# Operationalisation of Building Inspections and Repair: Systematisation-Based Approach
## 2 Materials and Methods
### Analysing the New Inter-Defects Three-Dimensional Correlation Matrices  
The information provided by the new inter-defects three-dimensional matrices was further analysed considering the prescription of diagnosis methods or repair techniques for different types of building elements/materials in roofs, facades, and floorings. For that purpose, the most relevant correlations in each layer of the new inter-defects three-dimensional correlation matrices had to be determined. With that goal in mind, hypothesis testing was applied [37]. The null hypothesis \(H_{0}\) was that by choosing threshold \(a\), up to 35% of the most relevant correlations (\(b\)) would be obtained. The test may be summarised as follows:  
\(H_{0}\): \(b\geq 35\%\);  
\(H_{1}\): \(b<35\%\);  
\(Statistic\): \(\overline{X}\);  
\(Rule\) form: accept \(H_{0}\) if \(\overline{X}\geq c\);  
\(Significance level\): \(\alpha=10\%\);  
\(Result\) of threshold \(a\): \(a=\overline{x}+\gamma\), where \(\overline{x}\) is the average of correlations higher than 0 in each layer of the matrix, and \(\gamma\) is the respective standard deviation. By applying the described methodology in each layer of the new three-dimensional inter-defects correlation matrices (selecting the most relevant correlations), an auxiliary matrix within each layer is built, indicating whether the correlation between the defects is higher than or equal to \(a\). If it is higher, the cell in the intersection between the given and the second defect in the auxiliary matrix is filled with 1. Then, the relevant correlations are compared according to the type of building component--roof claddings, facade elements/claddings, and floorings. Such a comparison is made by stacking the auxiliary matrices per layer category. That is, the layers of external claddings of pitched roofs and flat roofs are piled up (roof claddings); the layers of wall renders, ETICS, painted facades, architectural concrete surfaces, door and window frames, adhesive ceramic tiling, and natural stone claddings are piled up (facade elements/claddings); and the layers of wood floorings, epoxy resin industrial floor coatings, vinyl and linoleum floorings, adhesive ceramic tiling, and natural stone claddings are piled up (floorings). According to these categories, the comparable cells of each layer (representing a relevant correlation between the same types of defects) are summed up. Out of the obtained sums, the highest are highlighted. That means that, for roof claddings (Table 8), only sums equal to 2 were considered. For facade elements/claddings (Table 9), only sums higher than or equal to 3 were considered. For floorings (Table 10), only sums higher than or equal to 2 were considered. This applies to both correlations based on diagnosis methods and repair techniques. The described process allows us to select the most relevant correlations per category--roof claddings, facade elements/claddings, and floorings. But it is also necessary to understand the diagnosis methods or repair techniques (according to the base matrix used) that led to such relevance. For that purpose, a survey of the diagnosis methods/repair techniques \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & & & **Defects** & & \\ \hline
**Defects** & **A-A1** & **A-A2** & **A-A3** & **A-A4** & **A-A5** & **A-A6** \\ \hline
**A-A1** & - & 17\% & 37\% & 30\% & 0\% & 0\% \\
**A-A2** & 50\% & - & 10\% & 0\% & 0\% & 0\% \\
**A-A3** & 61\% & 6\% & - & 28\% & 0\% & 0\% \\
**A-A4** & 56\% & 0\% & 31\% & - & 0\% & 0\% \\
**A-A5** & 0\% & 0\% & 0\% & 0\% & - & 0\% \\
**A-A6** & 0\% & 0\% & 0\% & 0\% & 0\% & - \\ \hline \hline \end{tabular} Note: refer to Table 3 for the designation of each defect.  
\end{table} 
Table 7: Excerpt of the door and window frames’ layer of the new inter-defects correlation matrix based on the defects–repair techniques correlation matrix from the global inspection system developed by [PERSON] [12]. associated with the relevant correlations was performed.