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## Multiscale Principal Components Analysis

The purpose of this example is to show the features of multiscale principal components analysis (PCA) provided in the Wavelet Toolbox™.

The aim of multiscale PCA is to reconstruct a simplified multivariate signal, starting from a multivariate signal and using a simple representation at each resolution level. Multiscale principal components analysis generalizes the PCA of a multivariate signal represented as a matrix by simultaneously performing a PCA on the matrices of details of different levels. A PCA is also performed on the coarser approximation coefficients matrix in the wavelet domain as well as on the final reconstructed matrix. By selecting the numbers of retained principal components, interesting simplified signals can be reconstructed. This example uses a number of noisy test signals and performs the following steps.

`Load the multivariate signal by typing the following at the MATLAB(R) prompt:`

``` load ex4mwden whos```
``` Name Size Bytes Class Attributes covar 4x4 128 double x 1024x4 32768 double x_orig 1024x4 32768 double ```

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 signal is "Blocks" which is irregular, and the second one is "HeavySine" which is regular, except around time 750. The other two signals are the sum and the difference of the two original signals, respectively. Multivariate Gaussian white noise exhibiting strong spatial correlation is added to the resulting four signals, which produces the observed data stored in `x`.

### Perform Simple Multiscale PCA

Multiscale PCA combines noncentered PCA on approximations and details in the wavelet domain and a final PCA. At each level, the most significant principal components are selected.

First, set the wavelet parameters:

```level = 5; wname = 'sym4';```

Then, automatically select the number of retained principal components using Kaiser's rule, which retains components associated with eigenvalues exceeding the mean of all eigenvalues, by typing:

`npc = 'kais';`

Finally, perform multiscale PCA:

`[x_sim, qual, npc] = wmspca(x ,level, wname, npc);`

### Display the Original and Simplified Signals

To display the original and simplified signals type:

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

We can see that the results from a compression perspective are good. The percentages reflecting the quality of column reconstructions given by the relative mean square errors are close to 100%.

`qual`
```qual = 1×4 98.0545 93.2807 97.1172 98.8603 ```

### Improve the First Result by Retaining Fewer Principal Components

We can improve the results by suppressing noise, because the details at levels 1 to 3 are composed essentially of noise with small contributions from the signal. Removing the noise leads to a crude, but efficient, denoising effect.

The output argument `npc` is the number of retained principal components selected by Kaiser's rule:

`npc`
```npc = 1×7 1 1 1 1 1 2 2 ```

For d from 1 to 5, `npc`(d) is the number of retained noncentered principal components (PCs) for details at level d. `npc`(6) is the number of retained non-centered PCs for approximations at level 5, and `npc` (7) is the number of retained PCs for final PCA after wavelet reconstruction. As expected, the rule keeps two principal components, both for the PCA approximations and the final PCA, but one principal component is kept for details at each level.

To suppress the details at levels 1 to 3, update the `npc` argument as follows:

```npc(1:3) = zeros(1,3); npc```
```npc = 1×7 0 0 0 1 1 2 2 ```

Then, perform multiscale PCA again by typing:

`[x_sim, qual, npc] = wmspca(x, level, wname, npc);`

### Display the Original and Final Simplified Signals

To display the original and final simplified signals type:

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

As we can see above, the results are improved.

### More about Multiscale Principal Components Analysis

More about multiscale PCA, including some theory, simulations and real examples, can be found in the following reference:

Aminghafari, M.; Cheze, N.; Poggi, J-M. (2006), "Multivariate denoising using wavelets and principal component analysis," Computational Statistics & Data Analysis, 50, pp. 2381-2398.

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