## 3 Development of ANN Model
### Principal Component Analysis  
Raw data \(\mathbf{X^{b}}\in\mathbb{R}^{m\times n}\) used for training the learning algorithm in this article consisted of \(m=151\) experimental sequences, where each experimental sequence comprises \(n=10\) variables. Because each variable varies within different ranges for the datasets used in this article, they are typically normalized before being supplied for training. However, because not all variables are mutually independent, none of them are necessary for training nor they are in the best form (uncorrelated) for training the algorithm. PCA is a mathematical tool used to reduce the number of correlated variables in a dataset to uncorrelated variables called the principal components [44]. Finding the principal components of the given dataset \(\mathbf{X^{b}}\) starts with column-wise normalization [45]:  
\[X^{a}_{i}=\frac{X^{b}_{i}-\overline{X}^{b}_{i}1}{\sigma\big{(}X^{b}_{i}\big{)}} \tag{9}\]  
were \(X^{b}_{i}\) is the \(i\)-th column vectors of \(X^{b}\), the over-bar indicates the mean of subtended term (therefore scalar), \(1\) is length-\(m\) column vector where all entries are unity, and \(\sigma(X^{b}_{i})\) represents the standard deviation of \(X^{b}_{i}\). Performing this operation for all column vectors of \(X^{b}\) yields a column-wise normalized matrix \(X^{a}\). The correlation matrix \(R^{b}\), which is the table of correlation coefficients between all variables (columns) in the raw dataset, \(X^{b}\), is equivalent to the covariance matrix of the column-wise normalized matrix \(X^{a}\). The correlation coefficient \(r^{a}_{ij}\) between column vector \(i\) and \(j\) of \(X^{b}\), namely \(X^{b}_{i}\) and \(X^{b}_{j}\), is calculated as:  
\[r^{a}_{ij}=\frac{1}{m-1}\frac{\Big{(}X^{b}_{i}-\overline{X}^{b}_{i}1\Big{)}^{ T}\Big{(}X^{b}_{i}-\overline{X}^{b}_{i}1\Big{)}}{\sigma\big{(}X^{b}_{i}\big{)} \sigma\Big{(}X^{b}_{j}\Big{)}} \tag{10}\]  
\[r^{a}_{ij}=\frac{1}{m-1}(X^{a}_{i})^{T}X^{a}_{j} \tag{11}\]  
Then, the correlation matrix \(R^{b}\) of \(X^{a}\) is simply the collection of \(r^{a}_{ij}\)'s as:  
\[R^{a}=\left[ \begin{array}{ccc}r^{a}_{11}&\cdots&r^{a}_{1n}\\ \vdots&\ddots&\vdots\\ r^{a}_{n1}&\cdots&r^{a}_{nn}\end{array} \right] \tag{12}\]  
To perform PCA, eigenvectors and eigenvalues of \(R^{b}\) are required. This is achieved by the eigenvalue decomposition as:  
\[R^{a}=V^{a}\Lambda^{a}V^{aT} \tag{13}\]  
where \(V^{a}\) is a matrix with the eigenvectors of \(R^{b}\) as its column vectors, and \(\Lambda^{a}\) is a diagonal matrix with the corresponding eigenvalues as its diagonal entries. Each eigenvector post-multiplied by the normalized data set \(X^{a}\) yields a principal component, and the significance of the computed principal component is represented by the corresponding eigenvalue,because it represents the variance of the designated principal component. The results are tabulated in Table 1.  
\[X=X^{a}V^{a} \tag{14}\]  
Variables (columns) of the preprocessed dataset \(X\) are now linearly independent; that is, \(X\) is de-correlated. To determine if this is true, one may show that the correlation matrix of \(X^{T}X\), namely \(R\), is a diagonal matrix. Using Equations (10), (13), and (14) sequentially yields:  
\[ \begin{array}{c}R=\frac{1}{m-1}X^{T}X=\frac{1}{m-1}(X^{a}V^{a})^{T}X^{a}V^{a} =\frac{1}{m-1}(V^{a})^{T}(X^{a})^{T}X^{a}V^{a}=(V^{a})^{T}R^{a}V^{a}\\ \hskip 14.226378 pt=(V^{a})^{T}V^{a}\Lambda^{a}{(V^{a})}^{T}V^{a}=I\Lambda^{a}I= \Lambda^{a}\end{array}  \tag{15}\]  
therefore, \(X\) is de-correlated. When the column vectors of \(V^{a}\) are sorted in descending order regarding the corresponding eigenvalues before being multiplied, the resulting principal components in \(X\) are significant in the order they appear. Figure 1 shows the significance of each principal component for the dataset used in this article. The percentages are the normalized eigenvalues and their sum equals 100. As indicated, including the first six principal components is sufficient to represent 96% of the variance in the original data set \(X^{b}\).