Advertisement

Modelling Stochastic Processes with Time Series Analysis

  • Yuri A. W. Shardt

Abstract

This chapter introduces the reader to the concept of time series analysis using transfer functions, state-space models, and spectral decomposition. Time series analysis is used to develop stochastic, or probabilistic, models. First, the theoretical properties of different model types, including standard autoregressive moving-average models, integrating models, and seasonal models, are examined and compared in both the time and frequency domains. The results obtained here can then be used to determine the appropriate model structure for a given data set. Spectral methods are also introduced at this point to assist in explaining various seasonal or periodic components in the data set. Next, the topic of parameter estimation is considered, and results are obtained for different methods and approaches, including the Yule–Walker for autoregressive models, the log-likelihood method for generalised autoregressive moving-average models, and the Kalman filter for state-space models. Finally, appropriate model validation methods are presented for time series analysis. Throughout this chapter, the Edmonton temperature data series is used to illustrate the concepts involved in time series analysis. By the end of the chapter, the reader should have a thorough understanding of the principles of time series analysis, including model structure determination, parameter estimation, and model validation.

References

  1. Bloomfield P (2000) Fourier analysis of time series: an introduction, 2nd edn. Wiley, New YorkCrossRefGoogle Scholar
  2. Box GE, Jenkins GM (1970) Time series analysis, forecasting, and control. Holden-Day, OaklandGoogle Scholar
  3. Franke J, Härdle WK, Hafner CM (2011) Statistics of financial markets: an introduction, 3rd edn. Springer, Heidelberg. doi: 10.1007/978-3-642-16521-4 CrossRefGoogle Scholar
  4. Hannan EJ, Deistler M (2012) The statistical theory of linear systems. Soc Ind Appl MathGoogle Scholar
  5. Harvey AC (1991) Forecasting, structural time series models and the Kalman filter. Cambridge University Press, CambridgeGoogle Scholar
  6. Hassler U (1994) The sample autocorrelation function of I(1) processes. Statistical Papers−Statistische Hefte 35:1–16Google Scholar
  7. Kalman RE (1960) A new approach to linear filtering and prediction problems. Trans ASME J Basic Eng 82:35–45CrossRefGoogle Scholar
  8. Kalman RE, Bucy RS (1961) New results in filtering and prediction theory. Trans ASME J Basic Eng 83:95–108CrossRefGoogle Scholar
  9. Khintchine A (1934) Korrelationstheorie der stationären stochastischen prozesse (correlation theory of stocastic processes). Math Ann 109(1):604–615. doi: 10.1007/BF01449156 CrossRefGoogle Scholar
  10. Priestley MB (1981) Spectral analysis and time series: Vol. 1: univariate series and vol 2: multivariate series, prediction, and control. Academic, New YorkGoogle Scholar
  11. Shardt Y (2012a) Data quality assessment for closed-loop system identification and forecasting with application to soft sensors. Doctoral thesis, University of Alberta, Department of Chemical and Materials Engineering, Edmonton, Alberta, Canada. doi: http://hdl.handle.net/10402/era.29018
  12. Shumway RH, Stoffer DS (2011) Time series analysis and its applications with R examples, 3rd edn. Springer, New York. doi: 10.1007/978-1-4419-7865-3 CrossRefGoogle Scholar
  13. Stoica P, Moses R (2005) Spectral analysis of signals. Prentice Hall, Upper Saddle RiverGoogle Scholar
  14. Welch PD (1967) The use of fast Fourier transform for the estimation of power spectra: a method based on time averaging over short, modified Periodograms. IEEE Trans Audio Elecotroacustics AU-15(2):70–73CrossRefGoogle Scholar
  15. Wichern DW (1973) The behaviour of the sample autocorrelation function for an integrated moving average process. Biometrika 60(2):235–239. doi:http://www.jstor.org/stable/2334535?origin=JSTOR-pdf

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Yuri A. W. Shardt
    • 1
  1. 1.Institute of Automation and Complex Systems (AKS)University of Duisburg-EssenDuisbergGermany

Personalised recommendations