One of the most important and challenging ``knowledge extraction' tasks in bioinformatics is the reverse engineering of gene regulatory networks (GRNs) from DNA microarray gene expression data. Indeed, as a result of the development of high-throughput data-collection techniques, biology is experiencing a data flood phenomenon that pushes biologists toward a new view of biology--systems biology--that aims at system-level understanding of biological systems. Unfortunately, even for small model organisms such as the yeast Saccharomyces cerevisiae, the number p of genes is much larger than the number n of expression data samples. The dimensionality issue induced by this ``small n, large p' data setting renders standard statistical learning methods inadequate. Restricting the complexity of the models enables to deal with this serious impediment. Indeed, by introducing (a priori undesirable) bias in the model selection procedure, one reduces the variance of the selected model thereby increasing its accuracy. Gaussian graphical models (GGMs) have proven to be a very powerful formalism to infer GRNs from expression data. Standard GGM selection techniques can unfortunately not be used in the ``small n, large p' data setting. One way to overcome this issue is to resort to regularization. In particular, shrinkage estimators of the covariance matrix--required to infer GGMs--have proven to be very effective. Our first contribution consists in a new shrinkage estimator that improves upon existing ones through the use of a Monte Carlo (parametric bootstrap) procedure. Another approach to GGM selection in the ``small n, large p' data setting consists in reverse engineering limited-order partial correlation graphs (q-partial correlation graphs) to approximate GGMs. Our second contribution consists in an inference algorithm, the q-nested procedure, that builds a sequence of nested q-partial correlation graphs to take advantage of the smaller order graphs' topology to infer higher order graphs. This allows us to significantly speed up the inference of such graphs and to avoid problems related to multiple testing. Consequently, we are able to consider higher order graphs, thereby increasing the accuracy of the inferred graphs. Another important challenge in bioinformatics is the prediction of gene function. An example of such a prediction task is the identification of genes that are targets of the nitrogen catabolite repression (NCR) selection mechanism in the yeast Saccharomyces cerevisiae. The study of model organisms such as Saccharomyces cerevisiae is indispensable for the understanding of more complex organisms. Our third contribution consists in extending the standard two-class classification approach by enriching the set of variables and comparing several feature selection techniques and classification algorithms. Finally, our fourth contribution formulates the prediction of NCR target genes as a network inference task. We use GGM selection to infer multivariate dependencies between genes, and, starting from a set of genes known to be sensitive to NCR, we classify the remaining genes. We hence avoid problems related to the choice of a negative training set and take advantage of the robustness of GGM selection techniques in the ``small n, large p' data setting.

L0 penalized likelihood procedures like Mallows' Cp, AIC, and BIC directly penalize for the number of variables included in a regression model. This is a straightforward approach to the problem of overfitting, and these methods are now part of every statistician's repertoire. However, these procedures have been shown to sometimes result in unstable parameter estimates as a result on the L0 penalty's discontinuity at zero. One proposed alternative, seamless-L0 (SELO), utilizes a continuous penalty function that mimics L0 and allows for stable estimates. Like other similar methods (e.g. LASSO and SCAD), SELO produces sparse solutions because the penalty function is non-differentiable at the origin. Because these penalized likelihoods are singular (non-differentiable) at zero, there is no closed-form solution for the extremum of the objective function. We propose a continuous and everywhere-differentiable penalty function that can have arbitrarily steep slope in a neighborhood near zero, thus mimicking the L0 penalty, but allowing for a nearly closed-form solution for the beta-hat vector. Because our function is not singular at zero, beta-hat will have no zero-valued components, although some will have been shrunk arbitrarily close thereto. We employ a BIC-selected tuning parameter used in the shrinkage step to perform zero-thresholding as well. We call the resulting vector of coefficients the ShrinkSet estimator. It is comparable to SELO in terms of model performance (selecting the truly nonzero coefficients, overall MSE, etc.), but we believe it to be more intuitive and simpler to compute. We provide strong evidence that the estimator enjoys favorable asymptotic properties, including the oracle property.

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One of the most important and challenging ``knowledge extraction' tasks in bioinformatics is the reverse engineering of gene regulatory networks (GRNs) from DNA microarray gene expression data. Indeed, as a result of the development of high-throughput data-collection techniques, biology is experiencing a data flood phenomenon that pushes biologists toward a new view of biology--systems biology--that aims at system-level understanding of biological systems.<p><p>Unfortunately, even for small model organisms such as the yeast Saccharomyces cerevisiae, the number p of genes is much larger than the number n of expression data samples. The dimensionality issue induced by this ``small n, large p' data setting renders standard statistical learning methods inadequate. Restricting the complexity of the models enables to deal with this serious impediment. Indeed, by introducing (a priori undesirable) bias in the model selection procedure, one reduces the variance of the selected model thereby increasing its accuracy.<p><p>Gaussian graphical models (GGMs) have proven to be a very powerful formalism to infer GRNs from expression data. Standard GGM selection techniques can unfortunately not be used in the ``small n, large p' data setting. One way to overcome this issue is to resort to regularization. In particular, shrinkage estimators of the covariance matrix--required to infer GGMs--have proven to be very effective. Our first contribution consists in a new shrinkage estimator that improves upon existing ones through the use of a Monte Carlo (parametric bootstrap) procedure.<p><p>Another approach to GGM selection in the ``small n, large p' data setting consists in reverse engineering limited-order partial correlation graphs (q-partial correlation graphs) to approximate GGMs. Our second contribution consists in an inference algorithm, the q-nested procedure, that builds a sequence of nested q-partial correlation graphs to take advantage of the smaller order graphs' topology to infer higher order graphs. This allows us to significantly speed up the inference of such graphs and to avoid problems related to multiple testing. Consequently, we are able to consider higher order graphs, thereby increasing the accuracy of the inferred graphs.<p><p>Another important challenge in bioinformatics is the prediction of gene function. An example of such a prediction task is the identification of genes that are targets of the nitrogen catabolite repression (NCR) selection mechanism in the yeast Saccharomyces cerevisiae. The study of model organisms such as Saccharomyces cerevisiae is indispensable for the understanding of more complex organisms. Our third contribution consists in extending the standard two-class classification approach by enriching the set of variables and comparing several feature selection techniques and classification algorithms.<p><p>Finally, our fourth contribution formulates the prediction of NCR target genes as a network inference task. We use GGM selection to infer multivariate dependencies between genes, and, starting from a set of genes known to be sensitive to NCR, we classify the remaining genes. We hence avoid problems related to the choice of a negative training set and take advantage of the robustness of GGM selection techniques in the ``small n, large p' data setting. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished