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Principal Components Analysis of Sampled Functions

Published online by Cambridge University Press:  01 January 2025

Philippe Besse  and
J. O. Ramsay
Philippe Besse
Affiliation:
University Paul Sabatier, Toulouse, France
J. O. Ramsay*
Affiliation:
McGill University
*
Requests for reprints should be sent to J. O. Ramsay, Department of Psychology, 1205 Dr. Penfield Ave., Montréal, Québec, H3A 1B1, Canada.
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Abstract

This paper describes a technique for principal components analysis of data consisting of n functions each observed at p argument values. This problem arises particularly in the analysis of longitudinal data in which some behavior of a number of subjects is measured at a number of points in time. In such cases information about the behavior of one or more derivatives of the function being sampled can often be very useful, as for example in the analysis of growth or learning curves. It is shown that the use of derivative information is equivalent to a change of metric for the row space in classical principal components analysis. The reproducing kernel for the Hilbert space of functions plays a central role, and defines the best interpolating functions, which are generalized spline functions. An example is offered of how sensitivity to derivative information can reveal interesting aspects of the data.

Keywords

reproducing kernelHilbert space of functionsspline functionsGreen's functionsinterpolationsmoothing
Type
Original Paper
Information
Psychometrika , Volume 51 , Issue 2 , June 1986 , pp. 285 - 311
Copyright
Copyright © 1986 The Psychometric Society

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Footnotes

This research was supported by Grant PA 0320 to the second author by the Natural Sciences and Research Council of Canada. We are grateful to the reviewers of an earlier version and to J. B. Kruskal and S. Winsberg for many helpful comments concerning exposition.

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