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基于beta回归的迎?号杨树树干密度混合效应模垊/p>

吴新卍/a>,苗铮,郝元朓/a>,董利虍/a>

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吴新? 苗铮, 郝元? 董利? 基于beta回归的迎?号杨树树干密度混合效应模型[J]. 北京林业大学学报. doi: 10.12171/j.1000-1522.20220450
引用本文: 吴新? 苗铮, 郝元? 董利? 基于beta回归的迎?号杨树树干密度混合效应模型[J]. 北京林业大学学报.doi:10.12171/j.1000-1522.20220450
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Wu Xinhua, Miao Zheng, Hao Yuanshuo, Dong Lihu. Mixed effect model of stem density of Populus nigra × P. simonii based on beta regression[J]. Journal of Beijing Forestry University. doi: 10.12171/j.1000-1522.20220450
Citation: Wu Xinhua, Miao Zheng, Hao Yuanshuo, Dong Lihu. Mixed effect model of stem density ofPopulus nigra × P. simoniibased on beta regression[J].Journal of Beijing Forestry University.doi:10.12171/j.1000-1522.20220450
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doi:10.12171/j.1000-1522.20220450

  • 东北林业大学林学院，黑龙 哈尔 150040

基金项目:中央高校基本科研业务费专项（2572020DR03），黑龙江头雁创新团队计划项目（森林资源高效培育技术研发团队）
详细信息
    作者简今

    吴新华。主要研究方向：木材密度研究。Email９a href="//www.inggristalk.com/j/article/doi/10.12171/mailto:wuxinhua10@163.com">wuxinhua10@163.com　地址?50040黑龙江省哈尔滨市香坊区和兴路26号东北林业大学林学院

    责任作耄

    董利虎，博士，教授。主要研究方向：林分生长与收获模型、生物量、碳储量。Email９a href="//www.inggristalk.com/j/article/doi/10.12171/mailto:donglihu2006@163.com">donglihu2006@163.com　地址：同三/span>

  • 中图分类叶S781.31；S792.11；S758.1

Mixed effect model of stem density ofPopulus nigra × P. simoniibased on beta regression

摘要
  • 摘要: 目的为了探究迎春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、/sec> 方法以黑龙江省尚志市90株迎?号杨树解析木数据为基础，构建迎?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（ R a 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、/sec> 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回归模型皃i>R a 2分别?.53?.52?.52?.63，MAE < 0.05 g/cm 3，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?.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?.0?.3?.0?.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?号杨树木材密度的显著影响因子、/sec> 结论beta回归模型可对?,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、/sec>

    Abstract: ObjectiveThis 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. MethodsMixed effect beta regression models for sapwood, heartwood, bark and stem density of P. nigra× P. simoniiwere established, which based on the analytical data of 90 trees of P. nigrax× P. simoniiplantation in Shangzhi City, Heilongjiang Province. 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 ?log-likehood value, akaike information criterion, bayesian information criterion, adjusted certainty coefficient ( R a 2) and likelihood ratio test. The leave-one-out-cross-validation was used to test the model, the indexes are mean absolute error (MAE) and mean absolute error percentage. Two sampling methods were combined (scheme ⅟ no relative height, scheme Ⅰ limit relative height below 0.1 to correct the model. ResultsThe density of sapwood, heartwood, bark and stem were not only affected by relative height, but also closely related to the average growth of breast diameter, age and breast diameter, respectively. R a 2of 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/cm 3. 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 increases and then decreases from the base of the stem to the top of the tree, and there is an inflection point at the relative height of 0.6. The stem density increases 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 is limited to 0.1 (2.0 m) or less, there is 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 breast diameter, age and breast diameter were significant influencing factors of wood density of P. nigra× P. simonii. ConclusionThe beta regression model can directly simulate the stem density of P. nigra × P. simoniiin 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.

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  • ?nbsp; 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＋i>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＋i>P< 0.05。Notes:ρsis the sapwood density,ρhis the heartwood density,ρbis the bark wood density,ρwis the the wood density of stem,tis the tree age,His the tree height,Dtis the average growth of diameter at breast height,Dgis the average diameter at breast height of stand,DBHis the average diameter at breast height, andHDis the average height of stand. Blue for negative correlation,and red for positive correlation,P< 0.05.

    Figure 1.Correlation between variables

    下载: 全尺寸图牆/a> 幻灯牆/a>

    ?nbsp; 2边材、心材、树皮和树干密度与变量间的关糺/p>

    横坐??.0分别代表树干基部和树梢。The abscissa numbers 0 and 1.0 stand the base of the stem and the treetops, respectively.

    Figure 2.Relation of sapwood, heartwood, bark and stem density versus variables

    下载: 全尺寸图牆/a> 幻灯牆/a>

    ?nbsp; 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 diameters

    下载: 全尺寸图牆/a> 幻灯牆/a>

    ?nbsp; 1各林场相关的立地因子

    Table 1.Site quality-related factors of each forest farm

    林场 Forest Farm 地形 Terrain 海拔 Altitude/m 坡度 Gradient 坡向 Exposure 坡位 Slope position
    尚志林场 Shangzhi Forest Farm 山坡 Slopes 201 ~ 319 0° ~ 5°?° ~ 15° 阳、阴 Sunny, shady 中下 Half down
    元宝林场 Yuanbao Forest Farm 平地 Flats 206 ~ 245 0° ~ 5° Flat
    老街基林 Laojieji Forest Farm 平地 Flats 300 0° ~ 5° Flat
    苇河林场 Weihe Forest Farm 山坡 Slopes 260 ~ 324 0° ~ 5°?° ~ 10° 阳、阴 Sunny, Shady 中下 Half down
    下载: 导出CSV

    ?nbsp; 2林分因子和解析木因子基本信息统计?/p>

    Table 2.Statistics of stand variables and sampling trees

    因子 Factors 最小 Min. 最大 Max. 平均 Mean SD
    林分因子 Stand factors 林分年龄/a Stand age/year 10 26 18 5
    平均胸径 Average diameter at breast height/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
    株数密度/（株·hm?）Tree density/(tree·ha?) 183 933 572 267
    树木因子 Tree factors 树木年龄/a Tree age/year 10 26 18 5
    胸径 Diameter at breast height/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?) 0.27 0.53 0.36 0.03
    心材密度 Heartwood density/(g·cm?) 0.26 0.50 0.33 0.03
    树皮密度 Bark wood density/(g·cm?) 0.20 0.55 0.37 0.07
    树干密度 Wood density of stem/(g·cm?) 0.27 0.51 0.36 0.03
    下载: 导出CSV

    ?nbsp; 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辂/p>

    Table 3.Fitting statistics test of mixed-effect beta regression model for 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 indexes
    Hr Hr2 Dt t DBH ?lnL AIC BIC Ra2 RL P
    边材 Sapwood 0 ∑/td> ∑/td> ∑/td> ? 988.27 ? 978.27 ? 950.60 0.31
    1 ∑/td> −△ ∑/td> ? 316.71 ? 304.71 ? 289.71 0.50
    2 −△ ∑/td> ∑/td> ? 417.79 ? 405.79 ? 390.79 0.53
    心材 Heartwood 0 ∑/td> ∑/td> ∑/td> ? 012.41 ? 002.41 ? 976.12 0.19
    1 ∑/td> −△ ∑/td> ? 237.85 ? 225.85 ? 210.85 0.41
    2 ∑/td> ∑/td> −△ ? 324.63 ? 312.63 ? 297.63 0.46
    3 −△ ∑/td> ∑/td> ? 291.24 ? 279.24 ? 264.24 0.44
    4 −△ ∑/td> −△ ? 383.05 ? 367.05 ? 347.05 0.52 58.42 < 0.001
    树皮 Bark 0 ∑/td> ∑/td> ∑/td> ? 887.39 ? 877.39 ? 850.11 0.29
    1 −△ ∑/td> ∑/td> ? 069.36 ? 057.36 ? 042.36 0.44
    2 ∑/td> ∑/td> −△ ? 106.94 ? 094.94 ? 079.94 0.46
    3 ∑/td> −△ ∑/td> ? 030.85 ? 018.85 ? 003.85 0.42
    4 −△ −△ ∑/td> ? 163.77 ? 147.77 ? 127.77 0.52 56.83 < 0.001
    树干 Stem 0 ∑/td> ∑/td> ∑/td> ? 377.79 ? 367.79 ? 340.53 0.36
    1 ∑/td> −△ ∑/td> ? 566.35 ? 554.35 ? 539.35 0.50
    2 ∑/td> ∑/td> −△ ? 736.63 ? 724.63 ? 709.63 0.56
    3 −△ ∑/td> ∑/td> ? 651.70 ? 639.70 ? 624.70 0.53
    4 ∑/td> −△ −△ ? 840.39 ? 824.39 ? 804.39 0.63
    5 −△ ∑/td> −△ ? 822.19 ? 806.19 ? 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、i>Hr代表相对高，Hr2代表相对高的平方＋i>Dt代表胸径平均生长量，t代表树龄＋i>DBH代表胸径，−2lnL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，Ra2代表调整确定系数＋i>RL代表似然比。Notes: stands adding random effects to this variable, stands the independent variables of the model fixed effect,Hrstands the relative height,Hr2stands the square of relative height,Dtstands the average growth of diameter at breast height,tstands the tree age,DBHstands diameter at breast height, ?lnLstands the ? log likelihood value, AIC stands the akaike information criterion, BIC stands the bayesian information criterion,Ra2stands the stands the adjusted certainty coefficient,RLstands the likelihood ratio.
    下载: 导出CSV

    ?nbsp; 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.Fixed effect parameter estimates, 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 Fitted parameters $ \;{\beta }_{0} $ ?.543 9***(0.016 3) ?.598 3***(0.028 9) ?.800 0***(0.039 6) ?.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} $ ?.201 2***(0.034 2) ?.082 1**(0.026 1) ?.320 6***(0.111 3) ?.003 6***(0.000 1)
    $ \;{\beta }_{3} $ ?.065 3***(0.012 8) ?.288 7***(0.046 9) ?.005 1*(0.001 9) 0.094 4**(0.031 7)
    随机效应方差协方差结 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} $ ?.478 6 ?.002 1 ?.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?3））中的截距? \;{\beta }_{1} $? \;{\beta }_{2} $? \;{\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 the intercept in the general formula of the fixed effect of the model (equation (13)), $ \;{\beta }_{1} $, $ \;{\beta }_{2} $ and $ \;{\beta }_{3} $ stands the coefficient before the independent variable in the fixed-effect general equation of the model, $ {\sigma }_{1}^{2}\;{\mathrm{a}\mathrm{n}\mathrm{d}\;\sigma }_{2}^{2} $ stands residual variance of model random, $ {\sigma }_{21} $ stands residual covariance of model random effects effects. ***standsP< 0.001, **standsP< 0.01, *standsP< 0.05.
    下载: 导出CSV

    ?nbsp; 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?) MAPE/% MAE/(g·cm?) 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 Barkwood 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.
    下载: 导出CSV

    ?nbsp; 6混合效应模型抽样方案的MAPE检验指标统讠/p>

    Table 6.MAPE test index statistics of sampling plans for mixed-effects model %

    方案−抽样数野br/> Plan-sampling size 边材 Sapwood 心材 Heartwood 树皮 Bark 树干 Stem
    ?0 6.069 1 6.582 3 13.275 2 5.949 2
    ?1 6.056 2 6.599 8 13.345 3 5.964 5
    ?2 6.055 2 6.563 9 13.303 0 5.946 1
    ?3 6.050 6 6.547 5 13.288 3 5.937 7
    ?4 6.046 6 6.536 1 13.272 3 5.933 4
    ?5 6.045 6 6.534 1 13.272 2 5.930 8
    ?6 6.045 4 6.530 6 13.266 7 5.930 1
    ?1 5.968 7?. 0（/td> 6.517 3?.3（/td> 13.232 9?.0（/td> 5.918 2?.0（/td>
    ?2 5.966 0?, 1.0（/td> 6.474 1?, 1.3（/td> 13.232 4?, 2.0（/td> 5.926 1?.0, 1.3（/td>
    ?3 5.968 7?, 1.0, 1.3（/td> 6.400 0?, 1.0, 1.3（/td> 13.238 9?, 2.0, 1.3（/td> 5.949 1?.0, 1.3, 2.0（/td>
    ?4 6.040 4?, 1.0, 1.3, 2.0（/td> 6.670 0?, 1.0, 1.3, 2.0（/td> 13.249 7?, 1.0, 1.3, 2.0（/td> 5.950 8?, 1.0, 1.3, 2.0（/td>
    注：表内括号中的内容代表取样圆盘高度（m）。Note: The content in parentheses in the table stands the height of the sampling disc (m).
    下载: 导出CSV
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