Parameter identification using intrinsic dimensionality
Gerard V. Trunk · IEEE Transactions on Information Theory · 1972
LetWbe anN-dimensional vector space andVbe aK- dimensional topological hypersurface inW. The intrinsic dimensionality problem can be stated as follows. GivenMrandomly selected points u_i, u_i \in V, estimateK, which is the dimensionality ofVand is called the intrinsic dimensionality of the points u_i. This problem has been attacked by several authors. If signals are represented by vectors in an abstract signal spaceW, rather than as time or frequency functions, the locus of signals that are generated by a hypothetical signal generator possessingKfree parameters is aK-dimensional hypersurfaceVin the signal space. The purpose of this paper is to identify some of the free parameters of the signals. LetHbe a one-parameter group that acts onW, HW = W. To say that a parameter associated with a groupHis a free parameter ofVmeans thatVis closed underH, i.e.,HV = V. To decide whetherHV = V, the estimated dimensionalities ofHVandVare compared. The method is illustrated by presenting several examples from the field of signal analysis. Finally, a slight modification is suggested so that parameters can he identified even when the vectors of interest do not represent time or frequency functions.