Data Compression and Regression Based on Local Principal Curves

Abstract

Publisher's copyright statement:he originl pulition is ville t wwwFspringerlinkFom Additional information: roeedings of the QPnd ennul gonferene of the qesellshft f¤ ur ulssi(ktion eFFD toint gonferene with the fritish glssi(tion oiety @fgA nd the huthGplemish glssi(tion oiety @ygAD relmutEhmidtEniversityD rmurgD tuly IT!IVD PHHVF Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Summary. Frequently the predictor space of a multivariate regression problem of the type y = m(x1, . . . , xp) + ǫ is intrinsically one-dimensional, or at least of far lower dimension than p. Usual modeling attempts such as the additive model y = m1(x1) + . . . + mp(xp) + ǫ, which try to reduce the complexity of the regression problem by making additional structural assumptions, are then inefficient as they ignore the inherent structure of the predictor space and involve complicated model and variable selection stages. In a fundamentally different approach, one may consider first approximating the predictor space by a (usually nonlinear) curve passing through it, and then regressing the response only against the one-dimensional projections onto this curve. This entails the reduction from a p− to a one-dimensional regression problem. As a tool for the compression of the predictor space we apply local principal curves. Taking things on from the results presented in [6], we show how local principal curves can be parametrized and how the projections are obtained. The regression step can then be carried out using any nonparametric smoother. We illustrate the technique using data from the physical sciences.

Some Asymptotics for Localized Principal Components and Curves

“…Access to these boundary regions can be of a special importance, for instance for time series data where the endpoints correspond to the most current observations. Furthermore, curves which are "too short" in the boundaries will result in projections clustered at the endpoints, which impacts negatively on the usability of the curve as a data compression tool, a problem which was observed by Einbeck, Evers & Hinchliff (2010) in the context of nonlinear compression of high-dimensional spectrographic data. In such situations, one may attempt extending the local principal curve beyond its natural endpoint in order to reach more data points at boundaries.…”

Implementation of a local principal curves algorithm for neutrino interaction reconstruction in a liquid argon volume

“…Note that the cumulative path lengths λ form a discrete parameterisation of the principal curve, which can be refined via a cubic spline interpolation towards a continuous parameterisation if necessary [6]. This can be useful since it allows the option to plot, and regress, physical quantities such as the amount of deposited energy as a function of distance along the particle trajectory covered by the local principal curve.…”

Using principal curves to analyse traffic patterns on freeways

“…Parameters between adjacent µ x are then retrospectively calculated through the arc length of a cubic spline function laid through them (Einbeck, Evers & Hinchliff, 2009). …”