Kernel Ellipsoidal Trimming

Ellipsoid estimation is an issue of primary importance in many practical areas such as control, system identification, visual/audio tracking, experimental design, data mining, robust statistics and novelty/outlier detection. This paper presents a new method of kernel information matrix ellipsoid estimation (KIMEE) that finds an ellipsoid in a kernel defined feature space based on a centered information matrix. Although the method is very general and can be applied to many of the aforementioned problems, the main focus in this paper is the problem of novelty or outlier detection associated with fault detection. A simple iterative algorithm based on Titterington's minimum volume ellipsoid method is proposed for practical implementation. The KIMEE method demonstrates very good performance on a set of real-life and simulated datasets compared with support vector machine methods

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

Dimensionality reduction of clustered data sets

We present a novel probabilistic latent variable model to perform linear dimensionality reduction on data sets which contain clusters. We prove that the maximum likelihood solution of the model is an unsupervised generalisation of linear discriminant analysis. This provides a completely new approach to one of the most established and widely used classification algorithms. The performance of the model is then demonstrated on a number of real and artificial data sets

Likelihood Ratio-Based Detection of Facial Features

One of the first steps in face recognition, after image acquisition, is registration. A simple but effective technique of registration is to align facial features, such as eyes, nose and mouth, as well as possible to a standard face. This requires an accurate automatic estimate of the locations of those features. This contribution proposes a method for estimating the locations of facial features based on likelihood ratio-based detection. A post-processing step that evaluates the topology of the facial features is added to reduce the number of false detections. Although the individual detectors only have a reasonable performance (equal error rates range from 3.3% for the eyes to 1.0% for the nose), the positions of the facial features are estimated correctly in 95% of the face images

Do unbalanced data have a negative effect on LDA?

For two-class discrimination, Xie and Qiu [The effect of imbalanced data sets on LDA: a theoretical and empirical analysis, Pattern Recognition 40 (2) (2007) 557–562] claimed that, when covariance matrices of the two classes were unequal, a (class) unbalanced data set had a negative effect on the performance of linear discriminant analysis (LDA). Through re-balancing 10 real-world data sets, Xie and Qiu [The effect of imbalanced data sets on LDA: a theoretical and empirical analysis, Pattern Recognition 40 (2) (2007) 557–562] provided empirical evidence to support the claim using AUC (Area Under the receiver operating characteristic Curve) as the performance metric. We suggest that such a claim is vague if not misleading, there is no solid theoretical analysis presented in Xie and Qiu [The effect of imbalanced data sets on LDA: a theoretical and empirical analysis, Pattern Recognition 40 (2) (2007) 557–562], and AUC can lead to a quite different conclusion from that led to by misclassification error rate (ER) on the discrimination performance of LDA for unbalanced data sets. Our empirical and simulation studies suggest that, for LDA, the increase of the median of AUC (and thus the improvement of performance of LDA) from re-balancing is relatively small, while, in contrast, the increase of the median of ER (and thus the decline in performance of LDA) from re-balancing is relatively large. Therefore, from our study, there is no reliable empirical evidence to support the claim that a (class) unbalanced data set has a negative effect on the performance of LDA. In addition, re-balancing affects the performance of LDA for data sets with either equal or unequal covariance matrices, indicating that having unequal covariance matrices is not a key reason for the difference in performance between original and re-balanced data

Parsimonious Mahalanobis Kernel for the Classification of High Dimensional Data

The classification of high dimensional data with kernel methods is considered in this article. Exploit- ing the emptiness property of high dimensional spaces, a kernel based on the Mahalanobis distance is proposed. The computation of the Mahalanobis distance requires the inversion of a covariance matrix. In high dimensional spaces, the estimated covariance matrix is ill-conditioned and its inversion is unstable or impossible. Using a parsimonious statistical model, namely the High Dimensional Discriminant Analysis model, the specific signal and noise subspaces are estimated for each considered class making the inverse of the class specific covariance matrix explicit and stable, leading to the definition of a parsimonious Mahalanobis kernel. A SVM based framework is used for selecting the hyperparameters of the parsimonious Mahalanobis kernel by optimizing the so-called radius-margin bound. Experimental results on three high dimensional data sets show that the proposed kernel is suitable for classifying high dimensional data, providing better classification accuracies than the conventional Gaussian kernel