基于Copula密度函数的毛竹胸径年龄结构特征二元联合分布

doi:10.11707/j.1001-7488.20211110

摘要/Abstract

摘要：

目的: 针对森林结构特征因子常用二元联合分布研究方法存在的缺陷，选择适用条件低、适应范围广、应用价值大的二元Copula分布（密度）函数构建毛竹胸径年龄二元联合分布模型，为精确测量森林结构特征因子联合分布提供参考。方法: 选取5种常用二元Copula密度函数、二元S_bb密度函数和二元Weibull分布函数对2009年浙江省177块毛竹连续清查固定样地的胸径年龄二元联合分布进行描述，对比分析各函数的测量精度；经二维频数直方图和AIC评定，选择胸径年龄最优二元Copula密度函数，并在此基础上构建浙江省毛竹胸径年龄二元Copula联合分布模型，采用柯尔莫哥洛夫检验各分布（密度）函数的拟合优度。结果: 二元Weibull分布函数的确定系数（R²）=0.990 1，二元S_bb密度函数的R²=0.736 2，二元Gumbel Copula密度函数的R²=0.984 1、AIC=19.519 6，为5种常用二元Copula密度函数中AIC最小的函数；二元Gumbel Copula密度函数、二元Weibull分布函数和二元S_bb密度函数累计值的最大偏差分别为0.015 8、0.007 0和0.078 1，在0.05显著性水平下的阈值为0.179 8。结论: 二元Gumbel Copula密度函数为测量毛竹胸径年龄联合分布的最优Copula密度函数；二元Weibull分布函数的测量精度最高，但参数比二元Copula密度分布（密度）函数多，迭代参数没有二元Copula密度（分布）函数易收敛，二元S_bb密度函数的测量精度最低；毛竹胸径年龄联合分布均服从3种分布；二元Copula分布（密度）函数适合于胸径年龄的任意边缘分布，即不需要确定边缘分布函数类型；二元Copula分布（密度）函数参数拟合不需要胸径年龄联合分布数据，只有胸径年龄边缘分布值时，应用二元Copula分布（密度）函数可得到相应的胸径年龄联合密度值，二元Copula分布（密度）函数比常用二元分布（密度）函数的适用条件低、适应范围广、应用价值大。

关键词: 毛竹, 二元分布, Copula密度函数, 生物量, 胸径年龄结构特征

Abstract:

Objective: In view of the shortcomings of the bivariate joint distribution function commonly used in the investigation of forest structure characteristic factors, a bivariate distribution(density)function with a low applicable condition, wide adaptation range and great application value was selected to provide a reference for accurately measuring the joint distribution of forest structure characteristic factors. Method: Five commonly used bivariate Copula density functions, bivariate S_bb functions and bivariate Weibull distribution functions, were selected to describe the bivariate joint density of DBH and age of moso bamboo(Phyllostachys edulis) based on 177 continuous inventory plots in Zhejiang Province in 2009, and the measurement accuracy of each function was compared and analyzed. The optimal Copula function of DBH and age was selected based on the metrics of bivariate frequency histogram and AIC. The bivariate Copula joint density model of DBH and age for moso bamboo in Zhejiang Province was established. The goodness of fit of the model was tested using Kolmogorov test. Result: The coefficient of determination(R²) of bivariate Weibull distribution function and the bivariate S_bb function was 0.990 1 and 0.736 2, respectively, and the R² of bivariate Gumbel Copula density function was 0.984 1 with the lowest AIC value of -19.519 6. The maximum deviations of the cumulative value of the bivariate Gumbel Copula function, the bivariate Weibull distribution function and the bivariate S_bb function were 0.007 0, 0.015 8 and 0.078 1. The critical significance value was 0.179 8. Conclusion: Bivariate Gumbel Copula probability density function is the best Copula function for representing the joint distribution of moso bamboo DBH and age. The measurement accuracy of bivariate Weibull distribution function is the highest, but the number of the parameters used is more than that of those used in the bivariate Copula function. Therefore, iterative parameter of bivariate Copula function is easier to be converged. The accuracy of bivariate S_bb function is the lowest. The joint distribution of DBH and age of moso bamboo all obey these three distribution functions. Bivariate Copula function is suitable for arbitrary edge distribution of DBH and age, that is, it does not need to determine the class of marginal distribution function. When there are only marginal distribution values of DBH and age, the joint density value of DBH and age could be obtained using the bivariate Copula function. Thus, the bivariate Copula function has a wider applicability and a higher application value than the commonly used binary distribution function.

Key words: moso bamboo, bivariate distribution, Copula density function, biomass, structure characteristics of DBH and age

中图分类号:

S758

刘恩斌,姚鸿文,任泽茜,周国模,杜华强. 基于Copula密度函数的毛竹胸径年龄结构特征二元联合分布[J]. 林业科学, 2021, 57(11): 94-104.

Enbin Liu,Hongwen Yao,Zexi Ren,Guomo Zhou,Huaqiang Du. Bivariate Joint Distribution of DBH and Age of Moso Bamboo Based on Copula Density Function[J]. Scientia Silvae Sinicae, 2021, 57(11): 94-104.

图/表 8

表1

图1

图2

表2

实测概率以及二元Gumbel Copula(GumCp)密度函数、二元Weibull分布函数和二元Sbb密度函数估计值"

胸径 DBH/cm	1度1du				2度2du
胸径 DBH/cm	实测概率 Measured probability	GumCp估计值 GumCp estimatedvalues	S_bb估计值 S_bb estimatedvalues	Weibull估计值 Weibull estimated values	实测概率 Measured probability	GumCp估计值 GumCp estimated values	S_bb估计值 S_bb estimated values	Weibull估计值 Weibull estimated values
5	0.014 1	0.016 6	0.017 2	0.015 2	0.031 5	0.029 6	0.016 1	0.031 2
6	0.015 1	0.017 5	0.031 0	0.015 5	0.034 2	0.031 9	0.029 0	0.031 0
7	0.022 9	0.024 1	0.041 6	0.022 8	0.048 2	0.045 2	0.039 1	0.044 7
8	0.028 0	0.027 9	0.046 2	0.028 4	0.055 5	0.053 7	0.043 4	0.055 2
9	0.028 7	0.027 7	0.044 2	0.029 5	0.055 1	0.054 8	0.041 6	0.057 1
10	0.028 0	0.024 9	0.036 9	0.025 0	0.048 8	0.050 8	0.034 8	0.048 4
11	0.016 2	0.014 2	0.026 7	0.016 6	0.026 1	0.029 4	0.025 2	0.032 6
12	0.009 7	0.007 3	0.016 3	0.008 3	0.013 9	0.015 4	0.015 4	0.016 7
13	0.002 9	0.002 3	0.007 9	0.003 0	0.004 6	0.004 8	0.007 4	0.006 2
14	0.000 7	0.000 4	0.002 6	0.000 7	0.001 0	0.000 9	0.002 5	0.001 6
15	0	0	0.000 4	1E-04	0	0.000 1	0.000 4	0.000 3

胸径 DBH/cm	3度3du				4度4du
胸径 DBH/cm	实测概率 Measured probability	GumCp估计值 GumCp estimated values	S_bb估计值 S_bb estimated values	Weibull估计值 Weibull estimated values	实测概率 Measured probability	GumCp估计值 GumCp estimated values	S_bb估计值 S_bb estimated values	Weibull估计值 Weibull estimated values
5	0.032 1	0.028 4	0.014 9	0.032 2	0.012 3	0.011 0	0.013 3	0.014 3
6	0.032 4	0.031 2	0.026 9	0.032 0	0.014 4	0.012 1	0.024 1	0.014 2
7	0.045 8	0.045 2	0.036 3	0.046 4	0.018 3	0.017 8	0.032 6	0.020 5
8	0.056 7	0.055 6	0.040 4	0.057 0	0.021 1	0.022 2	0.036 4	0.025 3
9	0.057 5	0.059 4	0.038 8	0.059 1	0.024 6	0.024 6	0.034 9	0.026 2
10	0.053 2	0.059 2	0.032 4	0.050 1	0.026 6	0.026 9	0.029 3	0.022 2
11	0.034 2	0.036 9	0.023 5	0.033 7	0.017 5	0.020 3	0.021 3	0.014 9
12	0.019 0	0.020 4	0.014 4	0.017 3	0.009 9	0.016 1	0.013 1	0.007 7
13	0.006 4	0.006 5	0.007 0	0.006 5	0.004 1	0.006 9	0.006 3	0.002 9
14	0.001 5	0.001 3	0.002 3	0.001 7	0.000 7	0.001 5	0.002 1	0.000 7
15	0.000 2	0.000 1	0.000 4	0.000 3	0.000 1	0.000 1	0.000 3	1E-04

胸径 DBH/cm	5度5du
胸径 DBH/cm	实测概率 Measured probability	GumCp估计值 GumCp estimated values	S_bb估计值 S_bb estimated values	Weibull估计值 Weibull estimated values
5	0.001 5	0.001 5	0.001 4	0.002 3
6	0.002 2	0.001 7	0.002 6	0.002 2
7	0.004 0	0.002 5	0.003 6	0.003 3
8	0.004 2	0.003 1	0.004 2	0.004 0
9	0.004 2	0.003 5	0.004 1	0.004 0
10	0.004 6	0.003 9	0.003 6	0.003 5
11	0.002 9	0.003 2	0.002 7	0.002 4
12	0.001 6	0.003 8	0.001 7	0.001 2
13	0.000 5	0.005 2	0.000 8	0.000 4
14	0.000 3	0.007 0	0.000 3	0.000 2
15	0	0.000 6	0.000 1	0

表2

图3

图4

图5

图6

参考文献 0

	范叶青, 周国模, 施拥军, 等. 地形条件对毛竹林分结构和植被碳储量的影响. 林业科学, 2013, 49 (11): 177- 182.
	Fan Y Q , Zhou G M , Shi Y J , et al. Effects of terrain on stand structure and vegetation carbon storage of Phyllostachys edulis forest. Scientia Silvae Sinicae, 2013, 49 (11): 177- 182.
	方晰, 田大伦, 项文化. 速生阶段杉木人工林碳素密度、贮量和分布. 林业科学, 2002, 38 (3): 14- 19.
	Fang X , Tian D L , Xiang W H . Density, storage and distribution of carbon in Chinese fir plantation at fast growing stage. Scientia Silvae Sinicae, 2002, 38 (3): 14- 19.
	葛宏立, 周国模, 刘恩斌, 等. 浙江省毛竹直径与年龄的二元Weibull分布模型. 林业科学, 2008, 44 (12): 15- 20.
	Ge H L , Zhou G M , Liu E B , et al. Diameter-age bivariate Weibull distribution model for moso bamboo forests in Zhejiang Province. Scientia Silvae Sinicae, 2008, 44 (12): 15- 20.
	李意德, 吴仲民, 曾庆波, 等. 尖峰岭热带山地雨林生态系统碳平衡的初步研究. 生态学报, 1998, 18 (4): 371- 378.
	Li Y D , Wu Z M , Zeng Q B , et al. Carbon pool and carbon dioxide dynamics of tropical mountain rain forest ecosystem at Jianfengling, Hainan Island. Acta Ecologica Sinica, 1998, 18 (4): 371- 378.
	刘恩斌, 周国模, 施拥军, 等. 测树因子二元概率分布——以毛竹为例. 林业科学, 2010, 46 (10): 29- 36. doi: 10.11707/j.1001-7488.20101005
	Liu E B , Zhou G M , Shi Y J , et al. Bivariate theoretic probability distribution of forest mensuration: a case of moso bamboo forest. Scientia Silvae Sinicae, 2010, 46 (10): 29- 36. doi: 10.11707/j.1001-7488.20101005
	阮宏华, 姜志林, 高苏铭. 苏南丘陵主要森林类型碳循环研究—含量与分布规律. 生态学杂志, 1997, 16 (6): 17- 21. doi: 10.3321/j.issn:1000-4890.1997.06.004
	Ruan H H , Jiang Z L , Gao S M . Preliminary studies of carbon cycling in three types of forests in the hilly regions of southern Jiangsu Province. Chinese Journal of Ecology, 1997, 6 (6): 17- 21. doi: 10.3321/j.issn:1000-4890.1997.06.004
	王岩, 隋思涟, 王爱青. 数理统计与MATLAB工程数据分析. 北京: 清华大学出版社, 2006.
	Wang Y , Sui S L , Wang A Q . Mathematical statistics and MATLAB engineering data analysis. Beijing: Tsinghua University Press, 2006.
	谢华, 罗强, 黄介生. 基于三维Copula函数的多水文区丰枯遭遇分析. 水科学进展, 2012, 23 (2): 186- 193.
	Xie H , Luo Q , Huang J S . Synchronous asynchronous encounter analysis of multiple hydrologic regions based on 3D Copula function. Advances in Water Science, 2012, 23 (2): 186- 193.
	张蕾. 基于Copula函数的岩土体抗剪强度参数二维分布模型研究. 武汉: 武汉大学, 2017,
	Zhang L . Bivariate distribution of shear strength parameters of soils and rocks using Copula. Wuhan: Wuhan University, 2017,
	张尧庭. 连接函数(Copula)技术与金融风险分析. 统计研究, 2002, (4): 48- 51. doi: 10.3969/j.issn.1002-4565.2002.04.011
	Zhang Y T . Connection function technology(Copula)and financial risk analysis. Statistical Research, 2002, (4): 48- 51. doi: 10.3969/j.issn.1002-4565.2002.04.011
	周国模. 毛竹林生态系统中碳储量、固定及其分配与分布的研究. 杭州: 浙江大学, 2006,
	Zhou G M . Carbon storage, fixation and distribution in mao bamboo stands ecosystem. Hangzhou: Zhejiang University, 2006,
	周国模, 姜培坤. 毛竹林的碳密度、碳贮量及其空间分布. 林业科学, 2004, 40 (6): 5- 11.
	Zhou G M , Jiang P K . Density, storage and spatial distribution of carbon in Phyllostachy pubescens forest. Scientia Silvae Sinicae, 2004, 40 (6): 5- 11.
	Cardil Forradellas A , Molina Terrén D M , Oliveres J , et al. Comparing Johnson's SBB, Weibull and Logit-Logistic bivariate distributions for modeling tree diameters and heights using Copulas. Forest Systems, 2016, 25 (1): 1- 5.
	Eskelson B N , Madsen L , Hagar J C , et al. Estimating riparian understory vegetation cover with beta regression and Copula models. Forest Science, 2011, 57 (3): 212- 221.
	Fang G , Pan R , Hong Y . Copula-based reliability analysis of degrading systems with dependent failures. Reliability Engineering & System Safety, 2020, 193, 106618.
	Gorgoso-Varela J J , Garcia-Villabrille J D , Rojo-Alboreca A , et al. Comparing Johnson's SBB, Weibull and Logit-Logistic bivariate distributions for modeling tree diameters and heights using Copulas. Forest Systems, 2016, 25 (1): 1- 5.
	Gorgoso-Varela J J , Ogana F N , Ponce R A . Evaluation of direct and indirect methods for modelling the joint distribution of tree diameter and height data with the bivariate Johnson's SBB function to forest stands. Forest Systems, 2019, 28 (1): 6.
	Hafley W L, Schreuder H T. 1976. Some non-normal bivariate distributions and their potential for forest application. International Union Forest Research Organizations, XVI World Congress Proceedings, Div. Ⅵ: 104-114.
	Kershaw Jr J A , Richards E W , McCarter J B , et al. Spatially correlated forest stand structures: a simulation approach using Copulas. Computers and Electronics in Agriculture, 2010, 74 (1): 120- 128. doi: 10.1016/j.compag.2010.07.005
	Li F , Zhang L , Davis C J . Modeling the joint distribution of tree diameters and heights by bivariate generalized Beta distribution. Forest Science, 2002, 48 (1): 47- 58.
	Mønness E . The bivariate power-normal distribution and the bivariate Johnson system bounded distribution in forestry, including height curves. Canadlian Journal of Forest Research, 2015, 45 (3): 307- 313. doi: 10.1139/cjfr-2014-0333
	Nelsen R B . An introduction to Copulas. Springer: New York, 1998,
	Ogana F N , Gorgoso-Varela J J , Osho J S A . Modelling joint distribution of tree diameter and height using Frank and Plackett Copulas. Journal of Forestry Research, 2020, 31 (5): 1681- 1690. doi: 10.1007/s11676-018-0869-1
	Petrauskas E, Rupšys P, Bartkevi Dč ius E. 2011. Volume modeling of individual trees using stochastic differential equations and trivariate Copula. 2011 International Conference on Circuits, System and Simulation(ICCSS2011), Bangkok, Thailand, Singapore: IACSIT Press.
	Rupšys P , Petrauskas E . Analysis of height curves by stochastic differential equations. International Journal of Biomathematics, 2012, 5 (5): 1250045. doi: 10.1142/S1793524511001878
	Sklar M . Fonctions de répartition àn dimensions et leurs marges. Publications de l'Institut de Statistique de L'Université de Paris, 1959, 8, 229- 231.
	Temesgen H , LeMay rm , Cameron I R . Bivariate distribution functions for predicting twig leaf area within hybrid spruce crowns. Canadian Journal of Forest Research, 2003, 33, 2044- 2051. doi: 10.1139/x03-127
	Wang M , Rennolls K . Bivariate distribution modelling with tree diameter and height data. Forest Science, 2007, 53 (1): 16- 24.
	Wang M , Rennolls K , Tang S . Bivariate distribution modeling of tree diameters and heights: dependency modeling using Copulas. Forest Science, 2008, 54 (3): 284- 293.
	Wang M , Upadhyay A , Zhang L . Trivariate distribution modeling of tree diameter, height, and volume. Forest Science, 2010, 56 (3): 290- 300.
	Wang S , Zhang X , Liu L . Multiple stochastic correlations modeling for microgrid reliability and economic evaluation using pair-Copula function. International Journal of Electrical Power & Energy Systems, 2016, 76, 44- 52.

胸径 DBH/cm	1度1du		2度2du		3度3du		4度4du		5度5du
胸径 DBH/cm	株数 Stem number/hm^-2	累计株数 Cumulative stem	株数 Stem number/hm^-2	累计株数 Cumulative stem	株数 Stem number/hm^-2	累计株数 Cumulative stem	株数 Stem number/hm^-2	累计株数 Cumulative stem	株数 Stem number/hm^-2	累计株数 Cumulative stem
5	529	529	1 182	1 711	1 203	2 914	461	3 375	57	3 432
6	565	1 094	1 283	3 559	1 215	5 977	541	6 979	84	7 120
7	858	1 952	1 804	6 221	1 716	10 355	686	12 043	148	12 332
8	1 048	3 000	2 081	9 350	2 123	15 607	790	18 085	157	18 531
9	1 074	4 074	2 066	12 490	2 156	20 903	922	24 303	158	24 907
10	1 050	5 124	1 829	15 369	1 995	25 777	996	30 173	173	30 950
11	608	5 732	978	16 955	1 283	28 646	655	33 697	108	34 582
12	363	6 095	520	17 838	712	30 241	370	35 662	61	36 608
13	109	6 204	171	18 118	240	30 761	152	36 334	20	37 300
14	27	6 231	37	18 182	58	30 883	25	36 481	10	37 457
15	1	6 232	0	18 183	6	30 890	2	36 490	0	37 466

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