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Multiscale Principal Component Analysis

`[X_SIM,QUAL,NPC,DEC_SIM,PCA_Params] = wmspca(X,LEVEL,WNAME,NPC)`

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

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

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

`[X_SIM,QUAL,NPC,DEC_SIM,PCA_Params] = wmspca(X,LEVEL,WNAME,NPC)`

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

returns a simplified version `X_SIM`

of the input
matrix `X`

obtained from the wavelet-based multiscale
principal component analysis (PCA).

The input matrix `X`

contains `P`

signals
of length `N`

stored columnwise (`N`

> `P`

).

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)`

.

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(d)`

is the number of retained noncentered principal components for details at level`d`

, for 1 <=`d`

<=`LEVEL`

.`NPC(LEVEL+1)`

is the number of retained non-centered principal components for approximations at level LEVEL.`NPC(LEVEL+2)`

is the number of retained principal components for final PCA after wavelet reconstruction.

`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).

Kaiser's rule keeps the components associated with eigenvalues greater the mean of all eigenvalues.

The heuristic rule keeps the components associated with eigenvalues greater than 0.05 times the sum of all eigenvalues.

If `NPC = 'nodet'`

, then the details are “killed”
and all the approximations are retained.

`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:

`PCA_Params(d).pc`

is a`P`

-by-`P`

matrix of principal components.The columns are stored in descending order of the variances.

`PCA_Params(d).variances`

is the principal component variances vector.`PCA_Params(d).npc = NPC`

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

Load the dataset consisting of 4 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 4 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 =
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

The multiscale principal components generalizes the usual PCA of a multivariate signal seen as a matrix by performing simultaneously a PCA on the matrices of details of different levels. In addition, a PCA is performed also on the coarser approximation coefficients matrix in the wavelet domain as well as on the final reconstructed matrix. By selecting conveniently the numbers of retained principal components, interesting simplified signals can be reconstructed.

Aminghafari, M.; Cheze, N.; Poggi, J-M. (2006), “Multivariate
de-noising using wavelets and principal component analysis,” *Computational
Statistics & Data Analysis*, 50, pp. 2381–2398.

Bakshi, B. (1998), “Multiscale PCA with application to
MSPC monitoring,” *AIChE J.*, 44, pp. 1596–1610.

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