The general theory of canonical correlation and its relation to functional analysis
- 1 October 1961
- journal article
- research article
- Published by Cambridge University Press (CUP) in Journal of the Australian Mathematical Society
- Vol. 2 (2) , 229-242
- https://doi.org/10.1017/s1446788700026707
Abstract
The classical theory of canonical correlation is concerned with a standard description of the relationship between any linear combination of ρ random variablesxs, and any linear combination ofqrandom variablesytinsofar as this relation can be described in terms of correlation. Lancaster [1] has extended this theory, forp=q= 1, to include a description of the correlation of any function of a random variablexand any function of a random variabley(both functions having finite variance) for a class of joint distributions ofxandywhich 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 wherepandqare 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.Keywords
This publication has 1 reference indexed in Scilit:
- Calculation of the amount of information about a random function contained in another such functionPublished by American Mathematical Society (AMS) ,1959