Time Series
Definition of ARIMA Models
ARIMA (auto-regressive integrated moving
average) models establish a powerful class of models which can be applied
to many real time series. ARIMA models are based on three parts: (1) an
autoregressive part, (2) a contribution from a moving average, and (3)
a part involving the first derivative of the time series:
The auto-regressive part (AR) of the model
has its origin in the theory that individual values of time series can
be described by linear models based on preceding observations. For
instance: x(t) = 3 x(t-1) - 4 x(t-2). The general formula for describing
AR[p]-models (auto-regressive models) is:
The order of the model is determined by p.
The consideration leading to moving average
models (MA models) is that time series values can be expressed as being
dependent on the preceding estimation errors. Past estimation or forecasting
errors are taken into account when estimating the next time series value.
The difference between the estimation x(t) and the actually observed value
x(t) is denoted e(t). For instance: x(t) = 3
e(t-1)
- 4 e(t-2).
The general description of MA[q]-models is:
When combining both AR and MA models, ARMA models
are obtained. In general, forecasting with an ARMA[p,q]-model is
described using the following equation:
After additional differentiation of the
time series, and integrating it after application of the model,
one speaks of ARIMA models. They are used when trend filtering is required.
The parameter d of the ARIMA[p,d,q]-model determines the number of differentiation
steps.
First, the time series is derived d times until
it is stationary.
Then, a suitable ARMA[p,q] model is fitted to
the resulting series.
Finally, the estimated forecasts have to be integrated
d times.
Many more variants of ARIMA models have been introduced
to treat specific cases. Here, the whole group of such models is subsumed
under the term ARIMA models. Since their characteristics are determined
by the three parameters p, d, and q, they are also referred to as ARIMA[p,d,q]-models.
The parameter p denotes the order of the auto-regressive part, the parameter
q the order of the moving average part, and d the number of differentiation
steps.
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