Non-Linear Regression with Repeated Data—A New Approach to Bark Thickness Modelling

• Published in: Forests Vol. 16; no. 7; p. 1160
• Main Authors: Ukalski, Krzysztof, Bijak, Szymon
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 14.07.2025
• Subjects: Analysis; Bark; Diameters; Methods; Regression analysis; Regression models; Root-mean-square errors; Software; Stems; Thickness; Timber; Trees; Variables; New York; United States
• ISSN: 1999-4907; 1999-4907
• DOI: 10.3390/f16071160

Broader use of multioperational machines in forestry requires efficient methods for determining various timber parameters. Here, we present a novel approach to model the bark thickness (BT) as a function of stem diameter. Stem diameter (D) is any diameter measured along the bole, not a specific one. The following four regression models were tested: marginal model (MM; reference), classical nonlinear regression with independent residuals (M1), nonlinear regression with residuals correlated within a single tree (M2), and nonlinear regression with the correlation of residuals and random components, taking into account random changes between the trees (M3). Empirical data consisted of larch (Larix sp. Mill.) BT measurements carried out at two sites in northern Poland. Relative root square mean error (RMSE%) and adjusted R-squared (R2adj) served to compare the fitted models. Model fit was tested for each tree separately, and all trees were combined. Of the analysed models, M3 turned out to be the best fit for both the individual tree and all tree levels. The fit of the regression function M3 for SITE1 (50-year-old, pure stand located in northern Poland) was 87.44% (R2adj), and for SITE2 (63-year-old, pure stand situated in the north of Poland) it was 80.6%. Taking into account the values of RMSE%, at the individual tree level the M3 model fit at location SITE1 was closest to the MM, while at SITE2 it was better than the MM. For the most comprehensive regression model, M3, it was checked how the error of the bark thickness estimate varied with stem diameter at different heights (from the base of the trees to the top). In general, the model’s accuracy increased with greater tree height.