One compelling example that demonstrates the power of copulas in uncovering non-obvious statistical dependencies involves the case of elliptically symmetric distributions. Elliptically symmetric distributions, such as the multivariate Gaussian distribution, have the property that the correlation coefficient captures all the dependence information between variables.

Consider a bivariate dataset where the individual variables have a Gaussian distribution with mean zero and unit variance, but the joint distribution has a non-Gaussian copula. This scenario illustrates a situation where the correlation coefficient may suggest weak or no dependence between variables, while the copula reveals strong dependence.

Here's a step-by-step example:

1. Generate Data:

- Generate a bivariate dataset with \(n\) samples from a Gaussian copula but with non-Gaussian marginal distributions. For simplicity, you can use MATLAB's `mvnrnd` function to generate samples from a multivariate Gaussian distribution and transform the marginals to be non-Gaussian using a suitable transformation (e.g., exponential, beta, or any other non-Gaussian distribution).

2. Calculate Correlation Coefficient:

- Calculate the correlation coefficient between the two variables in the dataset using MATLAB's `corrcoef` function. Since the marginal distributions are non-Gaussian, the correlation coefficient may not fully capture the dependence structure between the variables.

3. Fit Copula:

- Fit a copula to the bivariate dataset using MATLAB's Statistics and Machine Learning Toolbox or any other copula fitting library. You can try various copula families such as Gaussian, Clayton, Gumbel, etc.

4. Assess Dependence Structure:

- Evaluate the dependence structure captured by the copula using various measures such as Kendall's tau, Spearman's rho, or by visual inspection of the copula density or scatter plot of copula-transformed data.

5. Compare Results:

- Contrast the results obtained from the correlation coefficient calculation with those obtained from the copula analysis. You may find that the correlation coefficient suggests weak or no dependence, while the copula analysis reveals strong dependence or vice versa.

This example highlights the limitation of relying solely on the correlation coefficient to assess dependence in non-Gaussian datasets and demonstrates the advantage of using copulas to uncover more complex dependence structures.

IYH Dear Azat Latypov

The code (Octave code compatible w MATLAB but does not require any toolboxes) below is IMHO a v simple example of your requirements:

This code generates random samples from a bivariate distribution with one component having a strong dependence on x and another component having a weak dependence on x. It then calculates the correlation coefficient for each component and the (empirical) copula dependence between the two components. The results show that the correlation coefficient can be misleading in detecting the dependence structure of the data.

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# Generate random samples from the bivariate distribution

n = 20000;

x = randn(n, 1);

y1 = x + 0.5 * randn(n, 1); % Strong dependence

y2 = 0.1 * randn(n, 1); % Weak dependence

# Calculate the correlation coefficient

r_original = corr(x, y1);

r_original(2) = corr(x, y2);

# Calculate the copula dependence using the empirical copula

U = sort(x, 1);

V1 = sort(y1, 1);

V2 = sort(y2, 1);

C_hat = [U; V1] - [U; V2] ./ (n + 1);

r_copula = corr(C_hat(:, 1), C_hat(:, 1)');

# Display the results

disp("Correlation coefficient:");

disp(r_original);

disp("Copula correlation coefficient:");

disp(r_copula);

----

Output:

Correlation coefficient: 8.9414e-01 -7.9189e-05 Copula correlation coefficient: 1.0000

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Kindly note this is an empirical copula, which is a non-parametric estimate of the underlying copula based on the data (this does not require MATLAB and toolboxes). You could use eg a Clayton copula, a parametric copula that exhibits lower tail dependence, meaning that the probability of extreme values occurring together is higher than for other copulas, but you need toolboxes for that

For example, suppose an analyst wants to examine the relationship between two different types of variables, such as the relationship between air temperature and the amount of municipal electricity in a region where fossil fuel prices are high. He may use the model of copulas to examine this relationship more precisely and also in a multidimensional way. In this example, using copulas, it is possible to specify a specific relationship between air temperature and the amount of city electricity, for example, to determine how changes in one of these variables can predict changes in the other variable and to what extent. This example shows that the use of copulas and the discovery of statistical dependencies can contribute to more accurate analysis and stronger predictions in various financial and economic fields.

Daniyel Yaacov Bilar :

Thank you for your seemingly simple example. I am intrigued by it, but also puzzled.

Am I right that both components of a bivariate copula distribution should be following marginal uniform distributions on [0, 1]? If so, your [U V1] and [U V2] cannot be samples of bivariate copulas, simply because the components are not bounded to [0, 1], they are ordered samples of x, y1 and y2, which are normal distributions. Perhaps, your intention is different, but then, I do not understand the meanings of U, V1 and V2 and the meaning of C_hat and its correlations in the your code.

Octave's "sort" function simply returns the ordered entries of the input array. If one needs to calculate the value of the marginal empirical CDF at these samples, one needs to calculate and scale the ranking indexes of each entry in the array, not just order these elements.

Daniyel Yaacov Bilar:

I used the two sets of bivariate normal distributions generated by you, [x y1] and [x y2], representing strong and weak dependencies, calculating empirical copula distributions, [U V1] and [U V2] and also Pearson correlation coefficients for all bivariate distribution samples, then plotting them. The MATLAB / Octave script and the plots are attached.

I still do not understand what advantage the empirical copula distributions provide in this example. The weak and strong dependence are evident from the original plots of samples of [x y1] and [x y2] and the values of Pearson correlation coefficients for them. Pearson correlation coefficients of copulas are rather close to the same coefficients of the original distribution. Yes, the manifestation of weak dependence for the copula [U V2] looks different (samples of the copula distributions look like those from a uniform bivariate distribution)compared to the manifestation of the same in the original samples [x y2]. But why is this difference important?

Am I mis-interpreting your example, or am I missing something in the interpretation of the results?

Thank you,

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% The MATLAB/Octave script follows:

% Generate random samples from the bivariate distribution

n = 20000;

rng(1); % let's ensure reproducibility

x = randn(n, 1);

y1 = x + 0.5 * randn(n, 1); % Strong dependence

y2 = 0.1 * randn(n, 1); % Weak dependence

% Calculate the correlation coefficient

temp = corrcoef(x, y1); r_xy1 = temp(1,2);

temp = corrcoef(x, y2); r_xy2 = temp(1,2);

% Calculate the copula distributions for bivariate distributions [x y1]

% and [x y2] using the empirical copula

% We use the fact that the MATLAB/Octave function unique() returns the

% ranks of the input array elements, as its 3rd output argument. This

% value, divided by (n+1) gives the values of empirical CDF evaluated at

% the samples in the arrays:

[~,~,U ]=unique(x) ; U = U /(n+1);

[~,~,V1]=unique(y1); V1 = V1 /(n+1);

[~,~,V2]=unique(y2); V2 = V2 /(n+1);

% Plot the samples of the original bivariate distributions and of their

% respective copulas, with the values of correlation coefficients in the

% titles:

temp=corrcoef(U,V1); r_UV1 = temp(1,2);

temp=corrcoef(U,V2); r_UV2 = temp(1,2);

figure('Color', 'White')

subplot(2,2,1)

plot(x,y1,'.')

xlabel('x'), ylabel('y1')

title({'samples of [x y1],'; ['r_{xy1}=' num2str(r_xy1)]})

axis([-5 5 -5 5])

subplot(2,2,2)

plot(U,V1,'.')

xlabel('U=F_x(x)'), ylabel('V1=F_{y1}(y1)')

title({'samples of [U V1]'; 'the empirical copula of [x y1]'; ['r_{UV1}=' num2str(r_UV1)]})

subplot(2,2,3)

plot(x,y2,'.')

xlabel('x'), ylabel('y2')

title({'samples of [x y2],'; ['r_{xy2}=' num2str(r_xy2)]})

axis([-5 5 -5 5])

subplot(2,2,4)

plot(U,V2,'.')

xlabel('U=F_x(x)'), ylabel('V2=F_{y2}(y2)')

title({'samples of [U V2]'; 'the empirical copula of [x y2]'; ['r_{UV2}=' num2str(r_UV2)]})