Abstract
As an alternative to the batch means (BM) method in the stopping rule for symbolic dynamic filtering, this short paper presents an analytical procedure to estimate the variance parameter and to obtain a lower bound on the length of symbol blocks for constructing probabilistic finite state automata (PFSA). If the modulus of the second largest eigenvalue of the PFSA’s state transition matrix is relatively small or if the symbol block length is not too large, then the performance of the proposed stopping rule is superior to that of the stopping rule based on BM method. The algorithm of the proposed stopping rule is validated on ultrasonic data collected from a fatigue test apparatus for damage detection in the polycrystalline alloy 7075-T6.
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Wen Y., Ray A.: A stopping rule for symbolic dynamic filtering. Appl. Math. Lett. 23, 1125–1128 (2010)
Ray A.: Symbolic dynamic analysis of complex systems for anomaly detection. Signal Process. 84(7), 1115–1130 (2004)
Flegal J., Haran M.: Markov chain Monte Carlo: can we trust the third significant figure. Stat. Sci. 23(2), 250–260 (2008)
Jones G., Haran M., Caffo B., Neath R.: Fixed-width output analysis for Markov chain Monte Carlo. J. Am. Stat. Assoc. 101, 1537–1547 (2006)
Flegal, J., Jones, G.: Batch means and spectral variance estimators. In Markov chain Monte Carlo. technical report, University of Minnesota, Department of Statistics (2008)
Berman A., Plemmons R.: Nonnegative Matrices in the Mathematical Sciences. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1997)
Bapat R., Raghavan T.: Nonnegative Matrices and Applications. Cambridge University Press, Cambridge (1997)
Gupta S., Ray A.: Statistical mechanics of complex systems for pattern identification. J. Stat. Phys. 134(2), 337–364 (2009)
Bhattacharya R., Waymire E.: Stochastic Processes with Applications. Wiley-Interscience, New York (1990)
Garren S.T., Smith R.L.: Estimating the second largest eigenvalue of a Markov transition matrix. Bernoulli 6, 215–242 (2000)
Rosenthal J.: Convergence rates for Markov chains. Soc. Ind. Appl. Math. J. 37(3), 387–445 (1995)
Gupta S., Ray A., Keller E.: Symbolic time series analysis of ultrasonic data for early detection of fatigue damage. Mech. Syst. Signal Process. 21(2), 866–884 (2007)
Rajagopalan V., Ray A.: Symbolic time series analysis via wavelet-based partitioning. Signal Process. 86(11), 3309–3320 (2006)
Schervish M.: p-values: what they are and what they are not. Am Stat. 50, 203–206 (1996)
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Wen, Y., Ray, A. & Du, Q. A variance-estimation-based stopping rule for symbolic dynamic filtering. SIViP 7, 189–195 (2013). https://doi.org/10.1007/s11760-011-0215-y
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DOI: https://doi.org/10.1007/s11760-011-0215-y