A variance-covariance structure to take into account repeated measurements and heteroscedasticity in growth modeling

This study proposes a within-subject variance-covariance (VC) structure to take into account repeated measurements and heteroscedasticity in a context of growth modeling. The VC structure integrates a variance function and a continuous autoregressive covariance structure. It was tested on a nonlinear growth model parameterized with data from permanent sample plots. Using a stand-level approach, basal area growth was independently modeled for red spruce (Picea rubens Sarg.) and balsam fir [Abies balsamea (L.) Mill.] in mixed stands. For both species, the implementation of the VC structure significantly improved the maximum likelihood of the model. In both cases, it efficiently accounted for heteroscedasticity and autocorrelation, since the normalized residuals no longer exhibited departures from the assumptions of independent error terms with homogeneous variances. Moreover, compared with traditional nonlinear least squares (NLS) models, models parameterized with this VC structure may generate more accurate predictions when prior information is available. This case study demonstrates that the implementation of a VC structure may provide parameter estimates that are consistent with asymptotically unbiased variances in a context of nonlinear growth modeling using a stand-level approach. Since the variances are no longer biased, the hypothesis tests performed on the estimates are valid when the number of observations is large.     

On the validation of models of forest CO2 exchange using eddy covariance data: some perils and pitfalls

With the widespread application of eddy covariance technology, long-term records of hourly ecosystem mass and energy exchange are becoming available for forests around the world. These data sets hold great promise for testing and validation of models of forest function. However, model validation is not a straightforward task. The goals of this paper were to: (1) review some of the problems inherent in model validation; and (2) survey the tools available to modelers to improve validation procedures, with particular reference to eddy covariance data. A simple set of models applied to a data set of ecosystem CO2 exchange is used to illustrate our points. The major problems discussed are equifinality, insensitivity and uncertainty. Equifinality is the problem that different models, or different parameterizations of the same model, may yield similar results, making it difficult to distinguish which is correct. Insensitivity arises because the major sources of variation in eddy covariance data are the annual and diurnal cycles, which are represented by even the most basic models, and the size of the response to these cycles can mask effects of other driving variables. Uncertainty arises from three main sources: parameters, model structure and data, each of which is discussed in turn. Uncertainty is a particular issue with eddy covariance data because of the lack of replicated measurements and the potential for unquantified systematic errors such as flux loss due to advection. We surveyed several tools that improve model validation, including sensitivity analysis, uncertainty analysis, residual analysis and model comparison. Illustrative examples are used to demonstrate the use of each tool. We show that simplistic comparisons of model outputs with eddy covariance data are problematic, but use of these tools can greatly improve our confidence in model predictions.     

Allometric equations from a non-destructive approach for biomass prediction in natural forest and plantation in West Africa

Allometric equations are required for a rapid estimation of commercial timber volume and forest biomass stocks. In order to preserve the forest ecosystem, this study applied a non-destructive sampling approach to measure biophysical properties of living trees. From these measurements, volume and biomass models were developed for 11 dominant tree species in a semi-deciduous natural forest and for Acacia auriculiformis in a plantation located in southern Benin. The observations were combined to develop also generic models applicable to non-dominant tree species. Wood samples of the tree species were collected with an increment borer and analysed in the laboratory to determine species-specific wood densities. The sample size was composed of 243 trees in natural forest and 21 trees in plantation. The measurements were conducted in 30 plots of 50 m × 50 m. The graphical assessment of correlation between model outputs (biomass and volume) and variables (diameter and height) and the statistical analysis confirmed that the logarithmic model with two variables had the best predictions. The assessment also confirmed that the model using diameter only as a variable had good predictions when observations on height were unavailable. The comparative analysis of model predictions showed that the generic model in this study over-estimated biomass by up to 74.80% for certain species and under-estimated biomass by 21.18% for other species. The study shows that there are no statistically significant differences between the wood densities in this research and that published in previous studies.     

Using ergodic theory to assess the performance of ecosystem models

Ecosystem simulation models are designed to assess the flux of energy, water, carbon and nitrogen according to a given vegetation type. The reliability of the modeled results is determined by model validations. Model validations are typically done using classical statistical methods like regression analysis of predicted versus observed values, paired t statistics and error assessment procedures to characterize the quality of current and future model predictions. All these validation efforts concentrate on static aspects of the model but fail to describe the model dynamics. In this paper, we introduce methods from ergodic theory to analyze the dynamic behavior of ecosystem models. We describe (1) how the attractor representation of model behavior can be reconstructed from a time series of model outputs, and (2) what we can learn from the attractor to assess the model dynamics. As an application example, we provide simulation results for two important pine forest ecosystems in Austria, i.e., 23 Cembran pine and 16 Scots pine stands. These stands were simulated with three model parameterizations: one representing a generic, evergreen needle-leaf forest, and two species- specific parameter sets, one for Cembran pine and one for Scots pine. First, we applied standard validation methods to get static measures of model accuracy and precision. Next, we used ergodic theory to assess model dynamics. A comparison of both analyses reveals important issues related to model dynamics, such as the finding that the occurrence of instabilities in model behavior may not be detected by standard validation methods. Using ergodic theory, we were able to reconstruct the attractors of model behavior. In the attractor describing model dynamics for Cembran pine, simulated with the generic, evergreen needle-leaf forest parameter set, we detected instability in model behavior. We identified this instability as a riddled basin configuration, which is a strong indicator for the occurrence of a chaotic model behavior that may result in random predictions. Our results suggest that ergodic theory is a useful tool for assessing inconsistencies in the dynamics of ecosystem model simulations that have not been detected by standard statistical validation methods.     

Comparison of spatial prediction techniques for developing Pinus radiata productivity surfaces across New Zealand

Spatial interpolation is frequently used to predict values across a landscape enabling the spatial variation and patterns of a property to be quantified. Inverse distance weighting (IDW), ordinary kriging (OK), regression kriging (RK), and partial least squares (PLS) regression are interpolation techniques typically used where the region of interest's spatial extent is relatively small and observations are numerous and regularly spaced. In the current era of data ‘mining’ and utilisation of sparse data, the above criteria are not always fully met, increasing model uncertainties. Furthermore, regression modelling and kriging techniques require good judgement, experience, and expertise by the practitioner compared with IDW with its more rudimentary approach. In this study we compared spatial predictions derived from IDW, PLS, RK, and OK for Pinus radiata volume mean annual increment (referred to as 300 Index) and mean top height at age twenty (referred to as Site Index) across New Zealand using cross-validation techniques. Validation statistics (RMSE, ME, and R²) show that RK, OK, and IDW provided predictions that were less biased and of greater accuracy than PLS predictions. Standard deviation of rank (SDR) and mean rank (MR) validation statistics showed similar results with OK the most consistent (SDR) predictor, whereas RK had the lowest mean rank (MR), closely followed by IDW. However, the mean performance rankings for validation observations classified according to their distance to the nearest model data point indicate that although PLS provided the poorest predictions at relatively close separation distances (<2 km), in the medium range (∼4–8 km) performance was of similar ranking to that of the other techniques, and at greater separation distances PLS outperformed the other techniques. Maps illustrating the spatial variation of P. radiata forest productivity are provided.     

A flexible regression model for diameter prediction

We present a functional regression model for diameter prediction. Usually stem form is estimated from a regression model using dbh and height of the sample tree as predictor. With our model additional diameter observations measured at arbitrary locations within the sample tree can be incorporated in the estimation in order to calibrate a standard prediction based on dbh and height. For this purpose, the stem form of a sample tree is modelled as a smooth random function. The observed diameters are assumed as independent realizations from a sample of possible trajectories of the stem contour. The population average of the stem form within a given dbh and height class is estimated with the taper curves applied in the national forest inventory in Germany. Tree deviation from the population average is modelled with the help of a Karhunen–Loève expansion for the random part of the trajectory. Eigenfunctions and scores of the Karhunen–Loève expansion are estimated through conditional expectations within the methodological framework of functional principal component analysis (FPCA). In addition to a calibrated estimation of the stem form, FPCA provides asymptotic pointwise or simultaneous confidence intervals for the calibrated diameter predictions. For the application of functional principal component analysis modelling the covariance function of the random process is crucial. The main features of the functional regression model are discussed informally and demonstrated by means of practical examples.     

Quantitative validation and comparison of a range of forest growth model types

Predictions from a range of model types (simplified process-based, a statistical state space, statistical difference, and a hybrid model) were compared to 969 measurements of forest growth across an environmental gradient. The models compared were 3-PG, CANTY, CanSPBL(1.2), and CanSPBL(water). The study made an objective comparison and validation of model types, with the main criterion for comparison being each model's ability to match actual historical measurements of forest growth in an independent data set. A number of stand level forest growth variables were compared including basal area, mean top height, and stocking over 14,058 ha of plantation-grown Pinus radiata in south-eastern New Zealand. Stand variable predictions at 195 permanent plot locations covering a range of elevations from 0 to 660 m were highly correlated with field estimates derived from plot data. The hybrid model CanSPBL(water) on average was the most accurate model in the study where predictions of stocking, basal area, and mean top height were 96%, 96%, and 96% efficient. The statistical-difference equation model CanSPBL(1.2) was equally efficient but on average 3% less accurate and slightly more biased in predictions suggesting that the hybrid model explained differences in growth due to differences water availability and soil type. The process-based model 3-PG predicted stocking and basal area 89% and 88% efficiently. Finally, the statistical state-space model CANTY predicted stocking, basal area, and mean top height 96%, 87%, and 87% efficiently. Results quantify the amount of precision that can be expected from the three model types, and suggest that each approach has strengths and weaknesses.     

Modeling of heat and mass transfer during high temperature treatment of aspen

A three-dimensional and unsteady-state mathematical model, which accounts for simultaneous heat and mass transfer taking place during the high temperature treatment of wood, has been developed. It was validated by comparing the predictions with the experimental data. In the model, the coupled heat and mass transfer equations proposed by Luikov are solved, and the temperature and moisture content profiles within wood are predicted as a function of time for different heating rates. For the model validation, an experimental study was carried out with aspen under different operating conditions. The samples were heated to high temperatures using a thermogravimetric system. The weight loss and the temperature distribution within the sample were monitored and registered during the experiment. The model can use constant or variable thermo-physical properties. The temperature and moisture content of the wood predicted by the model using variable properties were compared with those predicted by the same model using constant properties as well as with the experimental data. The experimental and model results are in good agreement, and it was shown that the accuracy of the model depends on the accuracy of the properties. After the model validation was completed, a parametric study was carried out.     

Spatial prediction of forest stand variables

This study aims at the development of a model to predict forest stand variables in management units (stands) from sample plot inventory data. For this purpose we apply a non-parametric most similar neighbour (MSN) approach. The study area is the municipal forest of Waldkirch, 13 km north-east of Freiburg, Germany, which comprises 328 forest stands and 834 sample plots. Low-resolution laser scanning data, classification variables as well rough estimations from the forest management planning serve as auxiliary variables. In order to avoid common problems of k-NN-approaches caused by asymmetry at the boundaries of the regression spaces and distorted distributions, forest stands are tessellated into subunits with an area approximately equivalent to an inventory sample plot. For each subunit only the one nearest neighbour is consulted. Predictions for target variables in stands are obtained by averaging the predictions for all subunits. After formulating a random parameter model with variance components, we calibrate the prior predictions by means of sample plot data within the forest stands via BLUPs (best linear unbiased predictors). Based on bootstrap simulations, prediction errors for most management units finally prove to be smaller than the design-based sampling error of the mean. The calibration approach shows superiority compared with pure non-parametric MSN predictions.     

智能手机在林业调查工作中的应用探讨

对智能手机以及智能手机上的操作系统和开发工具作了简要介绍,分析智能手机在林业野外调查数据查询、数据采集、地块区划、样地定位和数据传输等方面的应用以及与传统方式相比的优势和缺点.     

落叶松云冷杉林单木树高曲线的研究

本研究以20块近天然落叶松云冷杉林为对象,基于4 309对实测树高-胸径数据,分树种(组)对31种常见的树高曲线进行了拟合.模型评价指标除考虑决定系数、均方误、平均误差、残差图外,还重点考虑模型的预测能力,即模型的预测区间和容许区间.结果表明:选出的树高曲线除落叶松和冷杉为线性模型外,其它均为三参数的Gompertzt和Logistic模型.研究给出了所选模型95%的预测区间及表示90%误差分布的容许区间,他们从统计上提供了模型将来用于预测的可靠性.     

Integrated Stand Growth Model of Mongolian Oak and Its Application

This paper established an integrated stand growth model of Mongolian oak(ISGM_oak) using the data from 61 permanent sample plots measured in 1997 and 2007.ISGM_ oak is a group of nonlinear simultaneous equations.The method of nonlinear error-in-variable simultaneous equations is used to estimate the parameters of ISGMoak with the statistical software Forstat 2.0,so the parameter estimation of the group of correlated equations in ISGMoak is unbiased and the equations are compatible.Model validation using bootstrap method showed that both the average relative error and square error are less than 15 percent.The ISGM_ oak model can be used to simulate the stand growth with different values of site index,stand density and to draw stand density management diagram for decision-making.     

蒙古栎林全林整体生长模型及其应用

利用汪清林业局1997年、2007年森林经理调查的61块蒙古栎林的固定样地数据建立了蒙古栎林全林整体模型。全林整体模型是一组非线性联立方程组,它由8个非线性模型组成。利用Forstat2.0软件中的"非线性误差变量联立方程组"方法求解模型参数,保证了模型的无偏性。"刀切法"检验结果表明各林分因子的平均相对误差和相对均方误差均在15%以下。应用全林整体模型可以进行不同初始条件林分的生长分析及制定不同立地条件的林分密度控制图,为蒙古栎林的经营提供参考。     

Challenges in developing and implementing a decision support systems (ETÇAP) in forest management planning: a case study in Honaz and Ibradı, Turkey

Decision support systems (DSSs) are indispensable tools in preparing a forest management plan for a better combination of multiple forest values. This study attempted to develop and explain a stand-based forest management DSS (Ecosystem-based multiple-use forest planning [ETÇAP]) comprising a traditional simulation, linear programming (LP), metaheuristics and geographic information system. The model consists of five submodels; traditional management approach to handle inventory data, an empirical growth and yield model, a simulation to conceptualize management actions, a LP technique to optimize resource allocation and a simulated annealing approach to directly create a spatially feasible harvest schedule. The ETÇAP model has been implemented in a comparative two case study areas; Denizli–Honaz and Akseki–Ibrad?. Both simulation and optimization models outperformed to the traditional management plan. The periodical change of growing stock, allowable cuts, carbon sequestration and water production are used as performance indicators. The results showed that more amount of wood could be harvested over time compared to traditional level of harvesting. It could be concluded that various management strategies allowed managers to stimulate more decision options for better outputs through intertemporal trade-offs of management interventions as the model provided tools to quantify forest dynamics over time and space. Challenges exist to establish the functional relationships between forest structure and values for better quantification and integration into the management plans.     

A state-space approach to stand growth modelling of European beech

Static models of forest growth, such as yield tables or cumulative growth functions, generally fail to recognize that forest stands are dynamic systems, subject to changes in growth dynamics due to silvicultural interventions or natural dynamics. Based on experimental data, covering a wide range of initial spacings and thinning practises, we developed a dynamic stand growth model of European beech in Denmark. The model entailed three equations for predicting dominant height growth, basal area growth, and mortality. The signs of the parameter estimates generally corroborated the anticipated growth paths of dominant height and basal area. Although statistical tests indicated significant systematic deviations between observed and predicted values, the deviations were small and of little practical importance. Cross validation procedures indicated that the model may be applied across a wide range of growth conditions and thinning practises without significant loss of precision.     

On the statistical and biological behaviors of nonlinear mixed forest models

This study examined both statistical and biological behaviors of two nonlinear mixed models fitted by the first-order (FO) and first-order conditional expectation (FOCE) methods. The simpler FO method was found to alter the mathematical forms of the base models and produce biologically unreasonable predictions for some subjects. This was not the case for the more complex FOCE method. Since the computations for predicting random parameters and for making subject-specific (SS) predictions were different for the two methods, mixing them would potentially produce large prediction biases and unexpected biological behaviors. This was demonstrated on two data sets (basal area–age and height–diameter data sets). For basal area–age data, accurate and precise basal area predictions were produced as long as a consistent method was used for predicting random parameters and for making SS predictions, regardless of the method used for model fitting. Otherwise, less accurate and precise predictions were produced, with some predictions totally against biological expectations. For height–diameter data, both consistent and inconsistent methods produced similar prediction statistics, but this does not mean that inconsistent methods should be adopted. Overall the FOCE method was found to be superior to the FO method in terms of producing biologically more reasonable predictions.     

Verification for generalizability and accuracy of a thinning-trees selection model with the ensemble learning algorithm and the cross-validation method

For the purpose of making a highly effective model in relation to the selection of trees for thinning for various forestry goals, the author examined the generalizability and accuracy of models using various ensemble learning algorithms and the m-fold cross-validation method. These techniques make it possible to improve discrimination accuracy by combining or integrating multiple learning results whose accuracies are not very high. WEKA, which is a machine learning tool for data mining programmed in Java machine language, was used to verify the results of the simulation models. The number of samples was 503. Pattern-recognition algorithms in this study used five classification-type models and one function-type model. It was found that: (1) without cross validation, two pattern-recognition algorithms can be classified as having comparatively high discrimination accuracy; (2) with cross validation, discrimination accuracy decreased as a whole, but was not very different from that without cross validation, and (3) from the viewpoint of generalizability, we constructed a model at around 70% discrimination accuracy. In order to construct more effective models, we need to design the model to utilize certain algorithms or to build in re-sampling methods such as ensemble learning and cross validation. Additionally, in the case of small sample datasets, ensemble learning is an effective method for constructing efficient models. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

赣南树木园引种试验系统分析研究

对赣南树木园内521个引种树种物候、生长动态长期观测所积累的资料进行多元统计分析.如用聚类分析揭示物种间物候期与生长的相似性并进行数量分类;用典范相关分析揭示两组物候期变量之间的相关性,为物候期预报与控制提供信息参考;对一些物种苗期进行生长节律分析与月生长建模,为物种苗期田间管理进行有价值技术储备;对物种测树指标进行数学建模,有利于了解物种生长特性,为引种与推广提供有价值的技术资料.统计分析手段以简取繁,除引用多元统计分析方法外,还藉助于灰色系统理论中的灰色建模与关联分析.     

A statistical thinning model for initialising large-scale ecosystem models

Abstract

Large-scale ecosystem models are important tools for carbon assessment at national scales. Many of these models are not initialised with known field data from any particular time, but simulate the growth of each stand from its estimated germination year up to the present or future. The models will overestimate current-day standing volume or biomass unless historic stand management (biomass removal due to thinning) is taken into account. The full management history of each stand is rarely known, and must be somehow estimated. One possibility is to build statistical thinning models based on data in a National Forest Inventory, which could then be integrated into the ecosystem models. If the harvesting model is constructed using only variables that are also used within the ecosystem model, then the management impacts can be included in the ecosystem model for the entire simulated life of the stand. In the case of most flux dynamics models, this precludes the use of the tree-level data that harvesting models have traditionally relied on. In this article, we develop a novel means to interrogate a subset of the Austrian National Forest Inventory based on deriving probability density functions for particular combinations of stand and site variables. We determine the parameters of a probabilistic model to estimate historic patterns of timber removals and validate it against inventory estimates. Our procedure can establish supportable estimates of historic management regimes suitable as input data for subsequent modelling of national-scale forest carbon stocks, sources and sinks.     

Quasi-non-linear fracture mechanics analysis of the splitting failure of single dowel joints loaded perpendicular to grain

A quasi-non-linear fracture mechanics model based on beam on elastic foundation theory is applied for analysis of dowel joints with a single dowel loaded perpendicular to grain. The properties of the elastic foundation are chosen so that the perpendicular-to-grain tensile strength and fracture energy properties of the wood are correctly represented. It is shown that this particular choice of foundation stiffness makes a conventional maximum stress failure criterion lead to the same solution as the compliance method of fracture mechanics. Results of linear elastic fracture mechanics are obtained as a special case by assuming an infinitely large value of the foundation stiffness. Results of tests on so-called plate joints are compared with theoretical predictions, showing good agreement for variations in initial crack length as well as edge distance.