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Life in the Fast Lane: Yates’s Alogrithm, Fast Fourier and Walsh Transforms

  • Paul J. Sanchez
  • John S. Ramberg
  • Larry Head
Part of the International Series in Operations Research & Management Science book series (ISOR, volume 46)

Abstract

Orthogonal functions play an important role in factorial experiments and time series models. In the latter half of the twentieth century orthogonal functions became prominent in industrial experimentation methodologies that employ complete and fractional factorial experiment designs, such as Taguchi orthogonal arrays. Exact estimates of the parameters of linear model representations can be computed effectively and efficiently using “fast algorithms.” The origin of “fast algorithms” can be traced to Yates in 1937. In 1958 Good created the ingenious fast Fourier transform, using Yates’s concept as a basis. This paper is intended to illustrate the fundamental role of orthogonal functions in modeling, and the close relationship between two of the most significant of the fast algorithms. This in turn yields insights into the fundamental aspects of experiment design.

Keywords

Fast Fourier Transform Discrete Fourier Transform Modeling Uncertainty Fast Algorithm Generate Vector 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Ahmed, N., and K. R. Rao. (1971). “The generalised transform.” Proc. Applic. Walsh Functions, Washington, D.C. AD727000, 60–67.MathSciNetCrossRefGoogle Scholar
  2. Beauchamp, K.G. (1984), “Applications of Walsh and related functions.” Aca-demic Press, London.Google Scholar
  3. Chatfield, C. (1984), “An Analysis of Time Series: An Introduction.” Chapman and Hall, New York.CrossRefGoogle Scholar
  4. Cooley, J.W., and J. W. Tukey. (1965), “An algorithm for the machine computation of complex Fourier series.” Math. Comput. 19, 297–301.CrossRefGoogle Scholar
  5. Good, I.J. (1958), “The interactive algorithm and practical Fourier analysis.” J. Roy. Stat. Soc. (London), B20, 361–372.zbMATHGoogle Scholar
  6. Heideman, M.T., D. H. Johnson, and C. S. Burrus. (1984), “Gauss and the History of the Fast Fourier Transform.” IEEE ASSP Magazine, Oct. 1984, 14–21.Google Scholar
  7. Hoadley, A.B., and J. R. Kettenring. (1990), “Communications Between Statisticians and Engineers/Physical Scientists.” Technometrics Vol 32No. 3,243–247.Google Scholar
  8. Kiefer, R., and J. Wolfowitz. (1959), “Optimal Designs in Regression Problems.” Ann. Math. Stat. Vol 30, 271–294.CrossRefGoogle Scholar
  9. Manz, J.W. (1972), “A sequency-ordered fast Walsh transform.” IEEE Trans. Audio Electroacoust. AV-20, 204–205.CrossRefGoogle Scholar
  10. Nelson, L.S. (1982), “Analysis of Two-Level Factorial Experiments.” JQT Vol 14,2, 95–98.Google Scholar
  11. Pratt, W.K., J. Kane, and H. C. Andrews. (1969), “Transform image coding.” Proc. IEEE, 57, 58–68.CrossRefGoogle Scholar
  12. Sanchez, P.J., and S. M. Sanchez. (1991). “Design of frequency domain experiments for discrete-valued factors.” Applied Mathematics and Computation, 42(1): 1–21.CrossRefGoogle Scholar
  13. Stoffer, David S. (1991). “Walsh-Fourier Analysis and Its Statistical Implications” J. American Statistical Association., June 1991, Vol. 86,#414, 461–485.MathSciNetCrossRefGoogle Scholar
  14. Walsh, J.L. (1923), “A closed set of orthogonal functions.” Ann. J. Math, 55, 5–24.MathSciNetzbMATHGoogle Scholar
  15. Yates, F. (1937), “The Design and Analysis of Factorial Experiments.” Technical Communication No. 35, Imperial Bureau of Soil Science, London.Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2002

Authors and Affiliations

  • Paul J. Sanchez
    • 1
  • John S. Ramberg
    • 2
  • Larry Head
    • 3
  1. 1.Operations Research DepartmentNaval Postgraduate SchoolMonterey
  2. 2.Systems and Industrial EngineeringUniversity of ArizonaTucson
  3. 3.Siemens Energy & Automation, Inc.Tucson

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