Skewness-based projection pursuit: A computational approach

Selected Payback Statistical Contributions to Matrix/Linear Algebra: Some Counterflowing Conceptualizations

Matrix/linear algebra continues bestowing benefits on theoretical and applied statistics, a practice it began decades ago (re Fisher used the word matrix in a 1941 publication), through a myriad of contributions, from recognition of a suite of matrix properties relevant to statistical concepts, to matrix specifications of linear and nonlinear techniques. Consequently, focused parts of matrix algebra are topics of several statistics books and journal articles. Contributions mostly have been unidirectional, from matrix/linear algebra to statistics. Nevertheless, statistics offers great potential for making this interface a bidirectional exchange point, the theme of this review paper. Not surprisingly, regression, the workhorse of statistics, provides one tool for such historically based recompence. Another prominent one is the mathematical matrix theory eigenfunction abstraction. A third is special matrix operations, such as Kronecker sums and products. A fourth is multivariable calculus linkages, especially arcane matrix/vector operators as well as the Jacobian term associated with variable transformations. A fifth, and the final idea this paper treats, is random matrices/vectors within the context of simulation, particularly for correlated data. These are the five prospectively reviewed discipline of statistics subjects capable of informing, inspiring, or otherwise furnishing insight to the far more general world of linear algebra.

Cumulants of Multivariate Symmetric and Skew Symmetric Distributions

This paper provides a systematic and comprehensive treatment for obtaining general expressions of any order, for the moments and cumulants of spherically and elliptically symmetric multivariate distributions; results for the case of multivariate t-distribution and related skew-t distribution are discussed in some detail.

Canonical Correlations and Nonlinear Dependencies

Canonical correlation analysis (CCA) is the default method for investigating the linear dependence structure between two random vectors, but it might not detect nonlinear dependencies. This paper models the nonlinear dependencies between two random vectors by the perturbed independence distribution, a multivariate semiparametric model where CCA provides an insight into their nonlinear dependence structure. The paper also investigates some of its probabilistic and inferential properties, including marginal and conditional distributions, nonlinear transformations, maximum likelihood estimation and independence testing. Perturbed independence distributions are closely related to skew-symmetric ones.

Skewness-Based Projection Pursuit as an Eigenvector Problem in Scale Mixtures of Skew-Normal Distributions

This paper addresses the projection pursuit problem assuming that the distribution of the input vector belongs to the flexible and wide family of multivariate scale mixtures of skew normal distributions. Under this assumption, skewness-based projection pursuit is set out as an eigenvector problem, described in terms of the third order cumulant matrix, as well as an eigenvector problem that involves the simultaneous diagonalization of the scatter matrices of the model. Both approaches lead to dominant eigenvectors proportional to the shape parametric vector, which accounts for the multivariate asymmetry of the model; they also shed light on the parametric interpretability of the invariant coordinate selection method and point out some alternatives for estimating the projection pursuit direction. The theoretical findings are further investigated through a simulation study whose results provide insights about the usefulness of skewness model-based projection pursuit in the statistical practice.

Copulaesque Versions of the Skew-Normal and Skew-Student Distributions

A recent paper presents an extension of the skew-normal distribution which is a copula. Under this model, the standardized marginal distributions are standard normal. The copula itself depends on the familiar skewing construction based on the normal distribution function. This paper is concerned with two topics. First, the paper presents a number of extensions of the skew-normal copula. Notably these include a case in which the standardized marginal distributions are Student’s t, with different degrees of freedom allowed for each margin. In this case the skewing function need not be the distribution function for Student’s t, but can depend on certain of the special functions. Secondly, several multivariate versions of the skew-normal copula model are presented. The paper contains several illustrative examples.

Inferring neural information flow from spiking data

The brain can be regarded as an information processing system in which neurons store and propagate information about external stimuli and internal processes. Therefore, estimating interactions between neural activity at the cellular scale has significant implications in understanding how neuronal circuits encode and communicate information across brain areas to generate behavior. While the number of simultaneously recorded neurons is growing exponentially, current methods relying only on pairwise statistical dependencies still suffer from a number of conceptual and technical challenges that preclude experimental breakthroughs describing neural information flows. In this review, we examine the evolution of the field over the years, starting from descriptive statistics to model-based and model-free approaches. Then, we discuss in detail the Granger Causality framework, which includes many popular state-of-the-art methods and we highlight some of its limitations from a conceptual and practical estimation perspective. Finally, we discuss directions for future research, including the development of theoretical information flow models and the use of dimensionality reduction techniques to extract relevant interactions from large-scale recording datasets.

MaxSkew and MultiSkew: Two R Packages for Detecting, Measuring and Removing Multivariate Skewness

The R packages MaxSkew and MultiSkew measure, test and remove skewness from multivariate data using their third-order standardized moments. Skewness is measured by scalar functions of the third standardized moment matrix. Skewness is tested with either the bootstrap or under normality. Skewness is removed by appropriate linear projections. The packages might be used to recover data features, as for example clusters and outliers. They are also helpful in improving the performances of statistical methods, as for example the Hotelling’s one-sample test. The Iris dataset illustrates the usages of MaxSkew and MultiSkew.

Projection pursuit in high dimensions

Projection pursuit is a classical exploratory data analysis method to detect interesting low-dimensional structures in multivariate data. Originally, projection pursuit was applied mostly to data of moderately low dimension. Motivated by contemporary applications, we here study its properties in high-dimensional settings. Specifically, we analyze the asymptotic properties of projection pursuit on structureless multivariate Gaussian data with an identity covariance, as both dimension p and sample size n tend to infinity, with [Formula: see text] Our main results are that (i) if [Formula: see text] then there exist projections whose corresponding empirical cumulative distribution function can approximate any arbitrary distribution; and (ii) if [Formula: see text], not all limiting distributions are possible. However, depending on the value of γ, various non-Gaussian distributions may still be approximated. In contrast, if we restrict to sparse projections, involving only a few of the p variables, then asymptotically all empirical cumulative distribution functions are Gaussian. And (iii) if [Formula: see text], then asymptotically all projections are Gaussian. Some of these results extend to mean-centered sub-Gaussian data and to projections into k dimensions. Hence, in the "small n, large p" setting, unless sparsity is enforced, and regardless of the chosen projection index, projection pursuit may detect an apparent structure that has no statistical significance. Furthermore, our work reveals fundamental limitations on the ability to detect non-Gaussian signals in high-dimensional data, in particular through independent component analysis and related non-Gaussian component analysis.