A Direct Estimation Approach to Sparse Linear Discriminant Analysis

This paper considers sparse linear discriminant analysis of high-dimensional data. In contrast to the existing methods which are based on separate estimation of the precision matrix \O and the difference \de of the mean vectors, we introduce a simple and effective classifier by estimating the product \O\de directly through constrained $\ell_1$ minimization. The estimator can be implemented efficiently using linear programming and the resulting classifier is called the linear programming discriminant (LPD) rule. The LPD rule is shown to have desirable theoretical and numerical properties. It exploits the approximate sparsity of \O\de and as a consequence allows cases where it can still perform well even when \O and/or \de cannot be estimated consistently. Asymptotic properties of the LPD rule are investigated and consistency and rate of convergence results are given. The LPD classifier has superior finite sample performance and significant computational advantages over the existing methods that require separate estimation of \O and \de. The LPD rule is also applied to analyze real datasets from lung cancer and leukemia studies. The classifier performs favorably in comparison to existing methods.Comment: 39 pages.To appear in Journal of the American Statistical Associatio

Optimal Clustering under Uncertainty

Classical clustering algorithms typically either lack an underlying probability framework to make them predictive or focus on parameter estimation rather than defining and minimizing a notion of error. Recent work addresses these issues by developing a probabilistic framework based on the theory of random labeled point processes and characterizing a Bayes clusterer that minimizes the number of misclustered points. The Bayes clusterer is analogous to the Bayes classifier. Whereas determining a Bayes classifier requires full knowledge of the feature-label distribution, deriving a Bayes clusterer requires full knowledge of the point process. When uncertain of the point process, one would like to find a robust clusterer that is optimal over the uncertainty, just as one may find optimal robust classifiers with uncertain feature-label distributions. Herein, we derive an optimal robust clusterer by first finding an effective random point process that incorporates all randomness within its own probabilistic structure and from which a Bayes clusterer can be derived that provides an optimal robust clusterer relative to the uncertainty. This is analogous to the use of effective class-conditional distributions in robust classification. After evaluating the performance of robust clusterers in synthetic mixtures of Gaussians models, we apply the framework to granular imaging, where we make use of the asymptotic granulometric moment theory for granular images to relate robust clustering theory to the application.Comment: 19 pages, 5 eps figures, 1 tabl

Design of an Adaptive Classification Procedure for the Analysis of High-Dimensional Data with Limited Training Samples

In a typical supervised classification procedure the availability of training samples has a fundamental effect on classifier performance. For a fixed number of training samples classifier performance is degraded as the number of dimensions (features) is increased. This phenomenon has a significant influence on the analysis of hyperspectral data sets where the ratio of training samples to dimensionality is small. Objectives of this research are to develop novel methods for mitigating the detrimental effects arising from this small ratio and to reduce the effort required by an analyst in terms of training sample selection. An iterative method is developed where semi-labeled samples (classification outputs) are used with the original training samples to estimate parameters and establish a positive feedback procedure wherein parameter estimation and classification enhance each other in an iterative fashion. This work is comprised of four discrete phases. First, the role of semi-labeled samples on parameter estimates is investigated. In this phase it is demonstrated that an iterative procedure based on positive feedback is achievable. Second, a maximum likelihood pixel-wise adaptive classifier is designed. Third, a family of adaptive covariance estimators is developed that combines the adaptive classifiers and covariance estimators to deal with cases where the training sample set is extremely small. Finally, to fully utilize the rich spectral and spatial information contained in hyperspectral data and enhance the performance and robustness of the proposed adaptive classifier, an adaptive Bayesian contextual classifier based on the Markov random field is developed

The tangent classifier

This is an Author's Accepted Manuscript of an article published in The American Statistician 66.3 (2012): 185-194 Copyright Taylor and Francis, available online at: http://www.tandfonline.com