The intrinsic dimensionality of signal collections

Abstract

In view of the trend toward the representation of signals as physical observables characterized by vectors in an abstract signal space, rather than as time or frequency functions, it is desirable that dimensionality be defined in a manner independent of the choice of basis on which the vectors are projected. The intrinsic dimensionality of a collection of signals is defined to be equal to the number of free parameters required in a hypothetical signal generator capable of producing a close approximation to each signal in the collection. Thus defined, the dimensionality becomes a relationship between the vectors representing the signals. This relationship need not be a linear one and does not depend on the basis onto which the vectors are projected in the signal-measuring process. A digital computer program for estimating this dimensionality from the signal coefficients on an arbitrary basis has been developed. The program makes use of some results obtained from a multi-dimensional scaling problem in experimental psychology and utilizes an inverse relationship between the variance in interpoint distances within a hypersphere and the dimensionality of the hypersphere. Using this method, the results are believed to be independent of the choice of orthogonal basis, and no prior knowledge of the analytical form of the signals is assumed. The validity of the program is tested and verified by using it to estimate the dimensionality of signals of known structure.