Rice's formula explained
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
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[8]
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]
P\left\{\supt\in[0,1]X(t)\gequ\right\}
as for large values of
u the probability that there is a level crossing is approximately the probability of reaching that level.
Notes and References
- Rychlik . I. . Extremes . 3 . 4 . 331–348 . 10.1023/A:1017942408501 . 2000 . Kluwer Academic Publishers. On Some Reliability Applications of Rice's Formula for the Intensity of Level Crossings. 115235517 .
- Book: Robert J. . Adler . Jonathan E. . Taylor. 10.1007/978-0-387-48116-6 . Random Fields and Geometry . Springer Monographs in Mathematics . 2007 . 978-0-387-48112-8 .
- Book: Grigoriu, Mircea . Stochastic Calculus: Applications in Science and Engineering . 2002 . 166 . 978-0-817-64242-6.
- Rice . S. O.. Stephen O. Rice. 1944 . Mathematical analysis of random noise . Bell System Tech. J. . 23. 3 . 282–332 . 10.1002/j.1538-7305.1944.tb00874.x .
- Rainal . A. J. . Origin of Rice's formula . 10.1109/18.21276 . IEEE Transactions on Information Theory . 34 . 6 . 1383–1387 . 1988 .
- Borovkov . K. . Last . G. . 10.1239/jap/1339878791 . On Rice's formula for stationary multivariate piecewise smooth processes . Journal of Applied Probability . 49 . 2 . 351 . 2012 . 1009.3885 .
- Book: Barnett, J. T. . Zero-Crossings of Random Processes with Application to Estimation Detection . Nonuniform Sampling: Theory and Practice . Farokh A.. Marvasti. Springer . 2001 . 0306464454.
- Ylvisaker . N. D. . The Expected Number of Zeros of a Stationary Gaussian Process . 10.1214/aoms/1177700077 . The Annals of Mathematical Statistics . 36 . 3 . 1043–1046 . 1965 . free .
- Book: Robert J. . Adler . Jonathan E. . Taylor. 10.1007/978-0-387-48116-6_4 . Excursion Probabilities . Random Fields and Geometry . Springer Monographs in Mathematics . 75–76 . 2007 . 978-0-387-48112-8 .
