Stochastic Modelling of Time Series - Hydrologic Time Series Analysis: Theory and Practice - page 91

Order of Seasonal ARIMA Model : A seasonal ARIMA is a generalization and
extension of the regular ARIMA process discussed earlier in this chapter. The
seasonal ARIMA or SARIMA model is used for a time series where a pattern
repeats seasonally over time. In addition to the non-seasonal or regular
parameters of the ARIMA model, seasonal model parameters for a specified
lag (selected in the identification step) need to be estimated. Analogous to the
simple or regular ARIMA model parameters, there are three seasonal model
parameters: seasonal autoregressive ( p s ), seasonal differencing ( d s ), and
seasonal moving average parameters ( q s ). The SARIMA model is usually
denoted as ARIMA( p , d , q )( p s , d s , q s ), which describes a model that includes ' p '
regular AR parameters and ' p s ' seasonal AR parameters, and, ' q ' regular MA
parameters and ' q s ' seasonal MA parameters, and these parameters for the
time series were computed when the series was differenced ' d ' times and ' d s '
time seasonally differenced. The seasonal lag used for the seasonal parameters
is usually determined during the identification phase and must be explicitly
specified.
The general guidelines for the selection of regular model parameters to be
estimated (based on ACF and PACF) also apply to seasonal model parameters.
The main difference is that in seasonal series, ACF and PACF will show
sizable coefficients at multiples of the seasonal lag (in addition to their overall
patterns reflecting the non-seasonal components of the series).
5.2.2 ARIMA: Estimation of Model Parameters
Once ARIMA model has been identified and selection of model order is over,
the next step is estimation of model parameters. The model parameters are
estimated by using function minimization procedures, in order to minimize
the sum of squared residuals. There are different methods for estimating the
ARIMA model parameters. It is supposed that all the estimation methods
should produce very similar values of the model parameters, but a particular
estimation method may be more or less efficient for any given ARIMA model.
Generally, the model parameter estimation make use of a function minimization
algorithm (e.g., quasi - Newton method for nonlinear estimation) to maximize
the likelihood/probability of the observed time series for the given values of
the model parameters. In practice, sum of squares of the residuals for the
given respective parameters are computed for the function minimization. The
sum of squares of residuals can be computed by any of three methods: (i) the
approximate maximum likelihood method (McLeod and Sales, 1983), (ii) the
approximate maximum likelihood method with backcasting, and (iii) the exact
maximum likelihood method (Melard, 1984).
All the methods for computing the sum of squares of residuals are equally
efficient in the most real-world time series applications. However, the method
of approximate maximum likelihood with no backcasts is the fastest among
three methods, and should particularly be used for estimating the model
parameters of very long time series with more than 30,000 data points. The