Residence time (statistics)

Template:Use American English Template:Use mdy dates Template:Short description 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.

Definition

Suppose Template:Math is a real, scalar stochastic process with initial value Template:Math, mean Template:Math and two critical values Template:Math}, where Template:Math and Template:Math. Define the first passage time of Template:Math from within the interval Template:Math as

$\tau (y_0)=\inf \{t\ge t_0:y(t)\in \{y_{\mathrm{avg}}-y_{\min },y_{\mathrm{avg}}+y_{\max }\}\},$

where "inf" is the infimum. This is the smallest time after the initial time Template:Math that Template:Math is equal to one of the critical values forming the boundary of the interval, assuming Template:Math is within the interval.

Because Template:Math proceeds randomly from its initial value to the boundary, Template:Math is itself a random variable. The mean of Template:Math is the residence time,Template:SfnTemplate:Sfn

$\overline{\tau }(y_0)=E[\tau (y_0)\mid y_0].$

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,Template:Sfn

$\overline{\tau }=N^{-1}(\min (y_{\min },y_{\max })),$

where the frequency of exceedance Template:Mvar is Template:NumBlk Template:Math is the variance of the Gaussian distribution,

$N_0=\sqrt{\frac{\underset{0}{\overset{\mathrm{\infty }}{\int }}{f}^2{\mathrm{\Phi }}_y(f)df}{\underset{0}{\overset{\mathrm{\infty }}{\int }}{\mathrm{\Phi }}_y(f)df}},$

and Template:Math is the power spectral density of the Gaussian distribution over a frequency Template:Mvar.

Generalization to multiple dimensions

Suppose that instead of being scalar, Template:Math has dimension Template:Mvar, or Template:Math. Define a domain Template:Math that contains Template:Math and has a smooth boundary Template:Math. In this case, define the first passage time of Template:Math from within the domain Template:Math as

$\tau (y_0)=\inf \{t\ge t_0:y(t)\in \partial \mathrm{\Psi }\mid y_0\in \mathrm{\Psi }\}.$

In this case, this infimum is the smallest time at which Template:Math is on the boundary of Template:Math rather than being equal to one of two discrete values, assuming Template:Math is within Template:Math. The mean of this time is the residence time,Template:SfnTemplate:Sfn

$\overline{\tau }(y_0)=\mathrm{E}[\tau (y_0)\mid y_0].$

Logarithmic residence time

The logarithmic residence time is a dimensionless variation of the residence time. It is proportional to the natural log of a normalized residence time. Noting the exponential in Equation Template:EqNote, the logarithmic residence time of a Gaussian process is defined asTemplate:SfnTemplate:Sfn

$\hat{\mu }=\ln \left(N_0\overline{\tau }\right)=\frac{\min (y_{\min },y_{\max }{)}^2}{2{\sigma }_y^2}.$

This is closely related to another dimensionless descriptor of this system, the number of standard deviations between the boundary and the mean, Template:Math.

In general, the normalization factor Template:Math can be difficult or impossible to compute, so the dimensionless quantities can be more useful in applications.

See also

• Cumulative frequency analysis
• Extreme value theory
• First-hitting-time model
• Frequency of exceedance
• Mean time between failures

References

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