en>Benejahnel at 13:59, 11 January 2014

2014-01-11T13:59:16Z

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In [[probability theory]], '''Rice's formula''' counts the average number of times an [[ergodic process|ergodic]] [[stationary process]] ''X''(''t'') per unit time crosses a fixed level ''u''.<ref>{{cite doi|10.1023/A:1017942408501}}</ref> Adler and Taylor describe the result as "one of the most important results in the applications of smooth stochastic processes."<ref name="adler">{{cite doi|10.1007/978-0-387-48116-6}}</ref>

==History==

The formula was published [[Stephen O. Rice]] in 1944,<ref>{{cite journal|last=Rice |first=S. O.|authorlink=Stephen O. Rice|year=1944 |title=Mathematical analysis of random noise |journal=Bell System Tech. J. | volume=23| pages=282–332 |url=http://www.alcatel-lucent.com/bstj/vol23-1944/articles/bstj23-3-282.pdf}}</ref> having previously been discussed in his 1936 note entitled "Singing Transmission Lines."<ref>{{cite doi|10.1109/18.21276}}</ref><ref>{{cite doi|10.1239/jap/1339878791}}</ref>

==Formula==

Write ''D''<sub>''u''</sub> 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>\mathbb E(D_u) = \int_{-\infty}^\infty |x'|p(u,x') \, \mathrm{d}x'</math>

where ''p''(''x'',''x''<nowiki>'</nowiki>) is the joint probability density of the ''X''(''t'') and its mean-square derivative ''X'''(''t'').<ref name="barnett">{{cite book|chapter=Zero-Crossings of Random Processes with Application to Estimation Detection | first = J. T. |last = Barnett | title=Nonuniform Sampling: Theory and Practice | editor-first=Farokh A.| editor-last= Marvasti|publisher= Springer | year = 2001 | isbn = 0306464454}}</ref>

If the process ''X''(''t'') is a [[Gaussian process]] and ''u'' = 0 then the formula simplifies significantly to give<ref name="barnett" /><ref>{{cite doi|10.1214/aoms/1177700077}}</ref>
::<math>\mathbb E(D_0) = \frac{1}{\pi} \sqrt{-\rho''(0)}</math>
where ''ρ''<nowiki>''</nowiki> 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]]<ref>{{cite doi|10.1007/978-0-387-48116-6_4}}</ref>
::<math>\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==

{{Reflist}}

[[Category:Stochastic processes]]

{{probability-stub}}

Interacting particle system - Revision history

en>Benejahnel at 13:59, 11 January 2014