Rice's formula

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In probability theory, Rice's formula counts the average number of times an ergodic stationary process X(t) per unit time crosses a fixed level u.[1] Adler and Taylor describe the result as "one of the most important results in the applications of smooth stochastic processes."[2]

History

The formula was published Stephen O. Rice in 1944,[3] having previously been discussed in his 1936 note entitled "Singing Transmission Lines."[4][5]

Formula

Write Du for the number of times the ergodic stationary stochastic process X(t) takes the value u in a unit of time (i.e. t ∈ [0,1]). Then Rice's formula states that

{\mathbb E}(D_{u})=\int _{{-\infty }}^{\infty }|x'|p(u,x')\,{\mathrm {d}}x'

where p(x,x') is the joint probability density of the X(t) and its mean-square derivative X'(t).[6]

If the process X(t) is a Gaussian process and u = 0 then the formula simplifies significantly to give[6][7]

{\mathbb E}(D_{0})={\frac {1}{\pi }}{\sqrt {-\rho ''(0)}}

where ρ'' is the second derivative of the normalised autocorrelation of X(t) at 0.

Uses

Rice's formula can be used to approximate an excursion probability[8]

{\mathbb P}\left\{\sup _{{t\in [0,1]}}X(t)\geq u\right\}

as for large values of u the probability that there is a level crossing is approximately the probability of reaching that level.

References

  1. ↑ Rychlik, I. (2000). "On Some Reliability Applications of Rice's Formula for the Intensity of Level Crossings". Extremes (Kluwer Academic Publishers) 3 (4): 331–348. doi:10.1023/A:1017942408501. 
  2. ↑ Adler, Robert J.; Taylor, Jonathan E. (2007). Random Fields and Geometry. Springer Monographs in Mathematics. doi:10.1007/978-0-387-48116-6. ISBN 978-0-387-48112-8. 
  3. ↑ Rice, S. O. (1944). "Mathematical analysis of random noise". Bell System Tech. J. 23: 282–332. 
  4. ↑ Rainal, A. J. (1988). "Origin of Rice's formula". IEEE Transactions on Information Theory 34 (6): 1383–1387. doi:10.1109/18.21276. 
  5. ↑ Borovkov, K.; Last, G. (2012). "On Rice's formula for stationary multivariate piecewise smooth processes". Journal of Applied Probability 49 (2): 351. doi:10.1239/jap/1339878791. 
  6. ↑ 6.0 6.1 Barnett, J. T. (2001). "Zero-Crossings of Random Processes with Application to Estimation Detection". In Marvasti, Farokh A. Nonuniform Sampling: Theory and Practice. Springer. ISBN 0306464454. 
  7. ↑ Ylvisaker, N. D. (1965). "The Expected Number of Zeros of a Stationary Gaussian Process". The Annals of Mathematical Statistics 36 (3): 1043. doi:10.1214/aoms/1177700077. 
  8. ↑ Adler, Robert J.; Taylor, Jonathan E. (2007). "Excursion Probabilities". Random Fields and Geometry. Springer Monographs in Mathematics. pp. 75–76. doi:10.1007/978-0-387-48116-6_4. ISBN 978-0-387-48112-8. 


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