Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-09T07:56:15.963Z Has data issue: false hasContentIssue false
  • English
  • Français

The general theory of canonical correlation and its relation to functional analysis

Published online by Cambridge University Press:  09 April 2009

E. J. Hannan
E. J. Hannan
Affiliation:
School of General Studies, Australian National University, Canberra.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The classical theory of canonical correlation is concerned with a standard description of the relationship between any linear combination of ρ random variables xs, and any linear combination of q random variables yt insofar as this relation can be described in terms of correlation. Lancaster [1] has extended this theory, for p = q = 1, to include a description of the correlation of any function of a random variable x and any function of a random variable y (both functions having finite variance) for a class of joint distributions of x and y which is very general. It is the purpose of this paper to derive Lancaster's results from general theorems concerning the spectral decomposition of operators on a Hilbert space. These theorems lend themselves easily to the generalisation of the theory to situations where p and q are not finite. In the case of Gaussian, stationary, processes this generalisation is equivalent to the classical spectral theory and corresponds to a canonical reduction of a (finite) sample of data which is basic. The theory also then extends to any number of processes. In the Gaussian case, also, the present discussion-is connected with the results of Gelfand and Yaglom [2] relating to the amount of information in one random process about another.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 2 , Issue 2 , October 1961 , pp. 229 - 242
Copyright
Copyright © Australian Mathematical Society 1961

References

[1]Lancaster, H. O., ‘The Structure of Bivariate Distributions’, Ann. Math. Statist., 29, (1958) 719–736.CrossRefGoogle Scholar
[2]Gelfand, I. M. and Yaglom, A. M., ‘Calculation of the Amount of Information about a Random Function Contained in Another Such Function’, American Mathematical Society Translations, Vol. 12 (1959) 199–246.Google Scholar
[3]Naimark, M. A., Normed Rings, Noordhoff, (1959).Google Scholar
[4]Kolmogoroff, A. N., Foundations of Probability Theory, Chelsea (1956).Google Scholar
[5]Cramér, H., Mathematical Methods of Statistics, Princeton (1946).Google Scholar
[6]Doob, J. L., Stochastic Processes, Wiley (1953).Google Scholar
[7]Riesz, F. R. and Sz-Nagy, B., Functional Analysis, Blackie (1956).Google Scholar
[8]Kampéde Fériet, J., ‘Analyse harmonique des fonctions aléatoires stationnaires d'ordre 2 définies sur un groupe abélien localement compact’, C. R. Acad. Sci. Paris, 226 (1948) 868–870Google Scholar
[9[Hannan, E. J., Time Series Analysis, Methuns (1960).Google Scholar
[10]Halmos, P. R., Measure Theory, Van Nostrand (1950).CrossRefGoogle Scholar