How to select k elements from two vectors of length n > k that maximize the correlation between the two vectors?

Hello everyone,

I have two numerical column vectors of length n, say vector A and vector B.

Consider a third column vector C which has k < n values equal to one, and zero otherwise.

I want C such that (A.*C)'*(B.*C) is maximized.

In other words, I need to decide which rows of A and B needs to remain (k of them needs to remain, n-k needs to me removed) to maximize the correlation between the vectors A and B. (In this sense, C can also be logical, 1 if row is in, zero if row is out)

Can you please help me to implement this maximization?

Many thanks for your help,

Marco

[John D'Errico]

First, what you asked to compute, thus a correlation, is not the same thing as a dot product. The formula you show is a dot product. So which is your real question?

Next, The search that you describe can be written in the form of an integer (binary) linear programming problem, at least it can if the problem is just a maximal dot product. In that case, your problem can be trivially solved using intlinprog. But is this really a question about correlation?

[Marco Brianti]

Dear John,

thanks for your reply. It is about the correlation. My point is that (A.*C)'*(B.*C) is somehow proportional to corr(A.*C,B.*C). Note that bewteen (A.*C)' and (B.*C) there is no dot-produc but only product. I am transposing the first vector such that the outcome is a scalar.

Does it make sense?

Many thanks,

Marco

[Marco Brianti]

By the way, I solve the issue by using intlinprog. Thanks for the help!