华北落叶松和白桦半参数树高曲线模型

Construction of Semiparametric Height Curve Model for Larch and Birch

【目的】运用半参数回归模型描述华北落叶松和白桦树高与胸径的关系,并与传统参数回归模型进行比较,为构建树高曲线模型和提高模型精度提供新方法。【方法】基于河北省张家口市崇礼冬奥核心区76块样地4921株华北落叶松和2833株白桦的胸径、树高等数据,按7∶3比例随机选取数据用于模型拟合与检验。半参数模型选择广义可加模型和单指标模型形式,自变量设为胸径和林分优势木高,分树种建模,其中广义可加模型将常数项作为参数部分,胸径、优势木高及二者交互作用作为非参数部分,单指标模型将胸径、优势木高或二者乘积的线性组合作为参数部分,联系函数作为非参数部分。选取4种常见的包含优势木高的标准树高曲线模型用于模型比较。为进一步构建可同时表示两树种的树高曲线模型,以广义可加模型和改进的Richard参数模型为基础模型,将树种组成作为参数部分引入广义可加模型,通过比较在改进Richard参数模型不同参数上添加树种哑变量的拟合效果筛选出最优参数模型,选择调整决定系数(Ra2)、均方根误差(RMSE)和赤池信息量(AIC)评价模型估计精度。【结果】分树种建模情况下,广义可加模型的拟合精度最高,华北落叶松和白桦的Ra2分别为88.98%和72.35%,较各参数模型提高3.13%~4.80%和7.37%~12.09%,RMSE分别为1.4413和2.0333,较各参数模型减少0.1904~0.2848和0.2529~0.4034。单指标模型对华北落叶松的拟合效果次之,Ra2和RMSE分别为85.99%和1.6241,对白桦的拟合效果居第4位,Ra2和RMSE分别为64.75%和2.2956。在对检验样本的预测精度方面,广义可加模型同样为最优模型,华北落叶松和白桦的RMSE分别为1.5804和2.1926,而单指标模型对两树种预测结果不佳。混合树种建模结果显示广义可加模型略优于参数模型,Ra2达83.00%,训练和检验样本的RMSE分别为1.7224和1.8075。无论是分别树种还是混合树种建模,广义可加模型的AIC均远小于参数模型,表现出显著的模型结构简洁性。【结论】在华北落叶松和白桦树高曲线模型构建中,半参数模型在参数模型基础上引入非参数回归方法,不仅可大大提高模型的灵活性和适用性,而且拟合精度通常会有所提高,其中广义可加模型表现出对数据拟合与预测的高精度,单指标模型可作为判断其他模型中联系函数选择是否合适的参考。随着更多林分变量引入树高胸径模型,半参数方法能够为复杂模型构建提供一种新思路。

【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 4921 larch trees(Larix principis-rupprechtii)and 2833 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(R2a),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 R2a 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 1.4413 and 2.0333,decreased by 0.1904-0.2848 and 0.2529-0.4034,respectively.The single index model ranked the second for fitting larch with a R2a of 85.99%and a RMSE of 1.6241,while ranked the fourth for fitting birch with a R2a of 64.75%and a RMSE of 2.2956.In predicting validating data,the generalized additive model had the lowest RMSE of 1.5804 for larch and 2.1926 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 R2a of 83.00%,a lower RMSE of 1.7224 for training data and 1.8075 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.