Hello everyone,

Since I am working with uncorrelated data but quite dependant, I had a look at how UQLAB perform statistical hypothesis test before finding an appropriate copula to the input data. What I have seen is that the “independent tests” that UQLAB uses such as *Kendall*, *Spearman* or *Pearson*, are actually tests for deciding whenever there is or not a significant correlation between variables.

Worse luck, uncorrelation is not the same as independence. There exist several easy counterexamples to illustrate this. For example, let X~U(-1,1) or X~N(0,1) and Y=X^2. Or a classical one could be to take X=cos(U), Y=sin(U), where U~(0,2*pi).

A way to test dependency is the classical **chi2** test, which can be done in Matlab using crosstab. This test is designed for categorical data but can be adapted to the continuous case grouping data in blocks.

- Is maybe UQLAB only performing correlation test, because copulas only remove correlation but not dependency between data?

For the sake of illustration, I let here the code to perform an “independence” test over uncorrelated dependent data with the UQLAB function **uq_test_block_independence()**

```
rng(1)
n=1e4;
u=(2*pi*rand(n,1));
x=sin(u);
y=cos(u);
corr(x,y) % Theoretically uncorelated. Compute E[XY], E[X]=E[Y]=0.
scatter(x,y); % Data has not even 2D support since they are "very" dependant
uqlab
uq_test_block_independence([x y], 0.05)
% -> indep. groups: 1, 2
%
% ans =
%
% 1×2 cell array
%
% {[1]} {[2]}
```