Auto-Regressive Integrated Moving Average (ARIMA) models are
used to model time series data and produce forecasts for future values.
The type of ARIMA model is usuually specified as
ARIMA(p,d,q). p is the autoregressive (AR)order,
or the number of autoregressive components.
q is the moving average (MA) order, or the number of moving
average components in the model. d is the degree of differencing.
When d is zero, the model is called an ARMA(p,q) model.
When in addition p or q is zero, the model is called
an MA(q) model or AR(p) model, respectively.
ARIMA models are implemented by the
Creating ARIMA Models
class has two constructors. The first takes three parameters and is used
to construct an ARMA(p,q) model. The first parameter is a
that contains the time series data.
The second and third parameters are the autoregressive and moving average order,
The second constructor takes four parameters and is used to construct
ARIMA(p,d,q) models. The first parameter is once again the
that contains the time series data. The remaining parameters are the
autoregressive order, degree of differencing, and the moving average order.
For an ARMA(p,q) model, the mean is usually not zero,
and it is estimated by default. When a series is differenced enough times,
the mean of the differenced series will be close to zero. For ARIMA models
with nonzero degree of differencing, the mean is not estimated by default.
In each case, the default behavior can be overridden by setting the
Computing the Model
method estimates the parameters of the model by minimizing the
exact maximum likelihood function.
The parameters of the model can be retrieved through the
collection. The first p parameters are the coefficients
of the autoregressive components. The next q parameters
are the coefficients of the moving average components. If the mean
was estimated also, it is returned as the last parameter.
The autoregressive and moving average coefficients can also be
retrieved separately through the
The members of these collections are of type
and can be used to obtain a wide range of information about the computed
values, including the standard error, significance tests and confidence
Verifying the Quality of the Model
Because an ARIMA model is computed by directly maximizing the
likelihood function, it does not have the same range of diagnostic values
available for linear regression models. Still, a number of standard measures
The GetLogLikelihood()()()() method
returns the logarithm of the likelihood of the computed model.
The GetAkaikeInformationCriterion()()()() method
returns the Akaike Information Criterion (AIC) value. This is commonly used to compare
The GetBayesianInformationCriterion()()()() method
returns the Bayesian Information Criterion (BIC) value, which is sometimes used instead of the AIC.
Once the model has been computed, the
method can then be used to forecast new values. This method has three overloads.
Without parameters, the method returns the one step ahead forecast based on
the computed model. With a single parameter, it computes a point forecast
the specified number of steps ahead. It returns a
that contains the point forecast for the specified range of periods.