Comparison of Fixed- and Mixed-Effects Approaches to Taper Modeling for Scots Pine in West Poland

Abstract

Diameter measurements along the stem, which are the basis for taper models, usually have a hierarchical structure. Mixed-effects models, where fixed and random effects are distinguished, are a possible solution for this type of data. However, in order to fully absorb the potential of this method, random effects prediction, which requires additional measurements (diameter along stem), is recommended. This article presents a comparison of various fitting methods (mixed- and fixed-effects model approaches) of the variable-exponent taper model created by Kozak for determining the outside bark diameter along the stem and predicting the tree volume of Scots pine trees in west Poland. During the analysis, it was assumed that no additional measured data were available for practical use; therefore, for the mixed-effects model approach, fixed effects prediction without random effects was applied. Both fitting strategies were compared based on modeling and an independent validation data set. The comparison of mixed- and fixed-effects fitting strategies for the diameter along the stem indicated that the taper model fitted using the mixed-effects model approach better fit the data. Moreover, the error rate for the total tree volume prediction for the independent data set was lower for the mixed-effects model solution than for the fixed-effects one.

Stochastic Models to Qualify Stem Tapers

“…In this study, the developed models are nondeterministic models that consider both the symmetric and asymmetric diffusions of stem diameter at a particular height [21]. Multiple regression models that describe stem tapers have been proposed, traditionally incorporating the diameter at breast height and tree height as the independent variables [14][15][16][17][18][19] or using additional independent variables such as the density [18], crown ratio [33], and many more [34,35]. The newly developed SDE models are affected by the probability density function of the relative diameter, which changes among the relative heights.…”

“…However, the most published models are deterministic regression models. It is advisable to divide taper equations into five categories, namely, (1) segmented polynomial taper equations [14], (2) variable exponential equations [15][16][17], (3) artificial neural network (ANN) equations [18], (4) fundamental mechanisms of stem profile formation equations [19,20], and (5) diffusion process equations [21]. The first four categories include deterministic taper models that describe the variation of the stem form by means of a single equation, where the diameter at any specified stem height is the dependent variable, while the diameter at breast height and overall height are the independent variables.…”

“…Studies from the last few decades have incorporated mixed effect (fixed and random effect) modeling techniques in the development of taper equations [26]. The theoretical investigation of this technique accounts for both the within-and between-tree variation in stem taper forms [17,27,28]. From the practical point of view, a mixed effect modeling technique needs additional prior information (measurements) for predicting the diameter at any specified height for a new stem.…”

Models for Tree Taper Form: The Gompertz and Vasicek Diffusion Processes Framework

“…In this study, we use a segmented SDE consisting of the two different SDEs defined by Equations (1) and (2). Many regression analysis techniques have recently been used for stem taper modeling, such as linear or nonlinear fixed-and mixed-effect models or generalized additive models [21]. The mixed-effect SDE models have the ability to model the stem variability and analyze data from several individual trees simultaneously.…”

Correction: Bronisz K. and Zasada M., Comparison of Fixed- and Mixed-Effects Approaches to Taper Modeling for Scots Pine in West Poland. Forests 2019, 10, 975

“…After publication of the research paper [1], the authors wish to make the following correction. The correction concerns the placement of information that the article was also financed by the Polish National Agency for Academic Exchange as part of the Foreign Promotion Program.…”