# Generalized correlation coefficients

Assume there are given two Gaussian random vectors $$\boldsymbol{x}$$ and $$\boldsymbol{y}$$ of equal length $$N$$ with corresponding means $$\boldsymbol{\mu}_x$$, $$\boldsymbol{\mu}_y$$ and covariance matrices $$\boldsymbol{C}_{xx}$$, $$\boldsymbol{C}_{yy}$$ as well as their cross covariance matrix $$\boldsymbol{C}_{xy} = \boldsymbol{C}_{yx}^T$$ which in the case of equal length is a square matrix.

Stacking them together yields a new random vector that is again Gaussian $$\boldsymbol{z} = \left( \begin{array}{rr} \boldsymbol{x} \\ \boldsymbol{y} \\ \end{array} \right)$$ with mean $$\boldsymbol{\mu}_z = \left( \begin{array}{rr} \boldsymbol{\mu}_x \\ \boldsymbol{\mu}_y \\ \end{array} \right)$$ and covariance matrix $$\boldsymbol{C}_{zz} = \left( \begin{array}{rr} \boldsymbol{C}_{xx} & \boldsymbol{C}_{xy} \\ \boldsymbol{C}_{xy}^T & \boldsymbol{C}_{yy} \\ \end{array} \right)$$.

I would now like to express $$\boldsymbol{C}_{xy}$$ by some sort of generalized correlation coefficients $$\rho_1, \cdots, \rho_N$$, $$\rho_i \in [-1, 1]$$ , that describe the correlation between $$x_i$$ and $$y_i$$. I.e. I´d be interested in something like $$\boldsymbol{C}_{xy} = f(\boldsymbol{C}_{xx}, \boldsymbol{C}_{yy}, \rho_1, \cdots, \rho_N)$$. This should be in analogy to the definition of the correlation coefficient for two scalar random variables $$\rho_{x,y}$$, where it is possible to write

$$\boldsymbol{z} = \left( \begin{array}{rr} x \\ y \\ \end{array} \right)$$, $$\boldsymbol{\mu}_z = \left( \begin{array}{rr} \mu_x \\ \mu_y \\ \end{array} \right)$$ and $$\boldsymbol{C}_{zz} = \left( \begin{array}{rr} \sigma^2_x & \rho_{x,y} \sigma_x \sigma_y \\ \rho_{x,y} \sigma_x \sigma_y & \sigma^2_y \\ \end{array} \right)$$.

However, I am not sure if this is possible, at least I have not found anything relevant yet in this regard.

Thanks for your help and constributions.

Let's take the simplest case of $$N=1$$. Just because $$X$$ and $$Y$$ are Gaussian random variables, it is not necessarily the case that $$X$$ and $$Y$$ have a jointly Gaussian distribution. See, for example, this answer on stats.SE for several examples of bivariate nonGaussian distributions whose marginal distributions are nonetheless Gaussian. (Full disclosure: one of the examples is attributed to an answer of mine on stats.SE). But, let's ignore such nitpicks and assume that $$X$$ and $$Y$$ do indeed enjoy a bivariate jointly Gaussian distribution and so one parameter $$\rho_{X,Y}$$ suffices to specify the $$2\times 2$$ covariance matrix. We note that $$N$$ equals $$1$$, and so we think that $$N$$ parameters will be needed in the general case also.
Generalizing to the case of $$N>1$$, we have the $$N\times N$$ cross-covariance matrix $$C_{X,Y} = E[(X-\mu_x)^T(Y-\mu_Y)]$$ which has $$N^2$$ entries and the only constraints on the entries are that $$|C_{X,Y}[i,j]| \leq \sigma_{X,i}\sigma_{Y,j}$$ for all choices of $$i$$ and $$j$$ and that the covariance matrix $$\left[\begin{matrix} C_{X,X} & C_{X,Y}\\ C_{Y,X} & C_{Y,Y}\end{matrix}\right]$$ be nonnegative definite. Thus, we have $$N^2$$ parameters $$\rho_{i,j}$$ that need to be specified; not $$N$$ as the OP wants. Put another way, we were misled into thinking that $$N$$ parameters were needed in general because we were generalizing from the false base case "number of parameters equals dimension" (which is true when $$N=1$$ but not for $$N>1$$) instead of the correct base case "number of parameters equals the square of the dimension" (which is true when $$N=1$$ and also when $$N>1$$).
Thank you for this information. I already had a feeling that N coefficients might not suffice. The reason I asked for this was that I wanted to find $$\boldsymbol{C}_{\boldsymbol{xy}}$$ based on a given degree of fit between $$\boldsymbol{x}$$ and $$\boldsymbol{y}$$. Meanwhile, I think I found a way to do it: Based on the relation $$\textrm{Cov}[\boldsymbol{x}-\boldsymbol{y}] = \boldsymbol{C}_{\boldsymbol{xx}} + \boldsymbol{C}_{\boldsymbol{yy}} - 2 \boldsymbol{C}_{\boldsymbol{xy}}$$ I can specify some $$\boldsymbol{C}_{\boldsymbol{dd}}$$ and then set $$\boldsymbol{C}_{\boldsymbol{xy}} = \frac{1}{2}(\boldsymbol{C}_{\boldsymbol{xx}} + \boldsymbol{C}_{\boldsymbol{yy}} - \boldsymbol{C}_{\boldsymbol{dd}})$$. Basically $$\boldsymbol{C}_{\boldsymbol{dd}}$$ now constains the information of the $$N^2$$ coerrelation coefficients. (still, I need to check that that the resulting covariance matrix $$C_{\boldsymbol{zz}}$$ is positive semidefinite though).