More Properties of the Performance Surface

For a quadratic performance surface Eq.1.9, computing the gradient and equating it to zero finds the value of the coefficients that minimize the cost, that is,

NEURAL AND ADAPTIVE SYSTEMS00000133.gif (1B.7)

or

NEURAL AND ADAPTIVE SYSTEMS00000134.gif (1B.8)

This solution is fundamentally the same as found in Eq.1.6 (b = 0 is equivalent to assuming that the average values of x and d are zero). Substituting this value of w* into Eq.1.9 , the minimum value of the error becomes

NEURAL AND ADAPTIVE SYSTEMS00000135.gif (1B.9)

Eq.1.9 can be rewritten in the form

NEURAL AND ADAPTIVE SYSTEMS00000136.gif (1B.10)

To verify this just sovle Eq. 1B.10 substituting Eq. 1B.8 for w* and Eq. 1B.9 for Jmin. Notice the following observations:

• The minimum value of the error Jmin Eq. 1B.9 depends on both the input signal (xi) and the desired signal (di)

• The location in coefficient space where the minimum w* occurs Eq. 1B.8 also depends on both xi, and di.

• The performance surface shape Eq. 1B.10 depends only on the input signal (xi)

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