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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`.

Algorithm

When the idnlarx property Focus is 'Prediction', wavenet uses a fast, noniterative technique for estimating parameters. Successive refinements after the first estimation use an iterative algorithm.

When the idnlarx property Focus='Simulation', wavenet uses an iterative technique for estimating parameters.

To always use noniterative or iterative algorithm, specify the IterWavenet algorithm property of the idnlarx class.

Examples

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

m=nlarx(Data,Orders,wavenet);