Research on the nonlinear mixed effect model of height-DBH of climatic sensitive Cunninghamia lanceolata
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摘要:
目的 建立基于林分优势高和气候因子的杉木树高−胸径非线性混合效应模型，为杉木生长研究和经营管理提供理论依据。 方法 基于2020年国家森林资源年度监测评价广西壮族自治区试点25块杉木样地的每木胸径和树高实测数据及样地位置对应的气候数据，选择7个常用的树高−胸径模型，筛选出模拟精度最高的模型作为基础模型，再引入代表林分竞争、立地条件和气候因子的变量构建广义非线性模型，并在此基础上，加入样地效应构建杉木非线性混合效应模型。最后，运用十折交叉验证法对3种模型进行检验。 结果 Chapman-Richards模型为最佳杉木树高−胸径关系基础模型，林分优势高、林分断面积和年平均降水量与树高生长显著相关，用于构建广义非线性模型，对比分析确定随机参数为3个的组合构造非线性混合效应模型。基础模型、广义非线性模型、非线性混合效应模型的调整决定系数分别为0.674 2、0.797 3和0.857 3，平均绝对误差分别为1.607 5、1.270 1和1.010 6 m，均方根误差分别为2.032 1、1.632 1和1.338 4 m，相对均方根误差分别为20.796 4%、16.703 3%和13.697 3%，混合效应模型呈现出更好的拟合效果。 结论 引入林分优势高和气候因子的杉木树高−胸径非线性混合效应模型可以较好地描述杉木树高胸径曲线，适用于大范围的树高预测。 Abstract:Objective The nonlinear mixed-effects tree height-DBH model of Cunninghamia lanceolata based on stand dominant height and climate factors is established, which provides theoretical basis for the research on growth and forest management. Method Based on the annual monitoring and evaluation of national forest resources of Guangxi Zhuang Autonomous Region, southern China in 2020, this study used the data of DBH and height of each tree, climate data of 25 Cunninghamia lanceolata sample plots, chose the basic model with the highest simulation accuracy among seven common height-DBH models. On this basis, stand competition, site condition and climatic factors were used to build generalized nonlinear model, then used the sample plot effect to build nonlinear mixed-effects model. The 10-fold cross-validation method was applied to the test of three models. Result Chapman-Richards model was the basic height-DBH model with the highest accuracy. The stand dominant height, basal area of forest stands and the mean annual precipitation was significantly related to the tree height growth, which were used to build generalized nonlinear model. Through comparative analysis, the study selected three random parameters to build nonlinear mixed-effects model. The adjustment determination coefficient of basic model, generalized nonlinear model and nonlinear mixed-effects model were 0.674 2, 0.797 3 and 0.857 3, respectively, the mean absolute errors were 1.607 5, 1.270 1 and 1.010 6 m, the root-mean-square errors were 2.032 1, 1.632 1 and 1.338 4 m, and the relative root mean square errors were 20.796 4%, 16.703 3% and 13.697 3%, respectively. The nonlinear mixed-effects model showed the best fitting effect. Conclusion Using nonlinear mixed-effects tree height-DBH model based on stand dominant height and climatic factors can better describe the height-DBH curve of Cunninghamia lanceolata, which is suitable for the prediction of tree height on a large scale. -
表 1 样地因子和气候因子统计
Table 1. Statistical table of survey factors and climatic variables in sample plots
变量 Variable 变量符号
Variable symbol最小值
Min. value最大值
Max. value平均值
Mean标准差
SD胸径
Diameter at breast height/cmD 5.0 40.9 11.7 5.1 树高
Tree height/mH 3.0 21.8 9.8 3.6 林分断面积/（m2·hm−2）
Stand basal area/(m2·ha−1)SBA 5.61 52.88 28.11 12.49 林分密度/（株·hm−2）
Stand density/(tree·ha−1)NT 375 5 640 2 521 1 354 平均年龄/a
Average age/yearA 4 43 14 8 林分优势高
Stand dominant height/mHdom 9.8 21.0 14.7 2.4 年平均气温
Annual mean temperature/℃MAT 13.50 23.50 18.28 2.25 最热月平均气温
Mean temperature of the warmest month/℃MWMT 23.50 29.50 27.06 1.66 最冷月平均气温
Mean temperature of the coldest month/℃MCMT −1.10 17.40 5.34 4.14 平均温差
Mean temperature difference/℃TD 8.80 21.50 18.26 3.20 年平均降水量
Annual mean precipitation/mmMAP 486.00 2 083.00 1 592.84 374.27 湿热指数
HumidexAHM 11.40 62.00 20.43 11.02 0 ℃以下天数
Days below 0 ℃DD_0 0 25 5 8 5 ℃以上天数
Days above 5 ℃DD5 3 271 6 548 4 840 752 18 ℃以下天数
Days below 18 ℃DD_18 116 2 131 1 126 489 18 ℃以上天数
Days above 18 ℃DD18 482 1 990 1 221 358 
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表 2 树高−胸径候选模型
Table 2. Candidate models of tree height-DBH
模型名称
Model name模型公式
Model formula公式序号
Equation No.Curtis $H = 1.3 + {a_1}{\left[ {D/\left( {1 + D} \right)} \right]^{{a_2}}} + \varepsilon $ （1） Meyer $ H = 1.3 + {a_1}\left[ {1 - \exp \left( { - {a_2}D} \right)} \right] + \varepsilon $ （2） Wykoff $ H = 1.3 + \exp \left[ {{a_1} + {a_2}/(D + 1)} \right] + \varepsilon $ （3） Chapman-Richards $H = 1.3 + {a_1}{\left[ {1 - \exp \left( { - {a_2}D} \right)} \right]^{{a_3}}} + \varepsilon $ （4） Logistic $H = 1.3 + {a_1}/\left[ {1 + {a_2}\exp \left( { - {a_3}D} \right)} \right] + \varepsilon $ （5） Weibull $ H = 1.3 + {a_1}\left[ {1 - \exp \left( { - {a_2}{D^{{a_3}}}} \right)} \right] + \varepsilon $ （6） Gompertz $H = 1.3 + {a_1}\exp \left[ { - {a_2}\exp \left( { - {a_3}D} \right)} \right] + \varepsilon $ （7） 注：a1、a2、a3为模型系数，ε为误差项。Notes: a1, a2, a3 are the model coefficients, ε is the error term.
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表 3 基础模型拟合结果
Table 3. Fitting results of basic models
模型名称
Model name参数估计值 Estimate 平均绝对误差
MAE/m均方根误差
RMSE/m相对均方根误差
rRMSE/%调整决定系数
R2 adja1 a2 a3 Curtis 21.552 3*** 10.449 5*** 1.632 2 2.051 7 20.996 1 0.668 3 Meyer 22.296 5*** 0.043 3*** 1.715 5 2.102 9 21.521 4 0.573 0 Wykoff 3.106 3*** −11.399 5*** 1.634 6 2.052 2 21.002 4 0.664 3 Richards 15.478 3*** 0.124 7*** 1.988 9*** 1.607 5 2.032 1 20.796 4 0.674 2 Logistic 14.274 7*** 8.671 5*** 0.231 9*** 1.608 4 2.039 2 20.868 8 0.668 1 Weibull 14.798 3*** 0.022 9*** 1.516 9*** 1.612 9 2.039 7 20.874 3 0.673 1 Gompertz 14.954 7*** 3.171 4*** 0.158 7*** 1.612 4 2.039 2 20.869 3 0.671 3 注：***为表现显著（P < 0.001）。Notes: *** means significant at the P < 0.001 level.
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表 4 广义非线性模型参数估计结果
Table 4. Results of parameter estimation of generalized nonlinear models
参数
Parameter估计值
Estimate标准差
SDt P a1 6.226 0 0.452 2 13.766 < 0.000 1 a2 0.085 6 0.005 2 16.372 < 0.000 1 a3 1.507 0 0.086 0 17.514 < 0.000 1 a4 0.570 5 0.025 8 22.152 < 0.000 1 a5 0.001 1 0.000 1 13.432 < 0.000 1 a6 0.000 1 0.000 0 2.803 0.005 1
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表 5 混合效应模型随机参数组合的部分拟合结果
Table 5. Partial results of combinations of random parameters for mixed-effects model
模型
Model随机参数
Random parameterAIC BIC Loglik LRT P 广义非线性模型
Generalized nonlinear model无 None 11 841.2 11 883.5 −5 913.63 模型1 Model 1 a2 10 904.4 10 952.7 −5 444.20 模型2 Model 2 a2、a3 10 814.9 10 875.3 −5 397.49 93.41 < 0.000 1 模型3 Model 3 a2、a3、a6 10 811.8 10 890.3 −5 392.92 9.14 0.027 5 注：模型2的LRT和P值为模型2与1相比较而得；模型3的LRT和P值为模型3与2相比较而得。Notes: the LRT and P in the line of model 2 are caululated by model 2 and 1. The LRT and P in the line of model 3 are caululated by model 3 and 2.
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表 6 基于不同异方差函数的非线性混合效应模型比较
Table 6. Comparison of nonlinear mixed-effects models based on different variance functions
模型
Model异方差函数 Heteroscedasticity function AIC BIC Loglik LRT P 模型3
Model 3无 None 10 811.8 10 890.3 −5 392.92 模型3.1
Model 3.1常数加幂函数 ConstPower function 10 760.2 10 850.8 −5 365.12 55.60 < 0.000 1 模型3.2
Model 3.2幂函数 Power function 10 758.2 10 842.7 −5 365.12 55.60 < 0.000 1 模型3.3
Model 3.3指数函数 Exponent function 10 782.0 10 866.5 −5 377.00 31.84 < 0.000 1 注：LRT和P值为模型与模型3相比较而得。Note: LRT and P are caululated by model and model 3.
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表 7 不同模型比较
Table 7. Comparison of different models
模型 Model AIC BIC Loglik MAE/m RMSE/m rRMSE/% R2 adj 基础模型
Basic model13 223.7 13 247.8 −6 607.84 1.607 5 2.032 1 20.796 4 0.674 2 广义非线性模型 Generalized nonlinear model 11 841.2 11 883.5 −5 913.63 1.270 1 1.632 1 16.703 3 0.797 3 混合效应模型
Mixed-effects model1 0758.2 10 842.7 −5365.12 1.010 6 1.338 4 13.697 3 0.857 3
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