Copula its applications
Copula & its applications
What is Copula? Definition Copulas are statistical tools that factorize multivariate distributions into the product of its marginals and a function that captures any possible form of dependence among them(marginals). This function is referred to as the copula, and it links the marginals together into the joint multivariate model
What is Copula? • Definition Copulas are statistical tools that factorize multivariate distributions into the product of its marginals and a function that captures any possible form of dependence among them (marginals). This function is referred to as the copula, and it links the marginals together into the joint multivariate model
What is Copula? Mathematical formulation p(x)=I(x;)c(Px),…,P(za).( 1 copula P(Xi is the marginal cdf of the random variable Xi Interestingly, this density has uniform marginals, since P(z)U[0; 1] for any random variable Z When P(X);.; P(Xa are continuous, the copula c( )is unique
What is Copula? • Mathematical formulation: P(xi ) is the marginal cdf of the random variable xi . Interestingly, this density has uniform marginals, since P(z)~ U[0; 1] for any random variable z. When P(x1 ); … ; P(xd ) are continuous, the copula c(.) is unique (2)
Especially, When factorizing multivariate densities into a product of marginal distributions and bivariate copula functions (called as vines) Each of these factors corresponds to one of the building blocks that are assumed either constant or varying across different learning domains applicable to DA, TL and mtL
Especially, when factorizing multivariate densities into a product of marginal distributions and bivariate copula functions (called as vines). Each of these factors corresponds to one of the building blocks that are assumed either constant or varying across different learning domains. → applicable to DA, TL and MTL!
Characteristics Infinitely many multivariate models share the same underlying copula function Figure 1: Left, sample from a Gaussian copula with correlation p=0.8. Middle and right, two samples drawn from multivariate models with this same copula but different marginal distributions, depicted as rug plots
Characteristics Infinitely many multivariate models share the same underlying copula function!