## 2 Bounds Determination of Time-Invariant and Time-Variant Uncertainties under Insufficient Samples
### Basics of Static and Dynamic Interval Models  
In order to better conduct uncertainty quantification research, the basic concept and main characteristics of static and dynamic interval models are firstly stated. (a) With regard to static uncertain parameters (such as geometric dimensions and material properties, etc. ), the interval vector consisting of uncertain parameters denoted by \(\mathbf{x}\) can be expressed as  
\[\mathbf{x}\in\mathbf{x^{I}}=\left[x_{1}^{I},\ x_{2}^{I},\ldots,\ x_{m}^{I}\right]^{T} \tag{1}\]  
and  
\[x_{i}\in x_{i}^{I}=\left[x_{i}^{L},\ x_{i}^{U}\right],\ i=1,2,\ldots,m \tag{2}\]  
where \(m\) is the total amount of static uncertain parameters and the subscript '\(I\)' stands for the interval set. The subscripts '\(L\)' and '\(U\)' stand for the lower and upper bounds of the interval, respectively. Namely  
\[\mathbf{x^{L}}=\left[x_{1}^{L},\ x_{2}^{L},\ldots,\ x_{m}^{L}\right]^{T}\text{ and }\mathbf{x^{U}}=\left[x_{1}^{U},\ x_{2}^{U},\ldots,\ x_{m}^{U}\right]^{T} \tag{3}\]  
As per the interval mathematics, Equation (1) can also be rewritten as  
\[\mathbf{x^{I}}=\left[\mathbf{x^{L}},\ \mathbf{x^{U}}\right]=\left[\mathbf{x^{c}}-\mathbf{x^{r}},\ \mathbf{x^{c}}+\mathbf{x^{r}}\right]=\mathbf{x^{c}}+\mathbf{x^{r}}\circ\mathbf{\
abla} \tag{4}\]  
where the Hadamard operator '\(\circ\)' stands for multiplying the corresponding elements in two vectors, and the vector \(\mathbf{\
abla}=\left[\Psi_{1},\ \Psi_{2},\ldots,\ \Psi_{m}\right]^{T}\) is an \(m\)-dimensional standard interval set with \(\Psi_{i}\in[-1,1]\). Hence, the center value vector \(\mathbf{x^{c}}\) and the radius vector \(\mathbf{x^{c}}\) can be written as  
\[\mathbf{x^{c}}=\left(x_{i}^{c}\right)=\frac{\left(\mathbf{x^{U}}+\mathbf{x^{L}}\right)}{2 }\text{ and }\mathbf{x^{r}}=\left(x_{i}^{r}\right)=\frac{\left(\mathbf{x^{U}}-\mathbf{x^{L}}\right)}{ 2},\ i=1,2,\ldots,m \tag{5}\]  
(b) Apart from the static uncertainties, the time-variant uncertainties (e.g., the change of environmental conditions and the deviation of dynamic excitations) should also be taken into account for practical applications. Under such circumstances, the interval process model is then applied to quantify the time-varying uncertainty items. Here, the definition of an interval process is expounded. Considering an uncertain process expressed as \(\left\{y(t)\in y^{I}(t),\ t\in[0,\ T]\right\}\), one implementation \(y(t_{j})\) for each time instant \(t_{j^{\prime}}\) (\(j=1,2,\ldots\)) belongs to \(y^{I}(t_{j})\), and for \(n\) selected times \(t_{1},t_{2},\ldots,t_{n}\), the joint distribution domain of \(n\) intervals \(y^{I}(t_{1}),\ y^{I}(t_{2}),\ldots,\ y^{I}(t_{n})\) are formed as a hyper-rectangular domain. Certainly, for an \(n\)-dimensional time-varying uncertainty problem, a common vector format is also obtained by  
\[\mathbf{y}(t)\in\mathbf{y^{I}}(t)=\left[y_{1}^{I}(t),\,y_{2}^{I}(t),\ldots,\,y_{n}^{I}(t )\right]^{T} \tag{6}\]  
As revealed by the definition, \(\mathbf{y^{L}}(t)\) and \(\mathbf{y^{U}}(t)\) can be taken as the vectors of lower and upper boundary histories of \(\mathbf{y}(t)\), and hence the mean process vector \(\mathbf{y}^{e}(t)\) and the radius process vector \(\mathbf{y}^{e}(t)\) are respectively given by  
\[\mathbf{y}^{e}(t)=\frac{\mathbf{y^{U}}(t)+\mathbf{y^{L}}(t)}{2}\text{ and }\mathbf{y}^{e}(t)= \frac{\mathbf{y^{U}}(t)-\mathbf{y^{L}}(t)}{2} \tag{7}\]  
For convenience, we can define the variance process vector \(\mathbf{D_{y}}(t)\) as  
\[\mathbf{D_{y}}(t)=\mathbf{y}^{e}(t)\circ\mathbf{y}^{e}(t) \tag{8}\]  
Obviously, once \(\mathbf{y}^{e}(t)\) and \(\mathbf{y}^{e}(t)\) are known, the properties of all the interval processes within a specific time can be well reflected. However, unlike the case of static uncertainty, Equations (7) and (8) cannot handle the correlation problem between two interval variables in terms of the process \(y_{k}(t)\) (\(k=1,2,\ldots,n\)) at different times \(t_{1}\) and \(t_{2}\).