3.5 Step IV: Estimation of the Statistics of the Output Metrics

Once the coefficients used in the series expansion of model outputs are estimated, the statistical properties of the outputs, such as the density functions, moments; joint densities; joint moments; correlation between two outputs, or between an output and an input; etc., can be readily calculated. One way of accomplishing this is through the generation of a large number of realizations of the srvs, and the calculation of the values of inputs and outputs from the transformation equations. This results in a large number of samples of inputs and outputs. These samples can then be statistically analyzed using standard methods.

It must be noted that the calculation of model inputs and outputs involves evaluation of simple algebraic expressions, and does not involve model runs, and hence substantial savings in the computer time are accomplished.

As an example, if the inputs ${\ensuremath{\textstyle\mbox{\boldmath${x}$ }} }_i$'s are represented as ${\ensuremath{\textstyle\mbox{\boldmath${x}$ }} }_i = F_i(\xi_i)$, and if the outputs ${\ensuremath{\textstyle\mbox{\boldmath${y}$ }} }_j$'s are estimated as ${\ensuremath{\textstyle\mbox{\boldmath${x}$ }} }_i = G_i(\xi_1,\xi_2\ldots\xi_n)$, then the following steps are involved in the estimation of the statistics of the inputs and outputs.

• generation of a large number of samples of $(\xi_1\ \xi_2\ \ldots\ \xi_{n,i})$,
• calculation of the values of input and output random variables from the samples,
• calculation of the moments using Equation 3.13, and
• calculation of density functions and joint density functions using the sample values.

From a set of $N$ samples, the moments of the distribution of an output ${\ensuremath{\textstyle\mbox{\boldmath${y}$ }} }_i$ can be calculated as follows:

$$\begin{array}{rcccccl} \eta_{y_i} & = & \mbox{Mean}({\ensure... ...gma_{y_i}^4}\sum^{N}_{j=1}(y_{i,j}-\eta_{y_i})^4\\ \end{array}$$ (3.13)

Further, the correlation coefficient of an input ${\ensuremath{\textstyle\mbox{\boldmath${x}$ }} }_k$ and an output ${\ensuremath{\textstyle\mbox{\boldmath${y}$ }} }_i$ can be calculated using the following:

$$r_{x_k,y_i} = \displaystyle\frac{\ensuremath{\mbox{E}\{({\ens... ..._{y_i}} \sum^{N}_{j=1}(x_{k,j}-\eta_{x_k})(y_{i,j}-\eta_{y_i})$$ (3.14)

Similarly, higher moments of outputs, or the correlation between two outputs, or between an input and an output can be directly calculated.