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## A note on correlation and covariance matrices

In neural network literature, the matrix in equation 3 is often called a correlation matrix. This can be a bit confusing, since does not contain the correlations between the variables in a statistical sense, but rather the expected values of the products between them. The correlation between xi and xjis defined as

 (18)

see for example[1], i.e. the covariance between xiand xj normalized by the geometric mean of the variances of xiand xj ( ). Hence, the correlation is bounded, . In this tutorial, correlation matrices are denoted .

The diagonal terms of are the second order origin moments, E[xi2], of xi. The diagonal terms in a covariance matrix are the variances or the second order central moments, , of xi.

The maximum likelihood estimator of is obtained by replacing the expectation operator in equation 18 by a sum over the samples. This estimator is sometimes called the Pearson correlation coefficient after K. Pearson[16].

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Magnus Borga
1999-10-29