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

doi:10.12302/j.issn.1000-2006.202108050

JOURNAL OF NANJING FORESTRY UNIVERSITY ›› 2022, Vol. 46 ›› Issue (3): 194-202.doi: 10.12302/j.issn.1000-2006.202108050

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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 E-mail:ff696977@qq.com;fengrili@126.com

Abstract

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

CLC Number:

S758

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, 2022, 46(3): 194-202.

Figures/Tables 9

Table 1

Fig.1

Table 2

Table 3

Fit statistics and likelihood ratio tests (LRT) for models with different residual variance structures"

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

Table 3

Table 4

Parameter estimates, and regression coefficients and standard errors for the model (9) to (12)"

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

Table 4

Fig.2

Table 5

Table 6

Fig.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

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

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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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