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qdc

QDC

### Quadratic Bayes Normal Classifier (Bayes-Normal-2)

[W,R,S,M] = QDC(A,R,S,M)
[W,R,S,M] = A*QDC([],R,S,M)
[W,R,S,M] = A*QDC(R,S,M)

 Input A Dataset R,S Regularisation parameters, 0 <= R,S <= 1 (optional; default: no regularisation, i.e. R,S = 0) M Dimension of subspace structure in covariance matrix (default: K,  all dimensions), or fraction of variance to be preserved.

 Output W Quadratic Bayes Normal Classifier mapping R Value of regularisation parameter R as used S Value of regularisation parameter S as used M Value of regularisation parameter M as used

### Description

Computation of the quadratic classifier between the classes of the dataset  A assuming normal densities. R and S (0 <= R,S <= 1) are regularisation  parameters used for finding the covariance matrix by

G = (1-R-S)*G + R*diag(diag(G)) + S*mean(diag(G))*eye(size(G,1))

This covariance matrix is then decomposed as G = W*W' + sigma^2 * eye(K),  where W is a K x M matrix containing the M leading principal components  and sigma^2 is the mean of the K-M smallest eigenvalues.

The use of soft labels is supported. The classification A*W is computed by  NORMAL_MAP.

If R, S or M is NaN the regularisation parameter is optimised by REGOPTC.  The best result are usually obtained by R = 0, S = NaN, M = [], or by  R = 0, S = 0, M = NaN (which is for problems of moderate or low dimensionality  faster). If no regularisation is supplied a pseudo-inverse of the  covariance matrix is used in case it is close to singular. It is better  to avoid this and use always a small value for S, e.g. 1e-8.

### Example(s)

prex_mcplot, prex_plotc,

### Reference(s)

1. R.O. Duda, P.E. Hart, and D.G. Stork, Pattern classification, 2nd edition, John Wiley and Sons, New York, 2001.
2. A. Webb, Statistical Pattern Recognition, John Wiley && Sons, New York, 2002.