Modeling a single-tree biomass equation by seemingly unrelated regression and dummy variables with <i>Larix kaempferi</i>

SHEN Jiapeng; CHEN Dongsheng; SUN Xiaomei; ZHANG Shougong

doi:10.11833/j.issn.2095-0756.2019.05.005

Volume 36 Issue 5

Sep. 2019

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SHEN Jiapeng, CHEN Dongsheng, SUN Xiaomei, ZHANG Shougong. Modeling a single-tree biomass equation by seemingly unrelated regression and dummy variables with Larix kaempferi[J]. Journal of Zhejiang A&F University, 2019, 36(5): 877-885. doi: 10.11833/j.issn.2095-0756.2019.05.005

Citation:

SHEN Jiapeng, CHEN Dongsheng, SUN Xiaomei, ZHANG Shougong. Modeling a single-tree biomass equation by seemingly unrelated regression and dummy variables with Larix kaempferi[J]. Journal of Zhejiang A&F University, 2019, 36(5): 877-885. doi: 10.11833/j.issn.2095-0756.2019.05.005

Modeling a single-tree biomass equation by seemingly unrelated regression and dummy variables with Larix kaempferi

doi: 10.11833/j.issn.2095-0756.2019.05.005

Key Laboratory of Tree Breeding and Cultivation of National Forestry and Grassland Administration, Research Institute of Forestry, Chinese Academy of Forestry, Beijing 100091, China

Received Date: 2018-09-10
Rev Recd Date: 2018-11-26
Publish Date: 2019-10-20

Abstract

Developing generalized single-tree biomass models suitable for forest biomass estimation is an effective way to provide scientific approaches. To simplify biomass modeling and improve the accuracy of model estimation for better understanding of biomass, carbon stocks, and dynamics in large-scale forests and to precisely estimate tree biomass, this study used stem, bark, needle, branch, and root biomass of Larix kaempferi of 161 sample trees in Gansu, Hubei, and Liaoning Provinces to generalize single-tree biomass equations suitable for different organs and regions using seemingly unrelated regression and dummy variable modeling methods. Results showed that the generalized biomass equations not only solved compatibility problems with different components but also increased accuracy with an average increase of 0.28%-0.44% in R², decreased 0.40%-6.61% in the root mean square error (E_RMS), and decreased 1.63%-6.61% in the mean abosolute bias (B_MA). Effects due to region increased accuracy more than effects due to developmental stages. When both region and developmental stages were added to the dummy variable model, it was more accurate and produced the best equation. Therefore, we suggest that both regional and developmental stages be considered as dummy variables to establish generalized biomass equations in order to solve the compatibility problem with different components as well as for overcoming problems of generalizing with different regions.
- forest mensuration,
- Larix kaempferi,
- seeming unrelated regression,
- dummy variable,
- generalized equations

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森林是陆地生态系统的主要组成部分，其生物量约占陆地植被的85%~90%，在平衡全球碳循环、减缓全球气候变暖等方面发挥着重要作用^[1-6]。因此，各个国家越来越重视对森林生物量的研究，许多学者先后建立了适用不同尺度范围的单木生物量方程。JENKINS等^[7]建立了美国国家尺度不同树种的通用性生物量方程。随后，荷兰、法国、芬兰等国家也分别建立了区域尺度森林生物量方程。CASE等^[8]对加拿大10个树种生物量方程进行比较后发现，随着生物量方程通用性水平的提高将导致预估误差增大。同时，以往通用性生物量研究大多对林木不同组分器官生物量（树干、树枝、树叶、树根等）分别单独建立模型进行生物量的估算，从而造成林木总生物量与各组分器官生物量的不相容问题。近些年来，已有学者通过线性或非线性联合估计^[9]、比例平差法^[10]和似乎不相关回归^[11-12]等方法来解决总生物量与不同组分器官生物量的相容性问题。因此，在建立大尺度范围通用性单木生物量模型的同时，兼顾林木总生物量与各组分器官生物量的相容性，是亟须关注的问题。日本落叶松Larix kaempferi具有速生、丰产、适应性强等优点，在中国人工林生产建设中得到了广泛的应用。许多研究者已从不同区域或不同组分器官等方面建立了相关的生物量方程^[13-17]。但这些研究仅单独考虑了生物量方程的区域尺度通用性或不同组分器官生物量的相容性，很少同时考虑兼顾生物量方程的通用性和相容性。因此，本研究以日本落叶松人工林为例，基于似乎不相关回归和哑变量法构建既可反应不同组分器官间生物量相容性又可反映不同区域通用性的单木生物量方程，以期为区域尺度生物量的精确估算提供科学的参考。

1. 材料与方法

1.1. 研究区概况

本研究在甘肃省天水市、湖北省建始县、辽宁省清原县3个不同地理分布区分别设置日本落叶松人工林样地。分布区具体概况如下：甘肃省天水市小陇山林区，地处33°30′~34°49′N，104°22′~106°43′E，是中国暖温带南缘，气候温暖湿润，大多数地域属暖温带—中温带半湿润大陆性季风气候类型，海拔为700~3 330 m，年平均气温为7.0~12.0 ℃，年平均降水量为460.0~800.0 mm，无霜期为140.0~218.0 d。湖北省建始县长岭岗林场，地处30°48′N，110°03′E，该区属北亚热带季风性山地温润气候，海拔为1 600~1 900 m，年平均气温11.0~16.0 ℃，无霜期200.0~300.0 d，年平均降水量1 400.0~1 800.0 mm。辽宁省清原县大孤家林场，地处42°22′~44°16′N，124°47′~125°12′E，属中温带东亚大陆性季风气候，年平均气温为5.4~7.2 ℃，年平均降水量400.0~800.0 mm，无霜期125.0~150.0 d。

1.2. 数据收集

本研究于2012-2013年进行试验数据采集，综合考虑区域、龄组等影响因素，在甘肃、湖北、辽宁3个省份分别设置幼龄林（20年生）、中龄林（20~30年生）、近熟林（30~40年生）3个龄组的日本落叶松样地109块（甘肃33块、湖北40块、辽宁36块），每个样地面积0.08 hm²（28.3 m × 28.3 m）。对样地进行每木检尺，并根据检尺数据结果，选择161株解析木（优势木、平均木和劣势木）进行生物量测定（甘肃55株、湖北46株、辽宁60株）。将每株解析木伐倒，采用Monsic分层切割法测定解析木的树干、树皮、树枝、树叶的鲜质量，地下生物量采用全根挖掘法，分小根（≤2 cm），粗根（2~5 cm），根茎（≥5 cm）分别称取鲜质量。各器官分别取样，并将样品放置烘箱烘干至恒量，计算各样品含水率，通过换算得到各器官组分的生物量。本研究将161株解析木数据进行随机抽样，其中建模数据样本量为116株，检验数据样本量为45株。数据基本统计信息见表 1。

数据		胸径/cm	树高/m	树干/kg	树皮/kg	树叶/kg	树枝/kg	树根/kg	地上部分/kg	单木/kg
建模数据	最小	1.10	2.30	0.10	0.05	0.08	0.08	0.06	0.32	0.39
	最大	28.40	27.65	304.90	36.23	9.30	25.79	8.49	361.41	445.86
	平均	14.92	15.24	72.94	8.86	3.20	9.14	20.81	94.07	115.06
	标准误	0.66	0.60	6.95	0.73	0.22	0.66	1.86	8.35	10.15
检验数据	最小	1.90	2.72	0.24	0.0	0.20	0.27	0.28	0.81	1.09
	最大	28.60	27.40	340.36	31.50	8.87	22.87	71.52	399.41	464.05
	平均	14.68	15.14	73.38	8.59	3.03	8.68	18.10	93.78	111.85
	标准误	1.05	1.00	12.50	1.21	0.33	0.94	2.58	14.69	17.08

Table 1. Summary statistics of sample tree of Larix kaempferi for fitting and validation by provinces and developmental stages

1.3. 研究方法

1.3.1. 基础模型

本研究采用y=a×D_b作为日本落叶松生物量基础模型。目前，常采用非线性回归和对数转换的线性回归拟合上述生物量方程，其依据是模型方程的误差结构，因此，采用XIAO等^[18]提出的似然分析法，即比较方程的赤池信息准则（AICc）并判断方程的误差结构。赤池信息量准则（AICc）计算公式如下：

式（1）中：A为赤池信息准则，k为模型的参数个数，N为样本数，lnL是相加型误差项和相乘型误差项的对数似然值。通过计算非线回归得到的赤池信息量准则（A_norm）和对数转换的线性回归得到的赤池信息量准则（A_ln）进行对比，比较两者之间的差值来确定模型误差结构，主要有以下3种情况：（1）当A_norm-A_ln＞2时，模型误差项是相乘的，应采用对数转换的线性方程；（2）当A_norm-A_ln＜-2时，模型误差项是相加的，应采用非线性方程；（3）当∣A_norm-A_ln∣≤2时，则2种模型的误差假设都不合适，应采用模型平均值作为最优解。分别采用假设方程误差结构为相加型的非线性回归和假设方程误差结构为相乘型的对数转换的线性回归拟合日本落叶松不同组分及单株总生物量数据，并得到了2种方程误差结构模型赤池信息量准则的差值（表 2）。结果表明：日本落叶松人工林不同组分生物量方程模型赤池信息量准则的差值都大于2。因此，日本落叶松不同组分和单株总生物量模型的误差结构均是相乘型的，采用对数转换的线性回归模型更合适。

省份	A_norm-A_ln
省份	树干	树皮	树叶	树枝	树根	地上部分	单株
甘肃	390.28	161.09	11.50	186.76	230.69	411.71	420.31
湖北	403.90	162.45	106.12	203.88	277.24	432.24	452.87
辽宁	544.83	292.03	142.89	248.13	403.79	592.16	616.52

Table 2. Information statistics of likelihood analysis for the additive and multiplicative error structure of biomass models of L. kaempferi

1.3.2. 相容性生物量模型构建

为解决各组分生物量之和等于总生物量，需建立相容性生物量模型。采用似乎不相关回归法建立了相乘结构误差项的相容性模型，即方程组1：

方程组1中：y₁，y₂，…，y₇分别代表树干、树皮、树叶、树枝、树根、地上生物量、单株总体生物量，D为胸径，a₁₀，a₂₀，…，a₅₀及b₁，b₂，…，b₅分别为树干、树皮、树叶、树枝、树根生物量模型的参数。

1.3.3. 通用性兼顾相容性生物量模型构建

在相容性生物量模型的基础上，以甘肃、湖北、辽宁3个省份的3个发育阶段（幼龄林、中龄林、近熟林）日本落叶松人工林为例，通过引入哑变量来建立适合不同区域、不同发育阶段的通用性单木生物量模型。单独考虑区域作为哑变量时，z₁=1，z₂=0表示研究区域是甘肃省；z₁=0，z₂=1表示研究区域是湖北省；z₁=0，z₂=0表示研究区域是辽宁省。据此构建方程组2：

方程组2中：y₁，y₂，…，y₇；D；a₁₀，a₂₀，…，a₅₀；b₁，b₂，…，b₅参照方程组1的参数；a₁₁，a₂₁，…，a₅₁；a₁₂，a₂₂，…，a₅₂分别为哑变量z₁和z₂的特定参数。单独考虑发育阶段作为哑变量时，z₃=1，z₄=0表示幼龄林；z₃=0，z₄=1表示中龄林；z₃=0，z₄=0表示近熟林；据此构建方程组3：

方程组3中：y₁，y₂，…，y₇；D；a₁₀，a₂₀，…，a₅₀；b₁，b₂，…，b₅参照方程组1参数；a₁₃，a₂₃，…，a₅₃；a₁₄，a₂₄，…，a₅₄分别为哑变量z₃和z₄的特定参数。同时考虑区域和发育阶段作为哑变量时，z₁=1，z₂=0，z₃=1，z₄=0为甘肃省幼龄林; z₁=1，z₂=0，z₃=0，z₄=1为甘肃省中龄林; z₁=1，z₂=0，z₃=0，z₄=0为甘肃省近熟林; z₁=0，z₂=1，z₃=1，z₄=0为湖北省幼龄林; z₁=0，z₂=1，z₃=0，z₄=1为湖北省中龄林; z₁=0，z₂=1，z₃=0，z₄=0为湖北省近熟林; z₁=0，z₂=0，z₃=1，z₄=0为辽宁省幼龄林; z₁=0，z₂=0，z₃=0，z₄=1为辽宁省中龄林; z₁=0，z₂=0，z₃=0，z₄=0为辽宁省近熟林。据此构建方程组4：

方程组4中：y₁，y₂，…，y₇；D；a₁₀，a₂₀，…，a₅₀；b₁，b₂，…，b₅参照方程组1参数；a₁₁，a₂₁，…，a₅₁；a₁₂，a₂₂，…，a₅₂分别为哑变量z₁和z₂的特定参数，a₁₃，a₂₃，…，a₅₃；a₁₄，a₂₄，…，a₅₄分别为哑变量z₃和z₄的特定参数。

1.4. 模型拟合与评价

本研究采用R软件中“systemfit”软件包对方程组1、方程组2、方程组3和方程组4进行拟合。模型拟合和检验效果均采用复相关系数（R²）、均方根误差（E_RMS）和绝对偏差（B_MA）来评价。

式（2）~（4）中：y_i是观测值，${\hat y_l}$是预测值，y_i是观测平均值，n是样本数。

2. 结果与分析

2.1. 单木通用性生物量模型参数

日本落叶松人工林单木生物量方程组1、方程组2、方程组3和方程组4的参数估计值均具有统计学意义（P＜0.01）（表 3）。通过参数值大小发现，方程组1、方程组2、方程组3和方程组4不同组分的参数b为1.44~2.62，平均值为2.06。其中，树枝、树叶、树皮、树根和树干参数b的变异系数分别为8.75%，7.25%，5.03%，3.25%和2.29%，变异幅度都小于10%。结果表明：日本落叶松单木生物量模型中参数b具有相对稳定性。在区域分布方面上，通过对比区域参数值（a₁和a₂）大小表明：相同胸径条件下，辽宁地区树干、树皮、树叶生物量最大，而湖北地区树根生物量最大。不同发育阶段方面参数值（a₃和a₄）大小表明：相同胸径条件下，幼龄林树枝、树叶生物量最大，中龄林树根生物量最大，而近熟林树皮生物量最大（表 3）。同时引入区域参数值（a₁和a₂）和发育阶段参数值（a₃和a₄）2个哑变量构建的生物量通用模型，反映了不同区域、不同发育阶段相同胸径日本落叶松的生物量大小差异，相对于单独引入区域或发育阶段作为哑变量构建的生物量模型其可以反映的生物量信息更全面。

方程组	组分	参数
方程组	组分	a₀	b	a₁	a₂	a₃	a₄
	树干	0.037 8	2.627 0
	树皮	0.022 9	2.097 0
方程组1	树叶	0.055 6	1.441 2
	树枝	0.066 3	1.751 1
	树根	0.025 3	2.352 9
	树干	0.052 4	2.583 0	-0.012 0	-0.016 7
	树皮	0.023 5	2.117 3	-0.000 9	-0.004 9
方程组2	树叶	0.050 7	1.494 7	-0.003 0	-0.008 1
	树枝	0.049 7	1.822 9	0.009 0	0.004 7
	树根	0.025 1	2.341 4	-0.000 4	0.002 7
	树干	0.063 7	2.467 8			-0.016 6	0.001 0
	树皮	0.043 6	1.916 7			-0.013 4	-0.003 2
方程组3	树叶	0.025 9	1.655 7			0.014 7	0.002 5
	树枝	0.024 9	2.039 9			0.019 0	0.001 0
	树根	0.032 7	2.260 6			-0.004 6	0.001 6
	树干	0.072 8	2.469 0	-0.012 5	-0.020 6.	-0.014 9	0.002 3
	树皮	0.041 7	1.949 7	-0.002 0	-0.007 1	-0.011 0	-0.001 6
方程组4	树叶	0.025 2	1.665 5	0.001 7	-0.001 4	0.012 6	0.002 2
	树枝	0.017 4	2.103 1	0.007 5	0.005 1	0.014 7	-0.000 4
	树根	0.040 3	2.194 3	-0.001 6	0.004 6	-0.007 9	0.000 7

Table 3. Parameter estimates of different models of Larix kaempferi

2.2. 单木通用性生物量模型拟合效果评价

由表 4可知：地上生物量和总生物量的拟合效果最好，而树叶和树枝生物量的拟合结果最差，其R²均低于0.90，E_RMS大于0.40，B_MA均大于0.29。这可能是由于在外业中枝叶取样困难，容易产生取样误差造成的。与普通模型（方程组1）相比，单独引入区域哑变量（方程组2），R²平均增加0.40%，E_RMS平均减小0.40%，B_MA平均减小5.95%。单独引入发育阶段哑变量（方程组3），R²平均增加0.28%，E_RMS平均减小1.92%，B_MA平均减小1.63%。同时引入区域和发育阶段哑变量（方程组4），R²平均增加0.44%，E_RMS平均减小6.61%，B_MA平均减小6.94%（表 4）。无论单独考虑区域因素还是林分的发育阶段均会提高生物量模型的预测精度，但区域因素的加入对模型预估精度的提高效果优于发育阶段，表明区域因素（环境条件）对树木生物量的影响更显著，两者同时考虑的方程组4对生物量模型的预估精度提高最大，是最优的日本落叶松单木生物量模型（表 4）。通过残差分析发现：最优模型的残差分布随胸径无明显的变化规律，也进一步证实引入区域和发育阶段的哑变量方程组4有更好的拟合效果。

变量	方程组	树干	树皮	树叶	树枝	树根	地上部分	单株
R²	方程组1	0.9622	0.946 0	0.802 8	0.879 6	0.961 4	0.968 9	0.976 4
	方程组2	0.965 1	0.949 2	0.810 8	0.877 2	0.961 8	0.977 0	0.981 7
	方程组3	0.962 3	0.950 2	0.805 3	0.887 0	0.962 6	0.970 3	0.977 2
	方程组4	0.966 7	0.953 8	0.809 7	0.887 9	0.963 9	0.975 9	0.980 8
E_RMS	方程组1	0.331 6	0.318 7	0.454 6	0.412 4	0.300 1	0.265 0	0.230 1
	方程组2	0.318 5	0.309 2	0.445 3	0.416 5	0.298 7	0.227 9	0.202 8
	方程组3	0.330 9	0.306 1	0.451 8	0.399 5	0.295 4	0.259 0	0.226 2
	方程组4	0.311 1	0.294 9	0.446 5	0.389 2	0.290 3	0.232 9	0.207 5
B_MA	方程组1	0.268 3	0.246 7	0.350 9	0.309 5	0.237 6	0.208 2	0.184 4
	方程组2	0.250 2	0.237 6	0.347 7	0.306 4	0.238 4	0.177 7	0.156 7
	方程组3	0.264 5	0.228 2	0.362 8	0.306 4	0.237 3	0.204 3	0.179 1
	方程组4	0.245 7	0.219 6	0.358 5	0.298 0	0.239 3	0.177 7	0.159 1

Table 4. Comparison and evaluation of the fitting index of different models of Larix kaempferi

2.3. 单木通用性生物量模型验证

根据随机抽取的45株日本落叶松单木检验数据对生物量模型进行验证，结果见表 5。当引入区域或发育阶段哑变量后，生物量模型的E_RMS和B_MA总体呈现降低的趋势，检验数据的结果与模型拟合数据的结果基本一致。同时，检验数据的统计指标也表明同时引入区域和发育阶段的哑变量模型（方程组4）是最优模型。

	方程组	树干	树皮	树叶	树枝	树根	地上部分	单株
E_RMS	方程组1	0.353 6	0.278 8	0.463 8	0.435 5	0.432 8	0.257 7	0.220 9
	方程组2	0.357 0	0.267 9	0.472 2	0.441 5	0.418 3	0.267 2	0.239 2
	方程组3	0.324 8	0.272 5	0.453 3	0.438 3	0.442 1	0.226 4	0.199 9
	方程组4	0.307 4	0.244 1	0.456 6	0.415 5	0.422 2	0.215 0	0.198 9
B_MA	方程组1	0.297 6	0.211 9	0.389 0	0.366 3	0.371 3	0.224 6	0.185 4
	方程组2	0.283 2	0.215 7	0.392 6	0.367 2	0.353 5	0.220 1	0.195 0
	方程组3	0.271 4	0.202 5	0.395 4	0.354 8	0.364 3	0.194 8	0.165 7
	方程组4	0.247 6	0.196 0	0.393 6	0.355 8	0.353 1	0.181 0	0.162 7

Table 5. Comparison and evaluation of the validation index of different models of L. kaempferi

3. 结论与讨论

本研究以甘肃、湖北、辽宁3个省份日本落叶松单木各器官（树干、树皮、树叶、树枝、树根）生物量数据为研究对象，建立了适合不同区域、不同发育阶段的日本落叶松单木生物量模型。其中，关于生物量模型尺度参数b仍存在一定的争论。有研究认为：生物量模型中参数b约为2.67，且不受树种、立地等条件的影响^[19]。而ZIANIS等^[20]对全球范围的279个生物量异速生长方程研究后得到参数b平均值为2.36，认为不同的环境条件会影响参数b值的变化，一般为2~3。本研究中日本落叶松生物量方程中参数b平均值仅为2.06，低于以上研究结果，因此树种的差异会影响参数b值。不同组分间树枝、树叶、树皮、树根和树干生物量模型中参数b值的变异系数分别为8.75%，7.25%，5.03%，3.25%和2.29%，表明不同器官间参数b值存在一定的差异，但树根和树干的b值相对稳定，可采用固定参数b值计算其生物量。

不同地理区域间降水、温度等因素差异会影响区域尺度范围森林生物量的分配模式^[21-24]，因此，引入相关因素构建生物量方程以提高模型的精度是森林生物量模型的发展趋势。如KETTERINGS等^[25]发现：与普通模型相比，引入区域因素可以提高森林生物量模型的预估精度。BAO等^[22]建立了越南区域尺度生物量模型有效地降低了生物量估算的不确定性。同时，国内学者也证实：引入区域因素可以有效地改善生物量模型的预估效果^{[21, 26-27]}。本研究采用哑变量的方法引入区域因素时发现，相比于普通模型（方程组1）R²平均增加了0.40%，E_RMS平均减小了0.40%，B_MA平均减小了5.95%，也证实了加入区域因素会提高生物量模型的预估精度。同时，不同发育阶段林木对水分、营养等吸收利用存在较大差异，也会影响着生物量的积累与分配规律^[28-29]。在本研究中当引入发育阶段时，相比于普通模型R²平均增加了0.28%，E_RMS平均减小了1.92%，B_MA平均减小了1.63%，表明了发育阶段对生物量的积累分配有显著影响，加入发育阶段因素会提高生物量预估精度。由于区域和发育阶段2个因素对生物量模型的预估精度均有改善，因此本研究同时考虑将区域和发育阶段作为哑变量构建了生物量模型（方程组4），最终拟合效果比单独引入1个因素的哑变量模型的预估精度均高，该方程组成为日本落叶松最优生物量模型。

大多数森林生物量研究仅单一考虑区域之间的通用性^{[21, 30]}或不同组分间生物量的相容性^[10-12]，且以往的相容性生物量方程主要集中在小尺度范围（单个区域或省区）^[31-33]，而同时考虑大尺度通用性兼顾相容性生物量模型的研究相对较少。而本研究同时考虑将区域和发育阶段作为哑变量构建的生物量模型可以有效地解决通用性和相容性的问题。但在实际应用中，由于树枝、树叶和树根生物量的测定工作量较大，且耗时耗力，加之由于林分结构复杂或其他干扰等因素可能会导致测量数据任务繁重、存在较大误差等问题^[15]。因此，本研究提供了一种灵活可行的生物量预测方法，如果偏重方程的适用性，建议使用仅考虑不同区域或发育阶段的生物量模型（方程组2或方程组3），如果偏重方程的预测精度，建议使用同时考虑地域和发育阶段的生物量模型（方程组4）。

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Modeling a single-tree biomass equation by seemingly unrelated regression and dummy variables with Larix kaempferi

doi: 10.11833/j.issn.2095-0756.2019.05.005