Low-Level Nonlinearity Evaluation Using Wavenet (SISO System)

Hello!

I'm trying to do a low-level nonlinearity evaluation to a NARX model based on wavenet estimator. This manipulation is based on http://www.mathworks.com/help/toolbox/ident/ug/bq5o_xw-1.html#br7139m.

As we see below, only a subset of regressors (in case, [y1(t-2), u1(t-1)]) are input to nonlinear function

M5 = NLARX(estData, [2 2 1], wavenet('Num',5), 'nlr', [2 3],'Focus', 'prediction', 'Display', 'on')

NL = M5.Nonlinearity; NL.Parameters; r = NL.Parameters.RegressorMean; Q=NL.Parameters.NonLinearSubspace; P=NL.Parameters.LinearSubspace; L=NL.Parameters.LinearCoef; as=NL.Parameters.ScalingCoef; aw=NL.Parameters.WaveletCoef; bs=NL.Parameters.ScalingDilation; bw=NL.Parameters.WaveletDilation; cs=NL.Parameters.ScalingTranslation; cw=NL.Parameters.WaveletTranslation; d = NL.Parameters.OutputOffset;

Where am I wrong?

x = [estData.y(1),0,estData.u(1),0]; %regressor(y(t-1) y(t-2) u(t-1) u(t-2)) values that are input in the linear function at t=1.

xnln = [0,0,estData.u(1),0]; %regressor(y(t-2) u(t-1)) values that are input in the nonlinear function at t=1.

% Compute the linear portion of the response (plus offset):

yLinear = (x-r)*P*L+d;

% Compute the nonlinear portion of the response:

f = @(z)(exp(-0.5*z'*z));%scale function (father wavelet)

yNonlinear1 = 0;

for k = 1:length(cs)

finputS = (bs*(xnln-r))*Q - cs(k);

yNonlinear1 = yNonlinear1 + as*f(finputS);

end

g = @(zz)((length(zz)-zz'*zz))*(exp(-0.5*zz'*zz));%wavelet function ("Mexican hat" mother wavelet)

yNonlinear2 = 0;

for k = 1:length(aw)

finputW = (bw(k,:)*(xnln-r))*Q - cw(k);

yNonlinear2 = yNonlinear2 +aw(k)*g(finputW);

end

% Total response y = F(x) = yLinear + yNonlinear1 + yNonlinear2

y = yNonlinear1 + yNonlinear2 + yLinear

In this case, i got the following response: y =

    0.1126    0.1127
    0.1127    0.1128

I would have to obtain a real number, is not it? the results obtained from the above steps must be equal to evaluate(NL,x)

Thanks

Answer 1

[Rajiv Singh]

In the f(0 and g() formulas you need an inner product.

In your formula for f(), replace z'*z by z*z'. In the formula for g(), replace zz'*zz by zz*zz'.

This was a typo in the doc some time ago, now fixed. The online documentation shows the correct formula.

Rajiv

[Emanuel]

I do not understand, because in [Q. Zhang, “Using wavelet network in nonparametric estimation,” IEEE Transactions on Neural Network, Vol. 8, No. 2, March 1997]

||x||^2 = x'x

Other doubt,

In which article I can find the same wavelet network expansion function used in Matlab? because, the wavelet network algorithm proposed in

Zhang, Q., and A. Benveniste, "Wavelet networks." IEEE Transactions on Neural Networks. Vol. 3, Issue 6, 1992, pp. 889–898

and

Q. Zhang, “Using wavelet network in nonparametric estimation,” IEEE Transactions on Neural Network, Vol. 8, No. 2, March 1997.

is a network with one hidden layer consisting of wavelons. However, the wavelet network used in Matlab is a one hidden layer consisting of scalons and wavelons.

Can you point out an article that speaks about the same expansion function used in matlab? In case:

F(x) = (x-r)PL + As1f((Bs1(x-r)Q-Cs1)+...Asnf((BsnS(x-r)Q-Csn)+Aw1G(((Bw1(x-r))Q-Cw1)+...+Awnwg((BwnW(x-r))Q-Cwn)+d

Thanks

[Rajiv Singh]

The paper uses a different notation for a regressor "vector" as MATLAB. In MATLAB, the regressors for one time value are represented as a row vector (reserving the row dim for different time values, as in common practice for most time dependent data) while in the literature, a column vector is used.

For the wavelet network formulation used, see:

Juditsky, A., H. Hjalmarsson, A. Benveniste, B. Delyon, L. Ljung, J. Sjöberg, and Q. Zhang, "Nonlinear Black-Box Models in System Identification: Mathematical Foundations." Automatica. Vol. 31, Issue 12, 1995, pp. 1725–1750.

[Emanuel]

Perfect! Thank you Rajiv!