# What is this “maximum correlation coefficient”?

A typical image processing statistic is the use of Haralick texture features, which are 14.

I am wondering about the 14th of these features: Given an adjacency map $P$ (which we can simply view an the empirical distribution of two integers $i,j < 256$), it is defined as: the square root of the second eigenvalue of $Q$, where $Q$ is:

$Q_{ij} = \sum_k \frac{ P(i,k) P(j,k)}{ [\sum_x P(x,i)] [\sum_y P(k, y)] }$

Even after much googling, I could not find any references for this statistic. What are its properties? What is it representing?

(The value $P(i,j)$ above is the normalised number of times that a pixel of value $i$ is found next to a pixel of value $j$).

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I am guessing that matrix $Q$ is stochastic, hence the maximal eigenvalue is 1. Since $Q$ elements are correlations, the second eigenvalue will be a maximum correlation in analogue to principal components, where the squared eigenvalue corresponds to the variance of principal component, which in turn is the linear combination of the columns of the matrix, or something to that effect. – mpiktas Apr 20 '11 at 10:29
@mpiktas Almost. Actually, the rhs is in the form $P \cdot P^T$ where $P$ is stochastic. This is needed to make $Q$ positive definite. Now its maximum eigenvalue usually exceeds unity but its second one does not--and is guaranteed to lie between 0 and 1. $Q$ really is a covariance matrix with a constant added to each term. – whuber Apr 20 '11 at 14:00