## 2 Materials and Methods
### Analytical Part
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \(\boldsymbol{f}_{w}\) (MPa) & \(\boldsymbol{u}_{w}\) (mm) & \(\boldsymbol{f}_{wmax}\) (MPa) & \(\boldsymbol{u}_{wmax}\) (mm) & \(\boldsymbol{E}_{w}\) (MPa) & \(\boldsymbol{G}_{w}\) (MPa) \\ \hline \multicolumn{8}{c}{M3a-normal conditions} \\ Average & 4.12 & 0.18 & 11.89 & 0.75 & 3723 & 1432 \\ Median & 3.91 & 0.18 & 12.62 & 0.73 & 3677 & 1335 \\ Stdev & 0.68 & 0.02 & 1.56 & 0.06 & 534 & 261 \\ CoV & 16\% & 11\% & 13\% & 8\% & 14\% & 18\% \\ \hline \multicolumn{8}{c}{M3a-50 F/T cycles} \\ Average & 3.74 & 0.16 & 11.08 & 0.92 & 3852 & 1380 \\ Median & 3.91 & 0.15 & 10.98 & 0.84 & 3745 & 1471 \\ Stdev & 0.43 & 0.02 & 0.67 & 0.18 & 323 & 466 \\ CoV & 12\% & 9\% & 6\% & 20\% & 8\% & 34\% \\ \hline \multicolumn{8}{c}{M3a-150 F/T cycles} \\ Average & 4.86 & 0.31 & 10.74 & 0.96 & 2817 & 1115 \\ Median & 4.80 & 0.28 & 10.34 & 0.82 & 2795 & 1151 \\ Stdev & 1.47 & 0.11 & 0.49 & 0.22 & 112 & 211 \\ CoV & 30\% & 34\% & 5\% & 22\% & 4\% & 19\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Results of the compressive tests of the wallets M3a [9]. where \(\mathbf{Q}_{n}\) is complementary to vector \(\mathbf{R}_{n}\) and therefore their concatenation yields the complete data model, vector \(\mathbf{X}_{n}\). The prediction vector, too, is composed of two truncated vectors, i.e., the given truncated vector and the unknown complementary vector:  
\[\mathbf{Q}=\{q_{1},\ \ldots,\ q_{l},\ \ldots,\ q_{D}\}\overset{\mathsf{A}}{ \mathbf{R}}=\{\hat{r}_{1},\ \ldots,\ \hat{r}_{k},\ \ldots,\hat{r}_{M}\} \tag{3b}\]  
The problem now is how an unknown complementary vector \(\overset{\mathsf{A}}{\mathbf{R}}\) can be estimated from a given truncated vector \(\mathbf{Q}\) and the model vectors \(\{\mathbf{X}_{1},\ \ldots,\ \ \mathbf{X}_{n},\ \ldots,\ \mathbf{X}_{N}\}\), i.e., how, e.g., the ductility of wallet, \(\mu\), can be estimated from known input parameters (type of mortar, number of F/T cycles, Young's modulus etc.) and the available data collected in the database. In the case of a large amount of data (big databases), different methods can be applied. One of them is demonstrated in [22]. However, in the case of a small database, which is available for the F/T phenomenon discussed in this paper, a different approach should be applied. In order to obtain the typical trends with F/T cycles, for optimal estimation of \(\overset{\mathsf{A}}{\mathbf{R}}\) in terms of one or two mechanical properties, the standard statistical polynomial fit is used in this (first) step. Two output parameters (\(M=2\)), Young's modulus, \(E\), and ductility, \(\mu\), are separately estimated as a function of two input parameters, F/T cycles (\(D=2\)) any type of mortar _type_\(M\). In this particular case, Equation (3a,b) can be written as:  
\[\mathbf{Q}_{n}=\{n_{n},\ typeM_{n}\}\mathbf{R}_{n}=\{E_{n},\ \ \mu_{n}\} \tag{4a}\] \[\mathbf{Q}=\{n,typeM\}\overset{\mathsf{A}}{\mathbf{R}}=\{\hat{E},\ \hat{\mu}\} \tag{4b}\]  
In the presented research, the estimates (i.e., functional relationship) of the variables \(\hat{E}\) and \(\hat{\mu}\) are determined by using standard nonlinear regression (see Section 3). However, to obtain as much information as possible from the observed phenomenon, a combination of Latin hypercube sampling (LHS) and fuzzy logic approach is used next. While Latin hypercube sampling [23; 24] is applied to obtain a reliable estimate of the statistical parameters for the probabilistic model, the fuzzy logic approach [25] is used to exploit the expert knowledge about the observed phenomenon (i.e., the influence of two mortars on the investigated mechanical properties). The latter approach can deal with the concept of partial truth, which enables the incorporation of soft logic and/or engineering knowledge into the mathematical models.