## 4 Feature Extraction Techniques
### Principal Component Analysis (PCA)
#### 4.1.3 Eigenvector and Eigenvalue Computation  
The covariance matrix \(\sum\) is used to compute the eigenvectors \(\mathcal{U}\) and eigenvalues \(\mathcal{V}\) to derive the principal components of the continuous signal \(S_{i}\). The principal components are computed by finding the solution to Equation (5).  
\[\sum\mathcal{U}=\mathcal{V}\mathcal{U}. \tag{5}\]  
where \(\mathcal{V}\) are scalar values and \(\mathcal{U}\) are non-zero vectors called principal components. The direction of the PCA space is depicted by \(\mathcal{U}\) while the corresponding \(\mathcal{V}\) shows the length and scaling factor of \(\mathcal{U}\)[12; 27]. The principal components are arranged according to the variance of \(\mathcal{V}\) from the highest to the lowest. Thus, the \(\mathcal{U}\) with the highest \(\mathcal{V}\) is the first principal component because it has the highest variance and carries the maximum possible information. This arrangement enables a reduction in the dimension of the principal components with no or minimal information lost. It should be noted that these principal components can be reconstructed linearly as a combination of the original signal. Also, the number of derived principal components is a function of the dimension of the analysed signal. That is, the \(d\)-dimension matrix of Equation (2) will produce \(d\) principal components after complete decomposition.  
To solve for \(\mathcal{U}\) and \(\mathcal{V}\), Equation (5) is decomposed using the singular vector decomposition (SVD) technique into three matrices given as:  
\[\tilde{G}=XYZ^{\dagger}, \tag{6}\]  
where \(Y\) is the \(a\times a\) diagonal matrix, \(X\) is the \(a\times d\) matrix called the left singular vector, and \(Z\) is the \(d\times d\) matrix referred to as the right singular vector. The elements on the diagonal of \(Y\) arranged from highest to lowest are the \(\mathcal{V}\) while the principal components (\(\mathcal{U}\)) are the columns of the right singular vector, \(Z\). The principal components after complete SVD are usually of the form:  
\[Z=\begin{pmatrix}z_{1,1}&z_{1,2}&\dots&z_{1,d}\\ z_{2,1}&z_{2,2}&\dots&z_{2,d}\\ \vdots&\vdots&\ddots&\vdots\\ z_{d,1}&z_{d,2}&\dots&z_{d,d}\end{pmatrix}. \tag{7}\]  
Therefore, to derive a suitable feature vector \(f\) of dimensions \(1\times d\) that can be combined with HMM, Equation (7) is transformed as:  
\[f_{d}=\left[\left(\frac{1}{d}\sum_{1}^{d}|z_{r}|\right)_{1},\left(\frac{1}{d }\sum_{1}^{d}|z_{r}|\right)_{2},\dots,\left(\frac{1}{d}\sum_{1}^{d}|z_{r}| \right)_{d}\right]=[f_{1},f_{2},\dots,f_{d}]. \tag{8}\]  
Hence, given \(\alpha\) credit card transactions used for training HMM, the feature vector can be represented in the form of a matrix as:  
\[f_{a,d}=\begin{pmatrix}f_{1,1}&f_{1,2}&\dots&f_{1,d}\\ f_{2,1}&f_{2,2}&\dots&f_{2,d}\\ \vdots&\vdots&\ddots&\vdots\\ f_{\alpha,1}&f_{\alpha,2}&\dots&f_{\alpha,d}\end{pmatrix}. \tag{9}\]  
It should be noted that because of privacy concerns, each transaction in the dataset has been converted to principal components with dimensions \(1\times d\), where \(d=28\). Nonetheless, if given the original dataset, the principal components which serve as the feature vector can be derived using these three sequential steps. This proposed PCA feature extraction technique is summarised in Algorithm 1.