Challenging the curse of dimensionality in multivariate local linear regression

Taylor, James; Einbeck, Jochen

doi:10.1007/s00180-012-0342-0

Challenging the curse of dimensionality in multivariate local linear regression

Taylor, James; Einbeck, Jochen

Authors

James Taylor

Professor Jochen Einbeck jochen.einbeck@durham.ac.uk
Professor

Abstract

Local polynomial fitting for univariate data has been widely studied and discussed, but up until now the multivariate equivalent has often been deemed impractical, due to the so-called curse of dimensionality. Here, rather than discounting it completely, we use density as a threshold to determine where over a data range reliable multivariate smoothing is possible, whilst accepting that in large areas it is not. The merits of a density threshold derived from the asymptotic influence function are shown using both real and simulated data sets. Further, the challenging issue of multivariate bandwidth selection, which is known to be affected detrimentally by sparse data which inevitably arise in higher dimensions, is considered. In an effort to alleviate this problem, two adaptations to generalized cross-validation are implemented, and a simulation study is presented to support the proposed method. It is also discussed how the density threshold and the adapted generalized cross-validation technique introduced herein work neatly together.

Citation

Taylor, J., & Einbeck, J. (2013). Challenging the curse of dimensionality in multivariate local linear regression. Computational Statistics, 28(3), 955-976. https://doi.org/10.1007/s00180-012-0342-0

Journal Article Type	Article
Publication Date	Jun 1, 2013
Deposit Date	Sep 24, 2012
Publicly Available Date	Sep 23, 2014
Journal	Computational Statistics
Print ISSN	0943-4062
Electronic ISSN	1613-9658
Publisher	Springer
Peer Reviewed	Peer Reviewed
Volume	28
Issue	3
Pages	955-976
DOI	https://doi.org/10.1007/s00180-012-0342-0
Keywords	Multivariate smoothing, Density estimation, Bandwidth selection, Influence function.