Liyong Fu

Comparison of seemingly unrelated regressions with error-in-variable models for developing a system of nonlinear additive biomass equations

Key messageNonlinear error-in-variable models can advance the development of the systems of additive biomass equations and lead to much higher prediction accuracy of tree biomass than nonlinear seemingly unrelated regression.AbstractIn this study, the approach of nonlinear error-in-variable models (NEIVM) was compared with nonlinear seemingly unrelated regressions (NSUR) for developing a system of nonlinear additive biomass equations using the data collected in Southern China for Pinus massoniana Lamb. Various tree variables were assessed to explore their contributions to improvement of biomass prediction using the systems of equations. It was found that diameter at breast height (D), total tree height (H) and crown width (CW) significantly contributed to the increase of prediction accuracy. The combinations of D, H, and CW led to three sets of independent variables: (1) D alone; (2) both D and H; and (3) D, H and CW together, which were used for the development of one-predictor, two-predictor and three-predictor systems of biomass equations, respectively. The results showed that both NEIVM and NSUR had high prediction accuracy of biomass for all the systems of biomass equations. For the one-predictor systems of biomass equations, both NEIVM and NSUR led to very similar predictions. However, for the two-predictor and three-predictor systems of biomass equations, the prediction accuracy of NEIVM was much higher than that of NSUR. When the two-predictor system of equations was used, in particular, NEIVM with one-step procedure, that is, by directly partitioning total tree biomass into four basic components, showed a higher accuracy of biomass prediction than NSUR for all the one-predictor, two-predictor and three-predictor systems of equations. This study implies that the NEIVM approach could provide a greater potential to develop a system of biomass equations that are dependent on the predictors with significant measurement errors.

Generic linear mixed-effects individual-tree biomass models for Pinus massoniana in southern China

Quantification of forest biomass is important for practical forestry and for scientific purposes. It is fundamental to develop generic individual-tree biomass models suitable for large-scale forest biomass estimation. However, compatibility of forest biomass estimates at different scales may become a problem. We developed generic individual-tree biomass models using a mixed-effects modeling approach based on aboveground biomass data of Masson pine (Pinus massoniana Lamb.) from nine provinces in southern China. Mixed-effects modeling could provide an effective approach to solving the compatibility of forest biomass estimates at different scales. A simple allometric function requiring diameter at breast height was used as a base model to construct generic individual-tree mixed-effects biomass models. Two factors of tree origin (natural and planted forests) and geographic region (nine provinces or three subregions) were included as random effect factors in the models. The results showed that the mixed-effects model not only provided more accurate estimates, but also possessed good universality compared with the population average model. We, therefore, recommend the mixed-effects model 17 to estimate national and regional-scale biomass for Masson pine in southern China. The mixed-effects modeling approach is versatile and can also be applied to construct generic individual-tree models for other tree species and variables.

Multilevel Nonlinear Mixed-Effect Crown Ratio Models for Individual Trees of Mongolian Oak (Quercus mongolica) in Northeast China

In this study, an individual tree crown ratio (CR) model was developed with a data set from a total of 3134 Mongolian oak (Quercus mongolica) trees within 112 sample plots allocated in Wangqing Forest Bureau of northeast China. Because of high correlation among the observations taken from the same sampling plots, the random effects at levels of both blocks defined as stands that have different site conditions and plots were taken into account to develop a nested two-level nonlinear mixed-effect model. Various stand and tree characteristics were assessed to explore their contributions to improvement of model prediction. Diameter at breast height, plot dominant tree height and plot dominant tree diameter were found to be significant predictors. Exponential model with plot dominant tree height as a predictor had a stronger ability to account for the heteroskedasticity. When random effects were modeled at block level alone, the correlations among the residuals remained significant. These correlations were successfully reduced when random effects were modeled at both block and plot levels. The random effects from the interaction of blocks and sample plots on tree CR were substantially large. The model that took into account both the block effect and the interaction of blocks and sample plots had higher prediction accuracy than the one with the block effect and population average considered alone. Introducing stand density into the model through dummy variables could further improve its prediction. This implied that the developed method for developing tree CR models of Mongolian oak is promising and can be applied to similar studies for other tree species.

Parameter estimation of two-level nonlinear mixed effects models using first order conditional linearization and the EM algorithm

Multi-level nonlinear mixed effects (ML-NLME) models have received a great deal of attention in recent years because of the flexibility they offer in handling the repeated-measures data arising from various disciplines. In this study, we propose both maximum likelihood and restricted maximum likelihood estimations of ML-NLME models with two-level random effects, using first order conditional expansion (FOCE) and the expectation-maximization (EM) algorithm. The FOCE-EM algorithm was compared with the most popular Lindstrom and Bates (LB) method in terms of computational and statistical properties. Basal area growth series data measured from Chinese fir (Cunninghamia lanceolata) experimental stands and simulated data were used for evaluation. The FOCE-EM and LB algorithms given the same parameter estimates and fit statistics for models that converged by both. However, FOCE-EM converged for all the models, while LB did not, especially for the models in which two-level random effects are simultaneously considered in several base parameters to account for between-group variation. We recommend the use of FOCE-EM in ML-NLME models, particularly when convergence is a concern in model selection.

Parameter estimation of nonlinear mixed-effects models using first-order conditional linearization and the EM algorithm

Nonlinear mixed-effects (NLME) models are flexible enough to handle repeated-measures data from various disciplines. In this article, we propose both maximum-likelihood and restricted maximum-likelihood estimations of NLME models using first-order conditional expansion (FOCE) and the expectation–maximization (EM) algorithm. The FOCE-EM algorithm implemented in the ForStat procedure SNLME is compared with the Lindstrom and Bates (LB) algorithm implemented in both the SAS macro NLINMIX and the S-Plus/R function nlme in terms of computational efficiency and statistical properties. Two realworld data sets an orange tree data set and a Chinese fir (Cunninghamia lanceolata) data set, and a simulated data set were used for evaluation. FOCE-EM converged for all mixed models derived from the base model in the two realworld cases, while LB did not, especially for the models in which random effects are simultaneously considered in several parameters to account for between-subject variation. However, both algorithms had identical estimated parameters and fit statistics for the converged models. We therefore recommend using FOCE-EM in NLME models, particularly when convergence is a concern in model selection.

Development of a System of Compatible Individual Tree Diameter and Aboveground Biomass Prediction Models Using Error-In-Variable Regression and Airborne LiDAR Data

Estimating individual tree diameters at breast height (DBH) from delineated crowns and tree heights on the basis of airborne light detection and ranging (LiDAR) data provides a good opportunity for large-scale forest inventory. Generally, ground-based measurements are more accurate, but LiDAR data and derived DBH values can be obtained over larger areas for a relatively smaller cost if a right procedure is developed. A nonlinear least squares (NLS) regression is not an appropriate approach to predict the aboveground biomass (AGB) of individual trees from the estimated DBH because both the response variable and the regressor are subject to measurement errors. In this study, a system of compatible individual tree DBH and AGB error-in-variable models was developed using error-in-variable regression techniques based on both airborne LiDAR and field-measured datasets of individual Picea crassifolia Kom. trees, collected in northwestern China. Two parameter estimation algorithms, i.e., the two-stage error-in-variable model (TSEM) and the nonlinear seemingly unrelated regression (NSUR), were proposed for estimating the parameters in the developed system of compatible individual tree DBH and AGB error-in-variable models. Moreover, two model structures were applied to estimate AGB for comparison purposes: NLS with AGB estimation depending on DBH (NLS&DD) and NLS with AGB estimation not depending on DBH (NLS&NDD). The results showed that both TSEM and NSUR led to almost the same parameter estimates for the developed system. Moreover, the developed system effectively accounted for the inherent correlation between DBH and AGB as well as for the effects of measurement errors in the DBH on the predictions of AGB, whereas NLS&DD and NLS&NDD did not. A leave-one-out cross-validation indicated that the prediction accuracy of the developed system of compatible individual tree DBH and AGB error-in-variable models with NSUR was the highest among those estimated by the four methods evaluated, but, statistically, the accuracy improvement was not significantly different from zero. The main reason might be that, except for the measurement errors, other source errors were ignored in the modeling. This study implies that, overall, the proposed method provides the potential to expand the estimations of both DBH and AGB from individual trees to stands by combining the error-in-variable modeling and LiDAR data and improve their estimation accuracies, but its application needs to be further validated by conducting a systematical uncertainty analysis of various source errors in a future study.

Additive crown width models comprising nonlinear simultaneous equations for Prince Rupprecht larch ( Larix principis - rupprechtii ) in northern China

Key messageCrown width models developed using nonlinear simultaneous equations with a two-step procedure provided the best performance and are recommended to predict the crown components and crown width of Prince Rupprecht larch.AbstractCrown width (CW) is defined as an average of two crown diameters at two cardinal directions: east–west and south–north, obtained from measurements of four crown radii (crown components) at four directions: east, west, south, and north. CW is one of the important tree variables in forest growth and yield modeling, and forest management. Reliable estimates of CW are central elements of forest management. However, the additivity of CW and crown components and their inherent correlations have not been addressed in existing CW models. In this study, two alternative procedures for nonlinear simultaneous equations (NSE) were used to develop CW models. The procedures included a disaggregation model structure with one- and two-step, proportional weighting systems, and two commonly used additivity methods, adjustment in proportion (AP) and ordinary least squares with separating regression (OLSSR). These methods were compared using data from a total of 3369 Prince Rupprecht larch (Larix principis-rupprechtii Mayr.) trees in 116 permanent sample plots in northern China. It was found that these methods effectively ensured that the sum of the crown components was equal to twice the total CW. The NSE accounted for the inherent correlations among the crown components and CW. The CW models developed using the NSE with the two-step procedure provided the best performance, followed by the models developed with AP and OLSSR. This methodology can be adapted to develop a system of CW models for other tree species.

Developing, testing and application of rodent population dynamics and capture models based on an adjusted Leslie matrix-based population approach

Small rodents in general and the multimammate rat Apodemus agrarius in particular, damage crops and cause major economic losses in China. Therefore, accurate predictions of the population size of A. agrarius and an efficient control strategy are urgently needed. We developed a population dynamics model by applying a Leslie matrix method, and a capture model based on optimal harvesting theory for A. agrarius. Our models were parametrized using demographic estimates from a capture–mark–recapture (CMR) study conducted on the Qinshui Forest Farm in Northwestern China. The population dynamics model incorporated 12 equally balanced age groups and included immigration and emigration parameters. The model was evaluated by assessing the predictions for four years based on the known starting population in 2004 from the 2004–2007 CMR data. The capture model incorporated two functional age categories (juvenile and adult) and used density-dependent and density-independent factors. The models were used to assess the effect of rodent control measures between 2004 and 2023 on population dynamics and the resulting numbers of rats. Three control measures affecting survival rates were considered. We found that the predicted population dynamics of A. agrarius between 2004 and 2007 compared favorably with the observed population dynamics. The models predicted that the population sizes of A. agrarius in the period between 2004 and 2023 under the control measure applied in August 2004 were very similar to the optimal population sizes, and no significant difference was found between the two population sizes. We recommend using the population dynamics and capture models based on CMR-estimated demographic schedules for rodent, provided these data are available. The models that we have developed have the potential to play an important role in predicting the effects of rodent management and in evaluating different control strategies.