Online Analysis of Seismic Signals

  • Hernando Ombao
  • Jungeon Heo
  • David Stoffer
Conference paper
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 45)


Seismic signals can be modeled as non-stationary time series. Methods for analyzing non-stationary time series that have been recently developed are proposed in Adak [1], West, et al. [25] and Ombao, et al. [12]. These methods require that the entire series be observed completely prior to analyses. In some situations, it is desirable to commence analysis even while the time series is being recorded. In this paper, we develop a statistical method for analyzing seismic signals while it is being recorded or observed. The basic idea is to model the seismic signal as a piecewise stationary autoregressive process. When a block of time series becomes available, an AR model is fit, the AR parameters estimated and the Bayesian Information criterion (BIC) value is computed. Adjacent blocks are combined to form one big block if the BIC for the combined block is less than the sum of the BIC for each of the split adjacent blocks. Otherwise, adjacent blocks are kept as separate. In the event that adjacent blocks are combined as a Single block, we interpret the observations at those two blocks as likely to have been generated by one AR process. When the adjacent blocks are separate, the observations at the two blocks were likely to have been generated by different AR processes. In this Situation, the method has detected a change in the spectral and distributional parameters of the time series.

Simulation results suggest that the proposed method is able to detect changes in the time series as they occur. Moreover, the proposed method tends to report changes only when they actually occur. The methodology will be useful for seismologists who need to monitor vigilantly changes in seismic activities. Our procedure is inspired by Takanaini [23] which uses the Akaike Information Criterion (AIC). We report simulation results that compare the online BIC method with the Takanami method and discuss the advantages and disadvantages of the two online methods. Finally, we apply the online BIC method to a seismic waves dataset.

Key words

Non-stationary time series Autoregressive models Akaike information criterion Bayesian information criterion Time-frequency analysis Seismic signals. 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    S. Adak, Time dependent spectral analysis of non-stationary time series, Journal of the American Statistical Association, 93 (1998), pp. 1488–1501.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    H. Akaike, Information theory and an extension of the maximum likelihood principle, 2nd International Symposium on Information Theory (eds. B. Petrov and F. Csaki) (1973), pp. 267–281.Google Scholar
  3. [3]
    M. Basseville and I. Nikiforov, Detection of Abrupt Changes — Theory and Applications, Prentice-Hall, Englewood, Cliff, New Jersey, 1993.Google Scholar
  4. [4]
    J. Cavanaugh and A. Neath, Generalizing the derivation of of the Schwartz information criterion, Communications in Statistics — Theory and Methods, 28 (1999), pp. 49–66.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    R. Dahlhaus, Fitting time series models to nonstationary processes, Annals of Statistics, 25(1996), pp. 1–37.MathSciNetGoogle Scholar
  6. [6]
    R. Davis, D. Huang, and Y. Yao, Testing for a change in the parameter values and order of an autoregressive model, Annals of Statistics, 23 (1995), pp. 282–304.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    D. Haughton, On the choice of a model to fit data from an exponential family, Annals of Statistics, 6(1988), pp. 342–355.MathSciNetCrossRefGoogle Scholar
  8. [8]
    T. Inouye, H. Sakamoto, K. Shinosaki, S. Toi, and S. Ukai, Analysis of rapidly changing EEGs before generalized spike and wave complexes, Electroencephalography and clinical Neurophysiology, 76 (1990), pp. 205–221.CrossRefGoogle Scholar
  9. [9]
    G. Kitagawa and H. Akaike, Procedure for the Modeling of Non-Stationary Time Series, Annals of the Institute of Statistical Mathematics, 30 (1978), pp. 351–363.zbMATHCrossRefGoogle Scholar
  10. [10]
    G. Kitagawa and W. Gersch, Smoothness Priors Analysis of Time Series, Lecture Notes in Statistics #116, New York: Springer Verlag, 1996.Google Scholar
  11. [11]
    A. Neath and J. Cavanaugh, Regression and Time Series Model Selection Using Variants of the Schwarz Information Criterion, Communications in Statistics — Theory and Methods, 26 (1997), pp. 559–580.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    H. Ombao, J. Raz, R. von Sachs, and B. Malow, Automatic Statistical Analysis of Bivariate Non-Stationary Time Series, Journal of the American Statistical Association, 96 (2001), pp. 543–560.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    H. Ombao, J. Raz, R. Strawderman, and R. von Sachs, A simple generalised cross validation method of span selection for periodogram smoothing, Biometrika, 88 (2001), Vol. 4, pp. 1186–1192.CrossRefGoogle Scholar
  14. [14]
    H. Ombao, J. Raz, R. Strawderman, and R. von Sachs, A simple generalised cross validation method of span selection for periodogram smoothing, Biometrika, 88 (2001), Vol. 4, pp. 1186–1192.CrossRefGoogle Scholar
  15. T. Ozaki and H. Tong, On the fitting of non-stationary autoregressive models in the time series analysis, Proceedings of the 8th Hawaii International Conference on System Science, Western Periodical Hawaii (1975), pp. 224–226.Google Scholar
  16. [16]
    M. Pagani, Power spectral analysis of beat-to-beat heart and blood pressure variability as a possible marker of sympatho-vagal interaction in man and conscious dog, XS Circulation Research, 59 (1986), p. 178.Google Scholar
  17. [17]
    M. Priestley, Spectral Analysis and Time Series, London: Academic Press, 1981.zbMATHGoogle Scholar
  18. [18]
    K. Sato and K. Ono, Component activities in the autoregressive activity of physiological systems, International Order of Neuroscience, 7 (1977), pp. 239–249.CrossRefGoogle Scholar
  19. [19]
    G. Schwarz, Estimating the dimension of a model, Annals of Statistics, 6 (1978), pp. 461–464.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    R. Shumway and D. Stoffer, Time Series Analysis and Its Applications, New York: Springer, 2000.zbMATHGoogle Scholar
  21. [21]
    T. Takanami and G. Kitagawa, A new efficient procedure for the estimation of onset times of seismic waves, Journal of Physics of the Earth, 36 (1988), pp. 267–290.CrossRefGoogle Scholar
  22. [22]
    T. Takanami and G. Kitagawa, Estimation ofthe arrival times of seismic waves by multivariate time series model, Annals of the Institute of Statistical Mathematics, 43 (1991), pp. 403–433.CrossRefGoogle Scholar
  23. [23]
    T. Takanami, High Precision Estimation of Seismic Waves Arrival Times, The Practice of Time Series Analysis (eds. Akaike and Kitagawa ), New York: Springer-Verlag, 1999.Google Scholar
  24. [24]
    T. Wada, S. Sato, and N. Matuo, Applications of multivariate autoregressive modeling for analyzing chloride-potassium-bicarbonate relationship in the body, Med. Biol. Eng. Comput., 31(1993), pp. 99–107.CrossRefGoogle Scholar
  25. [25]
    M. West, R. Prado, and A. Krystal, Evaluation and Comparison of EEG Traces: Latent Structure in Non-Stationary Time Series, Journal of the American Statistical Association, 94(1999), pp. 1083–1094.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York, LLC 2004

Authors and Affiliations

  • Hernando Ombao
    • 1
  • Jungeon Heo
    • 2
  • David Stoffer
    • 3
  1. 1.Department of StatisticsUniversity of IllinoisChampaignUSA
  2. 2.Department of Statistics and Department of PsychiatryUniversity of PittsburghPittsburghUSA
  3. 3.Department of StatisticsUniversity of PittsburghPittsburghUSA

Personalised recommendations