Database Reference
In-Depth Information
8.2.4 ARMA and ARIMA Models
In general, the data scientist does not have to choose between an AR(p) and an
MA(q) model to describe a time series. In fact, it is often useful to combine these
two representations into one model. The combination of these two models for a
stationary time series results in an
Autoregressive Moving Average model,
ARMA(p,q)
, which is expressed as shown in
Equation 8.15
.
where is a constant for a nonzero-centered time series
is a constant for j = 1, 2, …, p
is a constant for k = 1, 2, …, q
for all t
If
, then the ARMA(p,q) model is simply an AR(p) model.
Similarly, if
, then the ARMA(p,q) model is an MA(q) model.
To apply an ARMA model properly, the time series must be a stationary one.
However, many time series exhibit some trend over time.
Figure 8.7
illustrates a
time series with an increasing linear trend over time. Since such a time series does
not meet the requirement of a constant expected value (mean), the data needs to be
adjusted to remove the trend. One transformation option is to perform a regression
analysis on the time series and then to subtract the value of the fitted regression
line from each observed y-value.
