María Teresa Gallegos

A robust method for cluster analysis

Let there be given a contaminated list of n R d -valued observations coming from g different, normally distributed populations with a common covariance matrix. We compute the ML-estimator with respect to a certain statistical model with n - r outliers for the parameters of the g populations; it detects outliers and simultaneously partitions their complement into g clusters. It turns out that the estimator unites both the minimum-covariance-determinant rejection method and the well-known pooled determinant criterion of cluster analysis. We also propose an efficient algorithm for approximating this estimator and study its breakdown points for mean values and pooled SSP matrix.

Diagnostics and characterisation of preocclusive stenoses and occlusions of the internal carotid artery with B-flow.

The purpose was to evaluate whether B-flow can improve the ultrasonographic diagnosis of preocclusive stenosis and occlusion of the internal carotid artery (ICA) compared with colour-coded Doppler and power Doppler. Ninety patients with occlusions or preocclusive stenoses of the ICA suspected by Doppler sonography were examined with B-flow in comparison with colour-coded Doppler sonography (CCDS), power Doppler (PD) and intra-arterial digital subtraction angiography (DSA). Intrastenotic flow detection and lengths of stenoses were the main criteria. Ulcerated plaques found by surgery in 42/90 patients were compared by ultrasonography (US). Diagnosis of ICA occlusion with CCDS, PD and B-flow was correct in all 42 cases. A preocclusive ICA stenosis in DSA was detected correctly in all 48/48 cases (100%) for B-flow, in 44/48 (92%) for PD and in 39/48 (81%) for CCDS. Surgical findings showed in 17/42 cases ulcerated plaques; 15/17 (89%) of these cases were detected with B-flow, 12/17 (71%) with PD, 10/17 (59%) with CCDS, and 8/17 (47%) with DSA. With B-flow the extent of stenosis was appraised more precisely than with PD and CCDS (P<0.0001). In conclusion, B-flow is a reliable method for preocclusive stenosis of the ICA with less intrastenotic flow artefacts. B-flow facilitates the characterization of plaque morphologies.

Trimming algorithms for clustering contaminated grouped data and their robustness

We establish an affine equivariant, constrained heteroscedastic model and criterion with trimming for clustering contaminated, grouped data. We show existence of the maximum likelihood estimator, propose a method for determining an appropriate constraint, and design a strategy for finding reasonable clusterings. We finally compute breakdown points of the estimated parameters thereby showing asymptotic robustness of the method.

Automatic context-sensitive karyotyping of human chromosomes based on elliptically symmetric statistical distributions

We introduce a statistical model of a metaphase cell consisting of independent chromosomes with elliptically symmetric feature vectors. From this model we derive the ML-classifier for classification in the 24 chromosomal classes, taking into account the correct number of chromosomes in each class. Experimental results show that error rates of the best of these classifiers are less than 2% with respect to chromosomes if applied to the large Copenhagen data set Cpr. Simulation studies suggest that there should be even more information contained in the features of this data set.

Using combinatorial optimization in model-based trimmed clustering with cardinality constraints

Statistical clustering criteria with free scale parameters and unknown cluster sizes are inclined to create small, spurious clusters. To mitigate this tendency a statistical model for cardinality-constrained clustering of data with gross outliers is established, its maximum likelihood and maximum a posteriori clustering criteria are derived, and their consistency and robustness are analyzed. The criteria lead to constrained optimization problems that can be solved by using iterative, alternating trimming algorithms of k-means type. Each step in the algorithms requires the solution of a @l-assignment problem known from combinatorial optimization. The method allows one to estimate the numbers of clusters and outliers. It is illustrated with a synthetic data set and a real one.

A Bayesian approach to object identification in pattern recognition

We present a new Bayesian approach to object identification: variants. By object identification we mean the detection of the member (regular variant) of a given statistical population (model) among a group of observations (variants). We present estimators for selecting the regular variant, which 1) depend on the knowledge of this population and on a suitable reference measure, only, 2) are simple to evaluate, and 3) are optimal, i.e. Bayesian, under certain conditions. Moreover, we combine variant selection with Bayesian classification considering the situation where we observe m/spl les/n objects belonging to n classes and each object (i) is observed by way of b/sub i/ variants, including the regular one. We present the classifier-selector based on distributions of the regular variants of all classes and on suitable reference measures. We thus simultaneously estimate the regular variants and classes using efficient algorithms.

Strong consistency of k-parameters clustering

Pollard showed for k-means clustering and a very broad class of sampling distributions that the optimal cluster means converge to the solution of the related population criterion as the size of the data set increases. We extend this consistency result to k-parameters clustering, a method derived from the heteroscedastic, elliptical classification model. It allows a more sensitive data analysis and has the advantage of being affine equivariant. Moreover, the present theory yields a consistent criterion for selecting the number of clusters in such models.

Balanced partitions for Markov chains

We call a finite partition of the state space of a (discrete-time) Markov chain balanced if the flows in both directions between any two of its classes are equal in equilibrium. If a Markov chain is reversible then any finite partition is balanced. We use this notion in order to gain insight into the structure of the stationary distributions of not necessarily reversible transition kernels. We illustrate our theory with an asymptotic analysis of a non-reversible Markovian star network with loss.1

Probabilistic clustering via Pareto solutions and significance tests

The present paper proposes a new strategy for probabilistic (often called model-based) clustering. It is well known that local maxima of mixture likelihoods can be used to partition an underlying data set. However, local maxima are rarely unique. Therefore, it remains to select the reasonable solutions, and in particular the desired one. Credible partitions are usually recognized by separation (and cohesion) of their clusters. We use here the p values provided by the classical tests of Wilks, Hotelling, and Behrens–Fisher to single out those solutions that are well separated by location. It has been shown that reasonable solutions to a clustering problem are related to Pareto points in a plot of scale balance vs. model fit of all local maxima. We briefly review this theory and propose as solutions all well-fitting Pareto points in the set of local maxima separated by location in the above sense. We also design a new iterative, parameter-free cutting plane algorithm for the multivariate Behrens–Fisher problem.