Multivariate Wavelet Denoising
This section demonstrates the features of multivariate denoising provided in the Wavelet Toolbox™ software.
This multivariate wavelet denoising problem deals with models of the form X(t) = F(t) + e(t), where the observation X is p-dimensional, F is the deterministic signal to be recovered, and e is a spatially correlated noise signal. This kind of model is well suited for situations for which such additive, spatially correlated noise is realistic.
Multivariate Wavelet Denoising
This example uses noisy test signals. In this section, you will
Load a multivariate signal.
Display the original and observed signals.
Remove noise by a simple multivariate thresholding after a change of basis.
Display the original and denoised signals.
Improve the obtained result by retaining less principal components.
Display the number of retained principal components.
Display the estimated noise covariance matrix.
Load a multivariate signal by typing the following at the MATLAB® prompt:
load ex4mwden whos
Name Size Bytes Class covar
4x4
128
double array
x
1024x4
32768
double array
x_orig
1024x4
32768
double array
Usually, only the matrix of data
x
is available. Here, we also have the true noise covariance matrix (covar
) and the original signals (x_orig
). These signals are noisy versions of simple combinations of the two original signals. The first one is “Blocks” which is irregular, and the second is “HeavySine,” which is regular except around time 750. The other two signals are the sum and the difference of the two original signals. Multivariate Gaussian white noise exhibiting strong spatial correlation is added to the resulting four signals, which leads to the observed data stored inx
.Display the original and observed signals by typing
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(['Observed signal ',num2str(i)]) kp = kp + 2; end
The true noise covariance matrix is given by
covar covar = 1.0000 0.8000 0.6000 0.7000 0.8000 1.0000 0.5000 0.6000 0.6000 0.5000 1.0000 0.7000 0.7000 0.6000 0.7000 1.0000
Remove noise by simple multivariate thresholding.
The denoising strategy combines univariate wavelet denoising in the basis where the estimated noise covariance matrix is diagonal with noncentered Principal Component Analysis (PCA) on approximations in the wavelet domain or with final PCA.
First, perform univariate denoising by typing the following to set the denoising parameters:
level = 5; wname = 'sym4'; tptr = 'sqtwolog'; sorh = 's';
Then, set the PCA parameters by retaining all the principal components:
npc_app = 4; npc_fin = 4;
Finally, perform multivariate denoising by typing
x_den = wmulden(x, level, wname, npc_app, npc_fin, tptr, sorh);
Display the original and denoised signals by typing
kp = 0; for i = 1:4 subplot(4,3,kp+1), plot(x_orig(:,i)); set(gca,'xtick',[]); axis tight; title(['Original signal ',num2str(i)]) subplot(4,3,kp+2), plot(x(:,i)); set(gca,'xtick',[]); axis tight; title(['Observed signal ',num2str(i)]) subplot(4,3,kp+3), plot(x_den(:,i)); set(gca,'xtick',[]); axis tight; title(['denoised signal ',num2str(i)]) kp = kp + 3; end
Improve the first result by retaining fewer principal components.
The results are satisfactory. Focusing on the two first signals, note that they are correctly recovered, but the result can be improved by taking advantage of the relationships between the signals, leading to an additional denoising effect.
To automatically select the numbers of retained principal components by Kaiser's rule (which keeps the components associated with eigenvalues exceeding the mean of all eigenvalues), type
npc_app = 'kais'; npc_fin = 'kais';
Perform multivariate denoising again by typing
[x_den, npc, nestco] = wmulden(x, level, wname, npc_app, ... npc_fin, tptr, sorh);
Display the number of retained principal components.
The second output argument gives the numbers of retained principal components for PCA for approximations and for final PCA.
npc npc = 2 2
As expected, since the signals are combinations of two initial ones, Kaiser's rule automatically detects that only two principal components are of interest.
Display the estimated noise covariance matrix.
The third output argument contains the estimated noise covariance matrix:
nestco nestco = 1.0784 0.8333 0.6878 0.8141 0.8333 1.0025 0.5275 0.6814 0.6878 0.5275 1.0501 0.7734 0.8141 0.6814 0.7734 1.0967
As you can see by comparing with the true matrix covar given previously, the estimation is satisfactory.
Display the original and final denoised signals by typing
kp = 0; for i = 1:4 subplot(4,3,kp+1), plot(x_orig(:,i)); set(gca,'xtick',[]); axis tight; title(['Original signal ',num2str(i)]); set(gca,'xtick',[]); axis tight; subplot(4,3,kp+2), plot(x(:,i)); set(gca,'xtick',[]); axis tight; title(['Observed signal ',num2str(i)]) subplot(4,3,kp+3), plot(x_den(:,i)); set(gca,'xtick',[]); axis tight; title(['denoised signal ',num2str(i)]) kp = kp + 3; end
The results are better than those previously obtained. The first signal, which is irregular, is still correctly recovered, while the second signal, which is more regular, is denoised better after this second stage of PCA.