Online-estimate of squared correlation coefficient for linear optimization

blue2script

New Member
Summary: Is there a way to directly estimate the squared correlation between two zero-mean Gaussian variables on-line and without storing more than two samples at any given time?

Dear all,
I am working on an optimization procedure to minimize the correlation between two zero-mean normally distributed variables x and y. The optimization has to happen online in every single time-step of the dynamical system that connect x and y and is not allowed to use any memory of past states (i.e. I can't store any long-term estimate of some parameter).

As an example, assume we want to maximize the variance of x. The objective to maximize would be

$$L = x^2$$

The expectation value of L (over the distribution of x) is the variance. The same, however, is not possible for the correlation between x and y. The analogue would be

$$L_c = (xy)^2$$

However, the expectation value of x^2y^2 is

$$E[x^2y^2] = E[x^2]*E[y^2] + 2E[xy]^2$$

The first term is the product of the individual variances of x and y whereas the last term is the (desired) squared correlation coefficient between xy. So, optimizing $L_c$ will usually lead to a reduction of the variances of x and y, which is not at all desired (in fact, they should stay constant). This would be solvable if we have a good estimate of the variance of x and y (so we can subtract its product from the equation above to get the true squared of the correlation). However, I assume that the only thing the optimization process can use are two subsequent samples (not independent!) of x and y (subsequent from the perspective of a dynamical system through which x and y evolve). So I can't have any long-term estimate of the individual variances.

Is there a way to directly estimate the squared correlation between two zero-mean Gaussian variables on-line and without storing more than two samples at any given time?

I apologize for not being extremely clear - I have a hard time to phrase this problem in hard mathematical terms.