Past exam of the mathematical and Computational Biology course of the University of Cambridge / 2024 / Paper 1 / B3 / b

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- Solution b

Solution

Apply the least-squares normal equations to the transformed matrix and vector:

(W A)^{T} (W A) x_{w} = (W A)^{T} Wb,

or equivalently

A^{T} W^{T} W A x_{w} = A^{T} W^{T} Wb .

Here

W^{T} W = diag (w_{1}^{2}, \dots, w_{m}^{2})

. A unique minimizing vector exists when the weighted design matrix has full column rank; the equations themselves remain valid without that assumption.

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