Branch density model for Pinus koraiensis plantation based on thinning effects
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摘要:
目的 分析抚育间伐对红松人工林枝条数量的影响，建立基于间伐效应的生物数学模型，为制定更加科学合理的间伐体制提供理论依据。 方法 基于黑龙江省林口林业局和东京城林业局不同林分条件及抚育间伐强度下的红松人工林49株解析木4 370组枝解析数据，利用R语言的nlme包，建立了基于抚育间伐效应的枝条密度单水平非线性混合模型，并利用调整决定系数（ $ {R}_{{\rm{a}}}^{2} $ ）、赤池信息准则（AIC）、贝叶斯信息准则（BIC）、对数似然值（Log-likelihood）以及似然比检验（LRT）等评价指标对所收敛的模型进行评价。结果 当地位指数和树木等级相近时，抚育间伐强度和冠长越大，枝条密度越大；当抚育间伐强度和树木等级相近时，地位指数和冠长越大，枝条密度越大；而抚育间伐强度和地位指数相近时，树木胸径与枝条密度呈负相关。基于样地效应的混合模型模拟精度均高于基础模型和基于样木效应的混合模型，最终选用含有总着枝深度（DINC）、相对着枝深度的自然对数（lnRDINC）、相对着枝深度的平方（RDINC2）、胸径（DBH）、抚育间伐强度与间伐年龄的比值（TI/TA）这5个随机效应参数的非线性混合模型为枝条密度最优预测模型，其 ${R}_{{\rm{a}}}^{2}$ 为0.825 7，均方根误差（RMSE）为2.171 4。结论 基于抚育间伐效应的红松枝条密度最优非线性混合效应模型，不但能提高模型精度，还能更加准确地体现抚育间伐对林木枝条产生的影响。 Abstract:Objective In order to analyze the influence of thinning on the number of branches for Pinus koraiensis plantation, this study constructed a biological mathematic model based on the thinning effect, and provided a theoretical basis for developing a scientific and reasonable thinning program. Method Based on the data of 4 370 branches from 49 sample trees in Pinus koraiensis plantation in Linkou and Dongjingcheng Forestry Bureau of Heilongjiang Province of northeastern China, this study established a single-level nonlinear mixed effect model of branch density with thinning effects using nlme package of R. The converged models were then evaluated by adjusted coefficient of determination ( $ {R}_{{\rm{a}}}^{2} $ ), Akaike information criterion AIC, Bayesian information criterion (BIC), log likelihood and likelihood ratio test (LRT).Result When site index and tree size were similar, branch density increased with the increase of thinning intensity and crown length. When thinning intensity and tree size were similar, branch density increased with the increase of site index and crown length. However, when thinning density and site index were similar, branch density was negatively correlated with DBH. Nonlinear mixed effect model with plot effect had higher fitting precision than that with tree effect and corresponding fixed effect model. Finally, the nonlinear mixed model with five random coefficients, including DINC (depth into crown), lnRDINC (natural logarithm of relative depth into crown), RDINC2 (square of relative depth into crown), DBH and TI/TA (thinning intensity over thinning age) was selected as the most optimized model for predicting branch density, whose $ {R}_{{\rm{a}}}^{2} $ was 0.825 7 and RMSE was 2.171 4.Conclusion The optimal nonlinear mixed effect model with thinning effect not only has higher precision, but also more accurately reflects the effect of thinning on tree branches than other models. -
Key words:
- thinning /
- branch density /
- nonlinear mixed effect model /
- Pinus koraiensis plantation
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图 1 红松人工林数据采样点位置分布
Figure 1. Location distribution of sampling points of Pinus koraiensis plantation
图 3 不同条件下的枝条密度分布规律图
Figure 3. Distribution patterns of branch density under different conditions
表 1 红松人工林抚育间伐因子及林分因子统计表
Table 1. Statistics of thinning and stand factors for Pinus koraiensis planation
统计量
Statistic间伐强度
Thinning intensity
(TI)/%间伐年龄/a
Thinning age
(TA)/year地位指数
Site index
(SI)/m林分平均胸径
Average stand
DBH/cm林分平均高
Mean stand
height/m林分断面积/(m2∙hm−2)
Stand basal area/
(m2∙ha−1)林分密度/(株∙hm−2)
Stand density/
(tree∙ha−1)最大值 Max. value 40.00 35.00 16.19 18.76 11.85 33.62 2 217 最小值 Min. value 0.00 28.00 9.15 11.62 6.90 12.55 716 平均值 Mean value 22.62 31.52 13.35 14.76 9.78 23.12 1 388 标准差 SD 13.26 1.53 1.65 1.83 1.31 5.27 371 变异系数 CV/% 58.64 4.84 12.38 12.40 13.37 22.79 26.72 
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表 2 红松人工林解析木和枝条分布统计表
Table 2. Statistics of sample trees and branch distribution for Pinus koraiensis planation
项目
Item统计量
Statistic年龄/a
Age/year胸径
DBH/cm树高
Tree height
(HT)/m冠长
Crown length
(CL)/m冠幅
Crown width
(CW)/m冠长率
Crown length
ratio (CR)枝条密度/（个∙m−1）
Branch density/
(number∙m−1)拟合数据
(样本容量 = 40)
Fitting data
(sample size = 40)最大值 Max. value 38 23.60 14.23 10.60 2.93 0.88 29 最小值 Min. value 31 8.00 7.05 4.30 1.13 0.48 1 平均值 Mean value 35.15 14.79 10.42 6.96 1.97 0.68 13.98 标准差 SD 1.53 4.60 1.79 1.34 0.45 0.11 5.20 变异系数 CV/% 4.35 31.09 17.17 19.24 22.80 16.40 37.23 检验数据
(样本容量 = 9)
Validation data
(sample size = 9)最大值 Max. value 37 25.50 13.37 8.69 3.13 0.82 31 最小值 Min. value 33 8.00 7.90 3.80 1.13 0.48 3 平均值 Mean value 34.96 16.09 10.84 7.69 2.22 0.71 14.17 标准差 SD 1.18 4.94 1.48 1.43 0.63 0.11 5.70 变异系数 CV/% 3.38 30.67 13.67 18.55 28.48 15.21 40.21
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表 3 不同间伐强度、不同地位指数枝条密度双因素方差分析
Table 3. Two-way ANOVA of different thinning intensities and site index for branch density
TI SI n 总活枝数量
Total number of
living branch平均活枝密度/(个∙m−1)
Average density of
living branch/
(number∙m−1)最大活枝密度/(个∙m−1)
Max. density of
living branch/
(number∙m−1)最大活枝密度的冠层深度
Crown depth of
Max. branch
density/m最大活枝密度相对冠层深度
Relative crown depth of
Max. branch
density/%CK Ⅰ 6 98 ± 33abcd 14.1 ± 1.84ab 19.59 ± 1.87ab 2.76 ± 0.48bc 41.97 ± 11.50b Ⅲ 3 73 ± 21d 12.3 ± 0.63ab 16.09 ± 1.16bc 2.99 ± 0.25abc 55.27 ± 24.59ab V 3 74 ± 7cd 11.61 ± 0.35b 16.21 ± 0.21bc 4.14 ± 0.42a 70.6 ± 2.4a L Ⅲ 4 108 ± 22abc 13.9 ± 0.95ab 23.51 ± 5.64a 3.53 ± 0.79ab 49.25 ± 8.6b Ⅳ 2 72 ± 6d 11.16 ± 0.29b 16.56 ± 3.36bc 1.90 ± 0.06c 33 ± 5.23b V 2 68 ± 1d 11.39 ± 0.19b 17.08 ± 0.8bc 3.46 ± 0.77abc 56.8 ± 14.01ab M Ⅱ 10 112 ± 15ab 14.52 ± 3.06a 19.34 ± 4.1b 3.34 ± 1.04abc 44.95 ± 14.29b Ⅳ 7 97 ± 29abcd 12.03 ± 1.91b 15.9 ± 2.34c 2.99 ± 1.35abc 39.4 ± 14.89b V 2 74 ± 21cd 12.28 ± 3.5ab 17.11 ± 2.43bc 2.64 ± 0.43bc 44.35 ± 6.15b H Ⅰ 2 132 ± 5a 15.5 ± 0.68a 21.51 ± 1.54ab 2.88 ± 0.05abc 34.1 ± 0.42b Ⅱ 4 119 ± 8a 14.97 ± 0.81a 20.15 ± 0.72ab 3.44 ± 0.88abc 48.5 ± 15.41b Ⅲ 4 87 ± 34bcd 13.68 ± 2.99ab 17.44 ± 4.7bc 3.04 ± 0.29abc 50.8 ± 14.28ab 注：n.解析木株数；地位指数等级(SI)：Ⅰ (15.03 ~ 16.19 m)、Ⅱ (14.01 ~ 14.81 m)、Ⅲ (13.15 ~ 13.99 m)、Ⅳ (11.94 ~ 12.91 m)、Ⅴ (9.15 ~ 11.09 m)。表中数值为平均值 ± 标准差，同一列数据后不同字母表示5%水平差异显著。Notes: n represents parse tree number; site index level (SI): Ⅰ (15.03−16.19 m), Ⅱ (14.01−14.81 m), Ⅲ (13.15−13.99 m), Ⅳ (11.94−12.91 m), Ⅴ (9.15−11.09 m). The data in table are mean ± SD. Different letters in the same column after data indicate significant differences at 5% level.
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表 4 最优基础模型参数拟合结果
Table 4. Fitting results of the best basic model
参数 Parameter 样本容量 Sample size 估计值 Estimation 标准误差 SE t P $\lambda $ 3 534 24.485 517 1.193 713 20.512 1 < 0.000 1 ${k_1}$ 3 534 0.155 000 0.012 954 11.965 6 < 0.000 1 ${k_2}$ 3 534 −0.616 205 0.017 812 −34.595 3 < 0.000 1 ${k_3}$ 3 534 1.279 126 0.062 997 20.304 6 < 0.000 1 ${k_4}$ 3 534 0.005 237 0.001 403 3.733 7 < 0.000 1 ${k_5}$ 3 534 −0.037 561 0.006 627 −5.667 9 < 0.000 1 ${k_6}$ 3 534 −0.048 652 0.003 168 −15.660 0 < 0.000 1 ${k_7}$ 3 534 −0.055 832 0.005 951 −9.381 7 < 0.000 1
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表 5 基于不同随机效应参数组合的枝条密度模型拟合精度比较
Table 5. Comparison of mixed branch density model based on different random effect parameters
随机效应
Random effect模型
Model随机效应参数
Random effect
parameter参数个数
Number of
parameterRa 2 AIC BIC Log-
likelihood似然比检验
Likelihood ratio
test (LRT)P 11 无 None 8 0.591 6 18 533.29 18 588.82 −9 257.64 样木效应
Sample tree effect11.1 ${k_3}$ 9 0.613 1 18 446.23 18 507.91 −9 213.12 89.04 < 0.001 11.2 ${k_3}$ 、${k_7}$ 11 0.629 6 18 300.37 18 374.38 −9 138.18 149.88 < 0.001 11.3 ${k_1}$ 、${k_4}$ 、${k_5}$ 14 0.640 2 18 245.35 18 337.87 −9 107.67 61.02 < 0.001 样地效应
Sample plot effect11.4 ${k_5}$ 9 0.675 0 17 910.10 17 971.78 −8 945.05 11.5 ${k_2}$ 、${k_4}$ 11 0.742 4 17 202.39 17 276.40 −8 589.19 711.72 < 0.001 11.6 ${k_1}$ 、${k_2}$ 、${k_4}$ 14 0.797 8 16 566.78 16 659.30 −8 268.39 641.60 < 0.001 11.7 ${k_1}$ 、${k_2}$ 、${k_4}$ 、${k_7}$ 18 0.808 2 16 316.18 16 433.37 −8 139.09 258.60 < 0.001 11.8 ${k_1}$ 、${k_2}$ 、${k_3}$ 、${k_4}$ 、${k_7}$ 23 0.825 7 16 125.71 16 273.74 −8 038.85 200.48 < 0.001
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表 6 不同参数数量模型方差组成、参数估计及拟合统计量
Table 6. Variance component, fixed parameters and fitting statistic estimates of models with different numbers of parameters
项目 Item 参数 Parameter 模型11 Model 11 模型11.3 Model 11.3 模型11.8 Model 11.8 固定效应参数
Fixed effect parameter$\lambda $ 24.485 5*** 25.595 1*** 22.221 1** ${k_1}$ 0.155 0*** 0.229 0*** 0.157 7 ${k_2}$ −0.616 2*** −0.701 5*** −0.656 9*** ${k_3}$ 1.279 1*** 1.019 4*** 1.598 9*** ${k_4}$ 0.005 2*** −0.020 7** 0.001 7 ${k_5}$ −0.037 6*** −0.060 6*** −0.100 8*** ${k_6}$ −0.048 7*** −0.026 7*** −0.050 1* ${k_7}$ −0.055 8*** −0.002 7 −0.350 5* 随机效应方差−协方差结构
Variance of random effect-covariance structure$\sigma _{{k_1}}^2$ 0.050 5 0.207 6 $\sigma _{{k_2}}^2$ − 0.183 8 $\sigma _{{k_3}}^2$ − 0.886 5 $\sigma _{{k_4}}^2$ 0.016 0 0.048 3 $\sigma _{{k_5}}^2$ 0.046 3 − $\sigma _{{k_7}}^2$ − 0.313 3 $\sigma _{{k_1}{k_2}}^2$ − −0.915 0 $\sigma _{{k_1}{k_3}}^2$ − −0.936 0 $\sigma _{{k_1}{k_4}}^2$ 0.105 0 −0.911 0 $\sigma _{{k_1}{k_5}}^2$ −0.580 0 − $\sigma _{{k_1}{k_7}}^2$ − 0.119 0 $\sigma _{{k_2}{k_3}}^2$ − 0.789 0 $\sigma _{{k_2}{k_4}}^2$ − 0.843 0 $\sigma _{{k_2}{k_7}}^2$ − 0.013 0 $\sigma _{{k_3}{k_4}}^2$ − 0.798 0 $\sigma _{{k_3}{k_7}}^2$ − −0.085 0 $\sigma _{{k_4}{k_5}}^2$ −0.791 0 − $\sigma _{{k_4}{k_7}}^2$ − −0.471 0 拟合统计量
Fitting statistics$R_{\rm{a}}^2$ 0.591 6 0.640 2 0.825 7 RMSE 2.990 4 3.120 7 2.171 4 注：***表示在P < 0.001水平上显著，**表示P < 0.01水平上显著，*表示P < 0.05水平上显著。Notes: *** represents significiance at P < 0.001 level; ** represents significiance P < 0.01 level; * represents significiance at P < 0.05 level.
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表 7 最优基础模型与最优混合模型检验结果
Table 7. Validation results of the best basic model and the best mixed-effects model
模型 Model 样本容量 Sample size 平均绝对偏差 MAE 平均相对偏差绝对值 RMAE 预估精度 Fp 基础模型11 Basic model 11 836 4.099 9 39.86 97.376 4 混合模型11.8 Mixed model 11.8 836 2.465 4 20.86 98.553 5
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