林业科学  2015, Vol. 51 Issue (2): 28-36 PDF
DOI: 10.11707/j.1001-7488.20150204
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#### 文章信息

Dong Lihu, Zhang Lianjun, Li Fengri

Error Structure and Additivity of Individual Tree Biomass Model

Scientia Silvae Sinicae, 2015, 51(2): 28-36.
DOI: 10.11707/j.1001-7488.20150204

### 作者相关文章

1. 东北林业大学林学院 哈尔滨 150040；
2. 美国纽约州立大学环境科学和林业学院 锡拉丘兹 13210

Error Structure and Additivity of Individual Tree Biomass Model
Dong Lihu1, Zhang Lianjun2, Li Fengri1
1. Forestry College, Northeast Forestry University Harbin 150040;
2. State University of New York, College of Environmental Science and Forestry Syracuse 13210
Abstract: [Objective]Forest biomass is a basic quantity character of the forest ecological system. Biomass data are the foundation of researching many forestry and ecology problems. Therefore, accurate quantification of biomass is critical for calculating carbon storage, as well as for studying climate change, forest health, forest productivity and nutrient cycling, etc. Directly measuring the actual weight of each component (i.e., stem, branch, foliage and root) is undoubtedly the most accurate method, but it is destructive, time consuming, and costly. Thus, developing biomass models is regarded as a better approach to estimating forest biomass. However, some issues are needed to take care when constructing and applying biomass models, such as: 1) some reported biomass models may not hold the additivity or compatibility among tree component models; 2) which model error structure is appropriate for biomass data, i.e., additive error structure versus multiplicative error structure; 3) few models are available for tree belowground (root) biomass. Researchers have been continuously working and debating on these issues over the last decades. Development of the additive system of biomass equations were reported in the literature. However, how to evaluate the model error structure of the biomass equation in forestry have not been well investigated so far. The present paper mainly deals with two parts: evaluating error structure of the biomass model and developing the additive system of biomass equations.[Method]The P. simonii×P. nigra plantation in the west of Heilongjiang Province of China is selected to ensure error structure by likelihood analysis. Nonlinear seemly unrelated regression (NSUR) of SAS/ETS module is used to estimate the parameters in the additive system of biomass equations. The biomass model validation is accomplished by Jackknifing technique.[Result]The multiplicative error structure was favored for the total and component biomass equations for P. simonii×P. nigra plantation by a likelihood analysis, and the additive system of log-transformed biomass equations should be applied. Overall, the Ra2 of all biomass models was between 0.92 and 0.99. The mean relative error and mean absolute relative error were smaller for most biomass models. All models for total and component biomass had the good prediction precision (85 % or more). The effect of total tree, aboveground and stem biomass models are better than root, branch, foliage and crown biomass models. Overall, all models for total and component biomass could be a good predict of the P. simonii×P. nigra biomass.[Conclusion]Although the significance of likelihood analysis is proposed by several studies, it has not been widely applied in forestry. When total biomass is divided into aboveground and belowground biomass, aboveround biomass is divided into stem and crown biomass, crown biomass is divided into branch and foliage biomass, and stem biomass is divided into bark and wood biomass, the additivity of total and component biomass should be taken into account. Overall, the error structure and additivity of biomass models are the two key issues, and should be taken into account when biomass models are constructed. If the two issues are well solved, the constructed biomass models will be more effective.
Key words: P. simonii×P. nigra plantation    error structure    likelihood analyses    additivity    jackknifing technique

1 数据调查与收集

2009年，在黑龙江省西部的甘南县和杜尔伯特蒙古族自治县小黑杨人工林内选取14，21，23，25，27，29年生6个年龄段的样地各3块，样地间立地质量差异不大，样地面积均为30 m×30 m，进行每木检尺。将每木测定的结果按径阶2 cm统计分组，分组后按等断面积径级标准木法将林木分为5级，计算各径级的平均直径，并以此为标准在每个年龄段选择5株不同大小的林木作为样木，此外再选取1株优势木进行解析，即每年龄段共选6株样木，全部共36株样木。伐倒选取的样木，按1 m区分段，测定各区分段树干的鲜质量，将树冠分为3层，每层选取3～5个标准枝称其枝、叶的鲜质量。每株样木各区分段的树干和每层的枝、叶都分别取样，带回室内测其含水率。用全挖法采集树根，测定大根(>5 cm)、粗根(2～5 cm)、细根(≤2cm)的鲜质量，并分别取样品测其含水率[由于小于5 mm树根很难获取(Wang，2006)，所以本研究中树根不包括此部分]。所采集样品带回室内在80 ℃下烘干至恒质量，根据样品鲜质量和干质量分别推算样木各部分干质量，并汇总得到树冠生物量、地上生物量和总生物量，其样木信息及生物量统计见表 1

2 研究方法 2.1 幂函数误差结构分析

 $Y\text{=}a{{X}^{b}}+\varepsilon \text{, }\varepsilon \tilde{\ }N(0\text{,}{{\sigma }^{2}});$ (1)

 $\ln Y=\ln a+b\ln X+\varepsilon \text{, }\varepsilon \text{ }\!\!\tilde{\ }\!\!\text{ }~N(0\text{,}{{\sigma }^{2}}).$ (2a)

 $Y=a{{X}^{b}}{{e}^{\varepsilon }}\text{, }\varepsilon \tilde{\ }N(0\text{,}{{\sigma }^{2}}).$ (2b)

Xiao等(2011)提出用似然分析法检验幂函数的误差结构。在该方法中，赤池信息量准则(AICc)被用来衡量一个统计模型的拟合优度。对于一个幂函数关系的数据，可以计算出原始数据的非线性回归与对数转换的线性回归的似然值和AICc值。AICc值可以通过以下协定规则进行比较：如果|ΔAICc|(2个模型的AICc不同)小于2，则2个模型没有明显区别；否则，拥有较小AICc的模型被认为有更好的数据支持。为了更好地运用似然分析法判断模型的误差结构，Xiao等(2011)给出了使用该方法的步骤：

1)首先，分别用非线性回归[式(1)]和线性回归[式(2a)]拟合数据，估计出每个模型的参数abσ2；然后，用以下2个公式分别计算相加型和相乘型误差结构幂函数的似然值：

 ${L_{{\rm{norm}}}} = \prod\limits_{i = 1}^n {\left({\frac{1}{{\sqrt {2\pi \sigma _{{\rm{NLR}}}^2} }}{e^{\frac{{ - {{[{y_i} -({a_{{\rm{NLR}}}}X_{i{\rm{NLR}}}^b)]}^2}}}{{2\sigma _{{\rm{NLR}}}^2}}}}} \right)} ;$ (3)
 ${L_{\lg n}} = \prod\limits_{i = 1}^n {\left({\frac{1}{{{y_i}\sqrt {2\pi \sigma _{{\rm{LR}}}^2} }}{e^{\frac{{ - {{[lg{y_i} - \lg({a_{{\rm{LR}}}}X_{i{\rm{LR}}}^b)]}^2}}}{{2\sigma _{{\rm{LR}}}^2}}}}} \right)} .$ (4)

 ${\rm{AICc}} = 2k - 2\lg L + \frac{{2k(k + 1)}}{{N - k - 1}}.$ (5)

2)如果AICcnorm - AICclgn＜-2，则幂函数的误差项是相加的，模型应该用非线性回归进行拟合；如果AICcnorm - AICclgn> 2，则幂函数的误差项是相乘的，模型应该用对数转换的线性回归进行拟合；如果|AICcnorm - AICclgn| ≤ 2，则2种误差结构的假设都不合适，此时模型求平均值可能是最好的办法。

2.2 可加性模型

1)假定误差结构是相加的，非线性可加性生物量模型如下：

 $\left\{ {\begin{array}{*{20}{l}} {{W_i} = {a_i} \cdot {D^{{b_i}}} + {\varepsilon _i}}\\ {{W_{\rm{c}}} = {W_{\rm{b}}} + {W_{\rm{f}}} + {\varepsilon _{\rm{c}}}}\\ {{W_{\rm{a}}} = {W_{\rm{s}}} + {W_{\rm{b}}} + {W_{\rm{f}}} + {\varepsilon _{\rm{a}}}}\\ {{W_{\rm{t}}} = {W_{\rm{r}}} + {W_{\rm{s}}} + {W_{\rm{b}}} + {W_{\rm{f}}} + {\varepsilon _{\rm{t}}}} \end{array}} \right.$ (6)

2)假定误差结构是相乘的，将地下、树干、树枝和树叶生物量模型进行对数转换，而地上、树冠和总生物量没有被线性化。对数转换的可加性生物量模型如下：

 $\left\{ {\begin{array}{*{20}{l}} {\ln {W_i} = \ln {a_i} + {b_i}\ln D + {\varepsilon _i} = a_i^ * + b_i^ * \ln D + {\varepsilon _i}}\\ {\ln {W_{\rm{c}}} = \ln({W_{\rm{b}}} + {W_{\rm{f}}})+ {\varepsilon _{\rm{c}}}}\\ {\ln {W_{\rm{a}}} = \ln({W_{\rm{s}}} + {W_{\rm{b}}} + {W_{\rm{f}}})+ {\varepsilon _{\rm{a}}}}\\ {\ln {W_{\rm{t}}} = \ln({W_{\rm{r}}} + {W_{\rm{s}}} + {W_{\rm{b}}} + {W_{\rm{f}}})+ {\varepsilon _{\rm{t}}}} \end{array}} \right.$ (7)

2.3 模型评价

$R_{\rm{a}}^2 = 1 -(1 - {R^2})\left({\frac{{N - 1}}{{N - p}}} \right)，$

 ${R^2} = 1 - \frac{{\sum\limits_{i = 1}^N {{{({Y_i} - {{\hat Y}_i})}^2}} }}{{\sum\limits_{i = 1}^N {{{({Y_i} - \bar Y)}^2}} }}.$ (8)

 ${\rm{RMSE}} = \sqrt {{\rm{MSE}}} = \sqrt {\frac{{\sum\limits_{i = 1}^N {{{({Y_i} - {{\hat Y}_i})}^2}} }}{{N - p}}} .$ (9)

“刀切法”残差：

 ${e_{i，- i}} =({Y_i} - {\hat Y_{i，- i}}).$ (10)

 ${\rm{ME}}\% = \frac{{\sum\limits_{i = 1}^N {\left({\frac{{{e_{i，- i}}}}{{\bar Y}}} \right)\times 100} }}{N}.$ (11)

 ${\rm{MAE}}\% = \frac{{\sum\limits_{i = 1}^N {\left({\frac{{|{e_{i，- i}}|}}{{{Y_i}}}} \right)\times 100} }}{N}.$ (12)

 ${\rm{MPE}}\% = \frac{{{t_\alpha }\sqrt {\frac{{e_{i，- i}^2}}{{N - p}}} }}{{\bar Y\sqrt N }} \times 100.$ (13)

3 结果与分析 3.1 生物量模型误差结构

 图 1 人工林小黑杨树枝生物量对数转换数据的观测值和预测值(A)及原始数据的观测值和预测值(B) Fig. 1 The observed data of branch biomass and model predictions on log-transformed scale(A), original scale(B) 实线代表对数转换回归[LR, 式(2a)]的预测值，虚线为非线性回归[NLR, 式(1)]的预测值。 The solid lines represent the model predictions by log-transformed model[LR, eq(2a)], and the dashed lines represents the model predictions by nonlinear or power model[NLR, eq(1)].
3.2 模型拟合与检验

 图 2 人工小黑杨总生物量、地上和地下生物量模型残差 Fig. 2 Residuals of total, aboveground and belowground biomass models of P. simonii×P. nigra plantation A. 总生物量 Total; B. 地上Aboveground; C. 地下 Belowground.

4 结论与讨论 4.1 生物量模型误差结构

4.2 可加性生物量模型