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.