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## Available Nonlinearity Estimators for Nonlinear ARX Models

System Identification Toolbox™ software provides several nonlinearity estimators F(x) for nonlinear ARX models. For more information about F(x), see Structure of Nonlinear ARX Models.

Each nonlinearity estimator corresponds to an object class in this toolbox. When you estimate nonlinear ARX models in the app, System Identification Toolbox creates and configures objects based on these classes. You can also create and configure nonlinearity estimators at the command line.

Most nonlinearity estimators represent the nonlinear function as a summed series of nonlinear units, such as wavelet networks or sigmoid functions. You can configure the number of nonlinear units n for estimation. For a detailed description of each estimator, see the references page of the corresponding nonlinearity class.

Wavelet network
(default)
`wavenet`

`$g\left(x\right)=\sum _{k=1}^{n}{\alpha }_{k}\kappa \left({\beta }_{k}\left(x-{\gamma }_{k}\right)\right)$`

where $\kappa \left(s\right)$ is the wavelet function.

By default, the estimation algorithm determines the number of units n automatically.
One layer sigmoid network`sigmoidnet`

`$g\left(x\right)=\sum _{k=1}^{n}{\alpha }_{k}\kappa \left({\beta }_{k}\left(x-{\gamma }_{k}\right)\right)$`

where $\kappa \left(s\right)={\left({e}^{s}+1\right)}^{-1}$ is the sigmoid function. ${\beta }_{k}$ is a row vector such that ${\beta }_{k}\left(x-{\gamma }_{k}\right)$ is a scalar.

Default number of units n is 10.
Tree partition`treepartition`Piecewise linear function over partitions of the regressor space defined by a binary tree.The estimation algorithm determines the number of units automatically.
Try using tree partitions for modeling data collected over a range of operating conditions.
F is linear in x`linear`This estimator produces a model that is similar to the linear ARX model, but offers the additional flexibility of specifying custom regressors.Use to specify custom regressors as the nonlinearity estimator and exclude a nonlinearity mapping function.
Multilayered neural network`neuralnet`Uses as a `network` object created using the Deep Learning Toolbox™ software.
Custom network
(user-defined)
`customnet`Similar to sigmoid network but you specify $\kappa \left(s\right)$.(For advanced use)
Uses the unit function that you specify.

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