index, F is the recruitment rate in a 10 x 10 m square (trees yr1), I is the diameter increment (cm yr1), M is the mortality rate (%), bd is the upper limit of the first diameter class (cm), and a0, a1, a2, a3, a4, r1, r2, r3, r4, m0, and m1 are the estimated parameters. For further details regarding variable definitions and equation forms, re fer to Phillips et al. (2002a, 2004). PAGE 28 28Table 23. Approaches and modified pa rameters used to implement assumptions Assumption ApproachesDescription Modified parameters DEA Recruitment Recruitment model modified in order to replace any killed or naturally dead tree by a new recruited tree. This new tree has the same tree attributes from the dead tree except for the DBH, which was set to 5 cm. Only recruitment model structure modified in source code. Undisturbed forests are in a dynamic equilibrium DEA Fine Tuning Simulation of an undisturbed fo rest was run for 100 years and model parameters were finetune d (modified iteratively within their 95% confidence interval) to exhibit a constant species composition (constant basal area and tree density per species group) during this time window. Table 24 MSA Mortality Mortality was increased to 100% for trees with DBH Dmax. Only mortality model structure modified in source code Tree species have a maximum attainable size MSA Growth Growth was reduced to zero for trees with DBH Dmax. Only growth model structure modified in source code. DEA and MSA stand for Dynamic Equilibrium Assump tion and Maximum Size Assumption, respectively. Dmax is the maximum diameter (cm) of a species group, based on the available data. PAGE 29 29Table 24. Species group characteristics (Phillips et al., 2004) and parameters used in DEA Fine Tuning model (modified parameters in bold) Species groups characteristics DEA Fine Tuning Group name (reference) D95 G r1 r4 m0 m1 1 Slow growing midcanopy 41.8 0.21 0.0290.006 2.3 2.0 2 Slow growing understory 15.9 0.09 0.0450.000 2.3 2.9 3 Medium growing midcanopy 57.2 0.29 0.0000.013 1.9 1.4 4 Slow growing lower canopy 27.7 0.18 0.3010.2382.9 2.7 5 Medium growing upper canopy 72.5 0.26 0.7010.6961.9 1.2 6 Fast growing upper canopy 76.0 0.54 0.0010.0083.3 2.0 7 Fast growing pioneers 35.8 0.54 0.0400.1005.0 3.8 8 Emergents climax 104.00.37 0.0000.005 3.0 0.9 9 Very fast growing pioneers 38.7 1.26 0.0300.10010.44.9 10 Very fast growing upper canopy 78.2 0.94 0.0000.004 4.3 3.9 D95 is the 95th percentile of the cumulative diameter frequency distribution (cm), G is the average growth rate (cm yr1), r1 and r4 are parameters from the recruitment submodel, and m0 and m1 are parameters from the mortality submodel. PAGE 30 30Table 25. Analysis of Variance used to determin e uncertainty due to plots and model stochasticity Source of variation DF E(MS) Plot nplot1 2 ,, 2 ,, ipyrepimsyn Error nplot *(nrep1) 2 ,, imsy Total nplot*nrep1 These parameters were estimated separately for each model (bas eline model plus one model for each of the four assumptions), f or every logging scenario (logged and unlogged fore st), and for every 10yr time step. This analysis corresponds to the nested des ign shown in Fig. 21A. 2 ,, imsy and 2 ,, ipyare the variances due to model stochasticity an d due to plots (i.e., due to different starting conditions), respectively, for the ith model; nplot and nrep are the number of plots (i.e., 15) and num ber of repetitions per plot (i.e., 20), respectively. Table 26. Analysis of Variance used to dete rmine parameter uncertainty for the baseline model Source of variation DF E(MS) Plot nplot1 ipy psrps ipuy rpsimsynn n,, 2 ,, 2 2 ,, Par. uncertainty nplot*( nps1) ipuy rpsimsyn,, 2 2 ,, M. stochast. nplot* nps*(nrps1) 2 ,, imsy Total nplot* nps*nrps1 This parameter was estimated separately for each logging scenario (logged and unlogged forest), and for every 10yr time step. This analysis corresponds to the nested design shown in Fig. 21B. ipy ,, 2, ipuy ,, 2, and 2 ,, imsy are the variances due to pl ots (i.e., due to different starting conditions), due to parameter uncertainty, and due to model stochasticity, respectively, for the ith model; nplot, nrps, and nps are the number of plots (i.e., 15), number of repetitions per parame ter set (i.e., 2), and the number of randomly drawn parameter s ets per plot (i.e., 250), respectively. PAGE 31 31 Figure 21. Nested design used for simulations R and PS stand for repetitions and parameter sets, respectively. A) Nested design used to determine uncertainty due to assumptions, to starting conditi ons effect and to model stochasticity. This design was used for the 5 models evaluated (baseline + 4 assumptions); B) Ne sted design used to determine parameter uncertainty. Only the baseline model was used for these simulations. PAGE 32 32 CHAPTER 3 RESULTS Description of the Baseline Simula tion Set for the Unlogged Forest Simulations of the unlogged forest from regi on 11 and region 14, without any of the dynamic equilibrium or maximum size assumpti ons (i.e., baseline model), predicted drastic changes over a period of 100 years in species gr oup composition, both in basal area and tree density, particularly for the pioneer species gr oups (species groups 7 and 9). Also, tree density and basal area of big trees (DBH D95 [the 95th percentile of the diam eter distribution in the beginning of the simulation]) increased for nearly all species groups (Table 31). These somewhat counterintuitive results for an undisturbed forest are shown to indicate why a forest modeler might want to adopt post hoc assumptions that force the model to behave as expected. The effect of these assumptions is analyzed below. Mean Modeling Results Models with different assumptions did not pr edict drastically differe nt stand dynamics for short simulation lengths, within the time span of available field data (Fig. 31). Furthermore, a visual comparison of the observed versus the simulated stand level results does not allow a clear cut determination of which assumptions best fitted the observed data, which might mislead one to conclude that uncertainty due to assumptions is negligible. Nevertheless, the average longterm (i.e., 100 ye ars) simulation result changed drastically according to the adopted assumptions, particul arly for the dynamic equilibrium assumptions, suggesting that assumptions are indeed a major s ource of uncertainty in model projections (Fig. 32). Unfortunately, empirical data covering this time span do not ge nerally exist and therefore it is not possible to discern which of the simulation results best reflects real ity. Again, as shown previously, the baseline scenario clearly indica tes a mean trend of overall basal area that is PAGE 33 33 outside the natural variability cu rrently found within the area (black circles in Fig. 32), both for the unlogged and logged scenarios. One might assume that the basal area of the unlogged simulations after 100 years should be within the range of the r ecently measured basal area (as often assumed in the forest modeling literature), in which case one woul d judge the model with the DEA Tuning assumption to be more biologically sound when compared to other models (i.e., baseline, MSA Growth, MSA Mortality, and DEA Recruitment). Another important aspect is that the effect of adding a given assumption to the baseline model may depend on the management scenario be ing simulated (see Fig. 32). For instance, when the projections of the DEA Recruitment m odel are compared to the projections from the baseline model, it is clear that the difference in their projections is larger in the logged scenario then in the unlogged scenario from year 10 to year 60. As a consequence, even if all simulations were carried out with a single set of assumptions (e.g., DEA Recruitment), it would not necessarily follow that the comparison of loggi ng scenarios (e.g., different cutting cycles, logging intensities, and/or harv esting systems) would remain unchanged since this assumption might have different effects depending on the specific logging scenario being simulated. Probability of Each Model and Comparison of Uncertainty Sources The dynamic equilibrium assumption (DEA) a nd the maximum size assumption (MSA) are often adopted after the model has been calibrated a nd some of its outputs assessed, typically as a way to avoid what is judged to be unrealistic model behavior. Also, field data are generally not used to evaluate these assumptions, particularly because model modifications due to assumptions are often seen as minor changes in paramete rs. Thus, to assess the uncertainty of adopting these assumptions without using field data to evaluate them, equal weights were set for all models (5 /1i ). PAGE 34 34 The effects of all sources of uncertainty tended to increase with time except for the effect of starting conditions. Also, the joint effect of all of the chosen assumptions (i.e., MSA Growth, MSA Mortality, DEA Recruitment, DEA Fine Tuni ng) was the greatest source of uncertainty (except in relation to commercial basal area in the unlogge d scenario), probably due to the fact that, in contrast to the other sources of uncertainty, between model variance does not decrease with an increased number of simulations or plots measured. In genera l, the starting condition effect was the second greatest s ource of uncertainty, possibly du e to the small number (i.e., 15) of 1 ha plots being simulated. Parameter uncerta inty and model stochasticity were of similar magnitude and considerably smaller than assu mption and starting conditions effects. Clearly, overall model uncertainty is often dominated by the uncertainty due to assumption effects (Fig. 33). When the data are used to estimate the probabi lity of each model (which implies that the assumptions are evaluated based on the field data), these results drastically changed, substantially reducing th e uncertainty due to assumptions. As expected, the submodel data equally supported the baseline, MSA Growth and MSA Mortality models (i.e., these three models had the same posterior model probabi lity; Table 32). Although commercial basal area was best predicted by the DEA Fine Tuning mode l (data not shown) and this assumption was implemented by carefully changing parameter values within their confidence intervals, the submodel data clearly conflicted with the DEA mo dels, yielding an approximately zero probability for this model. With equal weights for the MS A Growth, MSA Mortality and baseline models, the greatest source of uncertainty became the st arting conditions effect while all the remaining sources of uncertainty were of similar magnitudes (Fig. 34). PAGE 35 35 It is important to note that, over the 100yr simulation, using data to estimate the probability of each model resulted in a 13fold decrease in overall variance of the mean for the stand basal area, both for logged and unlogged simulations, and a 5fold and 2fold decrease in overall variance for the commercial basal area in the logged and unlogged simulations, respectively (note the different scales used in Fig. 33 when compared to Fig. 34). Furthermore, the multimodel average (average of the projectio ns from all models, weighted by the probability of each model) also substantially changed when da ta were used to estimate model probabilities (Fig. 35). The reduction in overall variance of the mean and the change in the multimodel average have practical relevance from a forest ma nagement perspective. For instance, when data are taken into account in estimating the probability of each model, the ov erall basal area in the logged forest is expected to re turn to its prelogging levels wi thin 20 to 30 years, instead of within 20 to > 100 years, and th e postlogging recovery of comme rcial basal area after 100 years is 4659%, instead of 3767%. PAGE 36 36 Table 31. Percentage change in the unlogged forest All trees Large trees (DBH D95) Species groups Basal area Tree density Basal area Tree density 1 10 22 46 30 2 33 25 25 40 3 18 7 200 182 4 20 23 69 18 5 10 30 53 33 6 173 89 407 324 7 444 202 1092 836 8 7 45 18 25 9 1015 987 2209 2015 10 70 57 415 175 Total 29 2 141 82 Percentage change was determined as (100*[Y100Y0]/Y0), where Yi is the plot characteristic in year i and D95 is the 95th percentile of the diameter di stribution in the beginning of the simulation. Table 32. Posterior probability of each model, given the recruitment data, the mortality data, and both datasets combined Model Recruitment data Mortality data Both datasets combined Baseline 0.333 0.25 0.333 MSA mortality 0.333 0.25 0.333 MSA growth 0.333 0.25 0.333 DEA recruitment 0.000 0.25 0.000 DEA tuning 0.000 0.00 0.000 The posterior probability was estimated with Bayes theorem by combining the prior probability (each model had an equal prior probability, 1/5) and the likelihood (estim ated using WinBUGS). Recruitment and mortality data came from logged and unlogged forests (regions 11, 12, and 14). PAGE 37 37 Figure 31. Comparison of obser ved versus simulated stand leve l data. Observed stand level results are shown in black (empty black circ les are 1ha plots and black line is their average) and average simulation results from models with differe nt assumptions, over the time window of the observed data, are shown by grey lines. Overall basal area is shown in left panels (A,C,E) and commerc ial basal area is shown in right panels (B,D,F). Top panels (A and B) show regi on 11 (logged forest), middle panels (C and D) show region 12 (logged forest), and lower panels (E and F) show region 14 (unlogged forest). PAGE 38 38 Figure 32. Mean simulation re sults from different models (b aseline + 4 assumptions) over a time window of 100 years. Simulations were in itiated with 15 onehectare plots from regions 11 and 14, and logging (in logged s cenario) occurred in year 0. Initial observed overall and commercial basal area values for each of the 15 plots are represented by black circles placed in y ear 100 to illustrate how mean simulation results, except from the DEA Tuning model, can extrapolate beyond the original data range. Left panels (A and C) show results from logged forest and right panels (B and D) show results from unlogged forest; upper panels (A and B) show overall basal area and lower panels (C and D) show commercial basal area. PAGE 39 39 Figure 33. Overall variance of the mean when data are not taken into account in estimating model probabilities (i.e., assu mptions are not evaluated in light of the data), partitioned between parameter uncertain ty, starting conditions effect, model stochasticity, and assumptions effect. Results from logged forest are shown in left panels (A and C) and results from unlogged forest are shown in right panels (B and D); upper panels (A and B) show results re garding overall basal area and lower panels (C and D) show results rega rding commercial basal area. PAGE 40 40 Figure 34. Overall variance of the mean when data are taken into account in estimating the probability of each model (i.e., assumptions are evaluated in light of the data), partitioned between parameter uncertain ty, starting conditions effect, model stochasticity, and assumptions effect. Results from logged forest are shown in left panels (A and C) and results from unlogged forest are shown in right panels (B and D); upper panels (A and B) show results re garding overall basal area and lower panels (C and D) show results rega rding commercial basal area. PAGE 41 41 Figure 35. Multimodel average (continuous line ) and 95% confidence interval (dashed line), shown when data are not used (grey) and wh en data are used (black) to estimate the probability of each model. Results for unlogged forest are shown in left panels (A and C) and results for logged forest are shown in right panels (B and D). Upper panels (A and B) show overall basal area and lower panels (C and D) s how commercial basal area. PAGE 42 42 CHAPTER 4 DISCUSSION Types of Assumptions Many alternative assumptions can be cons idered biologically reasonable. Model projections, however, are greatly influenced by the specific set of assumptions that are chosen, particularly when these assumptions are not evaluate d in light of the data (i.e., data are not used to estimate the probabilities of each model). For instance, data are not generally used to compare the baseline model to the fine tuned model. Fu rthermore, the same assumption can often be implemented in many different ways, potential ly resulting in very different longterm projections, as shown for the DEA models. As a c onsequence, it is crucial that the uncertainty associated with these assumptions be adequately reported. The use of the data to estimate the probability of each model revealed a clear distinction between the analyzed assumptions. The D ynamic Equilibrium Assumption (DEA, as implemented in this thesis) forced the model towards the desired model behavior, either by completely ignoring the recruitm ent data (as in the DEA Recruitment model), or by slightly changing the fitted parameters from the recruitment and mortality submodels (as in the DEA Fine Tuning model). Both DEA a pproaches directly conflicted with the data. Similarly, the Maximum Size Assumption (MSA) was originally implemented by fine tuning the mortality parameter of large trees so that the undisturbed forest would keep a somewhat constant large tree density per species group after 100 years of simu lation. Despite the high ly variable and small dataset for large trees, this assumption (as pr eviously implemented) nevertheless strongly conflicted with the mortality data. In contrast to the Dynamic Equilibri um Assumption (DEA), the Maximum Size Assumption (MSA, as implemented in this thesis) was used to fill a knowledge gap that exists in PAGE 43 43 the calibration dataset, therefor e not conflicting with the cali bration data. Indeed, a multicomponent iterative model can easily start extr apolating outside the da ta range without an obvious indication to the user that this extr apolation is occurring. Trees with DBH > Dmax are one of the most obvious model extrapolations an d assumptions regarding the dynamics of large trees are needed. Other extrapolat ions are far more subtle and frequently go unrecognized. Fig. 41, for instance, depicts the combinations of diameter and competition index for which the growth submodel of species gr oup 10 would extrapolate outside th e range of field observations. Knowledge gaps within the calibration dataset ar e indeed numerous and therefore assumptions must be used to decide what the model should do in these situations. When not acknowledged, the implicit assumption is that the statistical re lationship between the depe ndent variable and the covariates can extrapolate corr ectly outside the data range. Thus, two types of assumptions are recognized in my study: the firs t type of assumption changes the model despite its conflict with the empirical data (as in the dynamic equilibrium assumption), while the second type does not conf lict with the data since the changes to the original submodel (calibrated with the data) only result in different submodel extrapolations (as the maximum size assumption). I believe that only the second type is justified. Nevertheless, if the first type of assumption is to be used, then the uncertainty associat ed with it should be properly reported. For instance, if the data ar e ignored in estimating the probability of each model, uncertainty due to this assumption can be estimated as the squared difference between the average result from the projections with and wi thout the assumption (i.e ., this assumes both models have the same probability). Comparison of Uncertainty Sources The field of statistics has traditionally acknowledged parametric uncertainty once a particular model form has been chosen. The exclusion of model cons truction and selection PAGE 44 44 uncertainty has been shown, however, to result in overoptimistic predictive or inferential uncertainty, which can have serious implicati ons and result in overc onfident decision making (Draper 1995, Hoeting et al. 1999). Likewise, probably the most st udied source of uncertainty in the ecological modeling literature has been paramete r estimate uncertainty, ei ther with a local or global sensitivity analysis (Saltelli et al. 2000, Ellner and Fieberg 2003). The problem of ignoring model uncertainty is likely to be exacerbated in situations where model extrapolations from availa ble data are needed for decision making given that models that are very different mathematically can have similar fits to the data but wildly different predictions outside the data range (Chatfield 1995, Draper 1995). I have shown here that assumptions used when submodels are extrapolated (i.e., MSA) can have an effect of similar magnitude to other more traditional sources of uncertainty, such as parameter uncertainty. The uncertainty associated with the maximum size assumption, however, was relatively small. This observation requires further investigation, in that it begs another question: Could the effect of assumptions, when related to submodel extrapolation, be th e largest source of uncertainty? To compare the effect of adopting different stra tegies when the growth submode l extrapolated, I modified the growth submodel so that the speciesgroup mean diameter incr ement was used whenever the combination of covariates (diameter and compe tition index) extrapolated the data range (as depicted in Fig. 41). Simulations with this m odified growth submodel were then compared to simulations with the baseline model (which a ssumes that diameter increment is correctly estimated by the growth submodel when it is extrapolated). This comparison revealed that assumption effect, even when used only to deal with submodel extrapolations, can indeed be the largest source of uncertainty (Fig. 42). PAGE 45 45 Given that models are, by definition, simplif ications of reality containing numerous assumptions, how can all these assumptions be take n into account? One might argue that even if the simulations are biased, the comparison of di fferent logging scenarios (e.g., different cutting cycles and harvest intensity combinations) simu lated with the same set of assumptions would generally be unbiased (Phillips et al. 2003). Unfort unately, my results rev ealed that this might not always be true given that the assumption effect may great ly depend on the scenario being simulated (e.g., logged vs. unlogged fo rest; Fig. 32 and Fig. 42). To take these assumptions into account, I be lieve that models should be built with many redundant components (e.g., different equations to represent a given submodel or alternative assumptions when submodels are extrapolated ) and simulations performed using different subsets of these components. Logically, this ap proach is only possible for models built in a modular fashion. Then, different data sources could be used to assign updated pr obabilities to each model structure. Also, model averaging has been increasingly suggested as a method that can incorporate uncertainty due to model constr uction and selection (Wintle et al. 2003, Ellison 2004). However, to my knowledge, model averaging has rarely been extended to more complex iterative models, such as forest dynamic models. Uncertainty due to assumptions can only be re duced with carefully planned experiments or targeted collection of observat ional data. For instance, thinni ng experiments might help to determine growth patterns for combinations of diameter and competition index outside the range of the data currently available. Likewise, the us ual data collection for tr opical forest dynamics, through the monitoring of one h ectare permanent plots (Alder and Synnott 1992), are unlikely to generate sufficient data on large tree dynamics, due to their low density. In the absence of such a data, model projections that inte grate the results from models with different assumptions are PAGE 46 46 likely to be much more robust than projecti ons based on a single model (and consequently a single set of assumptions). PAGE 47 47 Figure 41. Combinations of co mpetition index and DBH contained in the data used to calibrate the growth submodel for sp eciesgroup 10. This figure re veals the regions (indicated by the question marks), in independent variable space, that will require an extrapolation of the growth submodel of speciesgroup 10. PAGE 48 48 Figure 42. Overall variance of the mean when data are taken into account in estimating the probability of each model (i.e., assumptions are evaluated in light of the data), including only the simulation sets from the baseline model and the model with the modified growth submodel, partitioned between parameter uncertainty, starting conditions effect, model stoc hasticity, and assumptions e ffect. Results from logged forest are shown in left panels (A and C) and results from unlogged forest are shown in right panels (B and D); uppe r panels (A and B) show re sults regarding overall basal area and lower panels (C and D) show results regarding commercial basal area. PAGE 49 49 CHAPTER 5 CONCLUSIONS It is crucial that available data be used to determine the probability of each model (i.e., data are used to evaluate the different assumptions adopted in the modeling process), to reduce assumption uncertainty in particular, a nd overall model uncertainty in general. Uncertainty due to modeling assumptions can be of greater or similar magnitude when compared to the other sources of uncerta inty that are more commonly assessed. Targeted experimental studies are needed to reduce the need to extrapolate submodel results and to reduce uncertainty due to modeling assumptions. Overall modeling uncertainty is likely to be underestimated when th ese four sources of uncertainty (i.e., model stochasticity, para meter estimation, starting conditions and modeling assumptions) are not jointly taken into account. I have shown how the overall uncertainty of the mean can be partitioned among the different sources of uncertainty, with particular emphasis on the uncertainty that arises due to the use of models with different assumptions. This method can potentially be applied to other types of models. Forest dynamic models have and wi ll continue to be used to pr edict the outcomes of direct or indirect humaninduced changes (e.g., logging, bur ning, fragmentation or carbon accumulation in the atmosphere), sometime s with millenniumlong time windows (Chambers et al., 2001). 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The use of Bayesian model averaging to better represent uncerta inty in ecological models. Conservation Biology 17:15791590. Wisdom, M. J., L. S. Mills, and D. F. Doak 2000. Life stage simulation analysis: estimating vitalrate effects on population growth for conservation. Ecology 81:628641. PAGE 57 57 BIOGRAPHICAL SKETCH Denis Ribeiro do Valle was born in So Paulo, Brazil, having com pleted his undergraduate studies in forestry at ESALQ, the Agriculture Sc hool of the University of So Paulo. During his undergraduate studies, his experien ce was mostly with evenaged forestry, with internships in cellulose companies such as International Paper and Arauco Forestal. Before finishing his undergraduate studies, however, he participated on an experimental logging of mahogany in the state of Acre, becoming fascinated with the challenges of unevenaged forestry. In August 2003, Denis started working as an assistant researcher at a nongovernmental organization (NGO) called Amazon Institute fo r the People and the Environment (IMAZON), within the Dendrogene Project (a cooperation between IMAZON, Embrapa Amazonia Oriental, Brazil, and the University of Edinburgh, Scotland ). Most of his work was related to forest dynamic modeling, focusing on questions regardi ng sustainable timber logging. In August 2006, Denis started his masters studies at the University of Florida. 