## Optimal Transport vs. Fisher-Rao distance between copulas for clustering multivariate time series

### 6 June 2016 by Gautier MARTI

Copulas are distributions which encode the dependence between random variables. Because it is complicated to work with the whole distribution, one usually extracts meaningful numbers from it called correlation / association / dependence coefficients. These numbers only measure particular aspects of the dependence structure: monotone association, tail-dependence, etc. When extracted from copulas that encode a more complex dependence pattern, they can even be misleading. It motivates the need to develop tools to deal with the whole distribution. We may need to:

- compare two dependence structures (distance between two copulas);
- summarize several dependence structures (barycenter of several copulas);
- extract the strength of the association between the variables (projection of the copula onto [0,1]).