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APPENDIX II: DISTRIBUTION OF THE LEAST-SQUARES SUM

We shall define the vector Zi ident yi / sigmai and the matrix Fij ident fj(xi) / sigmai.

Note that H = FT . F by Eq. (27),

Equation 33     (33)

Then

Equation 34     (34)

where the unstarred alpha is used for alpha0.

Equation

using Eq. (34). The second term on the right is zero because of Eq. (33).

Equation 35     (34)

Note that

Equation

If qi is an eigenvalue of Q, it must be equal qi2, an eigenvalue of Q2. Thus qi = 0 or 1. The trace of Q is

Equation

Since the trace of a matrix is invariant under a unitary transformation, the trace always equals the sum of the eigenvalues of the matrix. Therefore M of the eigenvalues of Q are one, and (p - M) are zero. Let U be the unitary matrix which diagonalizes Q (and also (1 - Q)). According to Eq. (35),

Equation

Thus

Equation

where S* is the square of the radius vector in (p - M)-dimensional space. By definition (see Section 16) this is the chi2 distribution with (p - M) degrees of freedom.

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