idWaveletNetwork

Wavelet network function for nonlinear ARX and Hammerstein-Wiener models

Description

An idWaveletNetwork object implements a wavelet network function, and is a nonlinear mapping function for estimating nonlinear ARX and Nonlinear Hammerstein-Wiener models. The mapping function, which is also referred to as a nonlinearity, uses a combination of linear weights, an offset and a nonlinear function to compute its output. The nonlinear function contains wavelet unit functions that operate on a radial combination of inputs.

Mathematically, idWaveletNetwork is a function that maps m inputs X(t) = [x₁(t),x₂(t),…,x_m(t)]^T to a scalar output y(t) using the following relationship:

$$y(t)=y_0+{\rm X}{(t)}^{T}PL+W({\rm X}(t))+S({\rm X}(t))$$

Here:

• X(t) is an m-by-1 vector of inputs, or regressors.

• y₀ is the output offset, a scalar.

• 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.

• W(X) and S(X) together constitute the nonlinear function of the wavelet network. W(X) is a sum of dilated and translated wavelets while S(X) is a sum of dilated and translated scaling functions (also known as scalelets). The total number of wavelet d_w and scaling functions d_s is referred to as the number of units of the network.

For definitions of the wavelet function term W(X) and the scaling function term S(X), see More About.

Use idWaveletNetwork as the value of the OutputFcn property of an idnlarx model or the InputNonlinearity and OutputLinearity properties of an idnlhw object. For example, specify idWaveletNetwork when you estimate an idnlarx model with the following command.

sys = nlarx(data,regressors,idWaveletNetwork)

When nlarx estimates the model, it essentially estimates the parameters of the idWaveletNetwork function.

You can configure the idWaveletNetwork function to disable components and fix parameters. To omit the linear component, set LinearFcn.Use to false. To omit the offset, set Offset.Use to false. To specify known values for the linear function and the offset, set their Value attributes directly and set the corresponding Free attributes to False. Use evaluate to compute the output of the function for a given vector of inputs.

Creation

Syntax

W = idWaveletNetwork

W = idWaveletNetwork(numUnits)

W = idWaveletNetwork(numUnits,UseLinearFcn)

W = idWaveletNetwork(numUnits,UseLinearFcn,UseOffset)

Description

W = idWaveletNetwork creates a idWaveletNetwork object W, for which the function computes the number of units automatically during model estimation.

W = idWaveletNetwork(numUnits) specifies the number of units numUnits. This syntax includes an option that allows you to interactively assess the relationship between the number of units and unexplained variance.

W = idWaveletNetwork(numUnits,UseLinearFcn) specifies whether the function uses a linear function as a subcomponent.

W = idWaveletNetwork(numUnits,UseLinearFcn,UseOffset) specifies whether the function uses an offset term.

Input Arguments

numUnits — Number of units

'auto' (default) | 'interactive' | positive integer

Number of units, specified as the string or character vector that represents 'auto' or 'interactive', or as a positive integer. numUnits determines the number of wavelets or scaling functions, or, if both elements are present, the combined number of wavelets and scaling functions. Typically, the wavelet network contains either wavelets or scaling functions, but not both. Specify numUnits as one of the following values:

• 'auto' — The software determines the number of units automatically during model estimation.

• 'interactive' — During model estimation, the software displays an interactive bar plot that relates unexplained variance to the number of units. Click on a bar to view the achievable fit to the estimation data for the selected number of units. A blue bar indicates the optimal choice, based on the generalized cross-validation (GCV) criterion. A general rule for the selection of the number of units is to use the smallest number of units that capture most of the variance.

• Positive integer — The software uses the specified value directly.

This argument sets the W.NonlinearFcn.NumberOfUnits property.

UseLinearFcn — Option to use linear function

true (default) | false

Option to use the linear function subcomponent, specified as true or false. This argument sets the value of the W.LinearFcn.Use property.

UseOffset — Option to use offset term

true (default) | false

Option to use an offset term, specified as true or false. This argument sets the value of the W.Offset.Use property.

Properties

Inputs — Input signal names

cell array

Input signal names for the inputs to the mapping object, specified as a 1-by-m cell array, where m is the number of input signals. This property is determined during estimation.

Outputs — Output signal name

cell array

Output signal name for the output of the mapping object, specified as a 1-by-1 cell array. This property is determined during estimation.

LinearFcn — Parameters of linear function

linear function property values (default)

Parameters of the linear function, specified as follows:

• Use — Option to use the linear function in the wavelet network, specified as a scalar logical. The default value is true. For an example of setting this option, see Exclude Linear Term from Wavelet Network Mapping Object.

• Value — Linear weights that compose L', specified as a 1-by-p vector.

• InputProjection — Input projection matrix P, specified as an m-by-p matrix, that transforms the input vector of length m into a vector of length p. For Hammerstein-Wiener models, InputProjection is equal to 1.

• Free — Option to update entries of Value during estimation, specified as a 1-by-p logical vector. The software honors the Free specification only if the starting value of Value is finite. The default value is true.

Offset — Parameters of offset term

offset property values

Parameters of the offset term, specified as follows:

• Use — Option to use the offset in the wavelet network, specified as a scalar logical. The default value is true.

• Value — Offset value, specified as a scalar.

• Free — Option to update Value during estimation, specified as a scalar logical. The software honors the Free specification of false only if the value of Value is finite. The default value is true.

NonlinearFcn — Parameters of nonlinear function

nonlinear function property values

Parameters of the nonlinear function, specified as follows:

• NumberOfUnits — Number of units, specified as 'auto', 'interactive', or a positive integer. NumberOfUnits determines the number of wavelets or scaling functions, or, if both elements are present, the combined number of wavelets and scaling functions. Typically, the wavelet network contains either wavelets or scaling functions, but not both. The options for NumberOfUnits are as follows:

  • 'auto' — The software determines the number of units automatically during model estimation.

  • 'interactive' — During model estimation, the software displays an interactive bar plot that relates unexplained variance to the number of units. Click on a bar to view the achievable fit to the estimation data for the selected number of units. A blue bar indicates the optimal choice, based on the generalized cross-validation (GCV) criterion. A general rule for the selection of the number of units is to use the smallest number of units that capture most of the variance.

  • Positive integer — The software uses the specified value directly.

• Structure — Advanced options that control the structure of the wavelet and scaling functions, specified as in the following table.

  Property	Description	Default
  FinestCell	Minimum number of data points in the smallest cell, specified as 'auto' or a positive integer. A cell is the area covered by the part of a wavelet that is significantly nonzero. The default setting of 'auto' specifies that the software determines this value during estimation.	'auto'
  MinimumCells	Minimum number of cells in the partition, specified as a positive integer.	16
  MaximumCells	Maximum number of cells in the partition, specified as a positive integer.	16
  MaximumLevels	Maximum number of wavelet levels, specified as a positive integer.	10
  DilationStep	Dilation step size, specified as a positive integer.	2
  TranslationStep	Translation step size, specified as a positive integer.	1

• Parameters — Parameters of idWaveletNetwork, specified as in the following table.

  Field Name	Description	Default
  InputProjection	Projection matrix Q, specified as an m-by-q matrix. Q transforms the detrended input vector $$(X-\overline{X})$$ of length m into a vector of length q. Typically, Q has the same dimensions as the linear projection matrix P. In this case, q is equal to p, which is the number of linear weights. For Hammerstein-Wiener models, InputProjection is equal to 1.	[]
  ScalingCoefficient	Scaling function coefficients si, specified as a ds-by-1 vector.	[]
  ScalingTranslation	Scaling translation matrix, specified as a ds-by-q matrix of scaling translation row vectors ei.	[]
  ScalingDilation	Scaling function dilation coefficients di, specified as an ds-by-1 vector.	[]
  WaveletCoefficient	Wavelet function coefficients wi, specified as a dw-by-1 vector.	[]
  WaveletTranslation	Wavelet translation matrix, ei, specified as a dw-by-q matrix of wavelet translation row vectors ci.	[]
  WaveletDilation	Wavelet dilation coefficients bi, specified as an dw-by-1 vector.	[]

• Free — Option to estimate parameters, specified as a logical scalar. If all the parameters have finite values, such as when the idWaveletNetwork object corresponds to a previously estimated model, then setting Free to false causes the parameters of the nonlinear functions W(X) and S(X) to remain unchanged during estimation. The default value is true.

Examples

Create Wavelet Network Mapping Object

MO = idWaveletNetwork;

View the idWaveletNetwork object.

disp(MO)

Wavelet Network

 Nonlinear Function: Wavelet network with number of units chosen automatically
 Linear Function: uninitialized
 Output Offset: uninitialized

         NonlinearFcn: '<Wavelet and scaling function units and their parameters>'
            LinearFcn: '<Linear function parameters>'
               Offset: '<Offset parameters>'
    EstimationOptions: '<Estimation options>'

Exclude Linear Term from Wavelet Network Mapping Object

Create the idWaveletNetwork mapping object MO.

MO = idWaveletNetwork;

Exclude the linear term from MO.

MO.LinearFcn.Use = false;

View the idWaveletNetwork object.

disp(MO)

Wavelet Network

 Nonlinear Function: Wavelet network with number of units chosen automatically
 Linear Function: not in use
 Output Offset: uninitialized

         NonlinearFcn: '<Wavelet and scaling function units and their parameters>'
            LinearFcn: '<Linear function parameters>'
               Offset: '<Offset parameters>'
    EstimationOptions: '<Estimation options>'

The linear function is not in use.

Estimate Nonlinear ARX Model with Specific Mapping Function

Load the estimation data.

load twotankdata y u;

Create an iddata object from the estimation data.

z = iddata(y,u,0.2);

Create a wavelet network mapping object with five units.

MO = idWaveletNetwork(5);

Estimate the nonlinear ARX model.

sys = nlarx(z,[4 4 1],MO)

sys =

Nonlinear ARX model with 1 output and 1 input
  Inputs: u1
  Outputs: y1

Regressors:
  Linear regressors in variables y1, u1

Output function: Wavelet network with 5 units
Sample time: 0.2 seconds

Status:                                         
Estimated using NLARX on time domain data "z".  
Fit to estimation data: 96.8% (prediction focus)
FPE: 3.553e-05, MSE: 3.515e-05

Estimate MIMO Hammerstein-Wiener Model

Load the estimation data.

load motorizedcamera;

Create an iddata object.

z = iddata(y,u,0.02,'Name','Motorized Camera','TimeUnit','s');

z is an iddata object with six inputs and two outputs.

Specify the model orders and delays.

Orders = [ones(2,6),ones(2,6),ones(2,6)];

Specify the same nonlinearity estimator for each input channel.

InputNL = idWaveletNetwork;

Specify different nonlinearity estimators for each output channel.

 OutputNL = [idDeadZone,idWaveletNetwork];

Estimate the Hammerstein-Wiener model.

sys = nlhw(z,Orders,InputNL,OutputNL);

To see the shape of the estimated input and output nonlinearities, plot the nonlinearities.

plot(sys)

Click on the input and output nonlinearity blocks on the top of the plot to see the nonlinearities.

Apply idWaveletNetwork Constraints When Estimating Nonlinear ARX Model

Load the estimation data.

load iddata7 z7;

Specify idWaveletNetwork for the model output function. Configure the idWaveletNetwork object to have a fixed offset value of 1 and to use five units in the nonlinear component.

w = idWaveletNetwork;
w.Offset.Value = 1;
w.Offset.Free = false;
w.NonlinearFcn.NumberOfUnits = 5;

Specify a linear model regressor set using a lag array that produces four consecutive output regressors and five consecutive input regressors.

reg = linearRegressor({'y1','u1'},{1:4,0:4});

Estimate the nonlinear ARX model.

sys = nlarx(z7,reg,w);

Examine the postestimation properties of the output function.

disp(sys.OutputFcn)

Wavelet Network
Inputs: y1(t-1), y1(t-2), y1(t-3), y1(t-4), u1(t), u1(t-1), u1(t-2), u1(t-3), u1(t-4)
Output: y1(t)

 Nonlinear Function: Wavelet network with 5 units
 Linear Function: initialized to [-1.12 0.469 1.25 -0.556 -0.81 0.261 -0.074 0.711 1.15]
 Output Offset: fixed to 1

         NonlinearFcn: '<Wavelet and scaling function units and their parameters>'
            LinearFcn: '<Linear function parameters>'
               Offset: '<Offset parameters>'
    EstimationOptions: '<Estimation options>'

disp(sys.OutputFcn.Input)

Function inputs

     Name: {'y1(t-1)'  'y1(t-2)'  'y1(t-3)'  'y1(t-4)'  'u1(t)'  'u1(t-1)'  'u1(t-2)'  'u1(t-3)'  'u1(t-4)'}
     Mean: [5.3829e-17 7.1772e-17 2.6914e-17 0 -4.4857e-18 -4.4857e-17 3.3643e-17 2.2429e-17 1.3457e-17]
    Range: [2x9 double]

disp(sys.OutputFcn.NonlinearFcn)

Wavelet and scaling function units and their parameters

    NumberOfUnits: 5
       Parameters: '<Wavelet parameters>'
             Free: 1
           Inputs: {'y1(t-1)'  'y1(t-2)'  'y1(t-3)'  'y1(t-4)'  'u1(t)'  'u1(t-1)'  'u1(t-2)'  'u1(t-3)'  'u1(t-4)'}
          Outputs: {'y1(t):Nonlinear'}

Apply idWaveletNetwork Constraints When Estimating Hammerstein-Wiener Model

Load the estimation data.

load throttledata

Specify idWaveletNetwork for the model output nonlinearity and set the arguments for NumUnits to 5 and UseLinearFcn and UseOffset to false.

w = idWaveletNetwork(5,false,false);

An alternative method for removing the linear function and offset is to use dot notation after creating the nonlinearity.

w.LinearFcn.Use = false;
w.Offset.Use = false;

Estimate the model, using an order specification of [4 4 1].

sys = nlhw(ThrottleData,[4 4 1],[],w);

Compare the simulated model response with the estimation data.

compare(ThrottleData,sys);

More About

Wavelet Nonlinear Function W(X)

The wavelet nonlinear function is a sum of the dilated and translated wavelets, and is described by the following equation:

$$W(X)={\displaystyle \sum _{i=1}^{d_w}{w}_i}{f}_w(b_i(X^{T}Q-c_i))$$

Here:

• Q is an m-by-q projection matrix, where m ≥ q

• w₁, w₂, …, w_dw are scalar coefficients called wavelet coefficients.

• b₁, b₂, …, b_dw are scalars called wavelet dilations that multiply the input matrix X.

• c₁, c₂, …, c_dw are 1-by-q row vectors called wavelet translations.

• $$f_w(x)=(m-x{x}^T)e^{-x{x}^T/2}$$ is a radial function that depends only upon the squared magnitude of the vector x. x is a row vector that is composed of a linear combination of inputs with an offset c_i. m is equal to the number of inputs, dim(x).

Scaling Nonlinear Function S(X)

The scaling nonlinear function is a sum of the dilated and translated scaling functions, and is described by the following equation:

$$S(X)={\displaystyle \sum _{i=1}^{d_s}{s}_i}{f}_s(d_i(X^{T}Q-e_i))$$

Here:

• Q is an m-by-q projection matrix, where m ≥ q.

• s₁, s₂, …, s_ds are scalar coefficients called scaling coefficients.

• d₁, d₂, …, d_ds are scalars called scaling dilations that multiply the input matrix X.

• e₁, e₂, …, e_ds are 1-by-q row vectors called scaling translations.

• $$f_s(x)=e^{-x{x}^T/2}$$ is a radial function that depends only upon the squared magnitude of the vector x. x is a row vector that is composed of a linear combination of inputs with an offset e_i.

Algorithms

You can use idWaveletNetwork in both nonlinear ARX and Hammerstein-Wiener models. The algorithms for estimating idWaveletNetwork parameters depend on which model you are estimating.

• In a nonlinear ARX model, idWaveletNetwork uses either a noniterative or an iterative technique for predicting the parameters, depending on option settings in nlarxOptions.

  • If the Focus option is set to 'prediction', then idWaveletNetwork uses a fast noniterative technique to estimate parameters [1]. Successive refinements after the first estimation use an iterative algorithm.

  • If the Focus option is set to 'simulation', then idWaveletNetwork uses an iterative technique to estimate parameters.

  • To always use either an iterative or a noniterative algorithm, specify the IterativeWavenet property of nlarxOptions as 'on' or 'off', respectively.

• In a Hammerstein-Wiener model, idWaveletNetwork uses iterative minimization to determine the parameters.

References

[1] Qinghua Zhang. “Using Wavelet Network in Nonparametric Estimation.” IEEE Transactions on Neural Networks 8, no. 2 (March 1997): 227–36. https://doi.org/10.1109/72.557660.

Version History

Introduced in R2007a

R2022a: Previous idnlarx data normalization information moved from mapping object properties to idnlarx Normalization property

Information related to data normalization was moved from the idWaveletNetwork mapping object level to the model level. The Normalization property of the idnlarx model contains the data centering and scaling information that the estimation process computes. In addition, the regressor-selection process for the mapping objects has also moved to the model level. The model now passes the actual regressor names rather than the selection indices to the mapping object, eliminating the need for an index property at the mapping object level.

The following table summarizes the mapping object subproperties that were eliminated. For more information, see the Normalization property of idnlarx.

Main Properties / Subproperties	Input	Output	LinearMdl	Offset	NonlinearMdl
Mean	X	X
Range	X	X
Minimum			X	X	X
Maximum			X	X	X
SelectedInputIndex			X		X

R2021b: Use of previous idnlarx and idnlhw mapping object names is not recommended.

Starting in R2021b, the mapping objects (also known as nonlinearities) used in the nonlinear components of the idnlarx and idnlhw objects have been renamed. The following table lists the name changes.

Pre-R2021b Name	R2021b Name
wavenet	idWaveletNetwork
sigmoidnet	idSigmoidNetwork
treepartition	idTreePartition
customnet	idCustomNetwork
saturation	idSaturation
deadzone	idDeadZone
pwlinear	idPiecewiseLinear
poly1d	idPolynomial1D
unitgain	idUnitGain
linear	idLinear
neuralnet	idFeedforwardNetwork

Scripts with the old names still run normally, although they will produce a warning. Consider using the new names for continuing compatibility with newly developed features and algorithms. There are no plans to exclude the use of these object names at this time.

R2021a: Use of previous nonlinearity estimator properties is not recommended

Starting in R2021a, the properties of the mapping objects, previously known as nonlinearity estimators, have been reorganized. These objects, which were renamed in R2021b, are idWaveletNetwork (W), idSigmoidNetwork (S), idTreePartition (T), idCustomNetwork (C), and idLinear (L). The property changes do not apply to idFeedforwardNetwork. The use of the pre-R2021a properties in the following table is discouraged. However, the software still accepts commands that set these properties. There are no plans to exclude such commands at this time.

Pre-R2021a Property	R2021a Property	Applicable Mapping Objects
NumberOfUnits	NonlinearFcn.NumberOfUnits	W,S,T,C
LinearTerm	LinearFcn.Use, Offset.Use	W,S,C
Parameters	Split into three pieces: LinearFcn.Value Offset.Value NonlinearFcn.Parameters	W,S,T,C,L linear (L) excludes NonlinearFcn.Parameters.
Options	NonlinearFcn.Structure	W,T