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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

Information

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.

References

Anderson, T. W. (1971). The statistical analysis of time series, New York: Wiley.Google Scholar
Aronszajn, N. (1950). Theory of reproducing kernels. Transactions of the American Mathematical Society, 68, 337–404.CrossRefGoogle Scholar
Aubin, J.-P. (1979). Applied functional analysis, New York: Wiley Interscience.Google Scholar
Besse, P. (1979). Etude descriptive des processus: Approximation et interpolation [Descriptive study of processes: Approximation and Interpolation], Toulouse: Université Paul-Sabatier.Google Scholar
Dauxois, J., Pousse, A. (1976). Les analyses factorielles en calcul des probabilité et en statistique: Essai d'étude synthètique [Factor analysis in the calculus of probability and in statistics], France: l'Université Paul-Sabatier de Toulouse.Google Scholar
Dauxois, J., Pousse, A., Romain, Y. (1982). Asymptotic theory for the principal component analysis of a vector random function: Some applications to statistical inference. Journal of Multivariate Analysis, 12, 136–154.CrossRefGoogle Scholar
Duc-Jacquet, M. (1973). Approximation des fonctionelles lineaires sur des éspaces hilbertiens auto-reproduisants [Approximation of linear functionals on reproducing kernel Hilbert spaces]. Unpublished doctoral dissertation, Grenoble.Google Scholar
Hunter, I. W., Kearney, R. E. (1982). Dynamics of human ankle stiffness: Variation with mean ankle torque. Journal of Biomechanics, 15, 747–752.CrossRefGoogle ScholarPubMed
Huxley, A. F. (1980). Reflections on Muscle, Princeton: Princeton University Press.Google ScholarPubMed
Keller, E., Ostry, D. J. (1983). Computerized measurement of tongue dorsum movements with pulsed-echo ultrasound. Journal of the Acoustical Society of America, 73, 1309–1315.CrossRefGoogle ScholarPubMed
Kimeldorf, G. S., Wahba, G. (1970). A correspondence between Bayesian estimation on stochastic processes and smoothing by splines. Annals of Mathematical Statistics, 41, 495–502.CrossRefGoogle Scholar
Kimeldorf, G. S., Wahba, G. A. (1971). Some results on Tchebycheffian spline functions. Journal of Mathematical Analysis and Application, 33, 82–94.CrossRefGoogle Scholar
Munhall, K. G. (1974). Temporal adjustment in speech motor control: Evidence from laryngeal kinematics. Unpublished doctoral dissertation, McGill University.Google Scholar
Parzen, E. (1961). An approach to time series analysis. Annals of Mathematical Statistics, 32, 951–989.CrossRefGoogle Scholar
Ramsay, J. O. (1982). When the data are functions. Psychometrika, 47, 379–396.CrossRefGoogle Scholar
Rao, C. R. (1958). Some statistical methods for comparison of growth curves. Biometrics, 14, 1–17.CrossRefGoogle Scholar
Rao, C. R. (1964). The use and interpretation of principal components analysis in applied research. Sankhya, 26(A), 329–358.Google Scholar
Rao, C. R. (1980). Matrix approximations and reduction of dimensionality in multivariate statistical analysis. In Krishnaiah, P. R. (Eds.), Multivariate analysis V (pp. 3–22). Amsterdam: North-Holland.Google Scholar
Roach, G. F. (1982). Green's Functions, Cambridge: Cambridge University Press.Google Scholar
Schumaker, L. (1981). Spline Functions: Basic Theory, New York: Wiley.Google Scholar
Shapiro, H. S. (1971). Topics in Approximation Theory, New York: Springer-Verlag.CrossRefGoogle Scholar
Stakgold, I. (1979). Green's Functions and Boundary Value Problems, New York: Wiley.Google Scholar
Tucker, L. R. (1958). Determination of parameters of a functional relation by factor analysis. Psychometrika, 23, 19–23.CrossRefGoogle Scholar
Wegmen, E. J., Wright, I. W. (1983). Splines in statistics. Journal of the American Statistical Association, 78, 351–365.CrossRefGoogle Scholar