基于非参数方法的落叶松树干削度方程

doi:10.3969/j.issn.1000-2006.201903038

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南京林业大学学报（自然科学版） ›› 2020, Vol. 44 ›› Issue (6): 184-192.doi: 10.3969/j.issn.1000-2006.201903038

基于非参数方法的落叶松树干削度方程

何培(), 夏宛琦, 姜立春^*()

东北林业大学林学院,森林生态系统可持续经营教育部重点实验室, 黑龙江哈尔滨 150040

收稿日期:2019-03-14 修回日期:2019-11-04 出版日期:2020-11-30 发布日期:2020-12-07
通讯作者: 姜立春
基金资助:
国家自然科学基金项目(31570624);黑龙江省应用技术研究与开发计划(GA19C006);中央高校基本科研业务费专项(2572019CP15)

Stem taper modeling equation for dahurian larch based on nonparametric regression methods

HE Pei(), XIA Wanqi, JIANG Lichun^*()

Key Laboratory of Sustainable Forest Ecosystem Management-Ministry of Education, School of Forestry, Northeast Forestry University, Harbin 150040, China

Received:2019-03-14 Revised:2019-11-04 Online:2020-11-30 Published:2020-12-07
Contact: JIANG Lichun

摘要/Abstract

摘要：

【目的】基于非参数理论,研究构建大兴安岭兴安落叶松(Larix gmelinii)非参数可加性树干削度方程,并与传统的Max and Burkhart 参数削度方程进行对比。【方法】利用树高、胸径、不同高度处直径、不同部位高度等变量及其变形构建非参数可加性削度方程。采用 7种光滑样条函数:薄板回归样条函数(TP)、 Duchon 样条函数(DS)、三次回归样条函数(CR)、P-样条函数(PS)、高斯过程平滑样条函数(GP)、B-样条函数(BS)和局部回归光滑函数(LO),基于R软件mgcv包的Gamm函数对非参数模型进行拟合。【结果】使用相对直径(d/D)作为因变量,胸径的平方(D²)、树高(H)、相对树高的算术平方根($\sqrt{h/H}$)作为自变量, 构建兴安落叶松最佳非参数可加性树干削度方程。拟合结果表明:基于CR和LO样条函数的可加性削度方程具有较小的R²(决定系数)和较大的赤池信息量准则(AIC)值,且CR和LO的残差图重心线略呈中间高、两头低的趋势。其他基于5种光滑样条函数的可加性削度方程表现出相似的拟合结果。可加性模型除了使用LO样条函数外,其他样条函数都优于Max and Burkhart参数削度方程的拟合结果。总体检验结果表明,除了CR样条函数模型外,其他各非参数模型(TP、 DS、 PS、GP 和BS)与拟合结果基本一致,即都优于Max and Burkhart参数削度模型的预测精度。基于树干不同高度直径预测的误差对比表明,除了CR模型外,非参数模型(TP、 DS、 PS、GP 和BS)在大多数树干高度处直径预测的平均误差和绝对平均误差都小于Max and Burkhart参数模型预测值。【结论】非参数模型(TP、 DS、PS、 GP和BS)在拟合统计量、残差分布图、总体和树干不同高度处直径的预测精度都表现出一致性,并优于林业上通常使用的Max and Burkhart参数削度方程。当模型以预测为目的时,所构建的非参数可加性削度方程可用于大兴安岭兴安落叶松干形和材积预测。

关键词: 非参数模型, 样条函数, 削度方程, 兴安落叶松

Abstract:

【Objective】Based on the nonparametric theory, the nonparametric additive taper equation was constructed for dahurian larch (Larix gmelinii) in Greater Khingan Mountains. The common parametric taper equation of Max and Burkhart, in forestry, was used for comparison.【Method】A nonparametric additive taper equation was constructed using the total height, diameter at breast height, diameter at different heights, height at different stems, and their transformation. The nonparametric model was fitted based on the Gamm function in the mgcv library of the R software. Specifically, we used 7 smooth splines: thin plate regression splines (TP), Duchon splines (DS), cubic regression splines (CR), P-splines (PS), Gaussian process smooths (GP), B-spline (BS), and local regression (LO).【Result】The optimal nonparametric form of taper equation was constructed using response variable relative diameter (d/D) and explanatory variables such as the square of diameter at breast height (D²), total height (H) and square root of relative height($\sqrt{h/H}$). Fitting results showed that the additive taper equations based on CR and LO had smaller R² and larger AIC values, and the residual trend lines of CR and LO were slightly higher in the middle and lower at both ends. The nonparametric additive taper equation fitting effects of the other 5 smooth splines were similar. The fitting results of these additive models are better than those of Max and Burkhart, except for LO model. The overall validation results showed that the nonparametric models (TP, DS, PS, GP and BS) were basically consistent with the fitting results, except for CR. That is, they were superior to the parametric taper equation of Max and Burkhart. The error comparison based on the prediction of different height diameters of stems showed that the average error and absolute mean error of the nonparametric models (TP, DS, PS, GP and BS) were less than those of Max and Burkhart at most heights, except for CR. 【Conclusion】The nonparametric models (TP, DS, PS, GP and BS) showed consistent accuracy in fitting statistics, residual distribution, and the prediction from overall and different heights. These models are superior compared to those commonly used parametric taper equation of Max and Burkhart in forestry. When prediction is the main purpose for building a model, the nonparametric additive taper equation constructed in this study can be used not only to predict the shape of the stem but also the volume of the dahurian larch in Daxing’anling.

Key words: nonparametric models, spline function, taper equation, Larix gmelinii

中图分类号:

S791.222

何培,夏宛琦,姜立春. 基于非参数方法的落叶松树干削度方程[J]. 南京林业大学学报（自然科学版）, 2020, 44(6): 184-192.

HE Pei, XIA Wanqi, JIANG Lichun. Stem taper modeling equation for dahurian larch based on nonparametric regression methods[J].Journal of Nanjing Forestry University (Natural Science Edition), 2020, 44(6): 184-192.DOI: 10.3969/j.issn.1000-2006.201903038.

图/表 10

表1

图1

表2

表3

表4

图2

图3

表5

非参数可加性模型拟合结果"

光滑样条 smooth splines	参数项 parametric terms					非参数项 nonparametric terms
光滑样条 smooth splines	截距 intercept	估计值 estimate	标准差 standard error	t	P	平滑项 smooth terms	自由度 df	F	P
	$α 1$	0.826 0	0.001 1	778.1	<0.001	$s 1 (D 2)$	6.317	23.260	<0.001
TP						$s 1 (H)$	3.683	8.861	<0.001
						$s 1 (h / H)$	8.222	9 140.065	<0.001
	$α 2$	0.826 0	0.001 1	778.4	<0.001	$s 2 (D 2)$	6.474	23.160	<0.001
DS						$s 2 (H)$	3.842	9.550	<0.001
						$s 2 (h / H)$	9.903	5 313.460	<0.001
	$α 3$	0.826 0	0.001 1	737.3	<0.001	$s 3 (D 2)$	5.591	23.205	<0.001
CR						$s 3 (H)$	3.775	7.351	<0.001
						$s 3 (h / H)$	7.205	9 294.653	<0.001
	$α 4$	0.826 0	0.001 0	777.8	<0.001	$s 4 (D 2)$	5.082	28.020	<0.001
PS						$s 4 (H)$	3.706	12.420	<0.001
						$s 4 (h / H)$	7.403	10 142.200	<0.001
	$α 5$	0.826 0	0.001 1	777.9	<0.001	$s 5 (D 2)$	6.367	23.040	<0.001
GP						$s 5 (H)$	3.788	8.930	<0.001
						$s 5 (h / H)$	8.331	9 015.160	<0.001
	$α 6$	0.826 0	0.001 1	778.0	<0.001	$s 6 (D 2)$	5.460	26.150	<0.001
BS						$s 6 (H)$	4.002	10.450	<0.001
						$s 6 (h / H)$	7.784	9 650.210	<0.001

表5

表6

图4

参考文献 27

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统计量 statistics	建模数据 fitting data		检验数据 validation data
统计量 statistics	胸径/cm DBH	树高/m tree height	胸径/cm DBH	树高/m tree height
最小值 min.	5.20	5.30	5.40	5.10
最大值 max.	63.40	26.50	56.60	29.50
平均值 mean	29.83	17.41	29.47	17.46
标准差 SD	13.22	5.14	13.61	5.39
变异系数 CV	44.31	29.54	46.18	30.87

模型 models	均方根误差 RMSE	R²	赤池信息量准则 AIC
(2)	4.313 2	0.897 0	14 250.45
(3)	4.316 5	0.896 8	14 257.98
(4)	4.310 5	0.897 1	14 244.44
(5)	4.313 9	0.897 0	14 252.01
(6)	2.169 8	0.973 9	7 552.07
(7)	2.167 5	0.973 9	7 541.62
(8)	2.164 2	0.974 1	7 526.58
(9)	2.161 9	0.974 1	7 516.15

统计量statistic	b₁	b₂	b₃	b₄	p	q
估计值 estimate	-3.271 1	1.427 1	-1.325 0	124.324 3	0.816 8	0.079 6
标准误差standard error	0.466 3	0.253 7	0.250 0	5.647 0	0.024 1	0.001 6
t值 t value	-7.015	5.623	-5.298	22.016	33.839	47.565
P值 P value	<0.001	<0.001	<0.001	<0.001	<0.001	<0.001

模型 models	均方根误差 RMSE	R²	赤池信息量准则 AIC
Max	2.293 6	0.970 8	8 105.11
TP	2.161 6	0.974 1	7 516.15
DS	2.160 1	0.974 1	7 509.20
CR	2.247 5	0.972 0	7 895.96
PS	2.163 4	0.974 0	7 524.11
GP	2.161 5	0.974 1	7 515.67
BS	2.162 5	0.974 1	7 519.94
LO	2.358 6	0.969 2	8 366.25

统计量 deviation of statistics	模型 models
统计量 deviation of statistics	Max	TP	DS	CR	PS	GP	BS
相对百分误差MAPE	8.974 4	8.049 5	8.035 5	9.113 9	8.064 5	8.047 7	8.060 4
均方根误差RMSE	2.050 7	2.037 2	2.037 4	2.152 1	2.038 3	2.038 1	2.039 1
均方根误差百分比RMSPE	1.089 8	0.056 7	0.051 0	0.332 7	0.057 1	0.054 3	0.052 9

基于非参数方法的落叶松树干削度方程

Stem taper modeling equation for dahurian larch based on nonparametric regression methods

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