# Non-Negative Matrix Factorization - Estimating the Mean

How do you estimate the mean in a Non-Negative Matrix Factorization framework?

It is obvious and well known how to estimate the covariance matrix, how ever I also need the estimated mean. I'm assuming there is a way to estimate the mean via regression as there is with similar algorithm's such as PCA and this should already be derived for Non-Negative Matrix Factorization. I just can't find it in the literature and common frameworks do not provide this and only return the matrix factorization (MatLab, R ,Python exc...).

Using the terminology from MatLab.

A represents an NxM Matrix (Could be 1 + Returns or Volumes or generally any positive matrix). One can compute nnmf(A) to get [W, H, D] where W*H is a lower-rank approximation to A.

Is this as simple as computing the direct mean and covariance of W*H or am I missing something?

• I am aware of the PCA of a covariance matrix and I have seen NNMF being applied to data. But how is NNMF applied to a covariance matrix? A covariance matrix is usually negative too. Is it transformed? In my experience NNMF is hard to interpret. – Richard Feb 24 '15 at 16:27
• Please clarify which mean you want to estimate - Thank you – vonjd Feb 24 '15 at 17:36
• @vonjd Sure I updated the question a bit, I want the mean from the factorized result set W, H , D above. – pyCthon Feb 24 '15 at 18:52