Multiple Variable Correlation Coefficient

The idea of the correlation coefficient is the same for oone or multiple dimensions. The equations get a little more complicated since we are now working with an ensemble of input vectors. So the nice form of Eq.1.7 has to be modified. An ensemble of vectors is better described as a matrix, so we are going to define a new matrix U as

where each column is one of the input samples. We are likewise going to define a column vector d with all the desired responses (this is a vector for the single-output regression, otherwise it also becomes a matrix)

The total error variance can be written as

where w* is the set of optimal coefficients. This expression can be easily derived if the output of the regressor is substituted in the definition of the error (Linear Models). The part of the error that is explained by the linear model is the second term. The variance of the output is expressed in the same way (just subtract the mean of the desired signal). Thus if we normalize this equation by the variance of the desired response we get

which leads to the correlation coefficient for the multivariate case.

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