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Abstract:
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We consider the problem of supervised classification with discriminant analysis for an ill-posed problem, i.e. when sample size of any class is less than the dimension of the measurement space ($n_k < p$). Usual discriminant analysis uses the plug-in maximum likelihood estimator for the covariance matrix which is known to perform poorly in poorly-posed problems or singular in ill-posed problems. We propose an orthogonally equivariant covariance matrix estimator which provides shrinked estimate of eigenvalues. To do so, we consider the Singular Wishart Distribution, then derive an approximation to the marginal likelihood (with respect to eigenvalues) and solve the Maximum Likelihood Equations based on this approximate marginal likelihood to estimate the eigenvalues. Our estimator is still singular, therefore, we regularize our estimator using additional parameters and find optimal solutions of these parameters via cross-validation under the usual mis-classification loss function. We evaluate our method using various standard metrics of classification via simulations and real data.
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