ARMAX, BJ and OE Models

The following linear difference equations are supported

ARX:
  A(q) y(t) = B(q) u(t-nk) + e(t)
ARMAX:
  A(q) y(t) = B(q) u(t-nk) + C(q) e(t)
OE: (Output Error)
  y(t) = [B(q)/F(q)] u(t-nk) + e(t)
BJ: (Box-Jenkins)
  y(t) = [B(q)/F(q)] u(t-nk) + [C(q)/D(q)] e(t)

These are all special cases of general linear input-output models. They correspond to linear difference equations relating the input to the output under various noise assumptions.

To estimate these models, select Parametric Models in the pop-up menu Estimate in the ident window, and then choose the desired structure in the Model Structure pop-up menu. The orders of the polynomials are selected by the pop-up menus in the Order editor dialog box, or by directly entering them in the edit box in the Parametric Models dialog.

The coefficients of the polynomials are estimated using a prediction error / maximum likelihood method, by minizing the size of the error term e in the expression above. Several options govern the minimization procedure. These are accessed by pressing the Iteration control... button and then pressing the Options button in the dialog that is opened.

The algorithms are further described in the manual under ARMAX, BJ, OE, PEM, and AUXVAR.

Help topics.

General Input-Output Models A general input-output linear model for a single-output system with input u and output y can be written:

A(q) y(t) = [B_i(q)/F_i(q)] u_i(t-nk_i) + [C(q)/D(q)] e(t)

Here u_i denotes input number i and there is an implied summation over all the inputs in the expression above.

A, B_i, C, D, and F_i are polynomials in the shift operator (z or q). The general structure is defined by giving the time delays nk and the orders of the polynomials (i.e., the number of poles and zeros of the dynamic model from u to y, as well as of the noise model from e to y).

Most often the choices are confined to one of the following special cases:
ARX: A(q) y(t) = B(q) u(t-nk) + e(t)
ARMAX: A(q) y(t) = B(q) u(t-nk) + C(q) e(t)
OE: y(t) = [B(q)/F(q)] u(t-nk) + e(t) (Output Error)
BJ: y(t) = [B(q)/F(q)] u(t-nk) + [C(q)/D(q)] e(t) (Box-Jenkins)

Note that A(q) corresponds to poles that are common between the dynamic model and the noise model (useful if noise enters system "close to" the input). Likewise F_i(q) determines the poles that are unique for the dynamics from input number i and D(q) the poles that are unique for the noise.

Help topics.

(file iduiio.htm)