Challenging the curse of dimensionality in multivariate local linear regression

Taylor, James; Einbeck, Jochen

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.
• Public URL: https://durham-repository.worktribe.com/output/1472834