Walsh-Fourier Analysis and its Statistical Applications

References

• Ahmed , N. and Rao , K. R. 1975 . Orthogonal Transforms for Digital Signal Processing , New York : Springer-Verlag .
   Google Scholar
• Excellent introduction to Fourier and Walsh–Fourier analysis from the engineering perspective. Discusses the fast Fourier and fast Walsh–Fourier transforms in a fair amount of detail, and FORTRAN computer subroutines for both transforms are provided
   Google Scholar
• Beauchamp , K. G. 1975 . Walsh Functions and Their Applications , London : Academic Press .
   Google Scholar
• Beauchamp , K. G. 1984 . Applications of Walsh and Related Functions , London : Academic Press .
   Google Scholar
• Excellent introductions to Walsh and related functions, also, and to the spectral decomposition of time series via the Walsh–Fourier transform from the engineering perspective. Comparisons between Walsh–Fourier and Fourier decompositions of simulated data, as well as waveform synthesis using sinusoids and Walsh functions, are discussed in detail. Provides a perspective on situations when Walsh based analysis, as opposed to trigonometric-based analysis, is relevant. The 1984 edition is essentially an updated version of the 1975 text and contains examples of many engineering applications of Walsh functions as well as a fairly up to date list of references
   Google Scholar
• Bloomfield , P. 1976 . Fourier Analysis of Time Series: An Introduction , New York : John Wiley .
   Google Scholar
• An introduction to statistical Fourier analysis that requires knowledge of calculus and an introductory course in mathematical statistics. Aimed at practitioners as well as students. Contains FORTRAN computer subroutines for analysis of data based on the introduced techniques
   Google Scholar
• Bramhall , J. N. Bibliography on Walsh and Walsh Related Functions . Proceedings of the Symposium on the Applications of Walsh Functions . pp. 416 – 460 . Springfield , VA : U.S. Department of Commerce .
   Google Scholar
• Extensive bibliography containing roughly 800 entries on Walsh and related functions published between 1900 and 1974. Entries are categorized into five main categories and appropriate subcategories. The major categories are: mathematical treatment, generation of Walsh functions, applications, comparison with other types of analysis, miscellaneous
   Google Scholar
• Breiman , L. and Friedman , J. H. 1985 . Estimating Optimal Transformation for Multiple Regression and Correlation . Journal of the American Statistical Association , 80 : 580 – 619 . (with comments)
   Web of Science ®Google Scholar
• Brillinger , D. R. 1980 . “ Analysis of Variance Problems Under Time Series Models ” . In Handbook of Statistics , Edited by: Krishnaiah , P. R. Vol. 1 , 237 – 278 . Amsterdam : North Holland .
   Google Scholar
• The definitive article on analysis of variance for time series in the spectral domain
   Google Scholar
• Brillinger , D. R. 1981 . Time Series: Data Analysis and Theory, , 2nd ed. , San Francisco : Holden-Day .
   Google Scholar
• Brockwell , P. J. and Davis , R. A. 1987 . Time Series: Theory and Methods , New York : Springer-Verlag .
   Google Scholar
• Chrestenson , N. E. 1955 . A Class of Generalized Walsh Functions . Pacific Journal of Mathematics , 5 : 17 – 31 .
   Google Scholar
• 1950 . Extends Fine's work on generalized Walsh functions See Fine
   Google Scholar
• Cryer , J. D. 1986 . Time Series Analysis , Boston : Duxbury Press .
   Google Scholar
• Day , N. L. , Wagener , D. K. and Taylor , P. M. 1985 . “ Measurement of Substance Use During Pregnancy: Methodological Issues ” . In Current Research on the Consequences of Maternal Drug Abuse , Edited by: Pinkert , T. M. 36 – 47 . Washington , DC : U.S. Government Printing Office . (National Institute on Drug Abuse Research Monograph 59)
   Google Scholar
• Fine , N. J. 1949 . On Walsh Functions . Transactions of the American Mathematical Society , 65 : 372 – 414 .
   Web of Science ®Google Scholar
• Fine , N. J. 1950 . The Generalized Walsh Functions . Transactions of the American Mathematical Society , 69 : 66 – 77 .
   Web of Science ®Google Scholar
• Fine , N. J. 1957 . Fourier–Stieltjes Series of Walsh Functions . Transactions of the American Mathematical Society , 86 : 246 – 255 .
   Google Scholar
• These articles deal with some of the mathematical properties of Walsh functions. The 1950 paper establishes the generalized Walsh functions by extending the definition of Walsh functions to W(x, y), where x, y ε R+. The 1957 paper essentially establishes a Walsh–Fourier spectral distribution function–see Kohn (1980a, Sect. 4) for details
   Google Scholar
• Fourier , J. L. 1822 . Theorie Analytique de la Chaleur, , English ed. , The Analytical Theory of Heat . (1878); Reprinted New York: Dover, 1955
   Google Scholar
• Frankel , H. D. Applications of Walsh Functions . Proceedings of the Symposium on the Applications of Walsh Functions . pp. 134 – 136 . Springfield , VA : U.S. Department of Commerce .
   Google Scholar
• Fuller , W. A. 1976 . Introduction to Statistical Time Series , New York : John Wiley .
   Google Scholar
• Good , I. J. 1958 . The Interaction Algorithm and Practical Fourier Analysis . Journal of the Royal Statistical Society , 20 : 361 – 372 . Ser. B
   Google Scholar
• The first appearance of the Walsh–Fourier transform in the statistics literature, but it makes no reference to the Walsh literature at the time. This work forms the basis of the sequency-order Walsh–Fourier transform by relating the calculations involved to Yates's algorithm for calculating effects in a 2n factorial design
   Google Scholar
• Good , I. J. 1971 . The Relationship Between Two Fast Fourier Transforms . IEEE Transactions on Computers , 20 : 310 – 317 .
   Web of Science ®Google Scholar
• Greenacre , M. J. 1984 . Theory and Applications of Correspondence Analysis , London : Academic Press .
   Google Scholar
• Haar , A. 1910 . Zur Theorie der Orthogonalen Funktionensysteme . Mathematische Annalen , 69 : 331 – 371 .
   Google Scholar
• Hauri , P. 1982 . The Sleep Disorders , Kalamazoo , MI : The Upjohn Company .
   Google Scholar
• Hannan , E. J. 1970 . Multiple Time Series , New York : John Wiley .
   Google Scholar
• Harmuth , H. F. 1969a . Applications of Walsh Functions in Communications . IEEE Spectrum , 6 : 82 – 91 .
   Web of Science ®Google Scholar
• Harmuth , H. F. 1969b . Transmission of Information by Orthogonal Functions, , 2nd Ed. , New York : Springer-Verlag . 1972
   Google Scholar
• Harmuth , H. F. 1977 . Sequency Theory: Foundations and Applications , New York : Academic Press .
   Google Scholar
• The engineering approach to the study and use of the Walsh functions was originated in the first two works. These were the seeds that begot the hundreds of articles listed in Bramhall's bibliography. Each work contains interesting applications of Walsh functions as well as a history of the uses of nonsinusoidal functions. The introductions are interesting to read, but the main texts will be difficult for statisticians to read
   Google Scholar
• Ito , T. Applications of Walsh Functions to Pattern Recognition and Switching Theory . Proceedings of the Symposium on the Applications of Walsh Functions . pp. 128 – 137 . Springfield , VA : U.S. Department of Commerce .
   Google Scholar
• Outlines a statistical theory of pattern recognition for multivariate binary variables based on the Walsh expansion of the density functions along the same lines as the techniques presented in Ott and Kronmal (1976) and discussed in Section 6.5
   Google Scholar
• Kimeldorf , G. , May , J. H. and Sampson , A. R. 1982 . “ Concordant and Discordant Monotone Correlations and Their Evaluations by Nonlinear Optimization ” . In Optimization in Statistics , Edited by: Zanakis , S. H. and Rustagi , J. S. 117 – 130 . Amsterdam : North Holland . (Studies in the Management Sciences Vol. 19)
   Google Scholar
• Kohn , R. 1980a . On the Spectral Decomposition of Stationary Time Series Using Walsh Functions I . Advances in Applied Probability , 12 : 183 – 199 .
   Web of Science ®Google Scholar
• Kohn , R. 1980b . On the Spectral Decomposition of Stationary Time Series Using Walsh Functions II . Advances in Applied Probability , 12 : 462 – 474 .
   Web of Science ®Google Scholar
• A rather complete set of articles establishing the Walsh–Fourier theory for real-time stationary time series summarized in Section 3. Shows that many of the results concerning the decomposition of stationary time series using trigonometric functions have their Walsh function analogies. Part I focuses on the asymptotic properties of the Walsh–Fourier transform and introduces a Walsh–Fourier spectral density function. Part II discusses estimation of the Walsh–Fourier spectrum
   Google Scholar
• Maqusi , M. 1981 . Applied Walsh Analysis , London : Heyden .
   Google Scholar
• Concentrates on theory based on the generalized Walsh functions. Contains some engineering applications and has an entire chapter devoted to Walsh series expansions of univariate and multivariate probability density functions and their uses in the computations of general moments of nonlinear transformations
   Google Scholar
• Morettin , P. A. 1972 . Walsh–Fourier Analysis of Time Series , University of California, Berkeley, Dept. of Statistics . unpublished Ph.D. dissertation
   Google Scholar
• Establishes a statistical theory for the analysis of dyadically stationary time series using the Walsh–Fourier transform
   Google Scholar
• Morettin , P. A. 1973 . A Note on a Central Limit Theorem for Dependent Random Variables . Bulletin of the Brazilian Mathematical Society , 4 : 47 – 49 .
   Google Scholar
• Morettin , P. A. 1974a . Limit Theorems for Stationary and Dyadic-Stationary Processes . Bulletin of the Brazilian Mathematical Society , 5 : 97 – 104 .
   Google Scholar
• These articles were the first to establish results for the analysis of realtime stationary time series based on the Walsh–Fourier transform
   Google Scholar
• Morettin , P. A. 1974b . Walsh-Function Analysis of a Certain Class of Time Series . Stochastic Processes and Their Applications , 4 : 183 – 194 .
   Google Scholar
• Morettin , P. A. 1981 . Walsh Spectral Analysis . SIAM Review , 23 : 279 – 291 .
   Web of Science ®Google Scholar
• Excellent review article on the statistical approach of Walsh spectral analysis. Discusses statistical results for dyadic- and real-time stationary time series. Provides a good opportunity to compare and contrast the dyadic- and real-time approaches to the problem. Includes many references
   Google Scholar
• Morettin , P. A. 1983 . A Note on a Central Limit Theorem for Stationary Processes . Journal of Time Series Analysis , 4 : 49 – 52 .
   Google Scholar
• Nishisato , S. 1980 . Analysis of Categorical Data: Dual Scaling and its Applications , Toronto : University of Toronto Press .
   Google Scholar
• Ott , J. and Kronmal , R. A. 1976 . Some Classification Procedures for Multivariate Binary Data Using Orthogonal Functions . Journal of the American Statistical Association , 71 : 391 – 399 .
   Web of Science ®Google Scholar
• The first application of Walsh functions in the statistics literature. Presents methods for prediction or classification for multivariate binary data based on a Walsh–Fourier expansion of the density. Good summary of Walsh functions and their generation in appendixes
   Google Scholar
• Paley , R. E. A. C. 1932 . A Remarkable Series of Orthogonal Functions . Proceedings of the London Mathematical Society , 34 : 241 – 279 .
   Google Scholar
• Reintroduces the Walsh functions to the scientific community by defining them as the product of Rademacher functions. This definition is better suited for analytical manipulations where one uses the Walsh functions in Paley order
   Google Scholar
• Priestley , M. B. 1981 . Spectral Analysis and Time Series , Vols. 1 and 2 , New York : Academic Press .
   Google Scholar
• Rademacher , H. 1922 . Einige Satze von Allgemeinen Orthogonal-funktionen . Mathematische Annalen , 38 : 122 – 138 .
   Google Scholar
• Rao , C. R. 1973 . Linear Statistical Inference and Its Applications, , 2nd ed. , New York : John Wiley .
   Google Scholar
• Robertson , T. , Wright , F. T. and Dykstra , R. L. 1988 . Order Restricted Statistical Inference , New York : John Wiley .
   Google Scholar
• Robinson , G. S. Discrete Walsh and Fourier Power Spectra . Proceedings of the Symposium on the Applications of Walsh Functions . pp. 298 – 309 . Springfield , VA : U.S. Department of Commerce .
   Google Scholar
• Robinson , G. S. 1972b . Logical Convolution and Discrete Walsh and Fourier Power Spectra . IEEE Transactions on Audio and Electroacoustics , 20 : 271 – 280 .
   Google Scholar
• Introduces the concept of the logical covariance function, which was the basis for establishing a statistical Walsh–Fourier theory for realtime stationary processes. Compares and contrasts Walsh-based and trigonometric-based spectral analysis
   Google Scholar
• Scheffé , H . 1959 . The Analysis of Variance , New York : John Wiley .
   Google Scholar
• Scher , M. S. , Richardson , G. A. , Coble , P. A. , Day , N. L. and Stoffer , D. S. 1988 . The Effects of Prenatal Alcohol and Marijuana Exposure: Disturbances in the Neonatal Sleep Cycling and Arousal . Pediatric Research , 24 : 101 – 105 .
   PubMed Web of Science ®Google Scholar
• Selfridge , R. G. 1955 . Generalized Walsh Transform . Pacific Journal of Mathematics , : 451 – 480 .
   Google Scholar
• Shumway , R. H. 1988 . Applied Statistical Time Series Analysis , Englewood Cliffs , NJ : Prentice-Hall .
   Google Scholar
• An excellent text that provides an introduction to the spectral analysis of time series. Written for students in the physical, biological, and social sciences, and for applied statisticians. Requires knowledge of one year of calculus and a course on elementary mathematical statistics. Only text to discuss analysis of variance and discriminant analysis for time series data. Comes with software
   Google Scholar
• 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 .
   Google Scholar
• Stoffer , D. S. 1987 . Walsh–Fourier Analysis of Discrete-Valued Time Series . Journal of Time Series Analysis , 8 : 449 – 467 .
   Google Scholar
• An approach to the analysis of discrete-valued time series based on the Walsh–Fourier transform is presented. A general signal-plus-noise model for discrete-valued processes is developed, and solutions to problems involving signal detection and time series regression for discrete-valued processes are discussed
   Google Scholar
• Stoffer , D. S. 1990 . Multivariate Walsh–Fourier Analysis . Journal of Time Series Analysis , 11 : 57 – 73 .
   Google Scholar
• Establishes statistical methodology for the spectral analysis of stationary multivariate time series via the Walsh–Fourier transform with an emphasis on cross-spectral analysis. Assumptions are fairly general so that techniques can apply to discrete-valued time series
   Google Scholar
• Stoffer , D. S. , Scher , M. S. , Richardson , G. A. , Day , N. L. and Coble , P. A. 1988 . A Walsh–Fourier Analysis of the Effects of Moderate Maternal Alcohol Consumption on Neonatal Sleep-State Cycling . Journal of the American Statistical Association , 83 : 954 – 963 .
   Web of Science ®Google Scholar
• The Walsh–Fourier transform is used in an analysis of variance of categorical time series to investigate the spectral components of EEG sleep-state patterns of infants whose mothers abstained from alcohol during pregnancy and infants whose mothers used moderate amounts of alcohol throughout pregnancy
   Google Scholar
• Texter , P. A. and Ord , J. K. 1989 . Forecasting Using Automatic Identification Procedures: A Comparative Analysis . International Journal of Forecasting , 5 : 209 – 215 .
   Web of Science ®Google Scholar
• Uses Walsh functions to develop a procedure to decide the order of differencing in automatic, time domain, forecasting procedures. Shows the relationship between the Walsh–Fourier transform and Yates's algorithm
   Google Scholar
• Tzafestas , S. G. 1985 . Walsh Functions in Signal and Systems Analysis and Design , New York : Van Nostrand .
   Google Scholar
• A diverse collection of articles–with editorial comments–that brings together major results of the engineering literature on Walsh functions
   Google Scholar
• Walsh , J. L. 1923 . A Closed Set of Orthogonal Functions . American Journal of Mathematics , 45 : 5 – 24 .
   Web of Science ®Google Scholar
• The article that first introduced Walsh functions to the scientific community
   Google Scholar