3.2 Step I: Representation of Stochastic Inputs

The first step in the application of the SRSM is the representation of all the model inputs in terms of a set of ``standardized random variables''. This is analogous to normalization process used in transforming deterministic variables. In this work, normal random variables are selected as srvs as they have been extensively studied and their functions are typically well behaved [25,49,83,160]. Here, the srvs are selected from a set of independent, identically distributed (iid) normal random variables, , where $n$ is the number of independent inputs, and each $\xi_i$ has zero mean and unit variance. When the input random variables are independent, the uncertainty in the $i$th model input $X_i$, is expressed directly as a function of the $i$th srv, $\xi_i$; i.e., a transformation of $X_i$ to $\xi_i$ is employed. Such transformations are useful in the standardized representation of the random inputs, each of which could have very different distribution properties.

One of two approaches may be taken to represent the uncertain inputs in terms of the selected srvs: (a) direct transformation of inputs in terms of the srvs, and (b) series approximations in srvs. Devroye [49] presents transformation techniques and approximations for a wide variety of random variables. Input random variables whose transformations are not found in literature can be approximated either by algebraic manipulation or by series expansion techniques. These techniques are described in detail in the following sections.

 

 

Table 3.1: Representation of common univariate distributions as functionals of normal random variables

Distribution Type	Transformation3.1
Uniform ($a, b$)	$a + (b-a)\left( \displaystyle\frac{1}{2} + \frac{1}{2}\mbox{erf}(\xi/\sqrt2)\right)$
Normal ($\mu$,$\sigma$)	$\mu+\sigma\xi$
Lognormal ($\mu$,$\sigma$)	$$\exp(\mu + \sigma\xi)$$
Gamma (a,b)	$$ab\left(\xi\sqrt{\frac{1}{9a}} + 1 - \frac{1}{9a}\right)^{3}$$
Exponential ($\lambda$)	$$-\frac{1}{\lambda} \log\left(\displaystyle\frac{1}{2} + \frac{1}{2}\mbox{erf}(\xi/\sqrt2)\right)$$
Weibull (a)	$$y^{\frac{1}{a}}$$
Extreme Value	$$-\log(y)$$

3.2.1 Direct Transformations

Direct transformations for expressing a given random variable as a function of another random variable are typically derived by employing the following relation [160]:

If $Y = h(X)$, where $X$ is a random variable with the pdf $f_X(x)$, then the pdf of $Y$, $f_Y(y)$, is given by

$$f_{Y}(y) = \frac{f_{X}(x_1)}{\left\vert h'(x_1)\right\vert} + \cdots + \frac{f_X(x_n)}{\left\vert h'(x_n)\right\vert} + \cdots$$ (3.1)

where $x_1,\ldots,x_n,\ldots$ are the real roots of the equation $y=h(x)$.

Using the above equation, given $f_Y(y)$, and $f_X(x)$, the transformation $Y = h(X)$ is identified by means of trial and error. Transformations are available in the literature for some common distributions. For example, if $X$ is uniform [0,1], then $$-\frac{1}{\lambda}\log{X}$$ is exponential ($\lambda$) distributed, and if $X$ is normal, the $X^2$ is gamma (1,1) distributed.

Although a large number of transformations for distributions exist in the literature, they are not directly applicable for the case of normal random variables because of the following reasons: (a) these transformations are optimized for computational speed as they are used as random number generators, and (b) a majority of these transformations are in terms of the uniform random numbers. However, some transformations in terms of the standard normal random variables are presented by Devroye [49]. In addition, some transformations were derived during the course of the development of the SRSM. Table 3.1 presents a list of transformations for some probability distributions commonly employed in transport-transformation modeling.

3.2.2 Transformation via Series Approximation

When a model input follows a non-standard distribution, direct transformation in terms of standard random variables is often difficult to obtain. This may be true for some distributions for which direct transformations are not available in the literature, and are difficult to derive. In such cases, the approximation of the random variable can be obtained as follows:

• the moments of the random variable are computed based on the given information (e.g., measurements or the given non-standard or empirical pdf),
• a series expansion in terms of an srv is assumed to represent the random variable,
• the moments of the corresponding series expansion are derived as functions of unknown coefficients,
• the error between the derived moments and the calculated moments is minimized (e.g., through least squares), thus resulting in a set of equations in the unknown coefficients, and
• the process is repeated using a higher order approximation, until the desired accuracy is obtained.

  
3.2.3 Transformation of Empirical Distributions

In the case of empirical distributions, the probability distribution function of an input is specified in terms of a set of measurements (i.e., a sample of random data points, or a frequency histogram). In some cases the uncertainty in a model input is specified by an empirical cumulative distribution function (cdf), in the form $\ensuremath{F_{{\ensuremath{\textstyle\mbox{\boldmath${x}$ }}}}(x)} = g(x)$, where $g(x)$ can be one of the following: (a) an algebraic function, or (b) a look-up table, or (c) a numerical subroutine. In such cases, can be approximated in terms of an srv $\xi$, as follows:

• Generate a sample of srv $\xi$
• Generate a uniform random variate from $\xi$ as
  
  $${\ensuremath{\textstyle\mbox{\boldmath ${z}$}} } = \left(\displaystyle\frac{1}{2} + \frac{1}{2}\mbox{erf}(\xi/\sqrt2)\right)$$
  
  
  ( corresponds to Uniform(0,1))
• For the calculated value of , obtain the value of the input , for which equals the sample value of
• Repeat the above steps for additional samples.

In this process, is sampled uniformly from the all its percentiles, and hence the samples represent the true distribution of . These steps can be represented by the following transformation equation:

$${\ensuremath{\textstyle\mbox{\boldmath${x}$ }} } = g^{-1}({\e... ...right), \mbox{\ and\ } g(x) \mbox{\ is the {\sl cdf\,}\ of } x$$ (3.2)

  
3.2.4 Transformation of Correlated Inputs

For cases in which the random inputs are correlated, the interdependence of a set of $n$ correlated random variables is often described by a covariance matrix. The covariance matrix, , as described in Appendix A, is a symmetric matrix with the following elements:

$$\Sigma_{i,j} = \ensuremath{\mbox{E}\{({\ensuremath{\textstyle... ...suremath{\textstyle\mbox{\boldmath ${x_j}$}}}-\eta_{x_j})\}} ,$$

where $\eta_{x_i}$ and $\eta_{x_j}$ are the means of and , respectively.

The transformation process for $n$ correlated random variables with zero mean, and similar distribution functions, is presented by Devroye [49], and is described briefly here:

To generate a random vector , with $n$ components, with mean 0 and covariance , we start with a vector comprising of $n$ iid unit random variables. Then, assuming a linear transformation ${\ensuremath{\textstyle\mbox{\boldmath${A}$ }} } = {\ensuremath{\textstyle\mbox{\boldmath${H}$ }} }{\ensuremath{\textstyle\mbox{\boldmath${B}$ }} }$, the following conditions are satisfied:

$$\ensuremath{\mbox{E}\{{\ensuremath{\textstyle\mbox{\boldmath${B}$ }}}{\ensuremath{\textstyle\mbox{\boldmath${B}$ }}}^T\}} = 1$$ (3.3)

$$\ensuremath{\mbox{E}\{{\ensuremath{\textstyle\mbox{\boldmath${A}$... ...} }\ensuremath{\mbox{E}\{{\ensuremath{\textstyle\mbox{\boldmath${B}$ }}}\}} = 0$$ (3.4)

$$\ensuremath{\mbox{E}\{{\ensuremath{\textstyle\mbox{\boldmath${A}$... ...e\mbox{\boldmath${H}$ }} }^T = {\ensuremath{\textstyle\mbox{\boldmath${C}$ }} }$$ (3.5)

The problem reduces to finding a nonsingular matrix such that ${\ensuremath{\textstyle\mbox{\boldmath${H}$ }} }{\ensuremath{\textstyle\mbox{\boldmath${H}$ }} }^T = {\ensuremath{\textstyle\mbox{\boldmath${C}$ }} }$. Since covariance matrices are positive semidefinite, one can always find a nonsingular .

The above method is generalized here for the case of transformation of an arbitrary random vector with a covariance matrix . If the $i$th random variable, ${\ensuremath{\textstyle\mbox{\boldmath${x}$ }} }_i$, has a probability density function $f_i(x)$, mean $\mu_i$, and standard deviation $\sigma_i$, the steps involved in the transformation into functions of srvs are as follows:

• construct a new vector such that ${\ensuremath{\textstyle\mbox{\boldmath${y}$ }} }_i = ({\ensuremath{\textstyle\mbox{\boldmath${x}$ }} }_i - \mu_i)/\sigma_i$
• construct a new covariance matrix such that $\Sigma^*_{i,j} = \Sigma_{i,j}/(\sigma_i\sigma_j)$
• solve for a nonsingular matrix such that ${\ensuremath{\textstyle\mbox{\boldmath${H}$ }} }{\ensuremath{\textstyle\mbox{\boldmath${H}$ }} }^T = {\ensuremath{\textstyle\mbox{\boldmath${\Sigma^*}$ }} }$
• obtain transformations $g_i(\xi_i)$ for the $i$th random variable ${\ensuremath{\textstyle\mbox{\boldmath${y}$ }} }_i$, using transformation methods for univariate random variables, as described in Table 3.1
• obtain the required vector , using ${\ensuremath{\textstyle\mbox{\boldmath${x}$ }} }_i = \mu_i + \sigma_i{\ensuremath{\textstyle\mbox{\boldmath${z}$ }} }_i$, where ${\ensuremath{\textstyle\mbox{\boldmath${Z}$ }} } = {\ensuremath{\textstyle\mbox{\boldmath${H}$ }} }{\ensuremath{\textstyle\mbox{\boldmath${Y}$ }} }$

Thus, given a random vector with $n$ elements and a covariance matrix , the sampling from involves the following steps:

• sampling of $n$ iid normal random variables, $\xi_i$'s,
• calculating the vector through $y_i$ = $g_i(\xi_i)$,
• calculating the vector through ${\ensuremath{\textstyle\mbox{\boldmath${Z}$ }} } = {\ensuremath{\textstyle\mbox{\boldmath${H}$ }} }{\ensuremath{\textstyle\mbox{\boldmath${Y}$ }} }$, and
• calculating the vector through ${\ensuremath{\textstyle\mbox{\boldmath${x}$ }} }_i = \mu_i + \sigma_i{\ensuremath{\textstyle\mbox{\boldmath${z}$ }} }_i$.

  
3.2.4.1 Specific Example: The Dirichlet Distribution

The Dirichlet distribution provides a means of expressing quantities that vary randomly, independent of each other, yet obeying the condition that their sum remains fixed. In environmental modeling, it can be used to represent uncertainty in chemical composition; for example, mole fractions vary independently, but their sum always equals unity. This distribution can also be used to represent uncertainty in compartmental mass fractions in physiological modeling.

Each component of a Dirichlet distribution is specified by a $\chi^{2}$ distribution, given by

$$f(x;a) = \left\{ \begin{array}{l} \displaystyle{\frac{1}{\Gam... ... > 0}\\ \displaystyle{0,\ \mbox{elsewhere}} \end{array}\right.$$

The mean of the $\chi^{2}$ distributions is $a$, and is related to the mean of the $i$th component, $x_{i,m}$, and the Dirichlet distribution parameter $\Theta$, as follows:

$$\mbox{E}[X_{i}] = x_{i,m}\Theta$$ (3.6)

The $i$th component can now be directly obtained as:

$$x_i = \frac{X_i}{X_1 + X_2 + \ldots + X_n}\ \ \mbox{where $n$ is the number of components in the Dirichlet distribution.}$$

Footnotes

... Transformation^3.1

 $\xi$ is normal (0,1) and $y$ is exponential (1) distributed