Computation of the Correlation Coefficient

It is important to remember (Eq. 1B.6) that with the optimal coefficients, the regression error, min interpreted as a vector, is perpendicular to the Adaline output y. This condition is called the orthogonality condition. In fact, from the figure below it is easy to see that the smallest error is obtained when the projection of d on y is the orthogonal projection.

During adaptation the error will always be larger than min, meaning that y can be larger than d, so Eq.1.14 may be larger than 1, which is misleading since |r| < 1. Using the fact that the minimum error is perpendicular to y, we can compute the dot product of with y and subtract it from the numerator of Eq.1. 7. We can prove that this new numerator is always smaller than d and that the dot product is zero at the optimal solution and so will not affect the final value of the correlation coefficient. This is exactly what is done in Eq.1.14 .

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