System Identification Toolbox™

wavenet - Class representing wavelet network nonlinearity estimator for nonlinear ARX and Hammerstein-Wiener models

Syntax

s=wavenet('NumberOfUnits',N) s=wavenet(Property1,Value1,...PropertyN,ValueN)

Description

wavenet is an object that stores the wavelet network nonlinear estimator for estimating nonlinear ARX and Hammerstein-Wiener models.

You can use the constructor to create the nonlinearity object, as follows:

s=wavenet('NumberOfUnits',N) creates a wavelet nonlinearity estimator object with N terms in the wavelet expansion.

s=wavenet(Property1,Value1,...PropertyN,ValueN) creates a wavelet nonlinearity estimator object specified by properties in wavenet Properties.

Use evaluate(s,x) to compute the value of the function defined by the wavenet object s at x.

Remarks

Use wavenet to define a nonlinear function , where y is scalar and x is an m-dimensional row vector. The wavelet network function is based on the following function expansion:

where f is a scaling function and g is the wavelet function. P and Q are m-by-p and m-by-q projection matrices, respectively. The projection matrices P and Q are determined by principal component analysis of estimation data. Usually, p=m. If the components of x in the estimation data are linearly dependent, then p<m. The number of columns of Q, q, corresponds to the number of components of x used in the scaling and wavelet function.

When used in a nonlinear ARX model, q is equal to the size of the NonlinearRegressors property of the idnlarx object. When used in a Hammerstein-Wiener model, m=q=1 and Q is a scalar.

r is a 1-by-m vector and represents the mean value of the regressor vector computed from estimation data.

d, a_s, b_s, a_w, and b_w are scalars. Parameters with the s subscript are scaling parameters, and parameters with the w subscript are wavelet parameters.

L is a p-by-1 vector.

c_s and c_w are 1-by-q vectors.

The scaling function f and the wavelet function g are both radial functions, as follows:

wavenet Properties

You can include property-value pairs in the constructor to specify the object.

After creating the object, you can use get or dot notation to access the object property values. For example:

% List all property values
get(w)
% Get value of NumberOfUnits property
w.NumberOfUnits

You can also use the set function to set the value of particular properties. For example:

h set(w, 'LinearTerm', 'on')

The first argument to set must be the name of a MATLAB variable.

Property Name	Description
`NumberOfUnits`	Integer specifies the number of nonlinearity units in the expansion. Default=`'auto'`. For example: wavenet('NumberOfUnits',5)
`LinearTerm`	Can have the following values: `'on'` — (Default) Estimates the vector L in the expansion. `'off'` — Fixes the vector L to zero and omits the term . For example: wavenet(H,'LinearTerm','on')
`Parameters`	Structure containing the parameters in the nonlinear expansion, as follows: `RegressorMean`: 1-by-`m` vector containing the means of `x` in estimation data, `r`. `NonLinearSubspace`: `m`-by-`q` matrix containing Q. `LinearSubspace`: `m`-by-`p` matrix containing P. `LinearCoef`: `p`-by-1 vector L. `ScalingDilation`: `ns`-by-1 matrix containing the values `bs_k`. `WaveletDilation`: `nw`-by-1 matrix containing the values `bw_k`. `ScalingTranslation`: `ns`-by-`q` matrix containing the values `cs_k`. `WaveletTranslation`: `nw`-by-`q` matrix containing the values `cw_k`. `ScalingCoef`: `ns`-by-1 vector containing the values `as_k`. `WaveletCoef`: `nw`-by-1 vector containing the values `aw_k`. `OutputOffset`: scalar `d`.
`Options`	Structure containing the following fields that affect the initial model: `FinestCell`: Integer or string specifying the minimum number of data points in the smallest cell. A cell is the area covered by the significantly nonzero portion of a wavelet. Default: `'auto'`, which computes the value from the data. `MinCells`: Integer specifying the minimum number of cells in the partition. Default: `16`. `MaxCells`: Integer specifying the maximum number of cells in the partition. Default: `128`. `MaxLevels`: Integer specifying the maximum number of wavelet levels. Default: `10`. `DilationStep`: Real scalar specifying the dilation step size. Default: `2`. `TranslationStep`: Real scalar specifying the translation step size. Default: `1`.

Examples

Use wavenet to specify the nonlinear estimator in nonlinear ARX and Hammerstein-Wiener models. For example: