3.7 An Illustration of the Application of the SRSM

As an illustration, consider a computational model with three independent random inputs $X_1, X_2,$ and $X_3$, and two outputs $Y_1$ and $Y_2$, where the input random variables $X_1$, $X_2$ and $X_3$ are given by

 

$$X_1 = \mbox{Uniform} (p_1,q_1)$$

$$X_2 = \mbox{Lognormal} (p_2,q_2)$$ (3.15)

$$X_3 = \mbox{Gamma} (p_3,q_3)$$

The input random variables can be represented by three srvs, ${\xi_1,\xi_2,\xi_3,}$ using Table 3.1 as follows:

 

$$X_1 = p_1 + (q_1 - p_1)\left(\displaystyle\frac{1}{2} + \frac{1}{2}\mbox{erf}(\xi_1/\sqrt2)\right)$$

$$X_2 = \displaystyle \exp(p_2 + q_2\xi_2)$$ (3.16)

$$X_3 = \displaystyle p_3q_3\left(\xi_3\sqrt{\frac{2}{9p_3}} + 1 - \frac{2}{9p_3}\right)^{3}$$

Where ${\xi_1,\xi_2,\xi_3}$ are iid N(0,1) random variables. A second order polynomial chaos approximation for $Y_1$ and $Y_2$ in terms of ${\xi_1,\xi_2,\xi_3}$ is given by

 

$$Y_1 = a_0 + a_1\xi_1 + a_2\xi_2 + a_3\xi_3 + a_4(\xi_1^2-1) + a_5(\xi_2^2-1) + a_6(\xi_3^2-1)$$

$$+ a_7\xi_1\xi_2 + a_8\xi_2\xi_3 + a_9\xi_1\xi_3$$ (3.17)

$$Y_2 = b_0 + b_1\xi_1 + b_2\xi_2 + b_3\xi_3 + b_4(\xi_1^2-1) + b_5(\xi_2^2-1) + b_6(\xi_3^2-1)$$

$$+ b_7\xi_1\xi_2 + b_8\xi_2\xi_3 + b_9\xi_1\xi_3$$

In order to estimate the 10 unknown coefficients (for each output) from the above equation, the selection of a set of $N$ sample points (equaling about twice the number of sample points, i.e., 20 in this case) is recommended for regression based SRSM, in the form

$$(\xi_{1,1}\ \xi_{2,1}\ \xi_{3,1}), (\xi_{1,2}\ \xi_{2,2}\ \xi_{3,2}), \ldots, (\xi_{1,N}\ \xi_{2,N}\ \xi_{3,N})$$

These sample points can be readily generated using a simple algorithm developed as a part of the SRSM implementation. The selection of the sample points is discussed in detail in Section 3.4.2, in the context of the ECM method.

These sample points correspond to the original model input samples, $(x_{1,1}\ x_{2,1}\ x_{3,1}) \ldots \\ (x_{1,N}\ x_{2,N}\ x_{3,N})$, as follows:

$$\begin{array}{cccc} \left(\begin{array}{c} \xi_{1,i}\\ \xi_{2... ...1 - \frac{2}{9p_3}\right)^{3}\\ \end{array}\right) \end{array}$$ (3.18)

for $i = 1 \ldots N$. After obtaining the original model input sample points, the model simulation is performed at the points given by $(x_{1,1}\ x_{2,1}\ x_{3,1}) \ldots \\ (x_{1,N}\ x_{2,N}\ x_{3,N})$. Then the outputs at these sample points, $y_{1,1} \ldots y_{1,N}$ and $y_{2,1} \ldots y_{2,N}$, are used to calculate the coefficients $a_1 \ldots a_{10}$ and $b_1 \ldots b_{10}$ by solving the following linear equations through singular value decomposition.

$$\hspace*{-0.5in}\begin{array}{cccc} {\ensuremath{\textstyle\m... ... & y_{2,N} \\ \end{array}\right] \mbox {, \ where} \end{array}$$ (3.19)

$${\ensuremath{\textstyle\mbox{\boldmath${Z}$ }} } = \left[\beg... ...,3}\xi_{3,3}&\ldots & \xi_{1,N}\xi_{3,N}\\ \end{array}\right]$$ (3.20)

In the above equations, can be calculated from the values of $\xi$'s at each sample point, whereas, $y_{1,i}$ and $y_{2,i}$ are the corresponding model outputs. Thus, the only unknowns in Equation 3.19, the coefficients $a_i$'s and $b_i$'s, can be readily estimated. Once the coefficients are estimated, the distributions of $Y_1$ and $Y_2$ are fully described by the polynomial chaos expansions as shown in Equation 3.18.

The next step involves determination of the statistical properties of the outputs, such as the individual moments, joint moments, correlations between the outputs and correlations between inputs and outputs. This process is done by numerically generating a large number of random samples of $(\xi_{1,i}\ \xi_{2,i}\ \xi_{3,i})$, calculating the values of model inputs and outputs for those samples, and computing the statistical properties from the values of inputs and outputs as described in Section 3.5.