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This paper proposes a new classification technique, called support feature machine (SFM), for multidimensional time-series data. The proposed technique was applied to the classification of abnormal brain activity represented in electroencephalograms (EEGs). First, the dynamical properties of EEGs from each electrode were extracted. These dynamical profiles were put in SFM, which is an optimization model that maximizes classification accuracy by selecting electrodes (features) that correctly classify unlabeled EEG samples based on the nearest-neighbor classification rule. The empirical studies were performed on the EEG data sets collected from 10 subjects. The performance of SFM was assessed and compared with the ones achieved by the traditional k-nearest-neighbor classifier and support vector machines (SVMs). The results show that SFM achieved, on average, over 90% correct classification and outperformed other classification techniques. In the validation step, SFM correctly classified unseen preseizure and normal EEGs with over 73% accuracy.

In this paper, we propose a new optimization framework for improving feature selection in medical data classification. We call this framework Support Feature Machine (SFM). The use of SFM in feature selection is to find the optimal group of features that show strong separability between two classes. The separability is measured in terms of inter-class and intra-class distances. The objective of SFM optimization model is to maximize the correctly classified data samples in the training set, whose intra-class distances are smaller than inter-class distances. This concept can be incorporated with the modified nearest neighbor rule for unbalanced data. In addition, a variation of SFM that provides the feature weights (prioritization) is also presented. The proposed SFM framework and its extensions were tested on 5 real medical datasets that are related to the diagnosis of epilepsy, breast cancer, heart disease, diabetes, and liver disorders. The classification performance of SFM is compared with those of support vector machine (SVM) classification and Logical Data Analysis (LAD), which is also an optimization-based feature selection technique. SFM gives very good classification results, yet uses far fewer features to make the decision than SVM and LAD. This result provides a very significant implication in diagnostic practice. The outcome of this study suggests that the SFM framework can be used as a quick decision-making tool in real clinical settings.

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Discrete k-median (DKM) clustering problems arise in many real-life applications that involve time-series data sets, in which nondiscrete clustering methods may not represent the problem domain adequately. In this study, we propose mathematical programming formulations and solution methods to efficiently solve the DKM clustering problem. We develop approximation algorithms from a bilinear formulation of the discrete k-median problem using an uncoupled bilinear program algorithm. This approximation algorithm, which we refer to as DKM-L, is composed of two alternating linear programs, where one can be solved in linear time and the other is a minimum cost assignment problem. We then modify this algorithm by replacing the assignment problem with an efficient sequential algorithm for a faster approximation, which we call DKM-S. We also propose a compact exact integer formulation, DKM-I, and a more efficient network design-based exact mixed-integer formulation, DKM-M. All of our methods use arbitrary pairwise distance matrices as input. We apply our methods to simulated single-variate and multivariate random walk time-series data. We report comparative clustering performances using normalized mutual information (NMI) and solution speeds among the DKM methods we propose. We also compare our methods to other clustering algorithms that can operate with distance matrices, such as hierarchical cluster trees (HCT) and partition around medoids (PAM). We present NMI scores and classification accuracies of our DKM algorithms compared to HCT and PAM using five different distance measures on simluated data, as well as public benchmark and real-life neural time-series data sets. We show that DKM-S is much faster than HCT, PAM, and all other DKM methods and produces consistently good clustering results on all data sets.

This chapter is focused on recent advances in mathematical programming methodologies in data mining research, which is a rapidly emerging interdisciplinary research area. The main focus of this review chapter lies on classification (supervised learning) and clustering (unsupervised learning), which are among the most studied data mining tasks. We give a thorough discussion on the mathematical modeling aspect of classification and clustering problems.

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