# Intervention model

Like the outlier and transfer function models, intervention models belong to the univariate time series models with which the occurrence of conspicuous observation values ​​can be modeled. The intervention model assumes that the time t at which the abnormal observation value occurs is known. A distinction can be made between interventions, which

• unique ,
• for an indefinite period or
• for a certain duration

occur.

This is modeled with the help of an indicator function I. In addition, the strength or duration of the effect must be modeled. This is done with a lag operator polynomial , which is also referred to as the impulse response function . It determines whether the effect of the intervention wears off over time, is intensified, or is constant. A possible formal notation would be:

${\ displaystyle Y_ {t} = \ nu (L) I_ {t} + {\ frac {\ Theta _ {q} (L)} {\ Phi _ {p} (L)}} Z_ {t}}$.

It is the ARMA part or the noise model , is the indicator function , and the impulse response function . The impulse-response function ${\ displaystyle {\ tfrac {\ Theta _ {q} (L)} {\ Phi _ {p} (L)}} Z_ {t}}$${\ displaystyle I_ {t}}$${\ displaystyle \ nu (L)}$

${\ displaystyle \ nu (L): = {\ frac {\ omega (L)} {\ delta (L)}} L ^ {d}}$

has the polynomial in the denominator , which models the permanent effect of the intervention . The polynomial in the numerator represents the expected initial effect. ${\ displaystyle \ delta (L)}$${\ displaystyle \ omega (L)}$

A time series can also be affected by several interventions of different types occurring at different times . This is known as the multiple intervention model. The estimate of the entire model can be estimated using the maximum likelihood method . The noise model as well as the impulse-response function must be identified beforehand. In doing so, one must fall back on expert knowledge of the observed time series . If there are several possible models to choose from, the suitable model can be selected using a selection criterion.