Calculate the conditional distribution using copulas

Hi all,

As I have determined the best fitted copula to a data set, I need to determine the conditional distributions of these copulas:

For this determination I want to calculate the PDF of the distributions of X and Y=y and determine het cumulative integral of the PDF's. The distribution I should use for the conditional distribution is 'Gumbel'.

gaus_cdf = copulacdf('Gaussian',[x(:),y(:)],0.6436);

clay_cdf = copulacdf('Clayton',[x(:),y(:)],a_clay);

gumb_cdf = copulacdf('Gumbel',[x(:),y(:)],d_gumb);

t_cdf = copulacdf('t',[x(:),y(:)],0.7055,nu);

frank_cdf = copulacdf('Frank',[x(:),y(:)],a_frank);

gaus_pdf = copulapdf('Gaussian',[x(:),y(:)],0.6436);

clay_pdf = copulapdf('Clayton',[x(:),y(:)],a_clay);

gumb_pdf = copulapdf('Gumbel',[x(:),y(:)],d_gumb);

t_pdf = copulapdf('t',[x(:),y(:)],0.7055,nu);

frank_pdf = copulapdf('Frank',[x(:),y(:)],a_frank);

surf(x,y,reshape(gumb_pdf,50,50))

%surf(x,y,ecdf)

xlabel('Temperature')

ylabel('Sunshine duration')

title('CDF Gumbel Copula')

Using the code above, the PDF's and the CDF's can be displayed in 3D, but I need the copulas depictered in 2D. Can anyone help me how to manage this correctly? The plot should like a bit like this:

In this plot, the conditional distribution of P(X=x | Y=y_50) (blue line) and P(X=x|Y=y_90) (pink line) should be plotted.

Thank you!

Answer 1

[Alan]

Hi Murielle,

From what I understand, you aim to plot the conditional probability derived from a Gumbel copula given values of Y. First, we can get the copula PDF using the “copulapdf” function which can be utilized as follows:

x = 0:0.1:1; 
y=x; 
[x, y] = meshgrid(x, y); 
gumb_pdf = copulapdf('Gumbel', [x(:), y(:)], 2);     % Using 2 as alpha 
gumb_pdf = reshape(gumb_pdf, 11, 11); 

Then, we can obtain the conditional probabilities by selecting the rows that correspond to the required Y values:

gumb_pdf_y_50 = gumb_pdf(6, :); % Selecting the 6th row for y=0.5  
gumb_pdf_y_90 = gumb_pdf(end-1, :); % Selecting the 10th row for y=0.9 

Finally, we can visualize the probabilities using “plot”:

plot(0:0.1:1, gumb_pdf_y_50);  
hold on;  
plot(0:0.1:1, gumb_pdf_y_90);  
legend(["Y=0.5", "Y=0.9"], 'Location','northwest');  
hold off; 

The same can be done with CDFs by using the "copulacdf" function.

I hope this helped.