## 3 Dynamic Response Analysis by Utilizing the Interval Quantitative Results
### Correlation Representation Achieved by Novel Optimization-Based Strategy  
As per the statements discussed in Section 3.1, the uncertainty bounds of the dynamic responses \(u_{i}(t),\ i=1,2,\ldots,r\) can be obtained, and indeed it can be regarded as interval processes. However, based on the definitions of the time-variant interval model cited in Section 2.1, the characteristics of the auto-correlation between \(u_{i}(t_{1})\) and \(u_{i}(t_{2})\) should be confirmed as well. In view of this, this study employs a novel optimization method based on the conception of the smallest parametric interval set to deal with this correlation issue. (a) Firstly, the common method for determining the smallest 'box' (i.e., one hyper-rectangle in essence) containing multi-dimensional sample data is investigated. Considering the case that the uncertain parameters \(\alpha_{i}\) (\(i=1,2,\ldots,s\)) constitute an \(s\)-dimensional parametric space with limited information containing \(S\) sample points, namely, \(\alpha_{i}^{(S)}=\{a_{i}(1),\ \alpha_{i}(2),\ldots,\ \alpha_{i}(S)\}\), the expression of the transformation matrix thus reads as  
\[\mathbf{T}(\mathbf{\theta})=(\delta_{1},\ \delta_{2},\cdots,\ \delta_{s}) \tag{35}\]  
where \(\mathbf{\theta}=(\theta_{i})\) (\(i=1,2,\ldots,s-1\)) is the vector of rotation angles relative to the original coordinate, and  
\[\delta_{j}=\left\{ \begin{aligned} & 0_{\frac{j-2}{\delta_{j}}}\\ &\delta_{j}\end{aligned} \right\} \tag{36}\]  
where \(0_{j-2}\) means a column vector with \(j-2\) zero items, and \(\tilde{\delta_{j}}\) is deduced by  
\[\tilde{\mathbf{\delta}_{j}}=\left( \begin{aligned} &-\text{sin}\theta_{j-1}\\ &\cos\theta_{j-1}\text{cos}\theta_{j}\\ &\vdots\\ &\cos\theta_{j-1}\text{sin}\theta_{j}\cdots\text{sin}\theta_{s-2} \text{cos}\theta_{s-1}\\ &\cos\theta_{j-1}\text{sin}\theta_{j}\cdots\text{sin}\theta_{s-2} \text{sin}\theta_{s-1}\end{aligned} \right) \tag{37}\]Each element of the original sample set \(\alpha_{i}^{(S)}\) will be transferred to \(\beta_{i}^{(S)}\) with the transformation matrix \(\mathbf{T}(\mathbf{\theta})\) (from \(\mathbf{\alpha}\)-space to \(\mathbf{\beta}\)-space). With the aim of determining the smallest interval set to envelope all the samples, the \(s\)-dimensional 'box' should be examined by  
\[|\mathbf{\beta}-\mathbf{\beta}^{c}|\leq\mathbf{\beta}^{r} \tag{38}\]  
where the center vector \(\mathbf{\beta}^{c}=\big{(}\beta_{1}^{c},\ \beta_{2}^{c},\cdots,\ \beta_{s}^{c} \big{)}^{T}\) and the semi-axis vector \(\mathbf{\beta}^{r}=\big{(}\beta_{1}^{r},\ \beta_{2}^{r},\cdots,\ \beta_{s}^{r} \big{)}^{T}\). The components \(\beta_{i}^{c}\) and \(\beta_{i}^{r}\) are given by  
\[\beta_{i}^{c}=\frac{1}{2}\bigg{(}\max_{S}\Big{(}\beta_{i}^{(S)}\Big{)}+\min_{ S}\Big{(}\beta_{i}^{(S)}\Big{)}\bigg{)}\ \text{and}\ \beta_{i}^{r}=\frac{1}{2}\bigg{(}\max_{S}\Big{(}\beta_{i}^{(S)}\Big{)}-\min_{ S}\Big{(}\beta_{i}^{(S)}\Big{)}\bigg{)} \tag{39}\]  
Then, the hyper-volume of the 'box' in Equation (38) is calculated by  
\[V_{hyper}=\prod_{i=1}^{s}\big{(}2\beta_{i}^{r}\big{)}^{s} \tag{40}\]  
which is a function of the rotation angles \(\theta_{i}\). The best interval set or the hyper-rectangle among all possible boxes should be the one that envelopes all the given sample points with the minimum volume, namely,  
\[V_{hyper}^{*}=\min_{\mathbf{\theta}^{*}}V_{hyper}(\theta_{1},\theta_{2},\ldots, \theta_{s-1}) \tag{41}\]  
To rapidly solve the optimization formulas as shown in Equation (41), either the gradient algorithm based on sensitivity information or the intelligent algorithm using population evolution has to be attached. After the minimum \(V_{hyper}^{*}\) is obtained by optimization, the optimal design parameters \(\theta_{i}^{*}\) can reflect the correlation between the uncertain parameters \(\alpha_{i}\) and \(\alpha_{i+1}\). (b) In fact, for the interval process \(u_{i}(t)\), the correlation properties between \(u_{i}(t_{1})\) and \(u_{i}(t_{2})\) with any instant times \(t_{1}\) and \(t_{2}\) make sense.