Fourier Analysis of Stationary Processes

  • David R. Brillinger
Chapter
Part of the Selected Works in Probability and Statistics book series (SWPS)

Abstract

This Paper begins with a description of some of the important Procedures of the Fourier analysis of real-valued stationary discrete time series.

Keywords

Entropy Mercury Torque Covariance Coherence 

References

  1. G. G. Stokes, “Note DR searching for pertodicittes,” Proc. Roy. Soc., vol. 29, p, 122, 1879.Google Scholar
  2. A. Schuster, “The periodogram of magnetic declination,” Cambridge Phil. Soc., vol. 18, p. 18, 1899.Google Scholar
  3. E. T. Whittaker and A. Robinson, The Calculus of Observations. Cambridge, England : Cambridge Univ. Press, 1944.Google Scholar
  4. I. W. Cooley and J. W. Tukey, “An a1,orilbm Cor the machine calculation of complex Fourier serie••” Math. Comput., vol. 19, pp. 297-301.1965.MathSciNetMATHGoogle Scholar
  5. A. A. Michelson, Light Wave••nd Their U.... Chica,o.IU.: Univ. Chicago Pre.., 1907.Google Scholar
  6. I. Newton, Optic,…. London, England: W.Jnny., 1730. M. I. Pupin, "Resonance analysis of alternating and polyphase Google Scholar
  7. currents," Tra1l.1. lEE. vol. 9. p. 523. 1894. H. L. Moore, Economic Cyclel Their Law and CauH. N~ York : Macmillan. 1914.Google Scholar
  8. H. A. Panofsky and R. A. McCormick, “Properti es or spectra of atmospheric turbulence at 100 metres,” Quan. J. Roy. M,teorol. Soc•• vol. 80. pp. 546-564, 1954.Google Scholar
  9. A. Leese and E. S. Epstein, “Application of two-dimensional spectral anuylis to the quantification of sateltte cloud photo. artPhs,” J. Appl. Meteorol., vet. 2. pp. 629-644, 1963.Google Scholar
  10. M. S. Butlett. “The spectral analysis of point proeesse…” J. Roy. Std. Soc.• vol. B 25, pp. 264--296, 1963. -, “The spectral analysis of two dimensional point processes,” Siom.triM. vet. 5 I. pp. 299-311. 1964.Google Scholar
  11. A. V. Oppenheim, R. W. Schafer, and T. G. Stockham. Jr., “Nen.. linear filtering of muJtiplied and convolved allnah/' Proc. IEEE. vol. 56. pp. 1264-1291, 1968.Google Scholar
  12. B. P. Bogert, M. J. Heale y, and J. W. Tukey. “The quefrency alanysis of time series for echoes: cepstrum, pseude-ecvartance, cross-cepstrum and sapbe Ctlckin&.” in Ttme Serles Analyns.Google Scholar
  13. M. Rosenblatt. Ed. New York: WOey, pp. 209-243, 1963. M. S. Bartlett, “Periodogram analysts and continuous spectra.” Biometrib, vol. 37, PI'. 1-16, 1950.Google Scholar
  14. R. H. Jones, “A reappraisal of the perlodogram in spectral analy, ts,'· Techllom.triCl, vol. 7. pp, 531-542. 1965.Google Scholar
  15. K. Hasselman, W. Munk, and G. J. F. MacDonald, “Bispectra of ocean waves,” in Time Series Analysis. M. Rosenblatt, Ed. New York: Wiley, pp. 125-139. 1963.Google Scholar
  16. N. Wiener, uGeneralized harmonic analYlts," Acta Math., vol. 55, pp. 117-258. 1930.MathSciNetMATHCrossRefGoogle Scholar
  17. H. Wold. SibltOfTtlPhy 011 Tim. SemI alld Stocluuttc PrOCCDCs. London. England: Oliver and Boyd. 1965.Google Scholar
  18. D. R. 8rillinger. Tim. S.rles:• Data AM/ym G1UI Th.ory. New York: Holt. Rinehart and Winston, 1974.Google Scholar
  19. A. Va. Khintc:hine. uKorrelaUonstheories der.tationiren stochastbchen Prozesse," Ma.th. Ann., vol. 109. pp. 604-615, 1934.Google Scholar
  20. H. Cramer. “On harmonic analy.is in certain functional.paces,” Ark. M.t. A.troll. Fy … vol. 288, pp. 1-7, 1942.Google Scholar
  21. V. P. Loonovand A. N. Shlryaev. “Some problems in the spectral theory of higher moments, II.” Th.ory Frob. App/. (USSR). vol. 5. pp. 460-464, 1960.Google Scholar
  22. B. Picinbono, uTendence vers Ie cuactere Gauasien par filtrage selectlf." C. R. Acad. ScL Parl8. vol. 248. p. 2280, 1959.Google Scholar
  23. M. Rosenblatt, “Some commenta on narrow band-paM roten,” Quart. AppL Math., vol. 18. pp. 387-393, 1961.Google Scholar
  24. D. R. Bri1li.n&er. “The frequency analyais of relations between stationlUY spatial series,” in hoc. 12th B{~nnlal Semina' of the C'41tadJ4n Math. COIIgre&l. R. Pyke, Ed. Montreal, P.Q•• Canada: Can. Math. Congr., pp. 39-81, 1970.Google Scholar
  25. E. J. Hannan and P. J. Thomson, “Spectral Inferenee over nanow banda,” J. AppL Prob., vel, 8, pp. 157-169. 1971.Google Scholar
  26. E. Parzen. “On consistent eatimates of the spectrum of stationary time series,” Ann. Mllth. Stati.rt.., vet, 28. pp. 329-348, 1957.Google Scholar
  27. M. Rosenblatt, “Statistical analy.ts of.toehutlc proe..... with.tationaIY reaidua1s,” In l'robabUity alld StatiltiCl. U. Grenander, Ed. New York: WOey. pp, 246-275. 19$.9.Google Scholar
  28. E. W. carpenter, “Explosions selsmolo,y.” Scknce, vol. 147, pp. 363-373. 1967..Google Scholar
  29. D. G. Lambert, E. A. Flinn. and C. B. Atthambeau. U A comparative study of the elutic wave radiation from earthquakes and underground explodons." Geophy& J. Roy. A.rtroPL Soc., vol. 29, pp. 40_32. 1972.Google Scholar
  30. W. H. Munk and G. I. F. MacDonald, Rotation of til. Earth.. Cambridge, England: Cambridge Univ. Pr.... 1960.Google Scholar
  31. G. 1. F. MacDonald and N. Nesa, IIA study of the free o&cillationl of the Earth," J. G.ophy.. R.... vol. 66, pp. 1865-1911. 1961.Google Scholar
  32. R. L. Wegel and C. R. Moore. “An electrical frequency analyzer.” BeU Syn. Tech. J.. vol. 3, pp. 299-323.1924.Google Scholar
  33. N. WIener, Tim. S.rle.. Cambridge. M.... : M.l.T. Pr.... 1964.Google Scholar
  34. U. Grenander and M. Rnsenblatt. Statiltlcal A1IGly.ll.r of StatiOMry Tim. S.ri… N.w York: WOey. 1957.Google Scholar
  35. E. Panen, '"An approach to empirical time series analysis." Radio Sci., vol. 68 D. pp. 551-565,1964.Google Scholar
  36. R. T. Lacou ~ UData adaptive.pectral anaJyais methods," Geo phyllc• • vol. 36, pp. 661-675. 1971.Google Scholar
  37. J. P. Burg. “The relationship between maximum entropy spectra and maximum likelihood spectra," G~ophync” vol. 31, PP. 375-376.1972.Google Scholar
  38. K.. N. Betk, UConsiltent autorepea:lve spectral estimates:' Ann. Star., vol. 2. pp. 489-502. 1974.Google Scholar
  39. V. E. Piaarenko, liOn the estimation of spectra by means of non ~ linear functions of the covariance matrix," Geophya. J. Roy. Anroll. Soc.• vol. 28. pp. 511-531. 1972.Google Scholar
  40. J. Capon, uInveatiption of long·period noise at the large apertuzt aeiamic array,” J. “Geophy s. Re& l vol. 74. pp. 3181-3194. 1969.Google Scholar
  41. E. J. Hannan, Mu/tipl. Tim. S.rle•. New York: WOey. 1970.Google Scholar
  42. T. W. Andenon. Th. Statl.rt/cal A1IGlym of Tim. S.m.. New York: WlIey. 1971.Google Scholar
  43. P. Whittle, uEatimation and information in stationary time seri..... Ark. Mat. Anroll. Fy … vol. 2. pp. 42_34. 1953.Google Scholar
  44. J. M. Cbamben, !tFitting nonlinear mode": numerical techniques/' BtDm,trik.a, vol. 60, pp. 1-14, 1973.Google Scholar
  45. D. R. Brillinger. uAn empirical invca1ia;ation of the Chandler wobble and two proposed exdtatlon proceases." Sull. I..t. Stat. Inn•• vol. 39, pp. 413-434.1973.Google Scholar
  46. P. Whittle. “Gaua.sia.n estimation in.taUonary time seriu,” BuU. lilt. Stat. Inn.. vol. 33, pp. 105-130. 1961.Google Scholar
  47. A. M. Walker, 14A.s.ymptotic properties of leest-squerea estimates of parameters of the spectrum of a stationary nondeterminbtic time-series." J. Australian klath. Soc., vol. 4, pp. 363-384, 1964.MATHCrossRefGoogle Scholar
  48. K. O. Dzaparidze, itA new method in estimatins spectrum parameten of a stationary regular time series," Teor, Veroyat. Er Prim ea., vol. 19, p, 130. 1974.Google Scholar
  49. L. H. Koopmens, "On the coefficient of coherence for weakly stationary stochutlc proeeeses," Ann. Math. StD.t., vol. 35. pp. 532-549. 1964.Google Scholar
  50. H. Akaike and Y. Yamanouchi, “On the statist ical estimation of frequency response function:' Ann. Inn. Stat. MQth., vol. 14, pp. 23-56. 1962.MathSciNetMATHGoogle Scholar
  51. L. 1. Tick. UEstimation of eoherenev, If in Advanced Snnlntu on Spectral AM/y.ll.r of Tim. Sorles, 8. Harris, Ed. New York: WlIey, 1967, pp. 133-152.Google Scholar
  52. R. J. Bhamali, uEatimation of the Wiener filter," in ContribuM'l Pap." 39th S.lIio.. t..t Stat. lnst., vol. I. pp. 82-88. 1973.Google Scholar
  53. A. E. Hoerl and R. W. Kennard, “Rldg. regretlion : biased..ti~· tion for nonorthogonal problems:' Technometrlu, vol. 12. PI'. 55-67.1970.Google Scholar
  54. B. R. Hunt. “Biased estimation for nonparametric identification of linear,yateDU,” Math. Bio.scl., vol. 10. pp - 215-237. 1971.Google Scholar
  55. S. Bochner and K. Chandruekhann, Fourier 1'rruufomu. Princeton, N.J.: Princeton Univ. Press. 1949.Google Scholar
  56. J. Mannol, '~A CllLU of fidelity criteria for the encoding ofviJuaI images," Ph.D. dissertation, Univ. California, Berkeley, 1912.Google Scholar
  57. L. J. Tick. “Conditlonalapectra, linear systems and coherency,” in Time Sems AnaJym, M. Rosenblatt. Ed. New York : Wiley. pp. 197 -203, 1963.Google Scholar
  58. C. W. I. Granger. Sp.ctral A1IGlym Of Eco..omte Tim. s.rle•. Princeton, N.J. : Princeton Univ. Press, 1964.Google Scholar
  59. N. R. Goodman, uMeasurement of matrix frequency response functions and multiple coherence functions. u Air Force Dynamics Lab., Wright Patterson AF8, Ohio, Tech. Rep. AFFDL·TR·65·56, 1965.Google Scholar
  60. J. S. Bendat and A. Piersot, M."",nm'lIt and A1IGlym ofRG1UIom Data. New York: Wiley, 1966.Google Scholar
  61. G. W. Groves and E. J. Hannan, “Time aeries regresaion of sea level on weather:' Rev. Geophy&. vol. 6. pp. 129-174, 1968.Google Scholar
  62. W. Gersch, “Causallty or drIviag In electrophya!ological signal analysis.” J. Math. SlencL. vet. 14, pp. 177-196. 1972.Google Scholar
  63. N. R. Goodman, “Eigenvalues and eiaenvecton of.pectral density matrices,' Tech. Rep. 179. Seismic Data Lab., Teledyne, 1n<:•• 1967.Google Scholar
  64. D. R. Bri11ine:er. liThe canonical analysiJ 01 time aeries:· in Multlvari< rte AM/ym-ll. P. R. Krlshnaiah. Ed. New York: Acad.mic, pp. 331-350. 1970.Google Scholar
  65. M. B. Priestley. T_ Subba Rao. and H. TonI. UIdentiflcatlOD of the structure of multivariable Itocbuttc systema," in MllltiYdrlate A1IGly~/ll. P. R. Krlshnaiah. Ed. New York: Academic. pp. 351-368.1973.Google Scholar
  66. M. Miyat., “Complex,eneraUution of Cl.l1onicalcorrelation and its applicatinn to a …·Iev.l,tudy.” J. MarI1te R.... vol. 28. pp. 202-214, 1970.Google Scholar
  67. W. S. Llgett, Jr., “Passive sonar : Fitting models to multiple time aeries: ' paper presented at NATO Advanced Study wtitute on Signal Processing, Loughborough. U. K.. 1972.Google Scholar
  68. D. R. Brilllnaer, uThe analya.. of time series coUected in an experimental design," Mu/tivarlat. A1IGly~/ll, P. R. KrlshnaIah, Ed. New York : Academic. pp. 241-256. 1973.Google Scholar
  69. D. R. Btillinger and M. Hatanaka. IIAn hatmonJc analysis o( non stationary multivariate economic processes," EconometrlCG. vol. 35. pp. 131 -141, 1969.Google Scholar
  70. E. J. Hannan and R. D. Terreil, “MUltiple equation.yateDU with stationary enon,” Econom~tri.t:uI..yol. 41, pp. 299-320, 1973.Google Scholar
  71. C. R. Rao. Lwar Statl.rt/cal I"fer."". G1UI IU AppUcatio.... New York: W~.y. 1965.Google Scholar
  72. D. R. Cox and P. A. W. Lewis. Th. Statiltlcal A1IGlym ofS.m. ofEvc.. u. London. England: Methuen. 1966.Google Scholar
  73. D. R. Brilllnaer, “lbe spectral analysis of stationary interval func· tions.” Proc. 6th Btrktley Symp. MGth. Stat. hob. Yolo 1,Google Scholar
  74. L. M. Le Cam, I. Neyman. and E. L. Scol!, Edt. Barkeley, Calif.: Univ. California Pr…, pp. 483-513.1972.Google Scholar
  75. D. 1. Daley and D. Vere-Jonel, HA summary of the theory of point processes." in Stochastic Point PrOCI!I:.MS, P. A. W. Lewis, Ed. New York: WlIey. pp. 299-383. 1972.Google Scholar
  76. L. Filher, itA survey of the mathematical theory of multidimenlional point procean:' in Stochatlc Point Proce»e~, P. A. W.Lewis, Ed. New York: Wil.y. pp. 468-513.1972.Google Scholar
  77. ~ G. Hawkes and L. Adamopouloa, “Cluster modell forearthquakea-reponaJ comparisons,” BuD. Int. Stat.lnIt.• vol. 39,pp. 454-460. 1973,Google Scholar
  78. A- M. Y_slom, uSome classes of random fields in n-dimensional space related to stationary random proceaes," 17J~OT)1 Prob. App/. (USSR), v91. 2. pp. 273-322.1959.Google Scholar
  79. L. Schwartz. Th.orle d.. Dl.rtrlbutio.... Voh 1.2. Paris. France: Hermann. 1957.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • David R. Brillinger
    • 1
  1. 1.Department of StatisticsUniversity of CaliforniaBerkeleyUSA

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