## 3 ERSA (Engineering Design Research Meta-Model Based Systematic Analysis) Process
### Phase 5-Discovering Similarities of the Methodologies  
To extract similarities and striking elements in the methodologies in order to identify and interpret meaningful patterns, trends, or insights, computation is required. Clustering algorithms, a subset of unsupervised machine learning techniques, group data points based on similarities. They provide a systematic way to identify commonalities and differences among methodologies. By organizing methodologies into clusters, researchers gain insights into the underlying structures of employed methods. Instead of analyzing an unstructured set of methodologies, coherent clusters sharing similar models can be identified. This simplifies trend identification, best practice determination, and the recognition of emerging paradigms. Clustering results can prioritize relevant methodologies for further investigation and can reveal gaps in existing research, aiding knowledge synthesis and new research directions. Careful consideration of data representation, algorithm choice, and result interpretation ensures the validity and reliability of clustering outcomes. To apply clustering algorithms, ERSA transform the matrices presented in Section 5 into vectors. Let us denote the transformation function as \(\tau\), where \(\tau\) represents the transformation from the matrix \(A\) to column vector X. Mathematically, we can define this function as:  
\[\tau:\ \mathbb{Z}^{4\times 4}\rightarrow\mathbb{Z}^{16} \tag{9}\]  
where \(\mathbb{Z}^{4\times 4}\) denotes the set of all \(4\times 4\) matrices with integer entries, and \(\mathbb{Z}^{16}\) represents the set of all 16-dimensional row vectors with integer values. The transformation function can be defined as:  
\[\tau(A)=X \tag{10}\]  
where \(A\) is a \(4\times 4\) matrix with integer entries representing the input from phase 5, and \(X\) is a \(1\times 16\) row vector with integer entries representing the output. The function \(\tau\) maps each integer element of the input matrix \(A\) to its corresponding position in the output column vector \(\tau\) while preserving the order of elements. Thus, the function \(\tau\) transforms a \(4\times 4\) matrix with integer entries into a 16-dimensional row vector with integer values. For instance, the transformation of the matrix (6) is represented below (11):  
\[\tau\left( \begin{bmatrix}1&1&1&0\\ 0&0&1&0\\ 0&0&1&0\\ 0&0&1&0\\ \end{bmatrix} \right)= \begin{bmatrix}1&1&1&0&0&0&1&0&0&0&1&0&0&0&1&0\\ 0&0&1&0&0&0&1&0&0&0&1&0\\ \end{bmatrix}  \tag{11}\]  
Thus, each article in the complete collection of articles \(\Sigma\) is represented by a vector \(X\). All these vectors are then assembled into a matrix \(Y\), where each row represents an article with employed methodology and each column represents 16 models. The assembly function \(f\) operates on \(n\) row vectors (\(n=card(\Sigma)\)), each with 16 integer elements, producing a resulting matrix \(Y\) with \(n\) rows and 16 columns. The assembly function \(f\) is defined as:  
\[f:\ \left(\mathbb{Z}^{4}\right)^{n}\rightarrow\mathbb{Z}^{n\times 16} \tag{12}\]  
\[f(X_{1},X_{2},\dots,X_{n})=Y \tag{13}\]  
where \(f\) is the function that assembles row vectors into matrix; \(X_{i}\) represents the \(i\)-th row vector with 16 integer elements; and \(Y\) is the resulting matrix, also with \(n\) rows and 16 columns. The elements of \(Y\) are obtained by assembling the row vectors \(X_{1}\), \(X_{2}\), \(\dots\), \(X_{n}\) as rows of the matrix \(Y\) with \(y_{ij}=x_{ij}\), where \(y_{ij}\) represents the element in the \(i\)-th row and \(j\)-th column of \(Y\), and \(x_{ij}\) represents the \(j\)-th element of the \(i\)-th row vector \(X_{i}\). The matrix \(Y= \begin{bmatrix}y_{ij}\end{bmatrix} \) (14) with dimension \(n\times 16\) illustrates the assembly of the vector \(X\) in the Equation (7). \[Y=