5 Principal component analysis (PCA)

In the case of threshold filtering, described in Sect. 3, the kurtosis criterion is used to find the optimum threshold which must be set to separate the deterministic component from the stochastic one. In a general case, with several components, or with two deterministic components the kurtosis criterion may be difficult to use. In that case components may be separated by performing the principal component analysis of the ampligram matrix. The principal component analysis is a method to resolve a data matrix into a number of orthogonal components. If the data matrix is a sample of a multivariate time series, the principal components, in which the matrix is resolved, reflect the independent, orthogonal constituents of the process described by the multivariate time series.

Since the ampligram may be considered as a multivariate time series (L=20, N=1024 for the above low-20 ampligram) the principal component analysis may be used to identify the number of independent modes in the data. A multivariate time series consisting of L variables measured at N equally separated instants forms a matrix Y. The next step of the analysis is to perform the principal component analysis (PCA) of the matrix Y. The results of PCA are:

• The vector of eigenvalues of the matrix (latent roots $\lambda_{l}$), telling how much of the total variance in the matrix may be explained by the consecutive principal components;
• The matrix of component loadings, being correlation coefficients between old variables and the principal components (new variables);
• The matrix a of component score coefficients, a transformation matrix between the old system of M variables and the principal components (the new coordinate system);
• The matrix S of component scores, with one column for each principal component, being a projection of old M variables upon the new coordinate axis (directions of principal components).

The matrix S of component scores is thus the new multivariate time series in the principal component space. It has been found by the present author that, in all cases when filtering was performed in the principal component space, a considerable improvement of signal-to-noise ratio has been obtained without distorting the signal. Each column of S is low-pass filtered using a simple filter. The result of filtering is matrix $S_{\rm f}$. After filtering an inverse transform:

$$Y_{\rm f} = S_{\rm f} \cdot a^{-1}$$ (8)

is performed resulting in a new version of the matrix Y. It is possible to combine the filtering procedure with a decomposition procedure. If one wants to know what the variations of the M-component vector would be with only one mechanism (or cause), corresponding to the principal component l active, it is possible to mask with zeros all other columns in $S_{\rm f}$, except column l and to perform a calculation of a new matrix $Y_{l{\rm f}}$:

$$Y_{l{\rm f}} = S_{\rm f} \cdot a^{-1}.$$ (9)

The operation may be repeated for each interesting component l. As the principal component transformation preserves the variance, the sum of all latent roots, $\lambda_{l}$, is equal to the total variance. If the data is standardized, i.e. normalized to standard deviation for each variable, the sum of latent roots is equal to number of variables. The magnitude of latent roots is usually expressed in percent of the total variance. If the data contains only pure noise, all variables will be uncorrelated, and the total variance will be evenly distributed between all latent roots:

$$\lambda_l \;\mbox{noise} (\%) = \frac{100\%}{L}.$$ (10)

The real data, measured or computer simulated, are never perfectly uncorrelated and the variance will not be evenly distributed between all latent roots. When all variables are related to the same common factor there will be one latent root (the first one, corresponding to the first principal component) significantly larger than the value indicated by (10).