Order of Integration

A Order of Integration transforms iterated integrals of functions into other.

• AKA: I(d)
• Context:
  • It can also be a statistic summary for a time series which represents the minimum number of differences a stochastic process required to achieve to stationarity. For instance, A stochastic process is said to be integrated of order d if it requires to be differentiated d times in order to achieve stationarity.
• See: Cointegration, ARIMA, ARMA, Random walk, Time Series, Lag Operator.

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References

2016

• (Wikipedia, 2016) ⇒ http://en.wikipedia.org/wiki/Order_of_integration Retrieved 2016-08-07
  • Order of integration, denoted I(d), is a summary statistic for a time series. It reports the minimum number of differences required to obtain a covariance stationary series (...)

Integration of order d: A time series is integrated of order d if

[math]\displaystyle{ (1-L)^d X_t \ }[/math]

is a stationary process, where [math]\displaystyle{ L }[/math] is the lag operator and [math]\displaystyle{ 1-L }[/math] is the first difference, i.e.

[math]\displaystyle{ (1-L) X_t = X_t - X_{t-1} = \Delta X. }[/math]

In other words, a process is integrated to order d if taking repeated differences d times yields a stationary process.

2016

• (Wikipedia, 2016) ⇒ http://en.wikipedia.org/wiki/Order_of_integration_(calculus) Retrieved 2016-08-07
  • In calculus, interchange of the order of integration is a methodology that transforms iterated integrals (or multiple integrals through the use of Fubini's theorem) of functions into other, hopefully simpler, integrals by changing the order in which the integrations are performed. In some cases, the order of integration can be validly interchanged; in others it cannot.

Problem statement: The problem for examination is evaluation of an integral of the form

[math]\displaystyle{ \iint_D \ f(x,y ) \ dx \,dy , }[/math]

where D is some two-dimensional area in the xy–plane. For some functions f straightforward integration is feasible, but where that is not true, the integral can sometimes be reduced to simpler form by changing the order of integration. The difficulty with this interchange is determining the change in description of the domain D.

The method also is applicable to other multiple integrals.

Sometimes, even though a full evaluation is difficult, or perhaps requires a numerical integration, a double integral can be reduced to a single integration, as illustrated next. Reduction to a single integration makes a numerical evaluation much easier and more efficient.

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