Abstract
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 Takanami [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.
The work of H. Ombao was supported in part by NIMH 62298 and NSF DMS-0102511.
The work of J. Heo was supported in part by NIMH 55123.
The work of D. Stoffer was supported in part by NSF DMS-0102511.
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References
S. Adak, Time dependent spectral analysis of non-stationary time series, Journal of the American Statistical Association, 93 (1998), pp. 1488–1501.
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.
M. Basseville and I. Nikiforov, Detection of Abrupt Changes — Theory and Applications, Prentice-Hall, Englewood, Cliff, New Jersey, 1993.
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.
R. Dahlhaus, Fitting time series models to nonstationary processes, Annals of Statistics, 25 (1996), pp. 1–37.
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.
D. Haughton, On the choice of a model to fit data from an exponential family, Annals of Statistics, 6 (1988), pp. 342–355.
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.
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.
G. Kitagawa and W. Gersch, Smoothness Priors Analysis of Time Series, Lecture Notes in Statistics #116, New York: Springer Verlag, 1996.
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.
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.
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.
H. Ombao, J. Raz, R. von Sachs, and W. Guo, The SLEX Model of a Non-Stationary Random Process, Annals of the Institute of Statistical Mathematics (2002), Vol. 1, in press.
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.
M. Pagani, Power spectral analysis of beat-to-beat heart and blood pressure variability as a possible marker of sympathovagal interaction in man and conscious dog, XS Circulation Research, 59 (1986), p. 178.
M. Priestley, Spectral Analysis and Time Series, London: Academic Press, 1981.
K. Sato and K. Ono, Component activities in the autoregressive activity of physiological systems, International Order of Neuroscience, 7 (1977), pp. 239–249.
G. Schwarz, Estimating the dimension of a model, Annals of Statistics, 6 (1978), pp. 461–464.
R. Shumway and D. Stoffer, Time Series Analysis and Its Applications, New York: Springer, 2000.
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.
T. Takanami and G. Kitagawa, Estimation of the arrival times of seismic waves by multivariate time series model, Annals of the Institute of Statistical Mathematics, 43 (1991), pp. 403–433.
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.
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.
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.
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Ombao, H., Heo, J., Stoffer, D. (2004). Online Analysis of Seismic Signals. In: Brillinger, D.R., Robinson, E.A., Schoenberg, F.P. (eds) Time Series Analysis and Applications to Geophysical Systems. The IMA Volumes in Mathematics and its Applications, vol 139. Springer, New York, NY. https://doi.org/10.1007/978-1-4684-9386-3_4
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DOI: https://doi.org/10.1007/978-1-4684-9386-3_4
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