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]

History

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

Formula

Write D_u 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" (PDF). 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. 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.