GCV COST

Computes the cost associated to the Generalized Cross Validation technique for a given value of lambda.

Contents

Function of MOBY-DIC TOOLBOX.

Description

Consider the regularized least squares problem:

$$ \min_w || G w \textrm{--} d ||^2 + || \lambda L w ||^2 \qquad (1) $$

and H = G'G, Q = L'L;

The choice of parameter lambda is a very critical task. One way to choose it is to minimize the following cost function:

$$\min_\lambda \sum_{k=1}^{n} [Gw_k(k) \textrm{--} d(k) ]^2 \qquad (2) $$

being n the number of elements of array d, $d(k)$ the k-th element of array d and $G w_k(k)$ the k-th element of array $G w_k$, with $w_k$ obtained by solving problem (1) considering all data d except for ith k-th element $d(k)$. It is a sort of prediction error on the data which has not been considered in the optimization problem.

The cost function (2) can be expressed in a more computationally convenient way in this way:

$$ \min_\lambda \frac{1}{n} \sum_{k=1}^n \left\{ \frac{G w(k) \textrm{--} d(k)} {1 \textrm{--} G G\# (k,k)} \right\}^2 $$

being $G G\# (k,k)$ the element in k-th row and k-th column of matrix $G G\#$, and $G\# = (H + \lambda Q)^{\textrm{--}1} G'$.

Syntax

V = GCVcost(H, Q, G, d, w, lambda)

Acknowledgements

Contributors:

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Published with MATLAB® 7.8