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Residence time (statistics)
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Residence time (statistics)
In statistics, the residence time is the average amount of time it takes for a random process to reach a certain boundary value, usually a boundary far from the mean.
Suppose y(t) is a real, scalar stochastic process with initial value y(t0) = y0, mean yavg and two critical values {yavg − ymin, yavg + ymax}, where ymin > 0 and ymax > 0. Define the first passage time of y(t) from within the interval (−ymin, ymax) as
where "inf" is the infimum. This is the smallest time after the initial time t0 that y(t) is equal to one of the critical values forming the boundary of the interval, assuming y0 is within the interval.
Because y(t) proceeds randomly from its initial value to the boundary, τ(y0) is itself a random variable. The mean of τ(y0) is the residence time,
For a Gaussian process and a boundary far from the mean, the residence time equals the inverse of the frequency of exceedance of the smaller critical value,
where the frequency of exceedance N is
σy2 is the variance of the Gaussian distribution,
and Φy(f) is the power spectral density of the Gaussian distribution over a frequency f.
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Residence time (statistics)
In statistics, the residence time is the average amount of time it takes for a random process to reach a certain boundary value, usually a boundary far from the mean.
Suppose y(t) is a real, scalar stochastic process with initial value y(t0) = y0, mean yavg and two critical values {yavg − ymin, yavg + ymax}, where ymin > 0 and ymax > 0. Define the first passage time of y(t) from within the interval (−ymin, ymax) as
where "inf" is the infimum. This is the smallest time after the initial time t0 that y(t) is equal to one of the critical values forming the boundary of the interval, assuming y0 is within the interval.
Because y(t) proceeds randomly from its initial value to the boundary, τ(y0) is itself a random variable. The mean of τ(y0) is the residence time,
For a Gaussian process and a boundary far from the mean, the residence time equals the inverse of the frequency of exceedance of the smaller critical value,
where the frequency of exceedance N is
σy2 is the variance of the Gaussian distribution,
and Φy(f) is the power spectral density of the Gaussian distribution over a frequency f.