Curve Estimation Based on Localised Principal Components - Theory and Applications

Abstract

In this work, basic theory and some proposed developments to localised principal components and curves are introduced. In addition, some areas of application for local principal curves are explored.

Only relatively recently, localised principal components utilising kernel-type weights have found their way into the statistical literature. In this study, the asymptotic behaviour of the method is investigated and 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 develop a method that lets the curve `delay' convergence if desired, gaining more access to boundary regions of the data.
Also, a method for automatic choice of the starting point to be one of the local modes within the data cloud is originated.

The modified local principal curves' algorithm is then used for fitting multi-dimensional econometric data. Special attention is given to the role of the curve parametrisation, which serves as a feature extractor and also as a prediction tool when properly linked to time as a probable underlying latent variable. Local principal curves provide a good dimensionality reduction and feature extraction tool for insurance industry key indicators and consumer price indices. Also, through `calibrating' it with time, curve parametrisation is used for the purpose of predicting unemployment and inflation rates.