Estimate parameters of ARX, ARIX, AR, or ARI model

specifies additional options using one or more name-value pair arguments. For instance,
using the name-value pair argument `sys`

= arx(`data`

,```
[na
nb nk]
```

,`Name,Value`

)`'IntegrateNoise',1`

estimates an ARIX or ARI
structure model, which is useful for systems with nonstationary disturbances.

specifies estimation options using the option set `sys`

= arx(`data`

,```
[na
nb nk]
```

,___,`opt`

)`opt`

. Specify
`opt`

after all other input arguments.

`[`

returns the estimated initial conditions as an `sys`

,`ic`

] = arx(___)`initialCondition`

object. Use this syntax if you plan to simulate or predict the model response using the same
estimation input data and then compare the response with the same estimation output data.
Incorporating the initial conditions yields a better match during the first part of the
simulation.

QR factorization solves the overdetermined set of linear equations that constitutes the least-squares estimation problem.

Without regularization, the ARX model parameters vector θ is estimated by solving the normal equation

$$\left({J}^{T}J\right)\theta ={J}^{T}y$$

where *J* is the regressor matrix and *y* is
the measured output. Therefore,

$$\theta ={\left({J}^{T}J\right)}^{-1}{J}^{T}y$$

Using regularization adds the regularization term

$$\theta ={\left({J}^{T}J+\lambda R\right)}^{-1}{J}^{T}y$$

where λ and R are the regularization constants. For more information on the regularization
constants, see `arxOptions`

.

When the regression matrix is larger than the `MaxSize`

specified in
`arxOptions`

, the data is segmented and QR factorization is performed iteratively
on the data segments.