Rice's formula
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] The formula is often used in engineering.[3]
History
The formula was published by Stephen O. Rice in 1944,[4] having previously been discussed in his 1936 note entitled "Singing Transmission Lines."[5][6]
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
- [math]\displaystyle{ \mathbb E(D_u) = \int_{-\infty}^\infty |x'|p(u,x') \, \mathrm{d}x' }[/math]
where p(x,x') is the joint probability density of the x(t) and its mean-square derivative x'(t).[7]
If the process x(t) is a Gaussian process and u = 0 then the formula simplifies significantly to give[7][8]
- [math]\displaystyle{ \mathbb E(D_0) = \frac{1}{\pi} \sqrt{-\rho''(0)} }[/math]
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[9]
- [math]\displaystyle{ \mathbb P \left\{ \sup_{t\in[0,1]} X(t) \geq u \right\} }[/math]
as for large values of u the probability that there is a level crossing is approximately the probability of reaching that level.
References
- ↑ 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.
- ↑ 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.
- ↑ Grigoriu, Mircea (2002). Stochastic Calculus: Applications in Science and Engineering. pp. 166. ISBN 978-0-817-64242-6.
- ↑ Rice, S. O. (1944). "Mathematical analysis of random noise". Bell System Tech. J. 23 (3): 282–332. doi:10.1002/j.1538-7305.1944.tb00874.x. http://www.alcatel-lucent.com/bstj/vol23-1944/articles/bstj23-3-282.pdf.
- ↑ Rainal, A. J. (1988). "Origin of Rice's formula". IEEE Transactions on Information Theory 34 (6): 1383–1387. doi:10.1109/18.21276.
- ↑ 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.
- ↑ 7.0 7.1 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.
- ↑ Ylvisaker, N. D. (1965). "The Expected Number of Zeros of a Stationary Gaussian Process". The Annals of Mathematical Statistics 36 (3): 1043–1046. doi:10.1214/aoms/1177700077.
- ↑ 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.
 |  |
