Some Asymptotics for Localized Principal Components and Curves

Abstract

The asymptotic behavior of localized principal components applying kernels as weights is investigated. In particular, we show that the first-order approximation of the first localized principal component at any given point only depends on the bandwidth parameter(s) and the density at that point. This result is extended to the context of local principal curves, where the characteristics of the points at which the curve stops at the edges are identified. This is used to provide a method which allows the curve to proceed beyond its natural endpoint if desired.

Keywords:

• Kernels
• mean shift
• PCA
• local principal curves

Mathematics Subject Classification:

• 62H25
• 62G20

Notes

For denotational convenience, we will from now on omit all “hats” on symbols denoting estimators—it is clear that , etc., are empirical and not theoretical quantities.

When using the term “first eigenvector”, we mean the eigenvector corresponding to the largest eigenvalue.

In economics, the curve representing the relationship between unemployment and inflation is well known as “Phillips Curve”(Phillips, 1958).

Figure 1 Local principal curve for unemployment-inflation data.