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I'm reading through this article (http://www.sciencedirect.com/science/article/pii/S0047259X06000662) where they have a population version of the Spearman rank correlation. … I'm trying to teach myself some basic copula theory so there are a few holes in my knowledge here and there. …

I read that for continuous margins, then Rank correlation only depend on copula and not on the margins. However, it is not the case for the non-continuous margins. … And why it then does not only depend on copula? Any idea? …

From the above, we can understand that a non-zero Spearman rank correlation coefficient implies that the rank-transformed variables $g(X)$ and $g(Y)$ are mutually dependent. … Thus ranking should not change the copula. If the copula was not the independence copula, then it is a copula that describes some sort of dependence. …

Use these quantiles on the CDF (or eCDF) of the desired target distribution Take a rank correlation of the resulting data to verify the correlation structure is preserved. … This article suggests it is the case -- under "Using Rank Correlation Coefficients": The correlation parameter, $\rho$, of the underlying bivariate normal determines the dependence between X1 and X2 …

Pair-copula is the way to model higher dimensional data with high complex dependencies, using only 2 varialbes at a time. Sampling from pair-copula is quite complex and need a careful understanding. … Also, we know that copula depends on rank correlation (e.g., kendall tau). Kendall tau can use to find the correlation between two varialbes. …

It is common to talk about the linear correlation, Pearson's $r$, between two random variables $\{(x_1,y_1),(x_2,y_2),\ldots,(x_n,y_n)\}$ as having two components: a) the copula and b) the marginal distributions … In contrast, the rank correlation, Spearman's $\rho$, depends only on the copula. …

I have two non-parametric rank correlations matrices emp and sim (for example, based on Spearman's $\rho$ rank correlation coefficient): library(fungible) emp <- matrix(c( 1.0000000, 0.7771328, 0.6800540 … Joël Bun, Jean-Philippe Bouchaud and Mark Potters (2016), Cleaning correlation matrices, Risk.net, April 2016 Li, David X., On Default Correlation: A Copula Function Approach (September 1999). …

For a start, with these distributions you'd have to be looking at specifying some copula or copulas, presumably with close to a linear relationship in the untransformed variables, and certainly with close … It would seem odd to discuss that particular test of the Pearson correlation without examining alternative tests - for example, permutation tests of the Pearson correlation, rank tests like Kendall's tau …

Note that R below is the correlation matrix for the multivariate normal distribution on which the copula is based. It is not the correlation of the copula itself, nor it is the rank correlation. … correlation as the copula. [~,J] = sort(U); Y = zeros(size(U)); Y(J(:,1),1) = sort(commodity1); Y(J(:,2),2) = sort(commodity2); Y(J(:,3),3) = sort(commodity3); imposed_rankcorr = corr(Y,'type','spearman …

You choose the copula by having the right rank correlation coefficient, for example Kendall's $\tau$. … You must use rank correlation because then the nonlinear (but monotone) transformation we are going to use, will preserve the correlation. …

These parameters wouldn't have been Spearman correlation coefficients. That's the reason I asked if you were using Gaussian copula and Spearman, which does use Spearman correlation matrix as input. … Matlab's t-copula uses correlation parameters which it gets from rank correlation coefficients, not Spearman. The other way to fit copulas is to fit them strauight to data with copulafit function. …

While Pearson correlation is not well defined in this case, because it depends on the marginal distribution (and might not even exist), the rank correlations of the generated variables are defined by the … Since most of the common copula families have a free parameter, you can adjust it to get the desired rank correlation. …

the trick used with copulas to estimate/describe correlation structure is to use rank correlations, they are preserved under (increasing) monotone transformations. … correlation but they usually won't correspond closely to a Pearson correlation. – Glen_b In general there is way (at least in principle) to find the Pearson correlation(s) for a copula - just as you …

Besides using Pearson correlation, you can also use rank correlation such as Spearman or Kendall correlation. … You can also display the scatterplot of the ranks whose distribution (called the empirical copula) is an estimator of the underlying copula encoding the `true' dependence between your time series. …

More generally, how do we choose which copula model to use in a given problem? The main guiding principle I learned is to choose a copula model based on the dependence structure of the variables. … A copula is called comprehensive if the copula allows for any dependence structure from full negative rank correlation to full positive rank correlation and also allows for independence. …