- Access by Xinjiang University
Zero-crossing statistics for non-Markovian time series
Phys. Rev. E 97, 032114 – Published 14 March, 2018
DOI: https://doi.org/10.1103/PhysRevE.97.032114
Abstract
In applications spanning from image analysis and speech recognition to energy dissipation in turbulence and time-to failure of fatigued materials, researchers and engineers want to calculate how often a stochastic observable crosses a specific level, such as zero. At first glance this problem looks simple, but it is in fact theoretically very challenging, and therefore few exact results exist. One exception is the celebrated Rice formula that gives the mean number of zero crossings in a fixed time interval of a zero-mean Gaussian stationary process. In this study we use the so-called independent interval approximation to go beyond Rice's result and derive analytic expressions for all higher-order zero-crossing cumulants and moments. Our results agree well with simulations for the non-Markovian autoregressive model.
Physics Subject Headings (PhySH)
Article Text
References (31)
- S. O. Rice, Bell Syst. Tech. J. 23, 282 (1944).
- D. Poggi and G. Katul, Boundary-Layer Meteorol. 136, 219 (2010).
- M. Mršnik, J. Slavič, and M. Boltežar, Int. J. Fatigue 47, 8 (2013).
- E. Barkai, E. Aghion, and D. A. Kessler, Phys. Rev. X 4, 021036 (2014).
- I. Rychlik, Extremes 3, 331 (2000).
- S. N. Majumdar, Phys. Rev. E 65, 035104 (2002).
- G. C. M. A. Ehrhardt, S. N. Majumdar, and A. J. Bray, Phys. Rev. E 69, 016106 (2004).
- M. Nyberg, T. Ambjörnsson, and L. Lizana, New J. Phys. 18, 063019 (2016).
- P. J. Brockwell and R. A. Davis, Introduction to Time Series and Forecasting (Springer Science & Business Media, New York, 2006).
- J. A. McFadden, IEEE Trans. Inf. Theory 4, 14 (1958).
- C. Sire, Phys. Rev. Lett. 98, 020601 (2007).
- L. Debnath and D. Bhatta, Integral Transforms and Their Applications (CRC Press, Boca Raton, FL, 2014).
- G. Lindgren, Stationary Stochastic Processes: Theory and Applications (CRC Press, Boca Raton, FL, 2012).
- A. T. A. Wood and G. Chan, J. Comput. Graphic. Stat. 3, 409 (1994).
- A. J. Bray, S. N. Majumdar, and G. Schehr, Adv. Phys. 62, 225 (2013).
- J. Abrahams, A survey of recent progress on level-crossing problems for random processes, in Communications and Networks, edited by I. F. Blake and H. V. Poor (Springer, New York, 2016), pp. 6–25.
- I. F. Blake and W. C. Lindsey, IEEE Trans. Inf. Theory 19, 295 (1973).
- J. M. Smith, K. I. Hopcraft, and E. Jakeman, Phys. Rev. E 77, 031112 (2008).
- R. R. Shenoy and S. C. Sekhar, IEEE Trans. Signal Process. 63, 3177 (2015).
- C. D. Iskander and P. T. Mathiopoulos, IEEE Trans. Commun. 50, 1301 (2002).
- L. Ding and A. Goshtasby, Pattern Recognit. 34, 721 (2001).
- A. Azadi and D. R. Nelson, arXiv:1708.08154.
- B. U. Kohler, C. Hennig, and R. Orglmeister, Appl. Genom. Proteom. 2, 138 (2003).
- P. C. Petrantonakis and L. J. Hadjileontiadis, IEEE Trans. Inf. Tech. Biomed. 14, 186 (2010).
- J. Hancock and J. J. Bommer, Earthquake Eng. Struct. Dyn. 34, 637 (2005).
- C. J. Huang, Y. J. Yang, D. X. Yang, and Y. J. Chen, Expert Syst. Appl. 36, 3737 (2009).
- F. Alías, J. C. Socoró, and X. Sevillano, Appl. Sci. 6, 143 (2016).
- C. Panagiotakis and G. Tziritas, IEEE Trans. Mult. 7, 155 (2005).
- H. M. Park and R. M. Stern, Speech Commun. 51, 15 (2009).
- D. B. Owen, Commun. Stat.-Simul. Comp. 9, 389 (1980).
- L. Råde and B. Westergren, Mathematics Handbook for Science and Engineering (Springer Science & Business Media, New York, 2013).
