## 3 Results
### Numerical Benchmark: Calibration of a SDoF Bouc-Wen Type Hysteretic System
Instead, the remaining ones were considered independent random variables with associated distributions given in Table 2 and constitute the vector of uncertain model parameters \(x_{M}\). It is worth specifying that the degradation effect of the BWBN model has been neglected on purpose, in order to introduce a discrepancy in the model used for emulating the reference response of the system. Besides, no uncertainties in input excitation have been considered herein, in order to address them only on the model side. The first moments of the prior probability density functions (PDFs) of model parameters \(k_{i}\) and \(\alpha\), namely \(\mu_{k_{i}}=7.42\) and \(\mu_{\alpha}=0.063\), were selected by two linear regressions on the reference restoring force curve for small and high displacements. The standard deviations of these parameters were chosen in order to take into account an amount of variation from these values of the 10%. It is worth noticing that in Table 2 a new random variable \(\gamma\)* was introduced in order to guarantee the bounded-input, bounded-output (BIBO) stability [40]. Indeed, assuming as prior knowledge on the parameter \(\gamma\) the required BIBO stability condition (i.e., \(\gamma\) being uniformly distributed between the bounds \(-\beta\leq\gamma\leq\beta\)), the conditional distribution \(\pi(\gamma|\beta)\) must be uniform. Figure 2: Montenegro (1979) ground motion record with peak ground acceleration (PGA) = 3.59 m/s\({}^{2}\). This can be achieved by transforming the input variables, namely removing the parameter \(\gamma\) and introducing an auxiliary variable \(\gamma*\sim U(-1,1)\). Hence, for a joint realization of the parameters \(\beta\) and \(\gamma*\), the actual parameter reads: \(\gamma\ =\ \gamma*\beta\) for \(\beta\
eq 0\). The standard deviations of the latter parameters were computed trivially once the supports of the uniform distributions were defined. To infer the unknown residual variances \(\sigma^{2}\) (Equation (9)), a uniform prior distribution was assumed:  
\[\tau\Big{(}\sigma^{2}\Big{)}\sim U\Big{(}0,max\big{|}i\big{|}^{2}\Big{)}, \tag{15}\]  
whose standard deviation was set equal to the maximum of the absolute value over the time of the experimental observation \(ii\). Markov Chain Monte Carlo (MCMC) simulations have been conducted using MATLAB-based Uncertainty Quantification framework UQLab-V1.3-113(R) [6] with 100 chains, 700 steps and AIES solver algorithm [28]. The number of the BWBN forward model \(\mathcal{M}^{BW}(x)\) calls in MCMC was 70'000. The system of ordinary differential equation (ODEs; Equations (10) and (12)) was solved with the MATLAB solver ode45 (explicit Runge-Kutta method with relative error tolerance \(1\times 10^{-3}\)). The Latin hypercube sampling (LHS) method was used to get the parameters prior distributions (Table 2) of the BWBN model, which are shown in Figure 4. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & **Distribution** & **Support** & **Mean** & **Std. Dev.** \\ \hline \(k_{i}(\mathrm{N/m})\) & Gaussian & \([4.27,\ 11.85]\) & 7.42 & 0.1 \\ \(\beta\left(\mathrm{m}^{-1}\right)\) & Uniform & \([55,\ 65]\) & 60 & 2.8 \\ \(\gamma*\) & Uniform & \([-1,\ 1]\) & 0 & 0.57 \\ \(\alpha\) & Lognormal & \([0,\ 1]\) & 0.063 & 0.01 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Uncertain parameters of the BWBN benchmark model. Figure 3: Simulated reference data: (**a**) response displacement of the Bouc–Wen–Baber–Noori (BWBN) oscillator; (**b**) response velocity of the BWBN oscillator; (**c**) hysteresis loop of the BWBN system; (**d**) total dissipated energy of the BWBN system (both elastic and hysteretic components of the restoring force are gathered), normalized with respect to the mass of the oscillator. The evolutions of the Markov chains are plotted instead in Figure 5 for each model parameter \(\mathbf{x}=\left\{k_{i},\beta,\gamma*,\alpha,\sigma^{2}\right\}\). These give valuable insights about convergence of the chains.