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 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

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

where p(x,x') is the joint probability density of the x(t) and its mean-square derivative x'(t).^[7]

If the process x(t) is a Gaussian process and u = 0 then the formula simplifies significantly to give^[7][8]

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

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^[9]

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

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. 3 (4). Kluwer Academic Publishers: 331–348. doi:10.1023/A:1017942408501. S2CID 115235517.
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. Grigoriu, Mircea (2002). Stochastic Calculus: Applications in Science and Engineering. p. 166. ISBN 978-0-817-64242-6.
4. Rice, S. O. (1944). "Mathematical analysis of random noise" (PDF). Bell System Tech. J. 23 (3): 282–332. doi:10.1002/j.1538-7305.1944.tb00874.x.
5. Rainal, A. J. (1988). "Origin of Rice's formula". IEEE Transactions on Information Theory. 34 (6): 1383–1387. doi:10.1109/18.21276.
6. Borovkov, K.; Last, G. (2012). "On Rice's formula for stationary multivariate piecewise smooth processes". Journal of Applied Probability. 49 (2): 351. arXiv:1009.3885. doi:10.1239/jap/1339878791.
7. Barnett, J. T. (2001). "Zero-Crossings of Random Processes with Application to Estimation Detection". In Marvasti, Farokh A. (ed.). Nonuniform Sampling: Theory and Practice. Springer. ISBN 0306464454.
8. 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.
9. 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.