*Principal Component Analysis in Financial Data Science DOI: http://dx.doi.org/10.5772/intechopen.102928*

between the original variables and the principal components correlate with the loadings. The variable loadings are contained in a loading matrix, which is created by multiplying the eigenvector matrix by a diagonal matrix containing the square root of each eigenvalue. The entries are determined by the component extraction method used. Non-standardized loadings show the covariance between mean-centered variables and standardized component values, regardless of whether the extraction is based on the singular value decomposition of the matrix or the eigenvalue decomposition of the covariance matrix.

The eigenvalue decomposition of the correlation matrix results in the standardized charges. The correlations between the original variables and the component scores are represented by these loadings. Because they always vary between 1 and 1 and are independent of the scale used, standardized charges are easy to read. In most cases, a threshold is set and only variables with loadings above this threshold are examined.

The total variance presents sum of variances of principal components. The ratio between the variance of principal component and the total variance is the fraction of variance explained by a principal component.

**Figure 1** shows total variance explained by using three methods of PCA. The steepest increase belongs to the PCA line, which cumulative explained variance is app. 87%. This line is almost parallel to the line from Sparse PCA which cumulative explained variance is 83%. However, when it comes to Robust PCA line it has been noticed that cumulative explained variance is only app. 26% and the increase of values is minimal.

PCA: The highest fraction of explained variance among these variables is 32%, and the lowest one is 5%. Cumulative explained variance is 86% (see **Table 2**).

**Figure 1.** *Total variance explained.*


#### **Table 2.**

*PCA total variance explained.*

Sparse PCA: The highest fraction of explained variance among these variables is 21%, and the lowest one is 5%. For instance, variables together explain 83% of the total variance (see **Table 3**).

Robust PCA: The highest fraction of explained variance among these variables is 21%, and the lowest one is 0%. For instance, variables together explain 25% of the total variance (see **Table 4**).

PCA is the best approach for this kind of data, regarding number of features.


#### **Table 3.**

*Sparse PCA total variance explained.*


#### **Table 4.**

*Robust PCA total variance explained.*
