Wavelet Decomposition Parameters

The wavelet decomposition is performed using the decomposition level LEVEL and the wavelet WNAME.

EXTMODE is the extended mode for the DWT (See dwtmode).

If a decomposition DEC obtained using mdwtdec is available, you can use

[...] = wmspca(DEC,NPC) instead of

[...] = wmspca(X,LEVEL,WNAME,'mode',EXTMODE,NPC).

Principal Components Parameter: NPC

If NPC is a vector, then it must be of length LEVEL+2. It contains the number of retained principal components for each PCA performed:

NPC must be such that 0 <= NPC(d) <= P for 1 <= d <= LEVEL+2.

If NPC = 'kais' (respectively, 'heur'), then the number of retained principal components is selected automatically using Kaiser's rule (or the heuristic rule).

If NPC = 'nodet', then the details are “killed” and all the approximations are retained.

Output Parameters

X_SIM is a simplified version of the matrix X.

QUAL is a vector of length P containing the quality of column reconstructions given by the relative mean square errors in percent.

NPC is the vector of selected numbers of retained principal components.

DEC_SIM is the wavelet decomposition of X_SIM

PCA_Params is a structure array of length LEVEL+2 such that:

Wavelet Principal Component Analysis of Noisy Multivariate Signal

Use wavelet multiscale principal component analysis to denoise a multivariate signal.

Load the dataset consisting of four signals of length 1024. Plot the original signals and the signals with additive noise.

load ex4mwden;
kp = 0;
for i = 1:4
    subplot(4,2,kp+1)
    plot(x_orig(:,i))
    axis tight
    title(['Original signal ',num2str(i)])
    subplot(4,2,kp+2)
    plot(x(:,i))
    axis tight
    title(['Noisy signal ',num2str(i)])
    kp = kp + 2;
end

Perform the first multiscale wavelet PCA using the Daubechies least-asymmetric wavelet with four vanishing moments, sym4. Obtain the multiresolution decomposition down to level 5. Use the heuristic rule to decide how many principal components to retain.

level = 5;
wname = 'sym4';
npc = 'heur';
[x_sim, qual, npc] = wmspca(x,level,wname,npc);

Plot the result and examine the quality of the approximation.

qual

qual = 1×4

   97.4372   94.5520   97.7362   99.5219

kp = 0;
for i = 1:4
    subplot(4,2,kp+1)
    plot(x(:,i))
    axis tight
    title(['Noisy signal ',num2str(i)])
    subplot(4,2,kp+2)
    plot(x_sim(:,i))
    axis tight
    title(['First PCA ',num2str(i)])
    kp = kp+2;
end

The quality results are all close to 100%. The npc vector gives the number of principal components retained at each level.

Suppress the noise by removing the principal components at levels 1�3. Perform the multiscale PCA again.

npc(1:3) = zeros(1,3);
[x_sim, qual, npc] = wmspca(x,level,wname,npc);

Plot the result.

kp = 0;
for i = 1:4
    subplot(4,2,kp+1)
    plot(x(:,i))
    axis tight
    title(['Noisy signal ',num2str(i)])
    subplot(4,2,kp+2)
    plot(x_sim(:,i))
    axis tight
    title(['Second PCA ',num2str(i)])
    kp = kp+2;
end