Development of Nonlinear Parsimonious Forest Models Using Efficient Expansion of the Taylor Series: Applications to Site Productivity and Taper

Abstract

The parameters of nonlinear forest models are commonly estimated with heuristic techniques, which can supply erroneous values. The use of heuristic algorithms is partially rooted in the avoidance of transformation of the dependent variable, which introduces bias when back-transformed to original units. Efforts were placed in computing the unbiased estimates for some of the power, trigonometric, and hyperbolic functions since only few transformations of the predicted variable have the corrections for bias estimated. The approach that supplies unbiased results when the dependent variable is transformed without heuristic algorithms, but based on a Taylor series expansion requires implementation details. Therefore, the objective of our study is to investigate the efficient expansion of the Taylor series that should be included in applications, such that numerical bias is not present. We found that five functions require more than five terms, whereas the arcsine, arccosine, and arctangent did not. Furthermore, the Taylor series expansion depends on the variance. We illustrated the results on two forest modeling problems, one at the stand level, namely site productivity, and one at individual tree level, namely taper. The models that are presented in the paper are unbiased, more parsimonious, and they have a RMSE comparable with existing less parsimonious models.

Nonlinear parsimonious forest modeling assuming normal distribution of residuals

“…The parsimonious framework that we are proposing in this study completes the work of Strimbu et al (2017) and Amarioarei et al (2020) by presenting not only the first-order expectations but also the second order. Our models expand the findings of Neyman and Scott (1960) by providing an approach for solving nonlinear models that is fast and simple.…”

“…In many instances, common algorithms based on heuristics (e.g., steepest descent, Gauss -Newton, or Marquardt) estimate the parameters of nonlinear relationship with opposite signs than the actual ones. Bayesian approaches (Gelman et al 2003) can lead to similar results, as proven by Amarioarei et al (2020). To prove the impact of the estimation procedure on the parameters to be estimated, we use an example.…”

“…Nevertheless, when an infinite number of terms are required but only few are used, the compounding from Eq. 5 could lead to computational errors (Amarioarei et al 2020). Significant deviation from the actual model can occur because of the truncation, even when only one transformation is present, like Eq.…”

“…All the computations were executed in R programming language (Gentleman and Ihaka 2014). We considered that an efficient Taylor series expansion is a series for which MAE < 1%, similarly to Amarioarei et al (2020).…”

“…However, the two approaches agreed, when the number of terms is chosen first time when the preset value is reached by the difference between the expectations computed with consecutive number of terms (i.e., one term apart), except for the hyperbolic tangent case (i.e., two terms apart). When the preset value was surpassed at least two times, the scree plot and the preset value approaches could differ by six terms (Amarioarei et al 2020). Considering that the expectation of the firstorder moment depends on the variance, the usage of a fewer terms should be avoided, as it can lead to biased results.…”