## 3 Uncertainty Quantification Model
### Probability Density Function Approximation  
The probability density function (PDF) is a mathematical function used to depict the likelihood of a random variable assuming a specific value. It is derived from the cumulative distribution function (CDF) and is obtained through a technique called \"Particle derivatives\", which involves employing Dirac's delta approximation [35]. This approach involves reviewing the derivation using Dirac's delta approximation to identify the PDF. The Dirac delta function \(\delta(y_{0})\) can be naturally approximated by a sequence of probabilities that converge to a distribution concentrated at \(y_{0}\).  
Let \(F(y)\) be a function defined by the following:  
\[F(y)=\delta(x-y)F(x)dx \tag{6}\]  
We have  
\[\frac{d}{dy}F(y)=F^{\prime}(y)=\frac{\int\delta^{\prime}(x-y)(F(x)-F(y))dx}{\int (x-y)\delta^{\prime}(x-y)dx} \tag{7}\]  
We may explore the possibility of employing the approximation technique, which is further elaborated in [35]:  
\[\delta(x-y)\approx f(x,y,n) \tag{8}\]  
so that  
\[F^{\prime}(y)\approx\frac{\sum_{j=1}^{kp}N_{J}\big{(}F_{j}-F(y)\big{)}\frac{d }{dy}f\big{(}p_{J},y,n\big{)}}{\sum_{j=1}^{kp}N_{J}\big{(}p_{J}-y\big{)}\frac{d }{dy}\big{(}P_{j},y,n\big{)}\big{)}};N_{J}=\int dx;\;F_{J}=F\big{(}p_{J}\big{)} \tag{9}\]  
In the realm of particle derivatives, we use \(\mathbf{p_{J}}\) to denote individual particles, while \(\mathbf{N_{J}}\) signifies the associated volume, and \(\mathbf{f}\) represents the kernel employed in the approximation process. Drawing an analogy with conventional physical parameters, we establish the equation \(\mathbf{N_{J}}=\mathbf{m_{j}}/\mathbf{\rho_{j}}\), where \(\mathbf{m_{j}}\) represents the mass of the particle and \(\mathbf{\rho_{j}}\) refers to its volumetric mass density, commonly referred to as \"density\". Consequently, Equation (9) can be derived as follows:  
\[F^{\prime}(y)\approx\frac{\sum_{j=1}^{kp}\frac{m_{j}}{\rho_{j}}\big{(}F_{j}-F( y)\big{)}\frac{d}{dy}f\big{(}p_{J},y,n\big{)}}{\sum_{j=1}^{kp}\frac{m_{j}}{ \rho_{j}}\big{(}p_{J}-y\big{)}\frac{d}{dy}\big{(}P_{J},y,n\big{)}\big{)}};N_{J }=\int dx \tag{10}\]  
To compute the partial derivatives, we follow a similar approach, introducing the approximations \(F_{J}=Fp_{J}\) which yields  
\[\frac{\partial F}{\partial y_{j}}(y)\approx\frac{\sum_{j=1}^{kp}N_{J}\big{(}F _{j}-F(y)\big{)}\frac{\partial}{\partial y_{j}}f\big{(}p_{J},y,n\big{)}}{\sum _{j=1}^{kp}V_{J}\big{(}p_{J}-y\big{)}\big{)}\frac{\partial}{\partial y_{j}}f \big{(}p_{J},y,n\big{)}};N_{J}=\int dx \tag{11}\]  
Then,  
\[\frac{\partial F}{\partial y_{j}}(y)\approx\frac{\sum_{j=1}^{kp}\frac{m_{j}} {\rho_{j}}\big{(}F_{j}-F(y)\big{)}\frac{\partial}{\partial y_{j}}f\big{(}p_{J },y,n\big{)}}{\sum_{j=1}^{kp}\frac{m_{j}}{\rho_{j}}\big{(}p_{J}-y\big{)}\big{)} \frac{\partial}{\partial y_{j}}f\big{(}p_{J},y,n\big{)}};N_{J}=\int dx \tag{12}\]  
It is important to highlight that a range of kernels can be found in the literature; the most commonly used are defined as follows:  
\[f(x,y,n)=\mu(n)\chi\bigg{(}\frac{\|x-y\|}{n}\bigg{)} \tag{13}\]  
where \(\mu(n)\) is a normalizing factor such that the integral on \(R^{n}\)\(\int f(x,y,n)dx=1\). In our particular study, we will adopt the quintic kernel, such that \(\mu(n)=\frac{1}{120};\)\(\frac{1}{478\pi};\frac{1}{120\pi}\) (dimension 1, 2, or 3, respectively)  
\[\chi=\left\{\begin{array}{c}(3-x)^{5}-6(3-x)^{5}+15(1-x)^{5},\;\;if\;0\leq x \leq 1\\ (3-5)^{5}-6(3-x)^{5},\;\;if\;1\leq x\leq 2\\ (3-x)^{5},\;\;if\;2\leq x\leq 3\\ 0,\;otherwise\end{array}\right. \tag{14}\]