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.