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When the data are functions

Abstract

A datum is often a continuous functionx(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 forp-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.

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Reference notes

  • Keller, E. & Ostry, D. J. Computerized measurement of tongue dorsum movements with pulsed echo ultrasound(a). Manuscript submitted for publication toJournal of the Acoustical Society of America, 1982.

  • 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.

  • Winsberg, S. & Ramsay, J. O. Monotone spline transformations for dimension reduction. Submitted for publication inPsychometrika.

  • Winsberg, S. & Ramsay, J. O. Monotone spline transformations for ordered categorical data. Paper presented at the Psychometric Society Annual Meeting, Montreal, Canada, 1982.

References

  • Aubin, J.-P.Applied Functional Analysis. New York: Wiley, 1979.

    Google Scholar 

  • Cailliez, F. & Pagès, J.-P.Introduction à l'Analyse des Données. Paris: Société de Mathématiques Appliquées et de Sciences Humaines, 9 rue Duban, 75016 Paris, 1976.

    Google Scholar 

  • Dauxois, J. & Pousse, A. Les analyses factorielles en calcul des probabilités et en statistique: Essai d'étude synthètique. Thèse d'état, l'Université Paul-Sabatier de Toulouse, France, 1976.

    Google Scholar 

  • Dieudonné, J.Foundations of Modern Analysis. New York: Academic Press, 1960.

    Google Scholar 

  • Doob, J. L.Stochastic Processes. New York: Wiley, 1953.

    Google Scholar 

  • Kreyszig, E.Introductory Functional Analysis with Applications. New York: Wiley, 1978.

    Google Scholar 

  • Schumaker, L.Spline Functions: Basic Theory. New York: Wiley, 1981.

    Google Scholar 

  • Tucker, L. R. Determination of parameters of a functional relationship by factor analysis.Psychometrika, 23, 1958, 19–23.

    Google Scholar 

  • Winsberg, S. & Ramsay, J. O. Monotonic transformations to additivity using splines.Biometrika, 67, 1980, 669–674.

    Google Scholar 

  • Winsberg, S. & Ramsay, J. O. Analysis of pairwise preference data using integratedB-splines.Psychometrika, 46, 1981, 171–186.

    Google Scholar 

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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.

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Ramsay, J.O. When the data are functions. Psychometrika 47, 379–396 (1982). https://doi.org/10.1007/BF02293704

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  • DOI: https://doi.org/10.1007/BF02293704

Key words

  • continuous data
  • functional analysis
  • duality diagram