zbMATH — the first resource for mathematics

Principal components analysis of sampled functions. (English) Zbl 0623.62048
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 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.

62H25 Factor analysis and principal components; correspondence analysis
41A05 Interpolation in approximation theory
41A15 Spline approximation
Full Text: DOI
[1] Anderson, T. W. (1971).The statistical analysis of time series. New York: Wiley. · Zbl 0225.62108
[2] Aronszajn, N., (1950). Theory of reproducing kernels.Transactions of the American Mathematical Society, 68, 337–404. · Zbl 0037.20701 · doi:10.1090/S0002-9947-1950-0051437-7
[3] Aubin, J.-P. (1979).Applied functional analysis. New York: Wiley Interscience.
[4] Besse, P. (1979). Etude descriptive des processus: Approximation et interpolation [Descriptive study of processes: Approximation and Interpolation]. Thèse de 3 cycle, Université Paul-Sabatier, Toulouse, 1979.
[5] 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]. Unpublished doctoral dissertation, l’Université Paul-Sabatier de Toulouse, France.
[6] 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. · Zbl 0539.62064 · doi:10.1016/0047-259X(82)90088-4
[7] 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.
[8] Hunter, I. W., & Kearney, R. E. (1982). Dynamics of human ankle stiffness: Variation with mean ankle torque.Journal of Biomechanics, 15, 747–752. · doi:10.1016/0021-9290(82)90089-6
[9] Huxley, A. F. (1980).Reflections on Muscle. Princeton: Princeton University Press. · Zbl 0429.10025
[10] 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. · doi:10.1121/1.389280
[11] 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. · Zbl 0193.45201 · doi:10.1214/aoms/1177697089
[12] Kimeldorf, G. S., & Wahba, G. A. (1971). Some results on Tchebycheffian spline functions.Journal of Mathematical Analysis and Application, 33, 82–94. · Zbl 0201.39702 · doi:10.1016/0022-247X(71)90184-3
[13] Munhall, K. G. (1974). Temporal adjustment in speech motor control: Evidence from laryngeal kinematics. Unpublished doctoral dissertation, McGill University.
[14] Parzen, E. (1961). An approach to time series analysis.Annals of Mathematical Statistics, 32, 951–989. · Zbl 0107.13801 · doi:10.1214/aoms/1177704840
[15] Ramsay, J. O. (1982). When the data are functions.Psychometrika, 47, 379–396. · Zbl 0512.62004 · doi:10.1007/BF02293704
[16] Rao, C. R. (1958). Some statistical methods for comparison of growth curves.Biometrics, 14, 1–17. · Zbl 0079.35704 · doi:10.2307/2527726
[17] Rao, C. R. (1964). The use and interpretation of principal components analysis in applied research.Sankhya, 26(A), 329–358. · Zbl 0137.37207
[18] Rao, C. R. (1980). Matrix approximations and reduction of dimensionality in multivariate statistical analysis. In P. R. Krishnaiah (Ed.),Multivariate analysis V (pp. 3–22). Amsterdam: North-Holland. · Zbl 0442.62042
[19] Roach, G. F. (1982).Green’s Functions, Cambridge: Cambridge University Press. · Zbl 0478.34001
[20] Schumaker, L. (1981).Spline Functions: Basic Theory, New York: Wiley. · Zbl 0449.41004
[21] Shapiro, H. S. (1971).Topics in Approximation Theory. New York: Springer-Verlag. · Zbl 0213.08501
[22] Stakgold, I. (1979).Green’s Functions and Boundary Value Problems. New York: Wiley. · Zbl 0421.34027
[23] Tucker, L. R. (1958). Determination of parameters of a functional relation by factor analysis.Psychometrika, 23, 19–23. · Zbl 0086.13404 · doi:10.1007/BF02288975
[24] Wegmen, E. J., & Wright, I. W. (1983). Splines in statistics.Journal of the American Statistical Association, 78, 351–365. · Zbl 0534.62017 · doi:10.2307/2288640
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.