## 3 Principal Component Analysis-Sample Reduction Process  
Principle component analysis (PCA) aims to minimise the dimensionality of a dataset with many connected variables while keeping as much variance as feasible. The conversion of the new collection of uncorrelated variables known as principal components (PCs) preserves most of the variance included in the original variables. A new set of dimensions or orthogonal measurements are linearly independent and ranked according to the data variance. This means the more crucial principal axis occurs first (more important = more variance/more spread-out data). In general, understanding the PCA, variance, covariance, eigenvalues, and eigenvectors has an essential role in this concept [41].  
Figure 3 shows large positive covariance means that X and Y are completely related, i.e., as X increases, Y also increases. Negative covariance portrays the exact opposite relation. Zero covariance means that X and Y have no relation.  
The visualisation of data is an excellent approach to understand the patterns that lie in the multidimensional datasets. When information is placed in the horizontal and vertical axes (two-dimensional plane), it is straightforward to understand and discern the pattern that lies in it; however, the difficulty of conceiving it visually in multidimensional data with many features to consider and computing the data analysis becomes complex. Principal component analysis prerequisites require the discovery of patterns between the datasets so that the data are distributed across each dimension by first analysing the contributions of each feature in providing information to the overall dataset through eigenvector analysis. It then performs dimension reduction by keeping the feature columns with the highest eigenvalues.  
To demonstrate the efficacy of PCA, Figure 4, which projects a set of multidimensional data onto a two-dimensional space, is used as an example. Due to the high dimensional nature of the data points, it can be challenging to identify a linear correlation between the data points. The points are represented as column vectors before being aggregated into a matrix \(M\). PCA is then applied to \(M\) as shown in Equation (1):  
Figure 3: Covariance and their values.  
\[PCA(M)=W\times V \tag{1}\]  
where \(W\) and \(V\) are the corresponding eigenvalues and eigenvectors, respectively.  
\(V_{PC_{1}}\) is generated using covariant eigenvalues \(V\) with the corresponding top \(k\) eigenvalue in \(W\), which best represents the data points, and are then selected, before plotting them as shown in Figure 5 [30]. The principal axes (\(PC_{1}\) and \(PC_{2}\)) are denoted by the red lines passing through the points that provide a graphical representation of the covariance amongst the points.  
Lastly, the vital part of the proposed PCA-sample reduction process (\(PCA_{SRP}\)) is as follows:  
\[PCA_{SRP}(M)=|M\times V_{PC_{1}}| \tag{2}\]  
where \(PCA_{SRP}(M)\) is the loading score of each sample in dataset \(M\) and \(V_{PC_{1}}\) is the \(PC_{1}\) eigenvectors of the samples in matrix \(M\).  
\[S=\sum_{0}^{D}\begin{cases}1&\text{ if }PCA_{SRP}(M)>Sc\times PCA_{SRP}(M)_{highest }\\ 0&\text{ otherwise}\end{cases} \tag{3}\]  
\(S\) is the number of samples above the loading score of set-biased \(Sc\), and \(S\leq D\), where \(D\) is the number of samples in the dataset, including the random and noise samples.  
\[passrate=S/D \tag{4}\]  
The _passrate_ is the rate of samples cleaned using PCA-SRP in dataset \(M\) based on Equation (4). Any samples above said threshold are accepted, and those below it will be rejected for the new process dataset \(D_{PS}\) as per Equation (3).  
Figure 4: Projection of data points in two dimensions.  
Figure 5: Projection of data points in the two newly constructed dimensions.