Méthode de la transformée inverse

La méthode de la transformée inverse est une méthode permettant d'échantillonner une variable aléatoire X de loi donnée à partir de l'expression de sa fonction de répartition F et d'une variable uniforme sur . Principe Cette méthode repose sur le principe suivant, parfois connu sous le nom de théorème de la réciproque : soient F une fonction de répartition, Q la fonction quantile associée, et U une variable uniforme sur . Alors, la variable aléatoire X = Q(U) a pour fonction de répartition F. Ainsi, pour échantillonner une variable aléatoire de loi donnée, il suffit de savoir échantillonner des variables uniformes sur et de pouvoir calculer la fonction quantile de la loi souhaitée. Le nom méthode de la transformée inverse provient du fait que la fonction quantile n'est autre que l'inverse généralisé à gauche de la fonction de répartition : :Q(u) = \inf \left{x : F(x) \geqslant u \right

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Tensor completion aims to reconstruct a high-dimensional data set where the vast majority of entries is missing. The assumption of low-rank structure in the underlying original data allows us to cast the completion problem into an optimization problem restricted to the manifold of fixed-rank tensors. Elements of this smooth embedded submanifold can be efficiently represented in the tensor train or matrix product states format with storage complexity scaling linearly with the number of dimensions. We present a nonlinear conjugate gradient scheme within the framework of Riemannian optimization which exploits this favorable scaling. Numerical experiments and comparison to existing methods show the effectiveness of our approach for the approximation of multivariate functions. Finally, we show that our algorithm can obtain competitive reconstructions from uniform random sampling of few entries compared to adaptive sampling techniques such as cross-approximation.

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MATHICSE Technical Report : Riemannian optimization for high-dimensional tensor completion

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Tensor completion aims to reconstruct a high-dimensional data set with a large fraction of missing entries. The assumption of low-rank structure in the underlying original data allows us to cast the completion problem into an optimization problem restricted to the manifold of fixed-rank tensors. Elements of this smooth embedded submanifold can be efficiently represented in the tensor train (TT) or matrix product states (MPS) format with storage complexity scaling linearly with the number of dimensions. We present a nonlinear conjugate gradient scheme within the framework of Riemannian optimization which exploits this favorable scaling. Numerical experiments and comparison to existing methods show the effectiveness of our approach for the approximation of multivariate functions. Finally, we show that our algorithm can obtain competitive reconstructions from uniform random sampling of few entries of compared to adaptive sampling techniques such as cross-approximation.

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Comparative study of joint analysis of microarray gene expression data in survival prediction and risk assessment of breast cancer patients

Haleh Yasrebi

Microarray gene expression data sets are jointly analyzed to increase statistical power. They could either be merged together or analyzed by meta-analysis. For a given ensemble of data sets, it cannot be foreseen which of these paradigms, merging or meta-analysis, works better. In this article, three joint analysis methods, Z-score normalization, ComBat and the inverse normal method (meta-analysis) were selected for survival prognosis and risk assessment of breast cancer patients. The methods were applied to eight microarray gene expression data sets, totaling 1324 patients with two clinical endpoints, overall survival and relapse-free survival. The performance derived from the joint analysis methods was evaluated using Cox regression for survival analysis and independent validation used as bias estimation. Overall, Z-score normalization had a better performance than ComBat and meta-analysis. Higher Area Under the Receiver Operating Characteristic curve and hazard ratio were also obtained when independent validation was used as bias estimation. With a lower time and memory complexity, Z-score normalization is a simple method for joint analysis of microarray gene expression data sets. The derived findings suggest further assessment of this method in future survival prediction and cancer classification applications.

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