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 $y(t_{0}) = y_{0}$, mean $y_{avg}$ and two critical values ${y_{avg} − y_{min}, y_{avg} + y_{max}}|undefined$, where $y_{min} > 0$ and $y_{max} > 0$. Define the first passage time of $y(t)$ from within the interval $(−y_{min}, y_{max})$ as


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

where "inf" is the infimum. This is the smallest time after the initial time $t_{0}$ that $y(t)$ is equal to one of the critical values forming the boundary of the interval, assuming $y_{0}$ is within the interval.

Because $y(t)$ proceeds randomly from its initial value to the boundary, $τ(y_{0})$ is itself a random variable. The mean of $τ(y_{0})$ is the residence time,
 * $$ \bar{\tau}(y_0) = E[\tau(y_0)\mid y_0].$$

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,


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

where the frequency of exceedance $N$ is

$σ_{y}^{2}$ 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 $$.

Generalization to multiple dimensions
Suppose that instead of being scalar, $y(t)$ has dimension $f$, 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


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

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 $y_{0}$ is within $Ψ$. The mean of this time is the residence time,


 * $$ \bar{\tau}(y_0) = \operatorname{E}[\tau(y_0)\mid y_0].$$

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 $p$, 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, $N_{0}$.

In general, the normalization factor $min(y_{min}, y_{max})/σ_{y}$ can be difficult or impossible to compute, so the dimensionless quantities can be more useful in applications.