Residence Time (statistics)
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In statistics, the residence time is the average amount of time it takes for a
random process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appe ...
to reach a certain boundary value, usually a boundary far from the mean.


Definition

Suppose is a real, scalar
stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
with initial value , mean and two critical values , where and . Define the first
passage time Passage, The Passage or Le Passage may refer to: Arts and entertainment Films * ''Passage'' (2008 film), a documentary about Arctic explorers * ''Passage'' (2009 film), a short movie about three sisters * ''The Passage'' (1979 film), starring ...
of from within the interval as : \tau(y_0) = \inf\, where "inf" is the
infimum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest low ...
. This is the smallest time after the initial time that is equal to one of the critical values forming the boundary of the interval, assuming is within the interval. Because proceeds randomly from its initial value to the boundary, is itself a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
. The mean of is the residence time, : \bar(y_0) = E tau(y_0)\mid y_0 For a
Gaussian process In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution, i.e. e ...
and a boundary far from the mean, the residence time equals the inverse of the
frequency of exceedance The frequency of exceedance, sometimes called the annual rate of exceedance, is the frequency with which a random process exceeds some critical value. Typically, the critical value is far from the mean. It is usually defined in terms of the number ...
of the smaller critical value, : \bar = N^(\min(y_,\ y_)), where the frequency of exceedance is is the variance of the Gaussian distribution, : N_0 = \sqrt, and is the
power spectral density The power spectrum S_(f) of a time series x(t) describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, ...
of the Gaussian distribution over a frequency .


Generalization to multiple dimensions

Suppose that instead of being scalar, has dimension , or . Define a domain that contains and has a smooth boundary . In this case, define the first passage time of from within the domain as : \tau(y_0) = \inf\. In this case, this infimum is the smallest time at which is on the boundary of rather than being equal to one of two discrete values, assuming is within . The mean of this time is the residence time, : \bar(y_0) = \operatorname tau(y_0)\mid y_0


Logarithmic residence time

The logarithmic residence time is a
dimensionless A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
variation of the residence time. It is proportional to the natural log of a normalized residence time. Noting the exponential in Equation , the logarithmic residence time of a Gaussian process is defined as :\hat = \ln \left(N_0 \bar \right) = \frac. This is closely related to another dimensionless descriptor of this system, the number of standard deviations between the boundary and the mean, . In general, the normalization factor can be difficult or impossible to compute, so the dimensionless quantities can be more useful in applications.


See also

*
Cumulative frequency analysis Cumulative frequency analysis is the analysis of the frequency of occurrence of values of a phenomenon less than a reference value. The phenomenon may be time- or space-dependent. Cumulative frequency is also called ''frequency of non-exceedance ...
*
Extreme value theory Extreme value theory or extreme value analysis (EVA) is a branch of statistics dealing with the extreme deviations from the median of probability distributions. It seeks to assess, from a given ordered sample of a given random variable, the pr ...
*
First-hitting-time model Events are often triggered when a stochastic or random process first encounters a threshold. The threshold can be a barrier, boundary or specified state of a system. The amount of time required for a stochastic process, starting from some initial ...
*
Frequency of exceedance The frequency of exceedance, sometimes called the annual rate of exceedance, is the frequency with which a random process exceeds some critical value. Typically, the critical value is far from the mean. It is usually defined in terms of the number ...
*
Mean time between failures Mean time between failures (MTBF) is the predicted elapsed time between inherent failures of a mechanical or electronic system during normal system operation. MTBF can be calculated as the arithmetic mean (average) time between failures of a system ...


Notes


References

* * * {{cite journal , last1=Richardson , first1=Johnhenri R. , last2=Atkins , first2=Ella M. , 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 Extreme value data Survival analysis Reliability analysis