Abstract
A nonparametric approach to analyzing a stationary binary time series X(n), n = 0, ±1, ±2, … taking values in 0, 1 is discussed. The analysis is accomplished in the spectral domain using the Walsh-Fourier transform which is based on Walsh functions. This seems to be a natural alternative to the trigonometric functions used in the usual spectral analysis since the Walsh functions take on only two values, +1 or − 1, (or “on” and “off”, as does the series X(n) itself). This approach enables the investigator to analyze a binary series in terms of square-waves and sequency (switches or changes per unit time) rather than sine-waves and frequency (cycles per unit time). We discuss (1) the basic theory of Walsh-Fourier analysis, (2) the computational aspects involved in calculating the discrete Walsh-Fourier transform, and (3) the analysis of simulated and real binary data in the sequency domain. We suggest that these methods would enhance the analysis of time series which take values in a discrete finite set.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ahmed, N., and K. R. Rao (1975), Orthogonal Transforms for Digital Signal Processing. New York: Springer-Verlag.
Andrews, H. C. (1970), Computer Techniques for Image Processing. New York: Academic Press.
Beauchamp, K. G. (1975), Walsh Functions and Their Applications. London: Academic Press.
Brillinger, D. R. (1975), Time Series: Data Analysis and Theory. New York: Holt, Rinehart and Winston.
Brillinger, D. R., and J. W. Tukey (1982), “Spectrum estimation and system identification relying on a Fourier transform.” Technical Report No. 5, Department of Statistics, Berkeley, California.
Fine, N. J. (1949), “On the Walsh functions.” Transactions of the American Mathematical Society 65, 372–414.
Fine, N. J. (1950), “The generalized Walsh functions.” Transactions of the American Mathematical Society 69, 66–77.
Fine, N. J. (1957), “Fourier-Stieltjes series of Walsh functions.” Transactions of the American Mathematical Society 86, 246–255.
Harmuth, H. (1972), Transmission of Information by Orthogonal Functions. Berlin: Springer-Verlag.
Jacobs, P. A., and P. A. W. Lewis (1978), “Discrete time series generated by mixtures. I: correlational and runs properties.” Journal of the Royal Statistical Society, Series B 40, 94–105.
Kedem, B. (1980), Binary Time Series. New York: Marcel Dekker.
Kohn, R. (1980a), “On the spectral decomposition of stationary time series using Walsh Functions, I.” Advances in Applied Probability 12, 183–199.
Kohn, R. (1980b), “On the spectral decomposition of stationary time series using Walsh Functions, II.” Advances in Applied Probability 12, 462–474.
Morettin, P. A. (1974a), “Limit theorems for stationary and dyadic-stationary processes.” Boletim de Sociedade Brasileira de Matematica 5, 97–104.
Morettin, P. A. (1974b), “Walsh-function analysis of a certain class of time series.” Stochastic Processes and their Applications 2, 183–193.
Morettin, P. A. (1981), “Walsh spectral analysis.” SIAM Review 23, 279–291.
Morettin, P. A. (1983), “A note on a central limit theorem for stationary processes.” Journal of Time Series Analysis 4, 49–52.
Ott, J., and R. A. Kronmal (1976), “Some classification procedures for multivariate binary data using orthogonal functions.” Journal of the American Statistical Association 71, 391–399.
Panchalingam, T. (1985), “Estimation of Walsh-Fourier spectral density for binary time series.” Masters Thesis, University of Pittsburgh.
Robinson, G. S. (1972), “Discrete Walsh and Fourier power spectra.” Proceedings of the Symposium on the Applications of Walsh Functions. Washington, DC. 298–309.
Skorokhod, A. V. (1956), “Limit theorems for stochastic processes.” Theory of Probability and its Applications 1, 261–290.
Stoffer, D. S. (1985), “Central limit theorems for finite Walsh-Fourier transforms of weakly stationary time series.” Journal of Time Series Analysis 6, 261–267.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1987 D. Reidel Publishing Company
About this chapter
Cite this chapter
Stoffer, D.S., Panchalingam, T. (1987). A Walsh-Fourier Approach to the Analysis of Binary Time Series. In: MacNeill, I.B., Umphrey, G.J., Carter, R.A.L., McLeod, A.I., Ullah, A. (eds) Time Series and Econometric Modelling. The University of Western Ontario Series in Philosophy of Science, vol 36. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-4790-0_12
Download citation
DOI: https://doi.org/10.1007/978-94-009-4790-0_12
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-8624-0
Online ISBN: 978-94-009-4790-0
eBook Packages: Springer Book Archive