Modelling conifer crown profiles as nonlinear conditional quantiles: An example with planted Korean pine in northeast China

Abstract

With few exceptions crown profile models for coniferous trees have conventionally been estimated within the least squares framework that is confined to describing the central trend of whorl radius along the stem. This article introduces a new approach of modelling the outer profiles of conifer tree crowns as nonlinear conditional quantiles using plantation grown Korean pine in northeast China as an example. The approach was demonstrated with two nonlinear model forms using data from detailed measurements of 83 sample trees with a total number of 1415 whorls and 5123 live branches. One model was based on a segmented polynomial function and the other was a modified variable-exponent function. For each model form, 11 nonlinear quantile curves were estimated using the branch radius data by taking 11 values of q, representing the qth quantile, from 0.5 to 0.95 with an even interval of 0.05 plus one extreme value of 0.99. Parameter estimates for a given value of q were obtained through repeated fitting by leaving one tree out each time in a simple form of jackknife model averaging. To select the best q values for delineating the outer or outmost contours of the exterior edges of tree crowns, three fix-effects (marginal) models of the same nonlinear form was fitted iteratively to the entire, about one-half and one-quarter of the whorl radius data after removing non-positive residuals at each iteration. Based on graphical evaluations and comparisons of benchmarking statistics, the 0.90th and 0.95th nonlinear quantiles were chosen for depicting the outer and outmost crown profiles of Korean pine trees for both model forms. The variable-exponent model appeared to be superior to the segmented polynomial model not only in delineating the outer crown profiles but also in estimating the largest crown radius of individual trees. In comparison to the marginal models or to the more traditional least squares regression, the new approach enables more accurate depictions of the outer or outmost crown profiles without subjective data selection and thus any loss of information. It also has the potential to profile the inner part of individual tree crowns through a family of nonlinear conditional quantial curves, therefore providing a greater insight into the crown structure than the conventional methods.