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.

Definition

Suppose y(t) is a real, scalar stochastic process with initial value y(t₀) = y₀, mean y_avg and two critical values {y_avg − y_min, y_avg + y_max}, where y_min > 0and y_max > 0. Define the first passage time of y(t) from within the interval (−y_min, y_max) as

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

where "inf" in 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,

$\bar{\tau}(y_0) = E[\tau(y_0)|y_0].$

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,

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

where the frequency of exceedance N is

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): N(y_{\max}) = N_0 e^{-\tfrac{y_{\max}^2}{2\sigma_y^2}},

(1)

σ_y² is the variance of the Gaussian distribution,

$N_0 = \sqrt{\frac{\int_0^\infty{f^2 \Phi_y(f) df}}{\int_0^\infty{\Phi_y(f) df}}},$

and Φ_y(f) is the power spectral density of the Gaussian distribution over a frequency f.

Generalization to multiple dimensions

Suppose that instead of being scalar, y(t) has dimension p, or y(t) ∈ ℝ^p. Define a domain Ψ ⊂ ℝ^p that contains y_avg and has a smooth boundary ∂Ψ. In this case, define the first passage time of y(t) from within the domain Ψ as

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

In this case, this infimum is the smallest time at which y(t) is on the boundary of Ψ rather than being equal to one of two discrete values, assuming y₀ is within Ψ. The mean of this time is the residence time,

$\bar{\tau}(y_0) = E[\tau(y_0)|y_0].$

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 (1), the logarithmic residence time of a Gaussian process is defined as

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

This is closely related to another dimensionless descriptor of this system, the number of standard deviations between the boundary and the mean, min(y_min, y_max)/σ_y.

In general, the normalization factor N₀ 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

• Meerkov 1987, pp. 1734-1735.
• Richardson 2014, p. 2027.
• Meerkov 1986, p. 494.
• Meerkov 1987, p. 1734.
• Richardson 2014, p. 2028.
• Meerkov 1986, p. 495, an alternate approach to defining the logarithmic residence time and computing N₀