MOTIVATION: Inner holes, artifacts and blank spots are common in microarray images, but current image analysis methods do not pay them enough attention. We propose a new robust model-based method for processing microarray images so as to estimate foreground and background intensities. The method starts with a very simple but effective automatic gridding method, and then proceeds in two steps. The first step applies model-based clustering to the distribution of pixel intensities, using the Bayesian Information Criterion (BIC) to choose the number of groups up to a maximum of three. The second step is spatial, finding the large spatially connected components in each cluster of pixels. The method thus combines the strengths of the histogram-based and spatial approaches. It deals effectively with inner holes in spots and with artifacts. It also provides a formal inferential basis for deciding when the spot is blank, namely when the BIC favors one group over two or three. RESULTS: We apply our methods for gridding and segmentation to cDNA microarray images from an HIV infection experiment. In these experiments, our method had better stability across replicates than a fixed-circle segmentation method or the seeded region growing method in the SPOT software, without introducing noticeable bias when estimating the intensities of differentially expressed genes. AVAILABILITY: spotSegmentation, an R language package implementing both the gridding and segmentation methods is available through the Bioconductor project (http://www.bioconductor.org). The segmentation method requires the contributed R package MCLUST for model-based clustering (http://cran.us.r-project.org). CONTACT: [email protected].

MOTIVATION: Clustering is a useful exploratory technique for the analysis of gene expression data. Many different heuristic clustering algorithms have been proposed in this context. Clustering algorithms based on probability models offer a principled alternative to heuristic algorithms. In particular, model-based clustering assumes that the data is generated by a finite mixture of underlying probability distributions such as multivariate normal distributions. The issues of selecting a 'good' clustering method and determining the 'correct' number of clusters are reduced to model selection problems in the probability framework. Gaussian mixture models have been shown to be a powerful tool for clustering in many applications. RESULTS: We benchmarked the performance of model-based clustering on several synthetic and real gene expression data sets for which external evaluation criteria were available. The model-based approach has superior performance on our synthetic data sets, consistently selecting the correct model and the number of clusters. On real expression data, the model-based approach produced clusters of quality comparable to a leading heuristic clustering algorithm, but with the key advantage of suggesting the number of clusters and an appropriate model. We also explored the validity of the Gaussian mixture assumption on different transformations of real data. We also assessed the degree to which these real gene expression data sets fit multivariate Gaussian distributions both before and after subjecting them to commonly used data transformations. Suitably chosen transformations seem to result in reasonable fits. AVAILABILITY: MCLUST is available at http://www.stat.washington.edu/fraley/mclust. The software for the diagonal model is under development. CONTACT: [email protected]. SUPPLEMENTARY INFORMATION: http://www.cs.washington.edu/homes/kayee/model.

We consider the problem of determining the structure of clustered data, without prior knowledge of the number of clusters or any other information about their composition. Data are represented by a mixture model in which each component corresponds to a different cluster. Models with varying geometric properties are obtained through Gaussian components with different parametrizations and cross-cluster constraints. Noise and outliers can be modelled by adding a Poisson process component. Partitions are determined by the expectation-maximization (EM) algorithm for maximum likelihood, with initial values from agglomerative hierarchical clustering. Models are compared using an approximation to the Bayes factor based on the Bayesian information criterion (BIC); unlike significance tests, this allows comparison of more than two models at the same time, and removes the restriction that the models compared be nested. The problems of determining the number of clusters and the clustering method are solved simultaneously by choosing the best model. Moreover, the EM result provides a measure of uncertainty about the associated classification of each data point. Examples are given, showing that this approach can give performance that is much better than standard procedures, which often fail to identify groups that are either overlapping or of varying sizes and shapes.

This paper concerns further developments in primal projective methods based on the potential function of De Ghellinck and Vial. Their method is unique in that feasibility and optimality are approached simultaneously from an interior starting point, without the addition of an artificial variable to handle feasibility, and without penalty or barrier terms. This behavior is obtained by incorporating the infeasibilities within the potential function. The primal algorithm requires an initial optimal objective estimate, and results are given demonstrating sensitivity of its value. We show that, by choosing a different initial search direction, feasibility and optimality can be treated in separate phases. Results are given, indicating that the two-phase method is often superior to the original single-phase method. We also describe a variant that selects the search direction on the basis of potential reduction, and a technique that allows the use of estimates that are not necessarily lower bounds for the optimal objective value. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.