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When the data are functions


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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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).

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Key words

  • continuous data
  • functional analysis
  • duality diagram