Gaussian Copula and the addition of an Identity matrix - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T07:35:50Z http://mathoverflow.net/feeds/question/96007 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/96007/gaussian-copula-and-the-addition-of-an-identity-matrix Gaussian Copula and the addition of an Identity matrix Moe 2012-05-04T18:40:41Z 2012-08-26T16:06:43Z <p>When I was looking at the Gaussian Copula Example @ <a href="http://en.wikipedia.org/wiki/Copula_(probability_theory" rel="nofollow">http://en.wikipedia.org/wiki/Copula_(probability_theory</a>)</p> <p>I realized the Gaussian Copula is stated as follow $$C^{Gauss}_\Sigma (u) = \frac{1}{\sqrt\det{\Sigma}} \exp{\Bigg ( -\frac{1}{2} \begin{pmatrix} \Phi^{-1}(u_1) \ \dots \ \Phi^{-1}(u_d)\end{pmatrix}^T. (\Sigma^{-1} - I).\begin{pmatrix} \Phi^{-1}(u_1) \ \dots \ \Phi^{-1}(u_d)\end{pmatrix} \Bigg) }$$ where $\Sigma$ is the correlation matrix, $\Phi^{-1}$ is the inverse cumulative distribution function of a standard normal and $I$ is the identity matrix.</p> <p>The question is, why is there an identity matrix in the exponential form?</p> <p>Thank you</p> http://mathoverflow.net/questions/96007/gaussian-copula-and-the-addition-of-an-identity-matrix/96009#96009 Answer by mike for Gaussian Copula and the addition of an Identity matrix mike 2012-05-04T18:55:43Z 2012-05-04T18:55:43Z <p>It's the jacobian.</p> http://mathoverflow.net/questions/96007/gaussian-copula-and-the-addition-of-an-identity-matrix/105548#105548 Answer by mathtick for Gaussian Copula and the addition of an Identity matrix mathtick 2012-08-26T16:06:43Z 2012-08-26T16:06:43Z <p>What's a good reference for this derivation (online). Wikipedia only has an advertisement for someone's book as a reference. </p> <p>I find this presentation very confusing, for example I had missing that this was the density (small c) as opposed to the CDF and was obvious not making any sense of it. I can imagine other readers doing the same thing.</p>