14,289 research outputs found
Brownian distance covariance
Distance correlation is a new class of multivariate dependence coefficients
applicable to random vectors of arbitrary and not necessarily equal dimension.
Distance covariance and distance correlation are analogous to product-moment
covariance and correlation, but generalize and extend these classical bivariate
measures of dependence. Distance correlation characterizes independence: it is
zero if and only if the random vectors are independent. The notion of
covariance with respect to a stochastic process is introduced, and it is shown
that population distance covariance coincides with the covariance with respect
to Brownian motion; thus, both can be called Brownian distance covariance. In
the bivariate case, Brownian covariance is the natural extension of
product-moment covariance, as we obtain Pearson product-moment covariance by
replacing the Brownian motion in the definition with identity. The
corresponding statistic has an elegantly simple computing formula. Advantages
of applying Brownian covariance and correlation vs the classical Pearson
covariance and correlation are discussed and illustrated.Comment: This paper discussed in: [arXiv:0912.3295], [arXiv:1010.0822],
[arXiv:1010.0825], [arXiv:1010.0828], [arXiv:1010.0836], [arXiv:1010.0838],
[arXiv:1010.0839]. Rejoinder at [arXiv:1010.0844]. Published in at
http://dx.doi.org/10.1214/09-AOAS312 the Annals of Applied Statistics
(http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics
(http://www.imstat.org
DISCO analysis: A nonparametric extension of analysis of variance
In classical analysis of variance, dispersion is measured by considering
squared distances of sample elements from the sample mean. We consider a
measure of dispersion for univariate or multivariate response based on all
pairwise distances between-sample elements, and derive an analogous distance
components (DISCO) decomposition for powers of distance in . The ANOVA F
statistic is obtained when the index (exponent) is 2. For each index in
, this decomposition determines a nonparametric test for the
multi-sample hypothesis of equal distributions that is statistically consistent
against general alternatives.Comment: Published in at http://dx.doi.org/10.1214/09-AOAS245 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
The Complexity of the Spherical -spin spin glass model, revisited
Some questions concerning the calculation of the number of ``physical''
(metastable) states or complexity of the spherical -spin spin glass model
are reviewed and examined further. Particular attention is focused on the
general calculation procedure which is discussed step-by-step.Comment: 13 pages, 3 figure
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