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Register a new userNon-Gaussian Component Analysis using Entropy Methods
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Added by
Abhishek Shetty
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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Non-Gaussian component analysis (NGCA) is a problem in multidimensional data analysis which, since its formulation in 2006, has attracted considerable attention in statistics and machine learning. In this problem, we have a random variable $X$ in $n$-dimensional Euclidean space. There is an unknown subspace $Gamma$ of the $n$-dimensional Euclidean space such that the orthogonal projection of $X$ onto $Gamma$ is standard multidimensional Gaussian and the orthogonal projection of $X$ onto $Gamma^{perp}$, the orthogonal complement of $Gamma$, is non-Gaussian, in the sense that all its one-dimensional marginals are different from the Gaussian in a certain metric defined in terms of moments. The NGCA problem is to approximate the non-Gaussian subspace $Gamma^{perp}$ given samples of $X$. Vectors in $Gamma^{perp}$ correspond to `interesting directions, whereas vectors in $Gamma$ correspond to the directions where data is very noisy. The most interesting applications of the NGCA model is for the case when the magnitude of the noise is comparable to that of the true signal, a setting in which traditional noise reduction techniques such as PCA dont apply directly. NGCA is also related to dimension reduction and to other data analysis problems such as ICA. NGCA-like problems have been studied in statistics for a long time using techniques such as projection pursuit. We give an algorithm that takes polynomial time in the dimension $n$ and has an inverse polynomial dependence on the error parameter measuring the angle distance between the non-Gaussian subspace and the subspace output by the algorithm. Our algorithm is based on relative entropy as the contrast function and fits under the projection pursuit framework. The techniques we develop for analyzing our algorithm maybe of use for other related problems.
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Functional principal component analysis is essential in functional data analysis, but the inferences will become unconvincing when some non-Gaussian characteristics occur, such as heavy tail and skewness. The focus of this paper is to develop a robust functional principal component analysis methodology in dealing with non-Gaussian longitudinal data, for which sparsity and irregularity along with non-negligible measurement errors must be considered. We introduce a Kendalls $tau$ function whose particular properties make it a nice proxy for the covariance function in the eigenequation when handling non-Gaussian cases. Moreover, the estimation procedure is presented and the asymptotic theory is also established. We further demonstrate the superiority and robustness of our method through simulation studies and apply the method to the longitudinal CD4 cell count data in an AIDS study.
We revisit the problem of estimating the mean of a real-valued distribution, presenting a novel estimator with sub-Gaussian convergence: intuitively, our estimator, on any distribution, is as accurate as the sample mean is for the Gaussian distribution of matching variance. Crucially, in contrast to prior works, our estimator does not require prior knowledge of the variance, and works across the entire gamut of distributions with bounded variance, including those without any higher moments. Parameterized by the sample size $n$, the failure probability $delta$, and the variance $sigma^2$, our estimator is accurate to within $sigmacdot(1+o(1))sqrt{frac{2logfrac{1}{delta}}{n}}$, tight up to the $1+o(1)$ factor. Our estimator construction and analysis gives a framework generalizable to other problems, tightly analyzing a sum of dependent random variables by viewing the sum implicitly as a 2-parameter $psi$-estimator, and constructing bounds using mathematical programming and duality techniques.
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