# Nanomaterials' Influence on the Performance of Thermal Insulating Mortars--A Statistical Analysis
## 3 Results and Discussion
### Cluster Analysis
#### 3.3.2 Discussion of the Cluster Analysis  
This research obtained equivalent results for different cluster methods. However, it should be noted that choosing only one method may lead to results that are not as optimised as desired. Figure 4: Graphics obtained for the cluster analysis. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Aggregation and Distance Methods** & **Cluster 1** & **Cluster 2** & **Cluster 3** \\ \hline \multirow{3}{*}{Complete linkage Euclidean} & _S1_, _S3_, _S4_, _I2_, _I3_, _I4_, _I5_, _I6_, _I7_, _I9_, _I10_, _I11_, _I12_, _I13_, _I14_, _I16_, _I17_, _I18_, _I19_, _I21_ & _S2_, _S5_, _S6_, _S7_, _S8_, _S9_, _I1_, _I7_, _I8_, _I15_, _I18_, _I19_, _I20_ & _L1, L2, L3, L4, L5_ \\ \hline \multirow{3}{*}{Ward Euclidean} & _S1_, _S3_, _S4_, _I2_, _I3_, _I4_, _I5_, _I6_, _I9_, _I10_, _I11_, _I12_, _I13_, _I14_, _I16_, _I17_, _I21_ & _S2_, _S5_, _S6_, _S7_, _S8_, _S9_, _I1_, _I7_, _I8_, _I15_, _I18_, _I19_, _I20_ & _L1, L2, L3, L4, L5_ \\ \hline \multirow{3}{*}{Ward City-block} & _S1_, _S3_, _S4_, _I2_, _I3_, _I4_, _I5_, _I6_, _I9_, _I10_, _I11_, _I12_, _I13_, _I14_, _I15_, _I20_ & _S2_, _S5_, _S6_, _S7_, _S8_, _S9_, _I1_, _I7_, _I8_, _I15_, _I18_, _I20_ & _L1, L2, L3, L4, L5_ \\ \hline \multirow{3}{*}{k-means} & _S1_, _S3_, _S4_, _I2_, _I3_, _I4_, _I5_, _I6_, _I9_, _I10_, _I11_, _I12_, _I13_, _I14_, _I16_, _I17_, _I19_ & _S2_, _S5_, _S6_, _S7_, _S8_, _S9_, _I1_, _I7_, _I8_, _I15_, _I18_, _I20_, _I21_, _I21_, _I22_, _I23_. \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Mixes composition of each cluster, for 3 clusters: hierarchical and non-hierarchical. From the various tested agglomeration and distance methods, a trend to obtain three different clusters was shown, following a division associated with the increase in the linkage distance, which is an indicator of a growth in the heterogeneity of groups, being a good indicator of separation. Moreover, when using the k-means method, the formulations' distribution remained stable. From the beginning, with the hierarchical methods, an interesting agglomeration in the dendrograms started to appear. These methods clearly show a quick agglomeration of formulations L1, L2, L3, and L4, this group being near to another one (containing I1, S2, and others), showing some degree of similarity. The other group (S1, I2, and others) show an evident heterogeneity from these two. All the groups show an early building of the clusters in all methods, and high stability in their agglomeration, independent of the agglomeration and distance methods applied, showing marked similarities within the same group. Further analysis found that L1 to L4 groups corresponded to the formulations more recently developed, including the nanomaterial silica aerogel, which allowed to obtain low \(\lambda\), high \(\Lambda\), and \(W\), but also low mechanical performance, thus presenting a characteristic behaviour. The group containing I1 and S2 comprised a mix of formulations. Although most of the members were related to industrial thermal mortars manipulated in the laboratory, with the inclusion of one or more thermal insulating aggregates, they also included some industrial formulations (e.g., 14, 18, and I15) despite only recently being introduced into the market. So, this cluster presented a mixture of behaviours and formulations, being a transitional group in terms of composition or performance. Figure 3 shows that the elements of this cluster, although presenting higher values of \(\lambda\), while lower than the average, also presented better mechanical performance. This cluster is related to the use of nanomaterials inserted as a second (or third) insulating aggregate in formulations already produced industrially, to further decrease their thermal conductivities. The third cluster is associated with industrial formulations. It is a group that has a more mature development, presenting both high mechanical performances and values of \(\lambda\).