residence time (statistics)

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

:$\backslash tau(y\_0)\; =\; \backslash inf\backslash \{t\; \backslash ge\; t\_0\; :\; y(t)\; \backslash in\; \backslash \{y\_\{\backslash operatorname\{avg\}\}-y\_\{\backslash min\},\backslash \; y\_\{\backslash operatorname\{avg\}\}+y\_\{\backslash max\}\backslash \}\; \backslash \},$

where "inf" is the infimum. This is the smallest time after the initial time {{math|t₀}} that {{math|y(t)}} is equal to one of the critical values forming the boundary of the interval, assuming {{math|y₀}} is within the interval.

Because {{math|y(t)}} proceeds randomly from its initial value to the boundary, {{math|τ(y₀)}} is itself a random variable. The mean of {{math|τ(y₀)}} is the residence time,{{sfn|Meerkov|Runolfsson|1987|pp=1734–1735}}{{sfn|Richardson|Atkins|Kabamba|Girard|2014|p=2027}}

:$\backslash bar\{\backslash tau\}(y\_0)\; =\; E[\backslash tau(y\_0)\backslash mid\; y\_0].$

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,{{sfn|Richardson|Atkins|Kabamba|Girard|2014|p=2027}}

:$\backslash bar\{\backslash tau\}\; =\; N^\{-1\}(\backslash min(y\_\{\backslash min\},\backslash \; y\_\{\backslash max\})),$

where the frequency of exceedance {{mvar|N}} is

{{NumBlk|:|$N(y\_\{\backslash max\})\; =\; N\_0\; e^\{-y\_\{\backslash max\}^2/2\backslash sigma\_y^2\},$|{{EquationRef|1}}}}

{{math|σ_y²}} is the variance of the Gaussian distribution,

:$N\_0\; =\; \backslash sqrt\{\backslash frac\{\backslash int\_0^\backslash infty\{f^2\; \backslash Phi\_y(f)\; \backslash ,\; df\}\}\{\backslash int\_0^\backslash infty\{\backslash Phi\_y(f)\; \backslash ,\; df\}\}\},$

and {{math|Φ_y(f)}} is the power spectral density of the Gaussian distribution over a frequency {{mvar|f}}.

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

:$\backslash tau(y\_0)\; =\; \backslash inf\backslash \{t\; \backslash ge\; t\_0\; :\; y(t)\; \backslash in\; \backslash partial\; \backslash Psi\; \backslash mid\; y\_0\; \backslash in\; \backslash Psi\; \backslash \}.$

In this case, this infimum is the smallest time at which {{math|y(t)}} is on the boundary of {{math|Ψ}} rather than being equal to one of two discrete values, assuming {{math|y₀}} is within {{math|Ψ}}. The mean of this time is the residence time,{{sfn|Meerkov|Runolfsson|1986|p=494}}{{sfn|Meerkov|Runolfsson|1987|p=1734}}

:$\backslash bar\{\backslash tau\}(y\_0)\; =\; \backslash operatorname\{E\}[\backslash tau(y\_0)\backslash mid\; y\_0].$

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 {{EqNote|1}}, the logarithmic residence time of a Gaussian process is defined as{{sfn|Richardson|Atkins|Kabamba|Girard|2014|p=2028}}{{sfn|Meerkov|Runolfsson|1986|p=495|ps=, an alternate approach to defining the logarithmic residence time and computing {{math|N₀}}}}

:$\backslash hat\{\backslash mu\}\; =\; \backslash ln\; \backslash left(N\_0\; \backslash bar\{\backslash tau\}\; \backslash right)\; =\; \backslash frac\{\backslash min(y\_\{\backslash min\},\backslash \; y\_\{\backslash max\})^2\}\{2\; \backslash sigma\_y^2\}.$

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

In general, the normalization factor {{math|N₀}} can be difficult or impossible to compute, so the dimensionless quantities can be more useful in applications.

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

{{reflist}}

• {{cite conference |title=Aiming Control |last1=Meerkov |first1=S. M. |last2=Runolfsson |first2=T. |year=1986 |conference=Proceedings of 25th Conference on Decision and Control |publisher=IEEE |pages=494–498 |location=Athens}}
• {{cite conference |last1=Meerkov |first1=S. M. |last2=Runolfsson |first2=T. |title=Output Aiming Control |year=1987 |conference=Proceedings of 26th Conference on Decision and Control |publisher=IEEE |pages=1734–1739 |location=Los Angeles}}
• {{cite journal |last1=Richardson |first1=Johnhenri R. |last2=Atkins |first2=Ella M.|author2-link=Ella Atkins |last3=Kabamba |first3=Pierre T. |last4=Girard |first4=Anouck R. |year=2014 |title=Safety Margins for Flight Through Stochastic Gusts |journal=Journal of Guidance, Control, and Dynamics |publisher=AIAA |volume=37 |issue=6 |pages=2026–2030 |doi=10.2514/1.G000299|hdl=2027.42/140648 |hdl-access=free }}

Category:Extreme value data

Category:Survival analysis

Category:Reliability analysis