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| Titel |
Information theoretic measures of dependence, compactness, and non-gaussianity for multivariate probability distributions |
| VerfasserIn |
A. H. Monahan, T. DelSole |
| Medientyp |
Artikel
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| Sprache |
Englisch
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| ISSN |
1023-5809
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| Digitales Dokument |
URL |
| Erschienen |
In: Nonlinear Processes in Geophysics ; 16, no. 1 ; Nr. 16, no. 1 (2009-02-06), S.57-64 |
| Datensatznummer |
250013087
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| Publikation (Nr.) |
 copernicus.org/npg-16-57-2009.pdf |
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| Zusammenfassung |
| A basic task of exploratory data analysis is the characterisation of
"structure" in multivariate datasets. For bivariate Gaussian distributions,
natural measures of dependence (the predictive relationship between
individual variables) and compactness (the degree of concentration of the
probability density function (pdf) around a low-dimensional axis) are
respectively provided by ordinary least-squares regression and Principal
Component Analysis. This study considers general measures of structure for
non-Gaussian distributions and demonstrates that these can be defined in
terms of the information theoretic "distance" (as measured by relative
entropy) between the given pdf and an appropriate "unstructured" pdf. The
measure of dependence, mutual information, is well-known; it is shown that
this is not a useful measure of compactness because it is not invariant under
an orthogonal rotation of the variables. An appropriate rotationally
invariant compactness measure is defined and shown to reduce to the
equivalent PCA measure for bivariate Gaussian distributions. This compactness
measure is shown to be naturally related to a standard information theoretic
measure of non-Gaussianity. Finally, straightforward geometric
interpretations of each of these measures in terms of "effective volume" of
the pdf are presented. |
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