Abstract
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.
Highlights
The key idea behind the term “projection pursuit” (PP) is concerned with the search of “interesting” low-dimensional representations of multivariate data, an idea dating back to Kruskal’s early work [1], which later on inspired its use as an exploratory data analysis tool for uncovering hidden patterns and structure in data as described in several of the foundational works [2,3,4,5]
It is shown that skewness PP can be described in terms of two eigenvector problems: the first problem stems from the third cumulant matrix of the scale mixtures of skew normal (SMSN) distribution and the second problem from the simultaneous diagonalization of the covariance and scale scatter matrices of the SMSN model; it can be shown that both approaches have an appealing interpretation in terms of the shape vector that regulates the asymmetry of the model
This paper has addressed the skewness-based PP problem when the underlying multivariate distribution of the input vector belongs to the family of SMSN distributions
Summary
The key idea behind the term “projection pursuit” (PP) is concerned with the search of “interesting” low-dimensional representations of multivariate data, an idea dating back to Kruskal’s early work [1], which later on inspired its use as an exploratory data analysis tool for uncovering hidden patterns and structure in data as described in several of the foundational works [2,3,4,5]. We will assume that the underlying multivariate distribution belongs to the flexible family of scale mixtures of skew normal (SMSN) distributions so the issue is delimited within the skewness model based PP framework. Under this assumption, it is shown that skewness PP can be described in terms of two eigenvector problems: the first problem stems from the third cumulant matrix of the SMSN distribution and the second problem from the simultaneous diagonalization of the covariance and scale scatter matrices of the SMSN model; it can be shown that both approaches have an appealing interpretation in terms of the shape vector that regulates the asymmetry of the model.
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