Inverse transform sampling

Summary

Inverse transform sampling (also known as inversion sampling, the inverse probability integral transform, the inverse transformation method, Smirnov transform, or the golden rule) is a basic method for pseudo-random number sampling, i.e., for generating sample numbers at random from any probability distribution given its cumulative distribution function. Inverse transformation sampling takes uniform samples of a number u between 0 and 1, interpreted as a probability, and then returns the smallest number x\in\mathbb R such that F(x)\ge u for the cumulative distribution function F of a random variable. For example, imagine that F is the standard normal distribution with mean zero and standard deviation one. The table below shows samples taken from the uniform distribution and their representation on the standard normal distribution. We are randomly choosing a proportion of the area under the curve and returning the n

Official source

https://en.wikipedia.org/wiki/Inverse_transform_sampling

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Siam Publications2016

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

Michael Maximilian Steinlechner

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.

MATHICSE2015

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.

Oxford Univ Press2016