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When the Data are Functions

Published online by Cambridge University Press:  01 January 2025

J. O. Ramsay
J. O. Ramsay*
Affiliation:
McGill University
*
Requests for reprints should be sent to: J. O. Ramsay, Dept. of Psychology, 1205 Dr. Penfield Ave., Montreal, Qurbec, Canada H3A 1B1.
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Abstract

A datum is often a continuous function x(t) of a variable such as time observed over some interval. One or more such functions are observed for each subject or unit of observation. The extension of classical data analytic techniques designed for p-variate observations to such data is discussed. The essential step is the expression of the classical problem in the language of functional analysis, after which the extension to functions is a straightforward matter. A schematic device called the duality diagram is a very useful tool for describing an analysis and for suggesting new possibilities. Least squares approximation, descriptive statistics, principal components analysis, and canonical correlation analysis are discussed within this broader framework.

Keywords

continuous datafunctional analysisduality diagram
Type
Original Paper
Information
Psychometrika , Volume 47 , Issue 4 , December 1982 , pp. 379 - 396
Copyright
Copyright © 1982 The Psychometric Society

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Footnotes

Presented as the Presidential Address to the Psychometric Society’s Annual Meeting, May, 1982. I wish to express my gratitude to my colleagues in France, especially at the University of Grenoble, for their warm hospitality during my sabbatical leave. Preparation of this paper was supported by Grant APA 0320 from the Natural Sciences and Engineering Research Council of Canada.

References

Reference Notes

Keller, E. & Ostry, D. J. Computerized measurement of tongue dorsum movements with pulsed echo ultrasound (a). Manuscript submitted for publication to Journal of the Acoustical Society of America, 1982.CrossRefGoogle Scholar
Pagès, J. P. & Tenenhaus, M. Geometry and duality diagram. An example of application: The analysis of qualitative variables. Paper presented at the Psychometric Society Annual Meeting, Montreal, Canada, 1982.Google Scholar
Winsberg, S. & Ramsay, J. O. Monotone spline transformations for dimension reduction. Submitted for publication in Psychometrika.Google Scholar
Winsberg, S. & Ramsay, J. O. Monotone spline transformations for ordered categorical data. Paper presented at the Psychometric Society Annual Meeting, Montreal, Canada, 1982.Google Scholar

References

Aubin, J.-P. Applied Functional Analysis, 1979, New York: Wiley.Google Scholar
Cailliez, F. & Pagès, J.-P. Introduction à l’Analyse des Données, 1976, Paris: Société de Mathématiques Appliquées et de Sciences Humaines.Google Scholar
Dauxois, J. & Pousse, A. Les analyses factorielles en calcul des probabilités et en statistique: Essai d’étude synthètique, 1976, France: l’Université Paul-Sabatier de Toulouse.Google Scholar
Dieudonné, J. Foundations of Modern Analysis, 1960, New York: Academic Press.Google Scholar
Doob, J. L. Stochastic Processes, 1953, New York: Wiley.Google Scholar
Kreyszig, E. Introductory Functional Analysis with Applications, 1978, New York: Wiley.Google Scholar
Schumaker, L. Spline Functions: Basic Theory, 1981, New York: Wiley.Google Scholar
Tucker, L. R. Determination of parameters of a functional relationship by factor analysis. Psychometrika, 1958, 23, 19–23.CrossRefGoogle Scholar
Winsberg, S. & Ramsay, J. O. Monotonic transformations to additivity using splines. Biometrika, 1980, 67, 669–674.CrossRefGoogle Scholar
Winsberg, S. & Ramsay, J. O. Analysis of pairwise preference data using integratedB-splines. Psychometrika, 1981, 46, 171–186.CrossRefGoogle Scholar