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Today, I will talk to you about how to analyze the choice of the optimal Copula function in MATLAB. Many people may not know much about it. In order to make you understand better, the editor has summarized the following for you. I hope you can get something according to this article.

The Copula function model explains the application of Copula function in real life. Copula function describes the correlation between variables. In fact, it is a kind of function that connects the joint distribution function with their respective edge distribution function, so some people call it connection function.

There are two main reasons why Copula function is favored by statisticians: the first is that Copula is a method to study the measure of dependence; the second is that Copula, as the starting point for constructing two-dimensional distribution families, can be used for multivariate model distribution and stochastic simulation. As a tool of dependence mechanism between variables, Copula function contains almost all the dependence information of random variables. Under the condition that it can not determine whether the traditional linear correlation coefficient can correctly measure the correlation between variables, Copula function is very useful for the analysis of the correlation between variables. The emergence of Copula function makes the description of dependence between variables more perfect. Since the Copula method was proposed, the Copula function has been widely used in the analysis of the dependence of returns on financial assets, financial risk, financial risk management and so on.

Application example of Copula function

In the previous analysis of selecting factors by Copula entropy method, the joint distribution of precipitation and air temperature is constructed from 9 factors that have influence on precipitation, and various types of Copula functions are compared to select the optimal Copula function. The analysis results are as follows: firstly, precipitation and air temperature are taken as two random variables X and Y, and the frequency histograms of the two random variables are made by using Matlab software, and their skewness and kurtosis are calculated, as shown in the following figure.

Frequency histogram

By calculating the kurtosis and skewness of two random variables, making the frequency histogram and analyzing the frequency histogram, skewness and kurtosis, it is obvious that the distribution of the two variables is asymmetric, and the normal-Copula function and t-Copula function can be excluded. For further analysis to determine the distribution types of precipitation and temperature, and select the appropriate Copula function. Using nonparametric approximation to estimate the type of population distribution: empirical distribution function graph and kernel distribution estimation graph

The binary histogram of edge distribution can determine the empirical distribution function and kernel distribution function to estimate their edge distribution. By analyzing their frequency and frequency distribution, we can clearly see that their tails are asymmetrical. Archimedes-Copula function can be used to describe the relationship between them. The parameter values of the three are calculated as follows: the expression of the three Copula functions is: with this expression, the density function diagram and distribution function diagram of the three Copula functions are as follows: bivariate Gumbel-Copula density function and distribution function diagram

Binary Clayton-Copula density function and distribution function graph binary Frank-Copula density function and distribution function diagram can be seen from the density function graph and distribution function diagram that the binary Frank-Copula function has a thicker tail and can better reflect the relationship between them. The tail correlation coefficient of this type of Copula function is obtained by calculation, and the rank correlation coefficient of three functions is calculated by using Copulastat function: using Corr function to calculate the rank correlation coefficient of Kendall and Spearman: the result of calculating square Euclidean distance:

The square Euclidean distance is calculated as follows: for Gumbel- Copula, the distance is 0.27, the distance is 0.1234, and the distance is 0.2159. Among the three, the square Euclidean distance of Clayton-Copula function is smaller, and the Clayton-Copula function can be used in the joint distribution of the two, which can better show the relationship between them.

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