# idLinear

Linear mapping object for nonlinear ARX models

## Description

An `idLinear`

object implements an affine function, and is a
mapping function for estimating nonlinear ARX models. The mapping function uses a combination
of linear weights and an offset. Unlike the other mapping objects for the nonlinear models,
the `idLinear`

object contains no accommodation for a nonlinear
component.

Mathematically, `idLinear`

is a linear function $$y=F(x)$$ that maps *m* inputs
*X*(*t*) =
[*x*(*t*_{1}),*x*_{2}(*t*),…,*x _{m}*(

*t*)]

^{T}to a scalar output

*y*(

*t*). .

*F*is a (affine) function of

*x*:

$$y(t)={y}_{0}+{({\rm X}(t)-\overline{X})}^{T}PL$$

Here:

*X*(*t*) is an*m*-by-1 vector of inputs, or*regressors*, with mean $$\overline{{\rm X}}$$.*y*is the output offset, a scalar._{0}*P*is an*m*-by-*p*projection matrix, where*m*is the number of regressors and is*p*is the number of linear weights.*m*must be greater than or equal to*p*.*L*is a*p*-by-1 vector of weights.

Set `idLinear`

as the value of the `OutputFcn`

property
of an `idnlarx`

model. For example, specify
`idLinear`

when you estimate an `idnlarx`

model with the
following
command.

sys = nlarx(data,regressors,idLinear)

`nlarx`

estimates the model, it also estimates the parameters of the
`idLinear`

function.
Use the `idLinear`

mapping object when you want to create nonlinear ARX
models that operate linearly on the regressors. The regressors themselves can be nonlinear
functions of the inputs and outputs. The `polynomialRegressor`

and `customRegressor`

commands allow you to create such regressors. When the `idnlarx`

model has no
custom regressors and the output function is set to `idLinear`

, the model is
similar to a linear ARX model. However, for the nonlinear ARX model, the offset is an
estimable parameter.

You can configure the `idLinear`

object to disable components and fix
parameters. Use `evaluate`

to compute the output of the function for
a given vector of inputs.

## Creation

### Syntax

### Description

creates an
`Lin`

= idLinear`idLinear`

object `Lin`

with unknown parameters.

## Properties

## Examples

## Compatibility Considerations

## See Also

`nlarx`

| `idTreePartition`

| `idSigmoidNetwork`

| `idWaveletNetwork`

| `idFeedforwardNetwork`

| `idCustomNetwork`

| `idnlarx`

| `evaluate`

| `linearRegressor`

| `polynomialRegressor`

**Introduced in R2007a**