armax (System Identification Toolbox)

Estimate the parameters of an ARMAX or ARMA model.

Syntax

m = armax(data,orders)
m = armax(data,'na',na,'nb',nb,'nc',nc,'nk',nk)
m = armax(data,orders,'Property1',Value1,...,'PropertyN',ValueN)

Description

armax returns m as an idpoly object with the resulting parameter estimates, together with estimated covariances.

armax estimates the parameters of the ARMAX model structure

using a prediction error method.

data is an iddata object containing the output-input data. The model orders can be specified as (...,'na',na,'nb',nb,...) or by setting the argument orders to

```
orders = [na nb nc nk]
```

The parameters na, nb, and nc are the orders of the ARMAX model, and nk is the delay. Specifically,

Alternatively, you can specify the vector as

```
orders = mi
```

where mi is an initial guess at the ARMAX model given in idpoly format. See Polynomial Representation of Transfer Functions in the "Tutorial" for more information.

For multi-input systems, nb and nk are row vectors, such that the k-th entry corresponds to the order and delay associated with the k-th input.

If data has no input channels and just one output channel (i.e., it is a time series), then

```
orders = [na nc] 
```

and armax calculates an ARMA model for the time series

The structure and the estimation algorithm are affected by any property name/property value pairs that are set in the input argument list. Useful properties are 'Focus', 'InitialState', 'Trace', 'MaxIter', 'Tolerance', 'LimitError', and 'FixedParameter'.

See Algorithm Properties, idpoly and idmodel for details of these properties and their possible values.

armax does not support multi-output models. Use the state-space model for this case (see n4sid and pem).

Algorithm

A robustified quadratic prediction error criterion is minimized using an iterative search algorithm, whose details are governed by the properties 'SearchDirection', 'MaxIter','Tolerance' and 'Advanced'. The iterations are terminated when MaxIter is reached, when the expected improvement is less than Tolerance, or when a lower value of the criterion cannot be found. Information about the search is contained in m.EstimationInfo.

The initial parameter values for the iterative search, if not specified in orders, are constructed in a special four-stage LS-IV algorithm.

The cut-off value for the robustification is based on the property LimitError as well as on the estimated standard deviation of the residuals from the initial parameter estimate. It is not recalculated during the minimization.

A stability test of the predictor is performed, so as to assure that only models corresponding to stable predictors are tested. Generally, both and (if applicable) must have all their zeros inside the unit circle.

Information about the minimization is furnished to the screen in case the property 'Trace' is set to 'On' or 'Full'. With 'Trace' = 'Full', current and previous parameter estimates (in column vector form, listing parameters in alphabetical order) as well as the values of the criterion function are given. The Gauss-Newton vector and its norm are also displayed. With 'Trace' = 'On' just criterion values are displayed.

References

Ljung (1999), Section 10.2.

ar arx