armax (System Identification Toolbox)

Estimate the parameters of an ARMAX or ARMA model.

th = armax(z,nn)
th = armax(z,nn,'trace')
[th, iter_info] = armax(z,nn,maxiter,tol,lim,maxsize,T,'trace')

Description

The parameters of the ARMAX model structure

are estimated using a prediction error method.

Matrix z contains the output-input data z = [y u] where y and u are column vectors (u is a matrix in the multi-input case). nn can be given either as

nn = [na nb nc nk]

or as

nn = thi

In the former case na, nb, and nc are the orders of the ARMAX model, and nk is the delay. In the latter case thi is an initial value, given in theta format. See Section 3 in the Tutorial for an exact definition of the orders.

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 z = y and nn = [na nc], armax calculates an ARMA model for y:

th is returned with the resulting parameter estimates, together with estimated covariances, stored in theta format.

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

If a last argument `trace' is supplied, information about the progress of the iterative search for the model will be furnished to the MATLAB command window.

The optional auxiliary variables iter_info, lim, maxiter, tol, maxsize, and T are explained under auxvar.

Algorithm

A robustified quadratic prediction error criterion is minimized using an iterative Gauss-Newton algorithm. The Gauss-Newton vector is bisected up to 10 times until a lower value of the criterion is found. If no such value is found, a gradient search direction is used instead, and the procedure is repeated. The iterations are terminated when maxiter is reached, when the Gauss-Newton vector has a norm less than tol, or when a lower value of the criterion cannot be found.

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

The cut-off value for the robustification is based on the parameter lim as well as on the estimated standard deviation of the residuals from the initial parameter estimate. It is not recalculated during the minimization. The value returned in element th(1,1) is the nonrobustified, quadratic criterion.

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. Note that if an initial parameter estimate is given in nn, its predictor stability is taken for granted (not tested).

Information about the minimization is furnished to the screen in case the argument `trace' is specified.. 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. The number in the upper left corner is the number of times the search vector has been bisected.