Abstract
Non-parametric assessment of extreme dependence structures between an arbitrary number of variables, though quite well-established in dimension $2$ and recently extended to moderate dimensions such as $5$, still represents a statistical challenge in larger dimensions. Here, we propose a novel approach that combines clustering techniques with angular/spectral measure analysis to find groups of variables (not necessarily disjoint) exhibiting asymptotic dependence, thereby reducing the dimension of the initial problem. A heuristic criterion is proposed to choose the threshold over which it is acceptable to consider observations as extreme and the appropriate number of clusters. When empirically evaluated through numerical experiments, the approach we promote here is found to be very efficient under some regularity constraints, even in dimension $20$. For illustration purpose, we also carry out a case study in dietary risk assessment.
Highlights
High dimension raises important issues in applied multivariate statistics; while sample sizes are finite, the set on which probability measures are defined can be so large that extrapolation is intricate
We propose to mimic a classical approach in statistical learning, namely Principal Components Analysis (PCA, Friedman tLeht2a-atnl.o,rre2ms0p0,e9wc)ti.tW hthueenwiintotrhrkiynposneicrtshhdepiseatranengcSleeds¡oi1nf siSntedR¡ad1d,.olTfikthheisetehrnaewaPbdlreaisntactihpaeanludsNseeeotsftea.dlÔg2SoÕrpaihtshetrmheess (PNS) technique developed by Jung et al (2012)
The highlighted row reports the number of trials where we managed to exactly recover the set of open faces intersecting with the support of the angular probability measure
Summary
High dimension raises important issues in applied multivariate statistics; while sample sizes are finite, the set on which probability measures are defined can be so large that extrapolation is intricate. Whereas a plethora of techniques has been developed in the field of statistical learning to overcome this issue (Friedman et al, 2009), multivariate extremes in dimensions larger that 2 are still handled with difficulty. Beyond a possible overall description of the tail dependence structure, when these classes are of small dimension, this method would enable further and more efficient assessment of multivariate tails.
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