# Prescriptive Maintenance: A Systematic Literature Review and Exploratory Meta-Synthesis
## 2 Methodology
### Content Data Analysis  
Factor Analysis (FA) is performed to uncover research clusters in the PsM domain. Although different FA methods exist, most are designed for continuous-type data (e.g., Principal Component Analysis). Given that content data is categorical, Multiple Correspondence Analysis (MCA) is used as the most suitable alternative for handling categorical data [44; 45]. The analysis is conducted in RStudio (v2024.04.2) of R (v.4.3.3) using _FactoMineR_ and _Factoshiny_ packages [46]. The first step includes forming an indicator matrix, such that rows correspond to studies \(n=61\) and Q = 32 (16 variables with outcomes [1; 0], i.e., [present, not present]). The indicator matrix \(Z\) of size \(n\times Q\) is constructed as:  
\[z_{iq}=\left\{\begin{array}{l}1\ study\ i\ has\ a\ category\ q\\ 0\ study\ i\ does\ not\ have\ q\end{array}\right.. \tag{1}\]  
After forming the \(Z\) matrix, the Correspondence matrix \(P\) is created to normalise \(Z\) by \(N=\sum_{id}z_{id}=16\cdot 61=976\), obtaining row and column mass. For row mass (studies), we calculate \(r_{i}=1/n\) (studies weighted equally), and for column mass (variables), we calculate \(c_{q}=\) category count \(q/N\). Before performing SVD (Singular Value Decomposition), a standardised residual matrix \(S\) needs to be obtained by computing deviations from independence:  
\[s_{iq}=\frac{p_{iq}-r_{i}c_{q}}{\sqrt{r_{i}c_{q}}}\, \tag{2}\]  
where \(p_{iq}\) is the element of \(P\). Performing SVD on \(S\) is as follows:  
\[S=U\Sigma V^{T}, \tag{3}\]where \(\Sigma\) contains singular values \(\sigma_{k}\) and \(U\) and \(V\) are left and right singular vectors, respectively. Lastly, estimating the coordination for studies from Principal Components (PCs) \(m\) is performed as follows:  
\[F=D_{r}^{-1/2}\text{U}\Sigma, \tag{4}\]  
where \(D_{r}\) is the diagonal matrix of row (studies) masses. Coordinates for categories is performed as follows:  
\[G=D_{c}^{-1/2}\text{V}\Sigma, \tag{5}\]  
where \(D_{c}\) is the diagonal matrix of column (categories) masses. Calculating total inertia \(T\) is as follows:  
\[T=\sum\sigma_{k}^{2}, \tag{6}\]  
while the first selected \(m\) (PCs) explains inertia (variance):  
\[m=\frac{\sum_{k=1}^{m}\sigma_{k}^{2}}{T}. \tag{7}\]  
Estimating total inertia is crucial to explain the dataset's variation (heterogeneity). It can be viewed as analogous to the PCA variance but adjusted for categorical data. Additionally, it is essential to consider this when analysing the quality representation of categories per inspected PC in discussing the factor map. As a final step, we utilise the first two m (PCs) components, as they capture the most variance in terms of performing cluster analysis and grouping similar studies. To do so, we use the Euclidean distance between \(i\) and \(j\) studies by computing the following:  
\[d_{ij}=\sqrt{\sum_{k=1}^{2}\left(f_{ik}-f_{jk}\right)^{2}}\, \tag{8}\]  
where \(f_{ik}\) is the coordinate of study \(i\) on PC \(k\). Lastly, to perform HCPCs (Hierarchical Clustering on Principal Components), [PERSON]'s linkage method is used to minimise within-cluster variance:  
\[\Delta(A,B)=\frac{n_{A}n_{B}}{n_{A}+n_{B}}\|\mu_{A}-\mu_{B}\|^{2}\, \tag{9}\]  
where \(\mu_{A}\), \(\mu_{B}\) are centroids of clusters A and B and \(n_{A}\), \(n_{B}\) are their sizes. To ensure transparency and replicability of the findings, we recommend that the reader perform the analysis using the _FactoMineR_ (v.2.12) R package to obtain the same results.  
The context of implementation (Application Scope), the functional objectives (Study Focus), the broader value-driven and systemic implications (Systemic and Sustainability Aspects), and the decision-making levels (Hierarchical Levels) are used to map a complex ecosystem in which PsM operates. To do so, we rely on the HCPC-MCA method. This enabled us to offer valuable insights about the evolving field of PsM research.