Abstract:

Objective: This study was implemented to provide a new method for modeling tree height curve. Semiparametric regression model was used to describe the relationships between tree height and diameter at breast height(DBH), and then compared with traditional parametric regression models. Method: The height and DBH of 4 921 larch trees(Larix principis-rupprechtii) and 2 833 birch trees(Betula platyphylla) were collected from 76 plots in the core area of Chongli Winter Olympics in Zhangjiakou city, Hebei Province. The data were randomly selected according to the ratio of 7∶3 for model fitting and validation. The generalized additive model and single index model were chosen as semiparametric models. Dominant tree height and DBH were selected as independent variables, and tree species was separated firstly. In the generalized additive model, the constant term was used as a parametric part, while DBH, dominant tree height and their interaction were set as nonparametric parts respectively. The parametric part in single index models of both species was linear combination of DBH and dominant tree height or their product, and the link function was considered as nonparametric. Four generalized height-diameter equations with dominant tree height were used for comparison. In order to further build a height curve model containing two tree species, the generalized additive model and modified Richard model were selected as basic models. Species composition was added as a parametric part to the generalized additive model. The optimal parametric model was chosen by comparing the fitting statistics of adding dummy variables of tree species to different parameters of modified Richard model. The evaluation indices included the adjusted coefficient of determination($R_{\mathrm{a}}^2$), root mean square error(RMSE) and Akaike information criterion(AIC). Result: Under the condition of modeling species separately, the fitting accuracy of the generalized additive model was the highest for training samples with a $R_{\mathrm{a}}^2$ of 88.98% and 72.35% for larch and birch, increased by 3.13%-4.80% and 7.37%-12.09% compared with parametric models, and with a RMSE of 1.441 3 and 2.033 3, decreased by 0.190 4-0.284 8 and 0.252 9-0.403 4, respectively. The single index model ranked the second for fitting larch with a $R_{\mathrm{a}}^2$ of 85.99% and a RMSE of 1.624 1, while ranked the fourth for fitting birch with a $R_{\mathrm{a}}^2$ of 64.75% and a RMSE of 2.295 6. In predicting validating data, the generalized additive model had the lowest RMSE of 1.580 4 for larch and 2.192 6 for birch. However, the single index model showed relatively poor prediction for the two species. In the case of modeling multi-species height curves, the generalized additive model presented a higher $R_{\mathrm{a}}^2$ of 83.00%, a lower RMSE of 1.722 4 for training data and 1.807 5 for validating data. In both cases, the AIC values of the generalized additive models were always the lowest, indicating their significant simplicity of model structure. Conclusion: In the processes of modeling tree height curve, semiparametric models, combined with the advantages of both parametric and nonparametric models could not only greatly improve the flexibility and applicability, but also increase the fitting accuracy in most cases. The generalized additive model might present a high precision in both data fitting and prediction, and the single index model could be used as a reference to judge whether the selections of link function in other models were appropriate or not. As more variables of stand levels were added into height curve equations, semiparametric models could be used to provide new ways for complex model construction.

Key words: height curve model, semiparametric model, larch(Larix principis-rupprechtii), birch(Betula platyphylla), dominant height