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.
Contents
Definition
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
τ
(
y
0
)
=
inf
{
t
≥
t
0
:
y
(
t
)
∈
{
y
avg
−
y
min
,
y
avg
+
y
max
}
}
,
{\displaystyle \tau (y_{0})=\inf\{t\geq 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 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,
τ
¯
(
y
0
)
=
E
[
τ
(
y
0
)
∣
y
0
]
.
{\displaystyle {\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,
τ
¯
=
N
−
1
(
min
(
y
min
,
y
max
)
)
,
{\displaystyle {\bar {\tau }}=N^{-1}(\min(y_{\min },\ y_{\max })),}
where the frequency of exceedance N is
σy2 is the variance of the Gaussian distribution,
N
0
=
∫
0
∞
f
2
Φ
y
(
f
)
d
f
∫
0
∞
Φ
y
(
f
)
d
f
,
{\displaystyle N_{0}={\sqrt {\frac {\int _{0}^{\infty }{f^{2}\Phi _{y}(f)\,df}}{\int _{0}^{\infty }{\Phi _{y}(f)\,df}}}},}
and Φy(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
{\displaystyle y(t)\in \mathbb {R} ^{p}}
. Define a domain
Ψ
⊂
R
p
{\displaystyle \Psi \subset \mathbb {R} ^{p}}
that contains yavg and has a smooth boundary ∂Ψ. In this case, define the first passage time of y(t) from within the domain Ψ as
τ
(
y
0
)
=
inf
{
t
≥
t
0
:
y
(
t
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
μ
^
=
ln
(
N
0
τ
¯
)
=
min
(
y
min
,
y
max
)
2
2
σ
y
2
.
{\displaystyle {\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(ymin, ymax)/σy.