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Recent Developments in Nonlinear Time-Series Modeling

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Abstract

Parametric analysis and modeling of signals using linear stationary time series models has found application in a variety of contexts, including speech and seismic signal processing, spectral estimation, process control, and others. The statistical theory of such models is well established and quite complete, and efficient computational algorithms exist in a number of cases.

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References

  1. K.K. Aase, Recursive estimation in non-linear time series models of autoregressive type, J. Roy. Stat. Soc. Ser. B, Vol. 45, pp. 228–237, 1983.

    MathSciNet  MATH  Google Scholar 

  2. H. Akaike, Information theory and an extension of the maximum likelihood principle, in 2nd International Symposium on Information Theory, B.N. Petrov and F. Czaki, Eds., Akademiai Kiado, Budapest, pp. 276–281, 1973.

    Google Scholar 

  3. T.W. Anderson, Statistical Analysis of Time Series, New York, NY: Wiley, 1971

    MATH  Google Scholar 

  4. V.E. Benes, Exact finite dimensional filters for certain diffusions with nonlinear drift, Stochastics, Vol. 5, pp. 65–92, 1981.

    MathSciNet  MATH  Google Scholar 

  5. M. Bhaskara Rao, T. Subba Rao and A.M. Walker, On the existence of strictly stationary solutions to bilinear equations, J. Time Series Anal., Vol. 4, pp. 95–110, 1983.

    Article  MATH  Google Scholar 

  6. D.R. Brillinger, An introduction to polyspectra, Ann. Math. Statist., Vol. 36, pp. 1351–1374, 1965.

    Article  MathSciNet  MATH  Google Scholar 

  7. D.R. Brillinger and M. Rosenblatt, Asymptotic theory of k-th order spectra, in Spectral Analysis of Time Series, B. Harris, Ed., New York, NY: Wiley, pp. 153–188, 1967.

    Google Scholar 

  8. R.W. Brockett, Volterra series and geometric control theory, Automatica, Vol. 12, pp. 167–172, 1976.

    Article  MathSciNet  MATH  Google Scholar 

  9. D.R. Cox, Statistical analysis of time series: Some recent developments. Scand. J. Statist., Vol. 8, pp. 93–115, 1981.

    MathSciNet  MATH  Google Scholar 

  10. M.H.A. Davis and S.I. Marcus, An introduction to nonlinear filtering, in Stochastic Systems: The Mathematics of Filtering and Identification and Applications, M. Hazewinkel and J.C. Willems, Eds., Doordrecht, Holland: D. Reidel, pp. 53–76, 1981.

    Google Scholar 

  11. M. Fujisaki, G. Kallianpur and H. Kunita, Stochastic differential equations for the nonlinear filtering problem, Osaka J. Math. Vol. 1, pp. 19–40, 1972.

    MathSciNet  Google Scholar 

  12. C.W.J. Granger and A.P. Andersen, An Introduction to Bilinear Time Series Models, Göttingen, Germany: Vanderhoeck and Ruprecht, 1978.

    Google Scholar 

  13. D. Guegan, Une condition d’ergodicité pour des modèles bilinéaires à temps discret, C. R. Acad. Sci. Paris, Vol. 297, pp. 537–540, 1983.

    MathSciNet  MATH  Google Scholar 

  14. V. Haggan and T. Ozaki, Modelling nonlinear random vibrations using an amplitude dependent autoregressive time series model, Biometrika, Vol. 68, pp. 189–196, 1981.

    Article  MathSciNet  MATH  Google Scholar 

  15. M.G. Hall, A.V. Oppenheim and A.S. Willsky, Time-varying parameter modelling of speech, Signal Processing, Vol. 5, pp. 267–285, 1983.

    Article  Google Scholar 

  16. M. Hallin, Spectral factorization of nonstationary moving average processes, Ann. Statist., Vol. 12, pp. 172–192, 1984.

    Article  MathSciNet  MATH  Google Scholar 

  17. M. Hallin and J. Fr. Ingenbleek, Nonstationary Yule-Walker equations, Statist, and Probab. Letters, Vol. 1, pp. 189–195, 1983.

    MathSciNet  MATH  Google Scholar 

  18. E.J. Hannan and B.G. Quinn, The determination of the order of an autoregression, J. Roy. Stat. Soc. Ser. B, Vol. 41, pp. 190–195, 1979.

    MathSciNet  MATH  Google Scholar 

  19. D.L. Hanson, A representation theorem for stationary Markov chains, Math. Mech,, Vol. 12, pp. 731–736, 1963.

    MathSciNet  MATH  Google Scholar 

  20. P.J. Harrison and C.F. Stevens, Bayesian forecasting (with discussion), J. Roy. Stat. Soc. Ser. B, Vol. 38, pp. 205–248, 1976.

    MathSciNet  MATH  Google Scholar 

  21. Y.C. Ho and R.C.K. Lee, A Bayesian approach to problems in stochastic estimation and control, IEEE Trans. Auto. Contr., Vol. AC-9, pp. 333–339, 1964.

    Google Scholar 

  22. D.A. Jones, Non-linear autoregressive processes, Proc. Roy. Soc. London, A, Vol. 360, pp. 71–95, 1978.

    Article  MATH  Google Scholar 

  23. G. Kallianpur, Some ramifications of Wiener’s ideas on nonlinear prediction, in Norbert Wiener Collected Works Volume III, P. Masani, Ed., Cambridge, MA: MIT Press, pp. 402–424, 1981.

    Google Scholar 

  24. R.E. Kalman, A new approach to linear filtering and prediction problems, Trans. ASME J. Basic Eng., Vol. 82, pp. 35–45, 1960.

    Google Scholar 

  25. R.E. Kaiman and R.S. Bucy, New results in linear filtering and prediction theory, Trans. ASME J. Basic Eng., Vol. 83, pp. 95–108, 1961.

    Google Scholar 

  26. L.A. Klimko and P.I. Nelson, On conditional least squares estimation for stochastic processes, Ann. Statist., Vol. 6, pp. 629–642, 1978.

    Article  MathSciNet  MATH  Google Scholar 

  27. T.L. Lai and C.Z. Wei, Least squares estimates in stochastic regression models with applications to identification and control of dynamic systems, Ann. Statist., Vol. 10, pp. 154–166, 1982.

    Article  MathSciNet  Google Scholar 

  28. T.L. Lai and C.Z. Wei, Asymptotic properties of general autoregressive models and strong consistency of least squares estimates of their parameters, J. Multivariate Anal., Vol. 13, pp. 1–23, 1983.

    Article  MathSciNet  MATH  Google Scholar 

  29. J. Ledolter, A recursive approach to parameter estimation in regression and time series models, Comm. Stat., Vol. A8, pp. 1227–1245, 1979.

    Article  MathSciNet  Google Scholar 

  30. J. Ledolter, Recursive estimation and adaptive forecasting in ARIMA models with time varying coefficients, in Adaptive Time Series Analysis II, D.F. Findlay, Ed., New York, NY: Academic Press, pp. 449–471, 1981.

    Google Scholar 

  31. K.S. Lim and H. Tong, A statistical approach to difference-delay equation modelling in ecology - two case studies, J. Time Series Anal., Vol. 4, pp. 239–268, 1983.

    Article  MathSciNet  Google Scholar 

  32. R.S. Liptser and A.N. Shiryayev, Statistics of Random Processes, Vol. 2, Applications, New York: Springer-Verlag, 1978.

    Google Scholar 

  33. J.I. Makhoul, Linear prediction: A tutorial review, Proc. IEEE, Vol. 32, pp. 561–581, 1975.

    Article  Google Scholar 

  34. S. Makridakis and S.C. Wheelwright, Adaptive filtering: An integrated autoregressive/moving average filter for time series forecasting, Op. Res., Vol. 28, pp. 425–437, 1977.

    Article  Google Scholar 

  35. H.B. Mann and A. Wald, On the statistical treatment of linear stochastic difference equations, Econometrica, Vol. 11, pp. 173–220, 1943a.

    Article  MathSciNet  MATH  Google Scholar 

  36. H.B. Mann and A. Wald, On stochastic limit and order relationships, Ann. Math. Statist., Vol. 14, pp. 217–226, 1943b.

    Article  MathSciNet  MATH  Google Scholar 

  37. J.D. Markel and A.H. Gray, Jr., Linear Prediction of Speech, New York, NY: Springer-Verlag, 1977.

    Google Scholar 

  38. R.R. Mohler, Bilinear Control Processes, New York, NY: Academic Press, 1973.

    MATH  Google Scholar 

  39. P. Newbold, Some recent developments in time series analysis, International Statistical Review, Vol. 49, pp. 53–66, 1981.

    Article  MathSciNet  MATH  Google Scholar 

  40. D.F. Nicholls and B.G. Quinn, Random Coefficient Autoregressive Models. An Introduction, Lecture Notes in Statistics Vol. 11, New York, NY: Springer-Verlag, 1982.

    Google Scholar 

  41. T.Ozaki, Non-linear time series models for non-linear random vibrations, J. Appl Prob., Vol. 17, pp. 84–93, 1980.

    Article  Google Scholar 

  42. T. Ozaki, Non-linear threshold autoregressive models for non-linear vibrations, J. Appl. Prob., Vol. 18, pp. 443–451, 1981.

    Article  MathSciNet  MATH  Google Scholar 

  43. D.P. Pham and L.T. Tran, On the first order bilinear time series model, J. Appl. Prob., Vol. 18, pp. 617–627, 1981.

    Article  MATH  Google Scholar 

  44. M. Pourahmadi, On the solution of a doubly stochastic model, Technical Report, Center for Stochastic Processes, Department of Statistics, University of North Carolina, Chapel Hill, N.C., 1984.

    Google Scholar 

  45. M.B. Priestley, State dependent models: A general approach to time series analysis, J. Time Series Anal., Vol. 1, pp. 47–71, 1980.

    Article  MathSciNet  MATH  Google Scholar 

  46. M.B. Priestley, Spectral Analysis and Time Series, New York, NY: Academic Press, 1981.

    Google Scholar 

  47. B.G. Quinn, A note on the existence of strictly stationary solutions to bilinear equations, J. Time Series Anal., Vol. 3, pp. 249–252, 1982a.

    Article  MathSciNet  MATH  Google Scholar 

  48. B.G. Quinn, Stationarity and invertibility of simple bilinear models, Stock Proc. Appl., Vol. 12, pp. 225–230, 1982b.

    Article  MathSciNet  MATH  Google Scholar 

  49. M. Rosenblatt, Stationary processes as shifts of independent random variables, J. Math. Mech., Vol. 8, pp. 665–681, 1959.

    MathSciNet  MATH  Google Scholar 

  50. G. Schwarz, Estimating the dimension of a model, Ann. Statist., Vol. 6, 1978.

    Google Scholar 

  51. W. Shou-Ren, A. Hong-Zhi and H. Tong, On the distribution of a simple stationary bilinear process, J. Time Series Anal, Vol. 4, pp. 209–216, 1983.

    Article  Google Scholar 

  52. W. Stout, Almost Sure Convergence, New York, NY: Academic Press, 1974.

    MATH  Google Scholar 

  53. R.A. Struble, Nonlinear Differential Equations, New York, NY: McGraw-Hill, 1962.

    MATH  Google Scholar 

  54. T. Subba Rao, On the theory of bilinear models, J. Roy. Stat. Soc. Ser. B, Vol. 43, pp. 244–255, 1981.

    MathSciNet  MATH  Google Scholar 

  55. T. Subba Rao and M.M. Gabr, A test for the linearity of stationary time series, J. Time Series Anal., Vol. 1, pp. 145–158, 1980.

    Article  MathSciNet  MATH  Google Scholar 

  56. D. Tjøstheim, Some doubly stochastic time series models, Technical Report, Center for Stochastic Processes, Department of Statistics, University of North Carolina, Chapel Hill, NC, 1983.

    Google Scholar 

  57. D. Tjøstheim, Estimation in nonlinear time series models, Technical Report, Center for Stochastic Processes, Department of Statistics, University of North Carolina, Chapel Hill, NC, 1984.

    Google Scholar 

  58. H. Tong, Discussion of a paper by A. J. Lawrance and T. Kottegoda, J. Roy. Stat. Soc. Ser. A, Vol. 140, pp. 34–35, 1977.

    Google Scholar 

  59. H. Tong and K.S. Lim, Threshold autoregression, limit cycles and cyclical data (with discussion), J. Roy. Stat. Soc. Ser. B, Vol. 42, pp. 245–292, 1980.

    MATH  Google Scholar 

  60. D.W. Trigg and A.G. Leach, Exponential smoothing with an adaptive response rate, Op. Res. Q., Vol. 18, pp. 53–59, 1967.

    Article  Google Scholar 

  61. R.L. Tweedie, Sufficient conditions for ergodicity and recurrence of Markov chains on a general state space, Stock. Proc. Appl., Vol 3, pp. 385–403, 1975.

    Article  MathSciNet  MATH  Google Scholar 

  62. N. Wiener, Non-linear Problems in Random Theory, Cambridge, MA: MIT Press, 1958.

    Google Scholar 

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Tjøstheim, D. (1986). Recent Developments in Nonlinear Time-Series Modeling. In: Blake, I.F., Poor, H.V. (eds) Communications and Networks. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-4904-7_9

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  • DOI: https://doi.org/10.1007/978-1-4612-4904-7_9

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