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In-Depth Information
Figure 8.10
Twice differenced series
Because the need to make a time series stationary is common, the differencing
can be included (integrated) into the ARMA model definition by defining the
Autoregressive Integrated Moving Average
model, denoted
ARIMA(p,d,q). The structure of the ARIMA model is identical to the expression in
Equation 8.15
,
but the ARMA(p,q) model is applied to the time series,
, after
applying differencing d times.
Additionally, it is often necessary to account for seasonal patterns in time series.
For example, in the retail sales use case example in Section 8.1, monthly clothing
sales track closely with the calendar month. Similar to the earlier option of
detrending a series by first applying linear regression, the seasonal pattern could
be determined and the time series appropriately adjusted. An alternative is to use
a
seasonal autoregressive integrated moving average model
, denoted
ARIMA(p,d,q)
(P,D,Q)
s
where:
• p, d, and q are the same as defined previously.
• s denotes the seasonal period.
• P is the number of terms in the AR model across the s periods.
• D is the number of differences applied across the s periods.
• Q is the number of terms in the MA model across the s periods.
For a time series with a seasonal pattern, following are typical values of s:
• 52 for weekly data
• 12 for monthly data
• 7 for daily data
