基于非线性分位数回归的落叶松树干削度方程

doi:10.11707/j.1001-7488.20191008

摘要/Abstract

摘要：

目的: 采用非线性分位数回归法构建落叶松树干削度方程，比较分析不同分位数（τ=0.1、0.2、0.3、0.4、0.5、0.6、0.7、0.8、0.9）及其组合分位数的拟合及检验结果，以提高模型预测精度。方法: 基于大兴安岭落叶松干形数据，采用Koenker和Bassett提出的分位数回归法，利用SAS软件的NLP拟合基于各分位数的Max-Burkhart分段削度方程，选取确定系数（R²）、平均误差（MAB）、均方根误差（RMSE）、相对误差（MPB）和预估精度（P%）对削度方程进行对比分析。结果: 1）Max-Burkhart分段削度方程在9个不同分位点（τ=0.1、0.2、0.3、0.4、0.5、0.6、0.7、0.8、0.9）均能收敛，说明分位数回归可以提供许多不同分位数的估计结果，进而可预测任意分位点处干形的变化趋势；2）基本模型和分位点处（τ=0.4、0.5、0.6）的分位数模型拟合结果相近，分位数组合（3、5、7、9）可提高模型拟合效果，其中基于3个分位数组合（τ=0.3、0.5、0.7）、5个分位数组合（τ=0.3、0.4、0.5、0.6、0.7）、7个分位数组合（τ=0.1、0.2、0.4、0.5、0.6、0.8、0.9）、9个分位数组合（τ=0.1、0.2、0.3、0.4、0.5、0.6、0.7、0.8、0.9）在分位数组合相同时分别表现最优；3）模型检验表明，大多数分位数回归组合的检验统计量都优于基本模型和各分位数模型，相对于基本模型，5个分位数组合（τ=0.3、0.4、0.5、0.6、0.7）模型的MPB、MAB、RMSE分别降低13.9%、13.9%、13%。结论: 分位数回归能够提高模型预测精度，基于5个分位数组合的Max-Burkhart分段削度方程在拟合及检验统计量等方面表现较好，适合于大兴安岭落叶松树干削度预测。

关键词: 落叶松, 非线性回归, 分位数回归, 削度方程

Abstract:

Objective: The aim of this study was to develop stem taper equation for Larix gmelinii in Daxing'anling based on quantile regression models, and the results were used to compare and analyze for fitting and validation of different quantiles (τ=0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9)and quantile groups. Method: Stem taper data of Larix gmelinii were collected from Daxing'anling. NLP method in SAS software was used to fit Max and Burkhart segmented taper equation which was based on quantile regression (Koenker and Bassett, 1978), and the performances of all models were evaluated by use of these evaluation statistics:coefficient of determination (R²), mean absolute bias (MAB), root mean square error (RMSE), mean percentage of bias (MPB), and prediction accuracy (P%).Result: 1) The results showed that Max and Burkhart segmented taper equations could converge based on nine different quantiles (τ=0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9), showing that quantile regression could provide many different quantile estimates, and predict stem profile for different quantiles. 2) The base model was similar to the models with quantile equal to 0.4, 0.5, and 0.6. Comparing with based model, we found that all of different quantile groups (3, 5, 7, 9)could improve the prediction precision. The quantile regression based on three quantiles group (τ=0.3, 0.5, 0.7), five quantiles group (τ=0.3, 0.4, 0.5, 0.6, 0.7), seven quantiles group (τ=0.1, 0.2, 0.4, 0.5, 0.6, 0.8, 0.9)and nine quantiles group (τ=0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9)were better than the others with the same quantile groups. The quantile regression with five quantiles group (τ=0.3, 0.4, 0.5, 0.6, 0.7)were the most accurate in predicting taper equation of Larix gmelinii based on fitting statistics. 3)Model validation showed that statistical criteria of most quantile regression groups were superior to those of the base model and the individual quantile model. Relative to the base model, MPB, MAB and RMSE of five quantile combination (τ=0.3, 0.4, 0.5, 0.6, 0.7)model reduced by 13.9%, 13.9%, 13% respectively. Objective: Quantile regression models could improve the prediction precision. Based on the five quantile group (τ=0.3, 0.4, 0.5, 0.6, 0.7), Max and Burkhart segmented taper equation showed good performances on fitting and validation statistics, and was suitable for predicting stem taper of Larix gmelinii in Daxing'anling.

Key words: Larix gmelinii, nonlinear regression, quantile regression, taper equation

中图分类号:

S757

马岩岩,姜立春. 基于非线性分位数回归的落叶松树干削度方程[J]. 林业科学, 2019, 55(10): 68-75.

Yanyan Ma,Lichun Jiang. Stem Taper Function for Larix gmelinii Based on Nonlinear Quantile Regression[J]. Scientia Silvae Sinicae, 2019, 55(10): 68-75.

图/表 5

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参考文献 0

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分组 Group	变量 Variable	样木数 Number	平均值 Mean	最小值 Min.	最大值 Max.	标准差 Std.	变动系数 CV
建模数据 Fitting data	直径DBH/cm	114	23.8	16.9	32.4	2.8	11.7
建模数据 Fitting data	树高Tree height/m	114	20.4	14.9	23.9	1.5	7.2
检验数据 Validation data	直径DBH/cm	28	23.9	17.1	29.9	3	12.4
检验数据 Validation data	树高Tree height/m	28	21	18.7	23	1.1	5.3

参数估计方法Parameter estimate method	β₁	β₂	β₃	β₄	a₁	a₂
非线性回归Nonlinear regression	-5.734 8	2.708 2	-2.717 7	119.097 6	0.860 4	0.078 6
分位数回归Quantile regression
τ=0.1	-5.349 0	2.620 4	-2.153 3	99.019 3	0.889 6	0.069 0
τ=0.2	-5.557 2	2.653 1	-2.229 9	99.310 7	0.911 2	0.070 0
τ=0.3	-7.553 0	3.707 8	-3.315 0	99.145 9	0.912 0	0.068 1
τ=0.4	-5.893 7	2.842 3	-2.789 7	100.075 9	0.853 8	0.079 2
τ=0.5	-5.793 2	2.643 3	-2.389 9	99.698 0	0.930 4	0.073 9
τ=0.6	-4.639 0	2.104 2	-2.178 6	99.125 7	0.833 4	0.083 8
τ=0.7	-4.987 2	2.254 1	-2.379 1	100.295 2	0.847 1	0.085 4
τ=0.8	-5.025 6	2.214 1	-2.407 2	99.468 4	0.860 3	0.088 7
τ=0.9	-4.220 2	1.719 9	-2.071 8	98.772 0	0.841 9	0.095 3

模型Model	MAB	RMSE	R²
基本模型Basic model	0.049 0	0.070 4	0.968 6
各分位数模型Individual quantile model
τ=0.1	0.081 9	0.110 4	0.922 8
τ=0.2	0.064 1	0.092 0	0.946 4
τ=0.3	0.056 0	0.084 0	0.955 3
τ=0.4	0.049 9	0.073 4	0.965 9
τ=0.5	0.050 4	0.074 1	0.965 3
τ=0.6	0.050 1	0.071 7	0.967 4
τ=0.7	0.054 5	0.075 9	0.963 5
τ=0.8	0.065 1	0.086 8	0.952 3
τ=0.9	0.090 4	0.115 6	0.915 4
分位数组合Quantile groups
τ=0.1，0.5，0.9	0.023 4	0.033 7	0.992 8
τ=0.2，0.5，0.8	0.023 9	0.040 5	0.989 6
τ=0.3，0.5，0.7	0.030 1	0.051 0	0.983 5
τ=0.4，0.5，0.6	0.037 2	0.060 1	0.977 1
τ=0.1，0.2，0.5，0.8，0.9	0.015 8	0.027 2	0.995 3
τ=0.1，0.3，0.5，0.7，0.9	0.015 7	0.027 0	0.995 4
τ=0.1，0.4，0.5，0.6，0.9	0.018 1	0.029 2	0.994 6
τ=0.3，0.4，0.5，0.6，0.7	0.028 8	0.050 7	0.983 7
τ=0.2，0.3，0.5，0.7，0.8	0.020 9	0.039 2	0.990 3
τ=0.2，0.4，0.5，0.6，0.8	0.021 2	0.039 3	0.990 2
τ=0.1，0.2，0.3，0.5，0.7，0.8，0.9	0.012 8	0.025 2	0.996 0
τ=0.1，0.3，0.4，0.5，0.6，0.7，0.9	0.014 2	0.026 3	0.995 6
τ=0.2，0.3，0.4，0.5，0.6，0.7，0.8	0.019 7	0.038 9	0.990 4
τ=0.1，0.2，0.4，0.5，0.6，0.8，0.9	0.013 1	0.025 4	0.995 9
τ=0.1，0.2，0.3，0.4，0.5，0.6，0.7，0.8，0.9	0.011 6	0.024 7	0.996 2

模型Model	MPB	MAB	RMSE	P%
基本模型Basic model	4.652 5	0.785 5	1.070 3	99.511 5
各分位数模型Individual quantile model
τ=0.1	8.936 7	1.508 9	1.852 4	99.078 8
τ=0.2	6.662 6	1.124 9	1.456 1	99.300 1
τ=0.3	5.710 4	0.964 2	1.296 5	99.387 4
τ=0.4	4.802 9	0.810 9	1.111 4	99.483 5
τ=0.5	4.900 5	0.827 4	1.095 3	99.501 9
τ=0.6	4.701 5	0.793 8	1.074 6	99.515 6
τ=0.7	5.130 4	0.866 2	1.153 0	99.488 1
τ=0.8	6.180 5	1.043 5	1.354 3	99.411 1
τ=0.9	8.592 7	1.450 8	1.785 7	99.246 9
分位数组合Quantile groups
τ=0.1，0.5，0.9	4.325 2	0.730 3	0.975 6	99.565 2
τ=0.2，0.5，0.8	5.031 2	0.849 5	1.126 1	99.501 5
τ=0.3，0.5，0.7	4.003 8	0.676 0	0.929 4	99.576 5
τ=0.4，0.5，0.6	8.536 9	1.441 4	2.374 6	98.962 6
τ=0.1，0.2，0.5，0.8，0.9	4.325 4	0.730 3	0.975 7	99.565 2
τ=0.1，0.3，0.5，0.7，0.9	4.336 6	0.732 2	0.977 9	99.564 3
τ=0.1，0.4，0.5，0.6，0.9	4.316 3	0.728 8	0.974 3	99.565 7
τ=0.3，0.4，0.5，0.6，0.7	4.003 7	0.676 0	0.929 0	99.576 6
τ=0.2，0.3，0.5，0.7，0.8	5.034 1	0.850 0	1.127 4	99.501 0
τ=0.2，0.4，0.5，0.6，0.8	5.025 9	0.848 6	1.125 5	99.501 7
τ=0.1，0.2，0.3，0.5，0.7，0.8，0.9	4.328 3	0.730 8	0.977 2	99.564 6
τ=0.1，0.3，0.4，0.5，0.6，0.7，0.9	4.336 5	0.732 2	0.977 5	99.564 4
τ=0.2，0.3，0.4，0.5，0.6，0.7，0.8	5.034 0	0.850 0	1.127 1	99.501 1
τ=0.1，0.2，0.4，0.5，0.6，0.8，0.9	4.320 1	0.729 4	0.975 0	99.565 4
τ=0.1，0.2，0.3，0.4，0.5，0.6，0.7，0.8，0.9	4.328 2	0.730 8	0.976 9	99.564 7

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