## 4 Interval Analysis Method for Consideration of Uncertainties  
The interval bounds can provide supports for structural health monitoring under uncertain conditions [36]. The general concept of interval mathematics is implemented by supplying the upper and lower boundaries of input parameters to generate the upper and lower boundaries of output parameters to consider the two uncertainties (epistemic and aleatory uncertainty). Artificial intelligence based surrogate models, such as the WWLS-SVM scheme, usually establish the relationship between the input and output via a black box procedure [37], therefore, the fundamental equation of interval analysis proposed by [PERSON] and [PERSON] [36] can be directly applied to the input parameters (extracted features) to produce the intervals of the output parameters (ESP values). It is noteworthy to indicate that, in black box-based methods, the algorithm's accuracy is checked by evaluating the output responses of the algorithm and comparing them with the training data. As shown in Equation (9), the stiffness reduction ratio (SRF) indicates the changes in the stiffness parameter for each member of the structure:  
\[SRF=1-\frac{\alpha_{d}}{\alpha_{u}}, \tag{9}\]  
where:  
\(\alpha_{d}\) = value of ESP in the damaged stat  
\(\alpha_{u}\) = value of ESP in undamaged (healthy) state. The intervals of the ESPs and extracted features for both states can be formulated as follows [9]:  
\[[\underline{a}]\approx\left[\underline{E_{p}}\right]=\text{ lower bound of ESP value}, \tag{10}\]  
\[[\overline{\alpha}]\approx\left[\overline{E_{p}}\right]=\text{upper bound of ESP value}, \tag{11}\]  
The upper and lower bounds of \(\alpha\) and \(E_{p}\) are denoted by the upper and lower bars, respectively. Therefore, the interval bounds for each parameter can be obtained as:  
\[E_{p}=\left[\underline{E_{p}},\overline{E_{p}}\right]=\left\{E_{p}^{1},E_{p} ^{2},\ldots,E_{p}^{s}\right\},E_{p}^{s}=\left[\underline{E_{p}^{s}},\overline {E_{p}^{s}}\right],. \tag{12}\]  
\[\alpha_{c}^{I}=\left[\underline{\alpha_{c}^{I}},\overline{\alpha_{c}^{I}} \right]=\left\{\alpha_{c1}^{I},\alpha_{c2}^{I},\ldots,\alpha_{ck}^{I}\right\} ^{T},\alpha_{ck}^{I}=\left[\underline{\alpha_{ck}^{I}},\overline{\alpha_{ck}^ {I}}\right], \tag{13}\]  
where:  
\(c\) = number of damage cases. \(k\) = number of segments of structures. \(I\) = interval number or vector representation  
\(s\) = number of sensor on structure  
\(p\) = is the acquiring number  
And the input and output middle values are indicated as:  
\[x^{c}=m(x)=\frac{(\underline{x}+\overline{x})}{2}, \tag{14}\]  
In the above equation, the variable \(x\) represents the exact values of the extracted features (input values) and the output of the algorithm (ESPs). The upper and lower bars show the upper and lower bounds of x, respectively. Thus, the training and testing functions of the WWLS-SVM are established based on Equations (10)-(14). The uncertainties are coupled with extracted features in terms of the interval bounds. These features are used as input parameters of the surrogate model, while the ESPs (\(\alpha\)) are used as outputs. Finally, two WWLS-SVM models, which include the lower bound and upper bound analyses, are provided, as shown in Table 1.  
The variable \(\omega\) indicates the uncertainty level in the acceleration signal. The interval variables of training and testing data are presented herein using superscripts \(lr\) and \(le\). The lower and the upper bounds of the input parameters are applied through the \(+\) and \(-\) values of the uncertainties in two surrogate models of WWLS-SVM 1 and WWLS-SVM 2. \(\underline{\alpha_{ck}}\) and \(\overline{\alpha_{ck}}\) are the outputs of the WWLS-SVM models and show the boundaries of the predicted ESPs of damage case c. Once the lower and upper boundaries of the ESPs are obtained, the possibility of damage existence (PoDE) are subsequently calculated and then the damage severity is measured using damage measure index (DMI). Definition of these indexes will be explained in the next section.