Basis Functions in the Analysis of Evoked Potentials
ER data may be reduced to a more compact form by approximating the original responses by a linear combination of functions. Two main classes of approximating functions are discussed. The first of these are fixed basis functions. These are functions that do not depend upon the data. These are sine-cosine (Fourier), Walsh-Hadamard (square waves), Haar functions (square wave impulse func tions) , and polynomials. The other functions are random in that they depend upon the data. These arise from various multivariate statistical procedures. The procedures discussed are principal components, factor analysis, union-intersection tests, and canonical correlation.
Some of the relative advantages of each of the methods are discussed. The fixed basis function methods are relatively fast and simple to compute. The multivariate methods are difficult to compute but tend to be better approximations because they depend upon the sample to select the functions. The multivariate test statistics based upon the union-intersection principle may be interesting because, in some cases, they both generate test statistics and characteristic waveforms. Most of these methods have not been used enough to offer any concrete recommendations.
Fourier, polynomial, Walsh-Hadamard, Haar, principal compon ents and factor analysis are compared in a least squares sense for goodness of fit for average evoked responses for 43 subjects. The principal components procedure will always result in a best fit. Factor analysis was the next best approximation with Fourier approximation next best and sometimes better. The Walsh-Hadamard and Haar were not as good but might well be satisfactory for some types of analysis, especially in view of their computational efficiency. Polynomial approximations are poor for low degree polynomials but improve more rapidly than the Walsh-Hadamard and the Haar approximations. Polynomial approximations are not as good as Fourier for these data and they are slower to compute.
The direct use of multivariate methods may be impractical due to the large matrices involved. However, hybrid methods that use Fourier, Walsh-Hadamard or Haar functions to reduce the dimensions and then use principal components or union-intersection tests may prove useful.
KeywordsCanonical Correlation Analysis Multivariate Method Evoke Potential Walsh Function Haar Function
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- Andrews, H.C. Multidimensional rotations in feature selection. IEEE Transactions, 1971, C-20(9), 1045–1051.Google Scholar
- Bass, CA. (ed.). Application of Walsh functions symposium. Springfield: National Tehcnical Information Service, 1970.Google Scholar
- Brigham, E.O. The fast Fourier transform. New Jersey; Prentice-Hall, 1974.Google Scholar
- Collatz, L. Functional analysis and numerical mathematics, New York: Academic Press, 1966.Google Scholar
- Dixon, W.J. (Ed.). Biomedical computer programs. Berkeley: University of California Press, 1974.Google Scholar
- Glasser, E.M., & Ruchkin, D.S. Principles of neurobiological signal analysis. New York: Academic Press, 1976.Google Scholar
- Harmon, H.H. Modern factor analysis. Chicago: University of Chicago Press, 1976.Google Scholar
- IEEE Transactions on audio and electroacoustics. 1969, AU-17, 66–76.Google Scholar
- IEEE Transactions on audio and electroacoustics. 1967, AU-15, 45–55.Google Scholar
- Jacoby, B.F. Walsh functions. Byte, 1977, _2(9), 190–198.Google Scholar
- Marriot, F.H.C. The interpretation of multiple observations, New York: Academic Press, 1974.Google Scholar
- Martin, D.C., Becker, J. & Buffington, V. An evoked potential study of endogenous affective disorders in alcoholics. Printed in this volume, Begleiter, H. (Ed.), 1977.Google Scholar
- Milne, W.E. Numerical calculus. Princeton: Princeton University Press, 1949.Google Scholar
- Morrison, D.F. Multivariate statistical methods. New York: McGraw-Hill, 1976.Google Scholar
- Ralston, A.A. A first course in numerical analysis. New York: McGraw-Hill, 1965.Google Scholar
- Shanks, J.L. Computation of the fast Walsh-Fourier transform. IEEE Transactions on Computers, 1969, May, 457–459.Google Scholar
- Wampler, R.H. A report on the accuracy of some widely used least squares computer programs. Journal of the American Statistical Association, 1970, 65(330), 549–565.Google Scholar
- Zeek, R.W., & Showalter, A.E. Applications of Walsh functions. Springfield: National Technical Information Service, 1971.Google Scholar
- Zeek, R.W., & Showalter, A.E. Applications of Walsh functions. Springfield: National Technical Information Service, 1972.Google Scholar