Mixed effect model of stem density of Populus nigra × P. simonii based on beta regression
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摘要:
目的 探究迎春5号杨树在树干纵向上的木材密度影响因子和变异规律，构建迎春5号杨树边材、心材、树皮和树干密度混合效应beta回归模型，为树干生物量预测和木材材性研究提供参考。 方法 以黑龙江省尚志市90株迎春5号杨树解析木数据为基础，构建迎春5号杨树边材、心材、树皮和树干密度的混合效应beta回归模型。采用相关性分析和最优子集法筛选beta回归基础模型的变量；利用负二倍的对数似然值、赤池信息准则、贝叶斯信息准则、调整确定系数（Ra 2）、似然比检验对收敛模型进行拟合优度的评价，利用留一交叉验证法对模型进行检验，指标为平均绝对误差（MAE）和平均绝对百分比误差；结合两种抽样方式（方案Ⅰ：不限定相对高；方案Ⅱ：限定相对高在0.1以下）对模型进行校正。 结果 边材、心材、树皮和树干密度不仅受到相对高的影响，还分别与胸径平均生长量、年龄、胸径密切相关，基于林木因子建立的混合效应beta回归模型的Ra 2分别为0.53、0.52、0.52、0.63，MAE < 0.05 g/cm3，与基础模型相比均提高了预测精度。边材和心材密度从树干基部往上先减小后增大，在相对高0.2处有拐点；树皮密度从树干基部到树梢先增大后减小，在相对高0.6处有拐点；树干密度沿着树干向上逐渐增大。固定相对高时，边材、心材密度都与胸径平均生长量呈负相关，树皮、树干密度分别与年龄、胸径呈负相关。在不限定相对高的情况下，沿着树干随机抽取4个圆盘的密度测量值来校准模型得到稳定的预测精度；限定取样高度在相对高0.1（2.0 m）以下时，对边材、心材、树皮和树干分别抽取一个圆盘（对应高度为1.0、1.3、2.0、1.0 m）的密度测量值，得到与最优抽样组合相似的预测精度。相对高、胸径平均生长量、年龄和胸径是迎春5号杨树木材密度的显著影响因子。 结论 beta回归模型可对（0，1）区间的迎春5号杨树树干密度直接模拟，引入随机效应可提高模型的预测精度。边材、心材、树皮和树干密度在树干纵向上的变化规律不同，构建的混合效应beta回归模型可为迎春5号杨树树干生物量估算和木材性质研究奠定基础。 Abstract:Objective This paper aims to explore the influencing factors and variation rules of wood density in the longitudinal stem of Populus nigra × P. simonii, so beta regression models with mixed effect of sapwood, heartwood, bark and stem density of the poplar were constructed, which was used as a reference for stem biomass prediction and wood timber properties. Method Mixed effect beta regression models for sapwood, heartwood, bark and stem density of P. nigra × P. simonii were established, which based on the analytical data of 90 trees of P. nigrax × P. simonii plantation in Shangzhi City, Heilongjiang Province of northeastern China. Using correlation analysis and optimal subset methods to screen the variables of the beta regression base model, and the goodness of fit of the convergence model was evaluated by −2log-likehood value, akaike information criterion, bayesian information criterion, adjusted certainty coefficient (Ra 2) and likelihood ratio test. The leave-one-out-cross-validation was used to test the model, the indexes were mean absolute error (MAE) and mean absolute error percentage. Two sampling methods were combined (scheme Ⅰ: no relative height; scheme Ⅱ: limit relative height below 0.1) to correct the model. Result The densities of sapwood, heartwood, bark and stem were not only affected by relative height, but also closely related to the average growth of DBH, age and DBH, respectively. Ra 2 of the mixed-effect beta regression model based on tree factors was 0.53, 0.52, 0.52, 0.63, respectively, and the MAE < 0.05 g/cm3. Sapwood density and heartwood density decreased first and then increased from the base to the top of the stem, with an inflection point at a relative height of 0.2. Bark density first increased and then decreased from the base of the stem to the top of the tree, and there was an inflection point at the relative height of 0.6. The stem density increased gradually along the stem. When fixed relative height, the densities of sapwood and heartwood were both negatively correlated with the average growth of DBH. The densities of bark and stem were negatively correlated with age and DBH, respectively. Without limiting the relative height, the wood density value corresponding to the height of 4 discs randomly sampled along the stem was calibrated to obtain stable prediction accuracy. When the sampling height was limited to 0.1 (2.0 m) or less, there was little difference in the prediction accuracy between the optimal sampling combination and the density values (1.0, 1.3, 2.0, 1.0 m, respectively) of sapwood, heartwood, bark and stem at a disc height. Relative height, average growth of DBH, age and DBH were significant influencing factors of wood density of P. nigra × P. simonii. Conclusion The beta regression model can directly simulate the stem density of P. nigra × P. simonii in the (0, 1) interval, and the random effect can improve the prediction accuracy of the model. The longitudinal variations of sapwood, heartwood, bark and stem density are different. The constructed mixed-effect beta regression model can lay a foundation for biomass estimation and wood property study of P. nigra × P. simonii. -
图 1 变量间的相关关系
ρs为边材密度，ρh为心材密度，ρb为树皮密度，ρw为树干密度，t为树龄，H为树高，Dt为胸径平均生长量，Dg为林分平均胸径，DBH为胸径，HD为林分平均高。蓝色代表负相关，红色代表正相关，P < 0.05。ρs is sapwood density, ρh is heartwood density, ρb is bark density, ρw is stem density, t is tree age, H is tree height, Dt is the average growth of DBH, Dg is the average DBH of stand, DBH is the diameter at breast height, and HD is the average height of stand. Blue represents negative correlation, and red represents positive correlation, P < 0.05.
Figure 1. Correlations among variables
图 2 边材、心材、树皮和树干密度与变量间的关系
横坐标0和1.0分别代表树干基部和树梢。Dt和t的单位分别为cm/a和a；The abscissa numbers 0 and 1.0 stand for the base of stem and the treetop, respectively. The units of Dt and t are cm/year and year.
Figure 2. Relationship of sapwood, heartwood, bark and stem density with variables
图 3 不同径阶的边材、心材、树皮和树干密度的平均绝对百分比误差
Figure 3. Mean absolute percentage error of sapwood, heartwood, bark and stem density of different diameter classes
表 1 各林场相关的立地因子
Table 1. Site factors related to each forest farm
林场 Forest farm 地形 Terrain 海拔 Altitude/m 坡度 Slope 坡向 Aspect 坡位 Slope position 尚志林场 Shangzhi Forest Farm 山坡 Hillside 201 ~ 319 0° ~ 5°、5° ~ 15° 阳、阴 Sunny, shady 中下 Lower-middle 元宝林场 Yuanbao Forest Farm 平地 Flat 206 ~ 245 0° ~ 5° 平 Flat 老街基林场 Laojieji Forest Farm 平地 Flat 300 0° ~ 5° 平 Flat 苇河林场 Weihe Forest Farm 山坡 Hillside 260 ~ 324 0° ~ 5°、5° ~ 10° 阳、阴 Sunny, shady 中下 Lower-middle 
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表 2 林分因子和解析木因子基本信息统计表
Table 2. Statistical table for basic information of stand factors and analytical wood factors
因子 Factor 最小值 Min. value 最大值 Max. value 平均值 Mean SD 林分因子
Stand factor林分年龄/a Stand age/year 10 26 18 5 平均胸径 Average DBH/cm 13.3 30.6 20.6 5.3 平均树高 Average tree height/m 11.9 26.2 20.3 4.1 林分密度/（株·hm−2） Stand density/(tree·ha−1) 183 933 572 267 树木因子
Tree factor树木年龄/a Tree age/year 10 26 18 5 胸径 DBH/cm 6.2 37.5 21.6 6.4 树高 Tree height/m 8.9 28.9 21.1 4.2 边材密度 Sapwood density/(g·cm−3) 0.27 0.53 0.36 0.03 心材密度 Heartwood density/(g·cm−3) 0.26 0.50 0.33 0.03 树皮密度 Bark density/(g·cm−3) 0.20 0.55 0.37 0.07 树干密度 Stem density/(g·cm−3) 0.27 0.51 0.36 0.03
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表 3 边材、心材、树皮和树干密度混合效应beta回归模型的拟合优度比较
Table 3. Goodness of fit comparison of beta regression models for mixed effects of sapwood, heartwood, bark and stem density
部位 Part 模型编号
Model No.随机效应 Random effect 评价指标 Evaluation index Hr Hr 2 Dt t DBH −2ln L AIC BIC Ra 2 RL P 边材
Sapwood0 − − − −7 988.27 −7 978.27 −7 950.60 0.31 1 − −△ − −8 316.71 −8 304.71 −8 289.71 0.50 2 −△ − − −8 417.79 −8 405.79 −8 390.79 0.53 心材
Heartwood0 − − − −6 012.41 −6 002.41 −5 976.12 0.19 1 − −△ − −6 237.85 −6 225.85 −6 210.85 0.41 2 − − −△ −6 324.63 −6 312.63 −6 297.63 0.46 3 −△ − − −6 291.24 −6 279.24 −6 264.24 0.44 4 −△ − −△ −6 383.05 −6 367.05 −6 347.05 0.52 58.42 < 0.001 树皮
Bark0 − − − −4 887.39 −4 877.39 −4 850.11 0.29 1 −△ − − −5 069.36 −5 057.36 −5 042.36 0.44 2 − − −△ −5 106.94 −5 094.94 −5 079.94 0.46 3 − −△ − −5 030.85 −5 018.85 −5 003.85 0.42 4 −△ −△ − −5 163.77 −5 147.77 −5 127.77 0.52 56.83 < 0.001 树干
Stem0 − − − −7 377.79 −7 367.79 −7 340.53 0.36 1 − −△ − −7 566.35 −7 554.35 −7 539.35 0.50 2 − − −△ −7 736.63 −7 724.63 −7 709.63 0.56 3 −△ − − −7 651.70 −7 639.70 −7 624.70 0.53 4 − −△ −△ −7 840.39 −7 824.39 −7 804.39 0.63 5 −△ − −△ −7 822.19 −7 806.19 −7 786.19 0.62 103.76 < 0.001 注：△代表在该变量上添加随机效应，−代表模型固定效应的自变量。Hr代表相对高，Hr 2代表相对高的平方，Dt代表胸径平均生长量，t代表树龄，DBH代表胸径，−2ln L代表负二倍的对数似然值，AIC代表赤池信息准则，BIC代表贝叶斯信息准则，Ra 2代表调整确定系数，RL代表似然比。Notes: △ stands for adding random effects to this variable, − stands for the independent variables of the model fixed effect, Hr stands for the relative height, Hr 2 stands for the square of relative height, Dt stands for the average growth of DBH, t stands for the tree age, DBH stands for diameter at breast height, −2ln L stands for the −2 log likelihood value, AIC stands for the akaike information criterion, BIC stands for the Bayesian information criterion, Ra 2 stands for the adjusted determination coefficient, RL stands for the likelihood ratio.
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表 4 边材、心材、树皮和树干密度最优混合效应模型的固定效应参数估计值、随机效应方差协方差结构
Table 4. Estimated values of fixed effect parameters, random effect variance covariance structure for optimal mixed-effects model of sapwood, heartwood, bark and stem density
参数 Parameter 边材 Sapwood 心材 Heartwood 树皮 Bark 树干 Stem 固定效应参数估计值
Estimated value of fixed effect parameter$ \;{\beta }_{0} $ −0.543 9***(0.016 3) −0.598 3***(0.028 9) −0.800 0***(0.039 6) −0.611 5***(0.025 9) $ \;{\beta }_{1} $ 0.471 8***(0.032 1) 0.709 0***(0.059 2) 1.678 4***(0.099 3) 0.241 6***(0.037 6) $ \;{\beta }_{2} $ −0.201 2***(0.034 2) −0.082 1**(0.026 1) −1.320 6***(0.111 3) −0.003 6***(0.000 1) $ \;{\beta }_{3} $ −0.065 3***(0.012 8) −0.288 7***(0.046 9) −0.005 1*(0.001 9) 0.094 4**(0.031 7) 随机效应方差协方差结构
Random effect variance covariance structure$ {\sigma }_{1}^{2} $ 0.016 1 0.423 4 0.003 3 0.020 7 $ {\sigma }_{2}^{2} $ 0.601 8 0.018 8 0.000 0 $ {\sigma }_{21} $ −0.478 6 −0.002 1 −0.000 3 模型方差 Model variance 0.010 4 0.010 0 0.044 3 0.008 8 $ {\mathrm{注}：\beta }_{0}\mathrm{代}\mathrm{表} $模型固定效应通式（式（13））中的截距，$ \;{\beta }_{1} $、$ \;{\beta }_{2} $和$ \;{\beta }_{3} $代表模型固定效应通式中自变量前的系数，$ {\sigma }_{1}^{2}{\mathrm{和}\sigma }_{2}^{2} $代表模型随机效应的残差方差，$ {\sigma }_{21} $代表模型随机效应的残差协方差，***代表P < 0.001，**代表P < 0.01，*代表P < 0.05。Notes: $ \;{\beta }_{0} $ stands for the intercept in the general formula of the fixed effect of the model (equation (13)); $ \;{\beta }_{1} $, $ \;{\beta }_{2} $ and $ \;{\beta }_{3} $ represent the coefficients before the independent variables in the fixed effects general formula of the model; $ {\sigma }_{1}^{2}\;{\mathrm{a}\mathrm{n}\mathrm{d}\;\sigma }_{2}^{2} $ stand for residual variance of model random effect, $ {\sigma }_{21} $ stands for residual covariance of model random effects. *** stands for P < 0.001, ** stands for P < 0.01, * stands for P < 0.05.
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表 5 基础模型与混合效应模型检验指标的比较
Table 5. Comparison of test indexes between base models and mixed effect models
部位 Part 基础模型 Base model 混合效应模型 Mixed effect model MAE/(g·cm−3) MAPE/% MAE/(g·cm−3) MAPE/% 边材 Sapwood 0.022 1 6.069 1 0.018 8 5.155 1 心材 Heartwood 0.022 3 6.584 6 0.019 6 5.728 1 树皮 Bark 0.046 7 13.271 2 0.041 0 11.751 0 树干 Stem 0.021 5 5.953 3 0.017 0 4.692 0 注：MAE为平均绝对误差，MAPE为平均绝对百分比误差。Notes：MAE is mean absolute error, and MAPE is mean absolute percentage error.
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表 6 混合效应模型抽样方案的MAPE检验指标统计
Table 6. MAPE test index statistics of sampling plans for mixed-effect model
% 方案−抽样数量
Plan-sampling size边材 Sapwood 心材 Heartwood 树皮 Bark 树干 Stem Ⅰ-0 6.069 1 6.582 3 13.275 2 5.949 2 Ⅰ-1 6.056 2 6.599 8 13.345 3 5.964 5 Ⅰ-2 6.055 2 6.563 9 13.303 0 5.946 1 Ⅰ-3 6.050 6 6.547 5 13.288 3 5.937 7 Ⅰ-4 6.046 6 6.536 1 13.272 3 5.933 4 Ⅰ-5 6.045 6 6.534 1 13.272 2 5.930 8 Ⅰ-6 6.045 4 6.530 6 13.266 7 5.930 1 Ⅱ-1 5.968 7（1.0） 6.517 3（1.3） 13.232 9（2.0） 5.918 2（1.0） Ⅱ-2 5.966 0（0, 1.0） 6.474 1（0, 1.3） 13.232 4（0, 2.0） 5.926 1（1.0, 1.3） Ⅱ-3 5.968 7（0, 1.0, 1.3） 6.400 0（0, 1.0, 1.3） 13.238 9（0, 2.0, 1.3） 5.949 1（1.0, 1.3, 2.0） Ⅱ-4 6.040 4（0, 1.0, 1.3, 2.0） 6.670 0（0, 1.0, 1.3, 2.0） 13.249 7（0, 1.0, 1.3, 2.0） 5.950 8（0, 1.0, 1.3, 2.0） 注：表内括号中的内容代表取样圆盘高度（m）。Note: content in parenth stands for the height of the sampling disc (m).
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