基于两水平非线性混合效应模型的长白落叶松削度方程构建

doi:10.12302/j.issn.1000-2006.202108050

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南京林业大学学报（自然科学版） ›› 2022, Vol. 46 ›› Issue (3): 194-202.doi: 10.12302/j.issn.1000-2006.202108050

基于两水平非线性混合效应模型的长白落叶松削度方程构建

聂璐毅(), 董利虎, 李凤日^*(), 苗铮, 谢龙飞

东北林业大学林学院,黑龙江哈尔滨 150040

收稿日期:2021-08-29 接受日期:2021-10-07 出版日期:2022-05-30 发布日期:2022-06-10
通讯作者: 李凤日
基金资助:
国家重点研发计划(2017YFD0600402);长白落叶松高效培育技术省级资助项目(GX18B041)

Construction of taper equation for Larix olgensis based on two-level nonlinear mixed effects model

NIE Luyi(), DONG Lihu, LI Fengri^*(), MIAO Zheng, XIE Longfei

Colleage of Forestry, Northeast Forestry University, Harbin 150040, China

Received:2021-08-29 Accepted:2021-10-07 Online:2022-05-30 Published:2022-06-10
Contact: LI Fengri

摘要/Abstract

摘要：

【目的】以长白落叶松(Larix olgensis)人工林干形指标为主要数据,利用非线性混合效应模型构建长白落叶松削度方程,研究不同的二次抽样方案对混合效应模型预估精度的影响。【方法】在不同类型的削度方程中选择拟合效果最好的Kozak方程作为基础模型,通过再参数化的方法引入树冠特征变量,分析树冠大小对干形的影响;在包含树冠变量的模型基础上,结合混合效应模型考虑样地、样木效应对于干形的影响,建立长白落叶松削度方程;采用2种抽样方案(方案Ⅰ,不限定抽样位置;方案Ⅱ,抽样位置限定在相对高0.1以下)对混合效应模型进行精度检验。【结果】树冠变量中冠长率与干形关系最为密切,将冠长率引入模型后,提升了模型的拟合精度,且模型参数在5%显著性水平上均显著。通过似然比检验,样地效应和样木效应极显著提升了模型拟合效果(P<0.01),并使用指数函数和一阶连续自回归结构[CAR(1)]来解决削度方程中普遍存在的异方差和自相关问题,最终的混合效应模型调整后决定系数( $R a 2$ )为0.994 1,均方根误差(RMSE)为0.623 1,拟合效果优于基础模型( $R a 2$ 提高了0.4%,RMSE减小了24.6%)。当使用不同的抽样方案进行预测时,方案Ⅰ表现出最高的预测精度,抽样数量为5时,平均绝对误差(MAE)为0.470 0,平均绝对误差百分比(MAPE)为4.62%;而方案Ⅱ不同抽样数量之间的预测效果差别不大(MAE的变化范围为0.520 3~0.536 6,MAPE的变化范围为5.14%~5.22%),但仍优于基础模型。【结论】将冠长率引入模型后,模型对干形的模拟更贴合实际中的林木生长情况;包含样地效应、样木效应的两水平混合效应模型可以很好地模拟长白落叶松的干形变化,为精准估计长白落叶松各材种和立木材积提供了依据。

关键词: 长白落叶松, 干形, 非线性混合效应模型, 混合模型校正, 削度方程

Abstract:

【Objective】 Based on the stem taper data of 177 artificial Larix olgensis trees from 31 permanent sample plots in Heilongjiang Province, a non-linear mixed effect model was used to construct a taper equation for L. olgensis. This will provide a theoretical basis for the precise prediction of the stem shape and tree volume. 【Method】 Among the different types of taper equations, the Kozak equation with the best fitting effect was selected as the basic model, and then the crown characteristic variables were introduced using reparameterization to analyze the influence of the crown size on the stem shape. Based on the model with the crown variables, the two-level mixed-effect model considering the impact of the sample plot effect and the sample tree effect on the stem shape was used to develop the taper equations for Larix olgensis trees. Prediction precision of the mixed-effect models was tested using two different sampling strategies. Strategy Ⅰ: the global optimal plan that does not limit the sampling position. Strategy Ⅱ: the plan with a relatively high limit of less than 0.1. 【Result】 Among the tree crown variables, the crown length rate has the closest relationship with the stem shape. Introducing the crown length rate into the taper equation, the fitting accuracy of the model was improved, and the model parameters were all significant at the 5% significance level. The results of the likelihood ratio test showed that the sample plot effect and the sample tree effect significantly (P < 0.01) improved the model accuracy. The exponential function and CAR (1) structure were used to solve the common heteroscedasticity and autocorrelation problems in the taper equation. The goodness-of-fit for the mixed-effect models for $R a 2$ was 0.994 1, RMSE was 0.623 1 which was better than the basic model ( $R a 2$ increased by 0.4%, RMSE decreased by 24.6%). When using a different sampling strategy for model prediction, Strategy Ⅰ had the best results (RMSE was 0.470 0, MAPE was 4.62%), when the sampling number was 5. For Strategy Ⅱ, there were little differences in predicting accuracy with different sampling numbers (the range of MAE was 0.520 3-0.536 6, the range of MAPE was 5.14%-5.22%), but still better than the basic model. 【Conclusion】 The two-level mixed-effects model, including plot effect and tree effect, was better than the basic model for the fitting effect and prediction accuracy, and it will be suitable for effectively predicting stem shape changes in L. olgensis. The results will provide a foundation for precisely predicting the timber volume and total tree volume of L. olgensis.

Key words: Larix olgensis, stem shape, nonlinear mixed-effect model, mixed model calibration, taper equation

中图分类号:

S758

聂璐毅,董利虎,李凤日,等. 基于两水平非线性混合效应模型的长白落叶松削度方程构建[J]. 南京林业大学学报（自然科学版）, 2022, 46(3): 194-202.

NIE Luyi, DONG Lihu, LI Fengri, MIAO Zheng, XIE Longfei. Construction of taper equation for Larix olgensis based on two-level nonlinear mixed effects model[J].Journal of Nanjing Forestry University (Natural Science Edition), 2022, 46(3): 194-202.DOI: 10.12302/j.issn.1000-2006.202108050.

图/表 9

表1

图1

表2

表3

不同方差-协方差结构下的模型拟合统计量及LRT检验指标"

模型 model	树内方差-协方差矩阵 R_i	参数 parameters	赤池信息准则 AIC	贝叶斯信息准则 BIC	对数似然值 LogLik	似然比值 LRT	P
(10)	σ²I_ni	19	5 864.76	5 978.42	-2 913.4
(11)	σ²Γ_iI_ni	20	5 586.55	5 706.18	-2 773.3	280.33	< 0.01
(12)	σ²G⁰^.⁵ⁱΓ_i $G i 0.5$	21	5 426.38	5 551.99	-2 692.2	162.06	< 0.01

表3

表4

模型(9)—(12)的参数估计及拟合指标统计表"

项目 items	参数 parameter	模型(9) model(9)	模型(10) model(10)	模型(11) model(11)	模型(12) model(12)
固定参数 fixed parameter	a₁	1.000 6	0.973 8	0.948 1	0.949 4
	a₂	1.004 3	1.013 5	1.024 9	1.025 2
	a₃	0.225 6	0.215 0	0.227 1	0.248 3
	a₄	-0.427 0	-0.399 3	-0.375 0	-0.443 3
	a₅	0.442 3	0.457 0	0.479 1	0.489 8
	a₆	1.125 6	0.704 3	0.390 7	0.768 7
	a₇	0.016 4	0.014 4	0.009 2	0.009 9
	a₈	-0.055 0	-0.041 2	-0.020 9	-0.031 5
	a₉	0.761 6	0.817 0	0.756 0	0.679 8
方差组成 variance components	σ²	0.472 2	0.325 4	0.391 4	0.257 0
拟合优度 goodness of fit	$R a 2$	0.989 7	0.995 6	0.994 8	0.994 1
	RMSE	0.826 5	0.535 6	0.582 8	0.623 1

表4

图2

表5

表6

图3

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统计量 statistics	林分因子 stand factors					样木因子 tree factors
统计量 statistics	树龄/a tree age	林分密度/ (株·hm^-2) stand density	胸径/cm DBH	每公顷断面积/ (m²·hm^-2) BAS	树高/m tree height	胸径/cm DBH	树高/m tree height	冠长/m CL	冠长率 CR	冠幅/m CW
平均值 mean	34	1 138.1	18.05	22.42	17.80	18.61	17.74	7.35	0.45	3.53
最小值 min	9	275.0	5.80	4.98	6.16	2.00	3.80	2.30	0.20	1.10
最大值 max	52	2 767.0	29.90	31.27	25.51	35.70	26.95	14.72	0.97	6.35
标准偏差 SD	13	754.1	6.17	6.38	5.25	7.00	5.40	2.13	0.18	1.05

模型 model	平均绝对误差 MAE	平均绝对误差百分比/% MAPE
(9)	0.572 6	5.32
(10)	0.395 7	4.09
(11)	0.442 8	4.40
(12)	0.457 2	4.50

模型 model	抽样方案 sampling schemes	平均绝对误差 MAE	平均绝对误差百分比/% MAPE	校准直径所在相对高 calibration diameter’s relative height
(9)	—	0.572 6	5.32	—
	Ⅰ-1	0.524 9	5.04	0.60
	Ⅰ-2	0.512 4	4.94	0,0.60
	Ⅰ-3	0.500 0	4.75	0,0.60,0.75
	Ⅰ-4	0.479 4	4.66	0,0.10,0.60,0.75
	Ⅰ-5	0.470 0	4.62	0,0.10,0.50,0.60,0.75
(12)	Ⅰ-6	0.467 7	4.61	0,0.10,0.40,0.50,0.60,0.75
	Ⅱ-1	0.536 6	5.22	0
	Ⅱ-2	0.529 7	5.18	0,0.10
	Ⅱ-3	0.524 8	5.15	0,0.04,0.10
	Ⅱ-4	0.521 3	5.14	0,0.04,0.08,0.10
	Ⅱ-5	0.520 3	5.14	0,0.04,0.06,0.08,0.10
	Ⅱ-6	0.526 6	5.15	0,0.02,0.04,0.06,0.08,0.10

基于两水平非线性混合效应模型的长白落叶松削度方程构建

Construction of taper equation for Larix olgensis based on two-level nonlinear mixed effects model

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参考文献 37

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模型 model	样木效应 tree effects	样地效应 plot effects	赤池信息准则 AIC	贝叶斯信息准则 BIC	对数似然值 LogLik	似然比值 LRT	P
(9)	无	无	7 198.29	7 258.10	-3 589.1
(10)	a₂_i、a₆_i、a₈_i	a₂_i_,_j、a₆_i_,_j	5 864.76	5 978.42	-2 913.4	13.05	< 0.01

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