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

Validation of Inference Procedures for Gene Regulatory Networks

The availability of high-throughput genomic data has motivated the development of numerous algorithms to infer gene regulatory networks. The validity of an inference procedure must be evaluated relative to its ability to infer a model network close to the ground-truth network from which the data have been generated. The input to an inference algorithm is a sample set of data and its output is a network. Since input, output, and algorithm are mathematical structures, the validity of an inference algorithm is a mathematical issue. This paper formulates validation in terms of a semi-metric distance between two networks, or the distance between two structures of the same kind deduced from the networks, such as their steady-state distributions or regulatory graphs. The paper sets up the validation framework, provides examples of distance functions, and applies them to some discrete Markov network models. It also considers approximate validation methods based on data for which the generating network is not known, the kind of situation one faces when using real data

On the Number of Close-to-Optimal Feature Sets

The issue of wide feature-set variability has recently been raised in the context of expression-based classification using microarray data. This paper addresses this concern by demonstrating the natural manner in which many feature sets of a certain size chosen from a large collection of potential features can be so close to being optimal that they are statistically indistinguishable. Feature-set optimality is inherently related to sample size because it only arises on account of the tendency for diminished classifier accuracy as the number of features grows too large for satisfactory design from the sample data. The paper considers optimal feature sets in the framework of a model in which the features are grouped in such a way that intra-group correlation is substantial whereas inter-group correlation is minimal, the intent being to model the situation in which there are groups of highly correlated co-regulated genes and there is little correlation between the co-regulated groups. This is accomplished by using a block model for the covariance matrix that reflects these conditions. Focusing on linear discriminant analysis, we demonstrate how these assumptions can lead to very large numbers of close-to-optimal feature sets

Incorporating prior knowledge induced from stochastic differential equations in the classification of stochastic observations

In classification, prior knowledge is incorporated in a Bayesian framework by assuming that the feature-label distribution belongs to an uncertainty class of feature-label distributions governed by a prior distribution. A posterior distribution is then derived from the prior and the sample data. An optimal Bayesian classifier (OBC) minimizes the expected misclassification error relative to the posterior distribution. From an application perspective, prior construction is critical

Identifying Genes Involved in Cyclic Processes by Combining Gene Expression Analysis and Prior Knowledge

Based on time series gene expressions, cyclic genes can be recognized via spectral analysis and statistical periodicity detection tests. These cyclic genes are usually associated with cyclic biological processes, for example, cell cycle and circadian rhythm. The power of a scheme is practically measured by comparing the detected periodically expressed genes with experimentally verified genes participating in a cyclic process. However, in the above mentioned procedure the valuable prior knowledge only serves as an evaluation benchmark, and it is not fully exploited in the implementation of the algorithm. In addition, partial data sets are also disregarded due to their nonstationarity. This paper proposes a novel algorithm to identify cyclic-process-involved genes by integrating the prior knowledge with the gene expression analysis. The proposed algorithm is applied on data sets corresponding to Saccharomyces cerevisiae and Drosophila melanogaster, respectively. Biological evidences are found to validate the roles of the discovered genes in cell cycle and circadian rhythm. Dendrograms are presented to cluster the identified genes and to reveal expression patterns. It is corroborated that the proposed novel identification scheme provides a valuable technique for unveiling pathways related to cyclic processes

Recovering Genetic Regulatory Networks from Chromatin Immunoprecipitation and Steady-State Microarray Data

<p/> <p>Recent advances in high-throughput DNA microarrays and chromatin immunoprecipitation (ChIP) assays have enabled the learning of the structure and functionality of genetic regulatory networks. In light of these heterogeneous data sets, this paper proposes a novel approach for reconstruction of genetic regulatory networks based on the posterior probabilities of gene regulations. Built within the framework of Bayesian statistics and computational Monte Carlo techniques, the proposed approach prevents the dichotomy of classifying gene interactions as either being connected or disconnected, thereby it reduces significantly the inference errors. Simulation results corroborate the superior performance of the proposed approach relative to the existing state-of-the-art algorithms. A genetic regulatory network for <it>Saccharomyces cerevisiae</it> is inferred based on the published real data sets, and biological meaningful results are discussed.</p

Spectral Preprocessing for Clustering Time-Series Gene Expressions

<p/> <p>Based on gene expression profiles, genes can be partitioned into clusters, which might be associated with biological processes or functions, for example, cell cycle, circadian rhythm, and so forth. This paper proposes a novel clustering preprocessing strategy which combines clustering with spectral estimation techniques so that the time information present in time series gene expressions is fully exploited. By comparing the clustering results with a set of biologically annotated yeast cell-cycle genes, the proposed clustering strategy is corroborated to yield significantly different clusters from those created by the traditional expression-based schemes. The proposed technique is especially helpful in grouping genes participating in time-regulated processes.</p