CN110287601B - Moso bamboo breast diameter age binary joint distribution accurate estimation method - Google Patents
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Abstract
The invention discloses a moso bamboo breast diameter age binary joint distribution accurate estimation method, which comprises the following steps: firstly, summarizing the diameter at breast height and age of a moso bamboo forest into two-dimensional statistical data according to the number of plants; secondly, drawing a three-dimensional frequency histogram of the diameter at breast height and age of the moso bamboos; thirdly, obtaining parameter vectors of a plurality of common binary Copula density functions; fourthly, drawing a plurality of binary Copula density function graphs of the diameter at breast height and age of the moso bamboo; fifthly, selecting a prepared binary Copula density function graph; sixthly, obtaining an optimal binary Copula density function model of the diameter at breast height and age of the moso bamboo; and seventhly, obtaining a moso bamboo breast diameter age binary Copula distribution estimation function, and further realizing the accurate estimation of the moso bamboo breast diameter age binary combined distribution. The invention can establish a binary Copula distribution (density) function of the diameter at breast height and age of the moso bamboo according to the edge distribution value of the diameter at breast height and the age; according to the continuous checking data of the moso bamboos in two adjacent stages, the advantage of the joint distribution (density) value of the breast diameter and the age in the middle year can be calculated.
Description
Technical Field
The invention relates to a method for establishing an accurate estimation model of regional forest biomass, in particular to a moso bamboo breast-height age binary joint distribution accurate estimation method.
Background
Mao bamboo is an important forest type in south China, is also the most important and typical bamboo forest resource type, and China is one of the most abundant countries of bamboo forest resources. The moso bamboo is also a special regional different-age forest plant, the diameter-at-breast growth and the high growth of the moso bamboo are completed in the first year, and the moso bamboo forest mostly has the habit of growing bamboo shoots every other year, namely the year and year, so the stand age structure of the moso bamboo forest is divided into 1 age grade according to 2 years, and therefore, the diameter-at-breast and the age of the moso bamboo forest can reflect the structural characteristics of the resources more completely. According to the research, the Mao bamboo forest stand also has a strong carbon sink function (Zhou national model, jiangbeyun, 2004; clearness, etc., 2002; lily, etc., 1998; ruan hong Hua, etc., 1997), and is an important biomass energy material in the future, so how to accurately estimate the regional scale Mao bamboo biomass is very important.
The binary probability distribution of the tree measuring factors plays an important role in accurate estimation of regional forest biomass and modern forest management activities, can accurately estimate the tree diameter steps, the tree number and accumulation change of each age step, growth amount measurement, yield estimation, loss amount calculation and the like, and also provides scientific basis for scientific design of tending thinning and accurate estimation of forest carbon banks. To date, binary generalized β distributions and Johnson's SBB distributions have been widely used to study the two-dimensional combination of breast diameter, tree height of tree-measuring factors (golden star ji, 2013, li et al, 2002 wang and rennols, 2007, hafley et al, 1976 h.t. schreuder et al, 1977 faherg li,2002 mingliang wang, 2007), binary Weibull distributions and binary Beta distribution functions to describe the two-dimensional structure of the area distribution of thin branches within the tree crowns (temesgene et al, 2003), and binary Weibull distributions and binary maximum entropy functions to describe the combination of moso bamboo breast diameter, age distribution (grand, 2008; liun, 2010.
The existing methods for researching the binary joint distribution of tree-measuring factors have the following defects: 1) The edge distribution model must be the same: the two-dimensional Beta distribution, the two-dimensional Weibull distribution and the two-dimensional Sbb edge distribution must be the Beta distribution, the Weibull distribution and the Sbb distribution, but in practical application, when the edge distributions of two tree measurement factors are different, a common binary distribution function cannot describe the joint distribution of the two tree measurement factors, so that the applicability of a common binary distribution model is not wide; 2) The corresponding joint distribution is not determined by the edge distribution: a commonly used method for researching binary joint distribution of tree measurement factors is to establish a joint distribution function of the tree measurement factors firstly and then deduce edge distribution (liun, 2010; pueraria hong, 2008) corresponding to the joint distribution, so the established joint distribution function is not determined by the corresponding edge distribution function, actually, the types and theories of the edge distribution functions are more and more perfect than those of the joint distribution function, but the theories and application values are far less than those of the joint distribution function, and therefore, the method for establishing binary joint distribution of the tree measurement factors according to the edge distribution of the tree measurement factors has very important significance; 3) The data of the middle year cannot be calculated according to the continuous checking sample plot of the moso bamboos in two adjacent periods: the continuous checking sample plot of the moso bamboos is checked once every 5 years, the joint distribution data of the diameter at breast height and the age of the moso bamboos needs to be given by parameter fitting of the binary joint distribution of the common tree-measuring factors, and the joint distribution data of the diameter at breast height and the age of the moso bamboos in any middle year cannot be determined only according to the continuous checking data in two adjacent periods, so that the continuous checking data of the moso bamboos in the middle 4 years cannot be accurately calculated by the common binary joint distribution function, and the dynamic change of the biomass of the moso bamboos in provincial scale cannot be accurately estimated.
With the continuous advance of scientific research, copula theory (Nelsen, 2006) rapidly developed in the field of mathematical research provides a new idea for solving the modeling problem of the related abnormal variables. The core idea of the Copula theory is that any binary joint distribution function can be split into a Copula function which can contain correlation size and related structural information and two corresponding edge distribution functions, so that the Copula theory has incomparable advantages in the aspects of flexibility and applicability during modeling, and according to a plurality of defects of a commonly-used research tree-measuring factor binary joint distribution model, the application of the Copula theory to the field of forestry and ecology is a great trend in current forestry research, and how to apply the Copula function to establish a moso bamboo breast diameter and age binary joint distribution model is a great problem to be solved urgently in current forestry research.
Disclosure of Invention
The invention aims to provide a method for establishing a moso bamboo breast-height age two-dimensional distribution model based on a Copula function. The method has the advantages that 1) a binary Copula distribution (density) function of the diameter at breast height and age of the moso bamboo can be established according to the edge distribution value of the diameter at breast height and the age; 2) And the advantage of the joint distribution (density) value of the breast diameter and age in the middle year can be calculated according to the continuous checking data of the moso bamboos in two adjacent periods.
The technical scheme of the invention is as follows: a moso bamboo breast-height age binary combined distribution accurate estimation method comprises the following steps:
step one, gathering continuous checking data of N moso bamboo sample plots in a research area into two-dimensional statistical data of the diameter at breast height and age of a moso bamboo forest according to the number of plants;
step two, respectively calculating an edge distribution value u and an age edge distribution value v of the diameter at breast height of the moso bamboos according to the two-dimensional statistical data in the step one, and drawing a three-dimensional frequency histogram of the diameter at breast height and age of the moso bamboos by using matlab software with u and v as independent variables;
step three, respectively fitting the two-dimensional statistical data of the step one by using a plurality of common binary Copula density functions by using the cumulative distribution values of the diameter at breast height and the age of the moso bamboo as independent variables and adopting a maximum likelihood method to obtain parameter vectors of the binary Copula density functions;
step four, drawing a plurality of moso bamboo breast diameter age binary Copula density function graphs by matlab software respectively;
step five, comparing all the moso bamboo chest diameter age binary Copula density function graphs obtained in the step four with the three-dimensional frequency histogram obtained in the step two, and selecting several Copula density function graphs with the shapes close to the three-dimensional frequency histogram as preparation binary Copula density function graphs;
step six, respectively calculating the AIC value and the determination coefficient R of the binary Copula density function of the breast diameter and age of the moso bamboos according to the AIC information criterion and the determination coefficient formula 2 Selecting R with minimum AIC value 2 Taking the largest binary Copula density function model which is closest to the three-dimensional frequency histogram in the step two as a moso bamboo breast diameter age optimal binary Copula density function model D (u, v);
and seventhly, obtaining a binary Copula distribution estimation function of the moso bamboo breast-diameter ages according to the conversion relation between the density function and the distribution function in the probability theory, and further realizing the accurate estimation of the two-dimensional distribution of the moso bamboo breast-diameter ages.
In the foregoing method for accurately estimating the binary joint distribution of the diameter at breast height and age of moso bamboos, the value of AIC in the sixth step is calculated as:
in the formula: d (u) i ,v i (ii) a θ) is the selected binary Copula density function; theta is a parameter vector of the selected binary Copula function; m is the parameter number of the selected binary Copula density function; l is a maximum likelihood function of the selected binary Copula density function; n is the sample size of the raw data to be fitted, m is equal to 1 when the chosen model is a two-dimensional Gaussian Copula, frank Copula, gumbel Copula, clayton Copula function, m is equal to 2,u when the chosen model is a two-dimensional t Copula function i ,v i (i =1,2, \ 8230; N) are the edge distribution values statistically obtained from the original sample data, respectively.
In the above moso bamboo breast diameter age binary joint distribution accurate estimation method, the coefficient R is determined in the sixth step 2 The calculation formula of (A) is as follows:
in the formula y i Is the ith measured value of the dependent variable,
is the ith estimated value of the dependent variable, N is the number of samples,
is the average of the dependent variables.
In the method for accurately estimating phyllostachys pubescens breast diameter age binary joint distribution, the plurality of binary Copula density functions in the third step, the fourth step and the sixth step include a two-dimensional normal Copula density function, a two-dimensional t-Copula density function, a two-dimensional Gumbel Copula density function, a two-dimensional Clayton Copula density function and a two-dimensional Frank Copula density function.
In the method for accurately estimating the binary joint distribution of the diameter at breast height and age of moso bamboos, the drawing of the binary Copula density function map specifically comprises the following steps:
a. calculating the edge distribution values of the diameter at breast height and the age of the moso bamboos according to the statistical data in the first step;
b. respectively fitting 5 binary Copula density function parameter vectors by taking the edge distribution values of the moso bamboo breast diameter and the age as independent variables and adopting a maximum likelihood method, and further estimating a moso bamboo breast diameter and age binary Copula density function value according to the edge distribution values of the moso bamboo breast diameter and the age;
c. and drawing a binary Copula density function graph of the diameter at breast height and age of the moso bamboos by matlab software by taking the edge distribution value of the diameter at breast height and the age of the moso bamboos as an independent variable and taking a binary Copula density function value of the diameter at breast height and age of the moso bamboos as a dependent variable.
Compared with the prior art, the method adopts a binary Copula function to describe the binary distribution of the diameter of breast of moso bamboo according to the defects of a common binary distribution (density) function, and establishes a two-dimensional distribution model of the diameter of breast of moso bamboo based on the Copula function, so that the method is suitable for any edge distribution of the diameter of breast and the age, and when the method is specifically applied, the type of the edge distribution function does not need to be determined, and the edge distribution function value of the diameter of breast and the age can be estimated according to the measured data; specifically, the establishment of the moso bamboo breast diameter and age binary Copula function comprises the steps of firstly, determining edge distribution values of breast diameter and age through continuous checking data of a moso bamboo sample plot, calculating the AIC values of the edge distribution values of the moso bamboo breast diameter and age under various prepared binary Copula density function models by adopting an AIC information criterion, finally, selecting the function model with the minimum AIC value as an optimal Copula function model, and fitting a parameter alpha of the optimal Copula function model to obtain a final binary Copula probability density estimation function.
The parameter fitting of the binary Copula function only needs measured values of chest diameter and age edge distribution and does not need joint distribution (density) values of the measured data of the chest diameter and the age, and the joint distribution (density) value of the chest diameter and the age in the middle year can be calculated according to the continuous checking data of the moso bamboos in two adjacent stages; the binary Copula distribution (density) function can describe the nonlinear correlation among the tree measuring factors, and further describe the correlation of different parts of the tree measuring factors.
In conclusion, the method has the advantages that 1) a binary Copula distribution (density) function of the diameter at breast height and the age of the moso bamboo can be established according to the edge distribution values of the diameter at breast height and the age; 2) And the advantage of the joint distribution (density) value of breast diameter and age in the middle year can be calculated according to the continuous checking data of the moso bamboos in two adjacent periods.
Drawings
FIG. 1 is a three-dimensional frequency histogram of cumulative distribution of the diameter at breast height and age of moso bamboo;
FIG. 2 is a two-dimensional normal Copula's joint density function graph;
FIG. 3 is a two-dimensional t-Copula joint density function graph;
FIG. 4 is a two-dimensional Gumbel Copula combined density function plot;
FIG. 5 is a two-dimensional Clayton Copula combined density function plot;
FIG. 6 is a two-dimensional Frank Copula's joint density function plot;
FIG. 7 is a graph of the distribution function of the binary Copula of the diameter-height and age of moso bamboo;
FIG. 8 is a two-dimensional probability density function plot of the diameter at breast height and age of Phyllostachys pubescens.
Detailed Description
The invention is further illustrated by the following figures and examples, which are not to be construed as limiting the invention.
Example (b): a moso bamboo breast-height age binary combined distribution accurate estimation method comprises the following steps:
step one, summarizing continuous checking data of N moso bamboo sample plots in a research area into two-dimensional statistical data of the diameter at breast height and age of the moso bamboo forest according to plant numbers, establishing a forest resource continuous checking system in 1979 in Zhejiang province, setting 4250 fixed sample plots in total by taking 5 years as a rechecking period, setting a sample point grid to be 4km multiplied by 6km, setting the shape of the sample plots to be square, the side length to be 28.28m and the area to be 800m 2 . 177 continuous checking sample plots of moso bamboos in 2009 are used for research, and the basic conditions of the moso bamboo sample plots are as follows: the number of the moso bamboo plants in each sample plot is 22-897, the diameter of the inner moso bamboo in each sample plot is 5-15 cm, and the age is more than 1-4 degrees (the current-year-old bamboo is 1 degree bamboo; the 2-3-year-old bamboo is 2 degree bamboo, and so on). The survey factors of the plots include soil thickness, slope direction (1: north, 2: north-east, 3: east, 4: south-east, 5: south, 6: south-west, 7: west,8: northwest, 9: no slope, i.e., mountain top), a slope (1: ridge, 2: upper portion, 3: middle part, 4: lower part, 5: valley portion, 6: flat ground), slope, elevation (10 m-1200 m), average breast diameter of sample plot, number of sample plots, breast diameter and age of each moso bamboo in sample plot, etc. The selection of the sample plot in this embodiment is a macro model of the total province region, so 245 sample plot data are merged and counted into one table, see table 1. The density values in the table are the number of moso bamboo plants of a certain diameter order and a certain age order, and the distribution values are the cumulative values of the number of moso bamboo plants of a certain age order from an initial diameter order and an initial age order;
TABLE 1 statistical data of diameter at breast height and age plot of Mao bamboo forest in Zhejiang province
Step two, respectively calculating an edge distribution value u and an age edge distribution value v of the diameter of the moso bamboo breast according to the two-dimensional statistical data of the step one, and drawing a three-dimensional frequency histogram of cumulative distribution of the diameter of the moso bamboo breast and the age by matlab software by taking u and v as independent variables, as shown in fig. 1;
step three, respectively fitting the two-dimensional statistical data of the step one by using a plurality of common binary Copula density functions by using the cumulative distribution values of the diameter at breast height and the age of the moso bamboo as independent variables and adopting a maximum likelihood method to obtain parameter vectors of the binary Copula density functions;
step four, drawing a plurality of moso bamboo breast diameter age binary Copula density function graphs by matlab software respectively, wherein common Copula functions comprise a normal Copula function, a t-Copula function and an archimedes Copula function, wherein Gumbel Copula function, clayton Copula function and Frank Copula function are the most common 3 archimedes Copula functions (Zhang-bloe, 2017), so that the 5 binary Copula functions are adopted to estimate the joint density of the moso bamboo breast diameter ages, and the obtained binary Copula density function graphs are respectively shown in fig. 2-6;
step five, comparing all the phyllostachys pubescens breast diameter age binary Copula density function graphs in the step four with the three-dimensional frequency histogram obtained in the step two, selecting several Copula density function graphs with the shapes close to the three-dimensional frequency histogram from the Copula density function graphs as preparation binary Copula density function graphs, and as can be seen from comparison between the graph 1 and the graphs 2-6, the two-dimensional Gumbel Copula and two-dimensional t Copula combined density graph is similar to the graph 1, and other binary Copula functions have larger difference from the graph 1, so that the Gumbel Copula and t Copula density function graphs are preferentially selected as the preparation binary Copula density function graphs;
step six, respectively calculating the AIC value and the determination coefficient R of the binary Copula density function of the breast diameter and age of the moso bamboos according to the AIC information criterion and the determination coefficient formula 2 Selecting R with minimum AIC value 2 The largest binary Copula density function model closest to the three-dimensional frequency histogram in the step two serves as a moso bamboo chest diameter age optimal binary Copula density function model D (u, v), and according to the value calculation formula of AIC, AIC values of two-dimensional t Copula and two-dimensional Gumbel Copula probability density functions and a determination coefficient R are calculated 2 The probability density functions of-14.3104, -19.5196, 0.9676 and 0.9841 respectively, so that the two-dimensional Gumbel Copula probability density function describes the binary combined distribution of the diameter and the age of the moso bamboo, and is an optimal Copula function model;
checking all binary Copula density function graphs again by using an AIC information criterion, wherein the AIC value of Gaussian Copula is 2, the AIC value of Copula is-14.3104, the AIC value of Clayton Copula is-1.2248, the AIC value of Frank Copula is-2.4059, and the AIC value of Gumbel Copula is-19.5196; determining a two-dimensional Gumbel Copula probability density function as an optimal Copula function model;
seventhly, according to the conversion relation between the density function and the distribution function in the probability theory, a moso bamboo breast diameter age binary Copula distribution estimation function can be obtained, and further accurate estimation of two-dimensional moso bamboo breast diameter age distribution is achieved, and at the moment, the estimation value of the moso bamboo breast diameter age combined density function is shown in the following table 2:
TABLE 2 actual measurement probability of Moso bamboo and estimated value of corresponding binary normal Copula probability density function in provincial domain scale
As can be seen from Table 2, the two-dimensional Gumbel Copula probability density function has high estimation precision on the breast diameter age joint density, and the estimation value at the lower tail part is slightly higher than the actual value.
The calculation formula of the value of the AIC in the step six is as follows:
in the formula: d (u) i ,v i (ii) a θ) is the selected binary Copula density function; theta is the parameter vector of the selected binary Copula function; m is the parameter number of the selected binary Copula density function; l is a maximum likelihood function of the selected binary Copula density function; n is the sample size of the raw data to be fitted, m is equal to 1 when the chosen model is a two-dimensional Gaussian Copula, frank Copula, gumbel Copula, clayton Copula function, m is equal to 2,u when the chosen model is a two-dimensional t Copula function i ,v i (i =1,2, \8230; N) are the edge distribution values statistically obtained from the original sample data, respectively.
Determining the coefficient R in the sixth step 2 The calculation formula of (c) is:
in the formula y i Is the ith measured value of the dependent variable,
is the ith estimated value of the dependent variable, N is the number of samples,
is the average of the dependent variables.
The plurality of binary Copula density functions in the third step, the fourth step and the sixth step comprise a two-dimensional normal Copula density function, a two-dimensional t-Copula density function, a two-dimensional Gumbel Copula density function, a two-dimensional Clayton Copula density function and a two-dimensional Frank Copula density function.
The drawing of the binary Copula density function graph specifically comprises the following steps:
a. calculating the edge distribution values of the diameter at breast height and age of the moso bamboos according to the statistical data in the step one;
b. respectively fitting 5 binary Copula density function parameter vectors by using the edge distribution values of the moso bamboo breast diameter and the age as independent variables and adopting a maximum likelihood method, and estimating a moso bamboo breast diameter and age binary Copula density function value by using the edge distribution values of the moso bamboo breast diameter and the age;
specifically, by the formula (1), a binary Copula distribution function value of the diameter at breast height and the age of the moso bamboo is calculated according to the edge distribution function value of the diameter at breast height and the age of the moso bamboo, and matlab software is used for drawing a corresponding graph, as shown in fig. 7; according to the formula (2), deducing a corresponding binary Copula density function value from the binary Copula distribution function values of the diameter at breast height and the age of the moso bamboo; finally, according to the formula (3), a two-dimensional density function value of the diameter at breast height and age of the moso bamboo can be obtained by the binary Copula density function value;
c. using the edge distribution value of the moso bamboo breast diameter and the age as an independent variable and the binary Copula density function value of the moso bamboo breast diameter and the age as a dependent variable, and drawing a binary Copula density function diagram of the moso bamboo breast diameter and the age by matlab software, as shown in fig. 8;
the formula (1) is H (x, y) = C (F (x), G (y)), wherein H (x, y) is a joint distribution of two-dimensional random variables (x, y), and the edge distribution of the two-dimensional random variables is F (x) and G (y) respectively;
the formula (2) is a probability density function D (u, v) of a Copula function C (u, v),
u＝F(x),v＝G(y)；
the formula (3) is a probability density function H (x, y) of H (x, y),
as a function of the edge probability density for the variable x,
an edge probability density function for variable y;
wherein the relation between the formula (2) and the formula (3) is as follows:
in the formula, D (u, v) is a binary Copula density function value, and h (x, y) is a moso bamboo breast diameter and age two-dimensional density function value.
The Copula function was originally proposed by Sklar (1959): a plurality of one-dimensional distributions are connected as a function of a multi-dimensional distribution, while each edge distribution must be a uniform distribution.
Copula function basic theory: by definition (Nelsen, 2006), the binary Copula function is defined as follows:
(1) C (·,. Cndot.) has a domain of definition in each dimension space of [0,1];
(2) C (·, ·) has zero base and is two-dimensionally increasing;
(3) For any variable u, v ∈ [0,1], C (u, 1) = u, C (1, v) = v are satisfied;
meanwhile, for any point (u, v) in the defined domain, 0 is less than or equal to C (u, v) is less than or equal to 1.
By definition, the Copula function has the following properties:
1. boundary conditions: for any binary variable (u, v) ∈ [0,1] × [0,1], C (u, 0) =0, C (0, v) =0, C (u, 1) = u, C (1, v) = v are satisfied, that is, as long as one variable is 0 and the copula function value is 0, as long as one variable is 1, the copula function value is determined by the other variable.
2. The increment is as follows: the Copula function is strictly monotonically non-decreasing within the domain, for variable u within the domain 1 ,v 1 ,u 2 ,v 2 And u is 1 ≤u 2 ,v 1 ≤v 2 Then C (u) 2 ,v 2 )-C(u 2 ,v 1 )-C(u 1 ,v 2 )+C(u 1 ,v 1 ) ≧ 0, i.e., if u, v ∈ [0,1 ∈ ≧ 0]If the value of (c) is increased at the same time, the Copula function value must be non-decreasing, and similarly, if a self-variation occursThe quantity is constant, and the Copula function value increases or does not change as the value of the other variable increases.
3. Frechet boundary: any Copula distribution has an upper Frechet bound C + (u, v) = max (u + v-1, 0) and lower bound C- (u, v) = min (u, v), i.e. C- (u, v) ≦ C + (u,v)。
Sklar's theorem: let the joint distribution of two-dimensional random variables (x, y) be H (x, y) and the edge distributions be F (x) and G (y), respectively, then there exists a Copula function C (1):
H(x,y)＝C(F(x),G(y))
if F (x) and G (y) are consecutive, C is unique, and conversely, for any probability distribution F (x) and G (y) and Copula function C, H (x, y) defined by equation (1) must be a joint distribution function with edges F (x) and G (y).
Let D (u, v) be the probability density function of the Copula function C (u, v), and obtain equation (2) from equation (1):
let u = F (x), v = G (y), then
Is an edge probability density function of the variable x,
as an edge probability density function of the variable y, the probability density function H (x, y) of H (x, y) is (3):
(3) The formula shows that a joint probability density function h (x, y) is divided into two parts, the former part D (x, y) is a Copula density function and can accurately and completely describe the correlation structure between random variables, and the latter part f (x) g (y) is the product of edge probability density functions.
The joint distribution of the moso bamboo breast-diameter ages has great significance for the management of the moso bamboo forest and the accurate estimation of the regional scale biomass, the Copula functions have more types and wide adaptability, and the application of the moso bamboo breast-diameter ages in forestry and ecology is very wide, and the following conclusion is obtained by combining the research of the text:
1) The method for establishing the moso bamboo breast-diameter age two-dimensional distribution model based on the Copula function is suitable for arbitrary edge distribution of the breast diameter and the age, and when the method is specifically applied, the type of the edge distribution function does not need to be determined, and the edge distribution function value of the breast diameter and the age, namely the independent variable of the Copula function, is estimated according to actually measured data.
2) The establishment of the moso bamboo breast diameter age binary Copula function is divided into two steps: the first step is to determine the edge distribution value of the chest diameter and the age, and the second step is to determine the optimal Copula function and the parameter for fitting the corresponding Copula function.
3) The measurement precision of the two-dimensional Gumbel Copula distribution (density) function on the combined distribution of the diameters at breast height and the ages of the moso bamboo is highest, R 2 =0.9841, which is the optimal Copula function.
The binary Copula distribution (density) function differs from the commonly used binary probability distribution (density) function for the diameter of breast of moso bamboo by:
1. the two independent variables are different: the mao bamboo breast diameter and age are commonly used with binary probability distribution (density) functions, such as binary Weibull, binary Sbb, and binary Beta functions, the independent variable is the breast diameter and age, the dependent variable is the joint distribution (density) value of the breast diameter and age, the independent variable of the binary Copula distribution (density) function is the marginal distribution function value of the mao bamboo breast diameter and age, and the dependent variable is the Copula connection function value.
2. The conditions required to fit the distribution (density) function parameters are different: the fitting of the parameters of the commonly used binary probability distribution (density) function requires the joint distribution (density) value of the breast diameter and age measured data, and the parameter fitting of the binary Copula distribution (density) function only requires the edge distribution value of the breast diameter and the age measured data of the moso bamboo, so that the condition required for fitting the parameters of the commonly used binary probability distribution (density) function is much higher than that of the binary Copula distribution (density) function, and the application of the binary Copula distribution (density) function is wider and the practicability is stronger.
3. Type of breast diameter and age margin distribution function: when the binary Beta distribution (density) function is applied to fit breast diameter and age combined distribution, the edge distribution function type of the breast diameter and the age must be required to be a unitary Beta distribution (density) function, and the binary Weibull and the binary Sbb functions also have similar requirements.
4. Establishing a joint distribution function: the corresponding edge distribution can be determined by the joint distribution of the tree-measuring factors usually, and the edge distribution is difficult to determine, so that the commonly used binary joint distribution (density) function is not determined by the corresponding edge distribution (density) function, and no matter in theory or practice, the types of the edge distribution of the tree-measuring factors are numerous and are easy to determine, but the value of the joint distribution is higher than that of the edge distribution, so that how to determine the corresponding joint distribution according to the edge distribution of the tree-measuring factors is significant.
5. And (3) testing the model: the joint distribution (density) value of the actually measured breast diameter and age is data necessary for fitting parameters of a commonly used binary probability distribution (density) function, the final purpose of the binary distribution (density) function is to obtain an estimation value of the joint distribution (density) of the breast diameter and the age, so in order to verify the reliability and the stability of the model, the actually measured data is divided into a modeling sample and a testing sample so as to carry out parameter fitting and testing on the commonly used binary probability distribution (density) function, and the fitting of the binary Copula distribution (density) function parameter only needs an edge distribution function value of the breast diameter and the age (the joint distribution (density) value of the actually measured breast diameter and the age is not needed), so the model is not needed to be tested by the testing sample.
6. Estimating the breast diameter and age joint distribution (density) of regional bamboo: the forest resource continuous checking is carried out in 1979 in Zhejiang province, the forest resource continuous checking is carried out once every 5 years, 7 times of continuous checking of resources are finished up to now, if the time prediction is carried out on the moso bamboo resource in the whole province, the data in the 7 th stage is far insufficient, the data in two adjacent stages are encrypted according to the year, when the prediction is carried out by using a common binary probability distribution (density) function, a function parameter is fitted firstly, the fitting of the function parameter needs to have a joint distribution (density) value of the breast diameter age of the moso bamboo, the joint distribution (density) value of the breast diameter age in the middle year cannot be calculated according to the continuous checking data of the moso bamboo in the two adjacent stages, namely, the common binary probability distribution (density) function cannot encrypt the forest continuous checking data, the breast diameter of the moso bamboo is very special with the growth of the moso bamboo, the moso bamboo is different from other plants, the edge distribution (density) value of the breast diameter and the age data in the two adjacent stages can be calculated, the edge distribution (density) value of the breast diameter and the moso bamboo can be estimated by calculating the edge distribution (density) of the breast diameter and the two adjacent bamboo regions, and the moso bamboo can be accurately predicted by carrying out the continuous checking on the moso bamboo area continuous checking of the accurate scale.
Comparison of the binary Copula distribution (density) functions:
comparing fig. 2, fig. 3 and fig. 6, the two-dimensional normal Copula, the two-dimensional t Copula and the two-dimensional Frank Copula density functions have symmetrical tails, which cannot describe asymmetrical tail correlation of the mangosteen breast diameter and the age, wherein the two-dimensional t Copula function has a thicker tail and is sensitive to asymmetrical tail correlation of the mangosteen breast diameter and can better describe symmetrical tail correlation of the mangosteen breast diameter and the age, the binary Clayton Copula and the binary Gumbel Copula function have asymmetrical tails and can describe asymmetrical tail correlation of the mangosteen breast diameter and the age, wherein an image of the binary Gumbel Copula function is in a J shape, which shows that the function is sensitive to upper tail variation of the mangosteen breast diameter and the age, and can better describe tail correlation of the upper tail height and the lower tail height of the mangosteen breast diameter and the age, and the density function image of the binary Clayton Copula function is in an L shape, which shows that the function is sensitive to upper tail variation of the mangosteen breast diameter and the tail correlation of the mangosteen breast diameter and the lower tail height of the mangosteen breast diameter and the tail height of the mangosteen Copula are better described.
Claims (5)
1. A moso bamboo breast-height age binary combined distribution accurate estimation method is characterized by comprising the following steps:
step one, gathering continuous checking data of N moso bamboo sample plots in a research area into two-dimensional statistical data of the diameter at breast height and age of a moso bamboo forest according to the number of plants;
step two, respectively calculating the edge distribution value of the diameter of the moso bamboo breast according to the two-dimensional statistical data of the step one
And age edge distribution value
To do so by
And with
Drawing a three-dimensional frequency histogram of the diameter at breast height and age of the moso bamboos by matlab software as independent variables;
step three, respectively fitting the two-dimensional statistical data of the step one by using a plurality of common binary Copula density functions by using the cumulative distribution values of the diameter at breast height and the age of the moso bamboo as independent variables and adopting a maximum likelihood method to obtain parameter vectors of the binary Copula density functions;
step four, drawing a plurality of moso bamboo breast diameter age binary Copula density function graphs by matlab software respectively;
step five, comparing all the moso bamboo chest diameter age binary Copula density function graphs in the step four with the three-dimensional frequency histogram obtained in the step two, and selecting several Copula density function graphs with the shapes close to the three-dimensional frequency histogram as prepared binary Copula density function graphs;
step six, respectively calculating the AIC values and determining the AIC values of a plurality of moso bamboo breast diameter age binary Copula density functions according to the AIC information criterion and the determining coefficient formulaCoefficient R 2 Selecting R with minimum AIC value 2 Taking the maximum binary Copula density function model which is closest to the three-dimensional frequency histogram in the step two as the optimal binary Copula density function model of the phyllostachys pubescens breast diameter age
；
And seventhly, obtaining a moso bamboo breast-diameter age binary Copula distribution estimation function according to the conversion relation between the density function and the distribution function in the probability theory, and further realizing the accurate estimation of the moso bamboo breast-diameter age binary distribution.
2. The moso bamboo breast diameter and age binary joint distribution accurate estimation method according to claim 1, characterized in that: the calculation formula of the value of the AIC in the step six is as follows:
in the formula:
selecting a binary Copula density function;
selecting a binary Copula function parameter vector; m is the parameter number of the selected binary Copula density function; l is the maximum likelihood function of the selected binary Copula density function; n is the sample size of the raw data to be fitted, m is equal to 1 when the chosen model is a binary Gaussian Copula, frank Copula, gumbel Copula, clayton Copula function, m is equal to 2 when the chosen model is a binary t Copula function,
the edge distribution values are respectively obtained by statistics of the original sample data.
3. The moso bamboo breast diameter and age binary joint distribution accurate estimation method according to claim 1, characterized in that: determining the coefficient R in the sixth step 2 The calculation formula of (c) is:
in the formula
Is the ith measured value of the dependent variable,
is the ith estimated value of the dependent variable, N is the number of samples,
is the average of the dependent variables.
4. The moso bamboo breast diameter and age binary joint distribution accurate estimation method according to claim 1, characterized in that: the plurality of binary Copula density functions in the third step, the fourth step and the sixth step comprise a binary normal Copula density function, a binary t-Copula density function, a binary Gumbel Copula density function, a binary Clayton Copula density function and a binary Frank Copula density function.
5. The moso bamboo breast diameter and age binary joint distribution accurate estimation method according to claim 4, characterized in that: the drawing of the binary Copula density function graph specifically comprises the following steps:
a. calculating the edge distribution values of the diameter at breast height and age of the moso bamboos according to the statistical data in the step one;
b. respectively fitting 5 binary Copula density function parameter vectors by using the edge distribution values of the moso bamboo breast diameter and the age as independent variables and adopting a maximum likelihood method, and estimating a moso bamboo breast diameter and age binary Copula density function value by using the edge distribution values of the moso bamboo breast diameter and the age;
c. and (3) drawing a binary Copula density function graph of the diameter at breast height and age of the moso bamboos by using matlab software by taking the edge distribution value of the diameter at breast height and the age of the moso bamboos as an independent variable and the binary Copula density function value of the diameter at breast height and age of the moso bamboos as a dependent variable.
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