2.3 Approaches for Representation of Uncertainty

Various approaches for representing uncertainty in the context of different domains of applicability are presented by Klir [125], and are briefly summarized in the following:

• Classical set theory: Uncertainty is expressed by sets of mutually exclusive alternatives in situations where one alternative is desired. This includes diagnostic, predictive and retrodictive uncertainties. Here, the uncertainty arises from the nonspecificity inherent in each set. Large sets result in less specific predictions, retrodictions, etc., than smaller sets. Full specificity is obtained only when one alternative is possible.
• Probability theory: Uncertainty is expressed in terms of a measure on subsets of a universal set of alternatives (events). The uncertainty measure is a function that, according to the situation, assigns a number between 0 and 1 to each subset of the universal set. This number, called probability of the subset, expresses the likelihood that the desired unique alternative is in this subset. Here, the uncertainty arises from the conflict among likelihood claims associated with the subsets of the universal set, each consisting of exactly one alternative. Since these alternatives are mutually exclusive, nonzero probabilities assigned to two or more events conflict with one another since only one of them can be the desired one. A detailed description of the probability theory is presented in Appendix A.
• Fuzzy set theory: Fuzzy sets, similar to classical sets, are capable of expressing nonspecificity. In addition, they are also capable of expressing vagueness. Vagueness is different from nonspecificity in the sense that vagueness emerges from imprecision of definitions, in particular definitions of linguistic terms. In fuzzy sets, the membership is not a matter of affirmation or denial, but rather a matter of degree.
• Fuzzy measure theory: This theory considers a number of special classes of measures, each of which is characterized by a special property. Some of the measures used in this theory are plausibility and belief measures, and the classical probability measures. Fuzzy measure theory and fuzzy set theory differ significantly: in the fuzzy set theory, the conditions for the membership of an element into a set are vague, whereas in the fuzzy measure theory, the conditions are precise, but the information about an element is insufficient to determine whether it satisfies those conditions.
• Rough set theory: A rough set is an imprecise representation of a crisp set in terms of two subsets, a lower approximation and upper approximation. Further, the approximations could themselves be imprecise or fuzzy.

Some of the widely used uncertainty representation approaches used in transport-transformation modeling include interval mathematics, fuzzy theory, and probabilistic analysis. These approaches are presented in the following sections.

2.3.1 Interval Mathematics

Interval mathematics is used to address data uncertainty that arises (a) due to imprecise measurements, and (b) due to the existence of several alternative methods, techniques, or theories to estimate model parameters. In many cases, it may not be possible to obtain the probabilities of different values of imprecision in data; in some cases only error bounds can be obtained. This is especially true in case of conflicting theories for the estimation of model parameters, in the sense that ``probabilities'' cannot be assigned to the validity of one theory over another. In such cases, interval mathematics can be used for uncertainty estimation, as this method does not require information about the type of uncertainty in the parameters [5,24].

The objective of interval analysis is to estimate the bounds on various model outputs based on the bounds of the model inputs and parameters. In the interval mathematics approach, uncertain parameters are assumed to be ``unknown but bounded'', and each of them has upper and lower limits without a probability structure [184]; every uncertain parameter is described by an interval. If a parameter $x_i$ of a model is known to be between $\overline{x_i}-\epsilon_i$ and $\overline{x_i}+\epsilon_i$, the interval representation of $x_i$ is given by [ $\overline{x_i}-\epsilon_i,\overline{x_i}+\epsilon_i$]. Correspondingly, the model estimates would also belong to another interval. Special arithmetic procedures for calculation of functions of intervals used in this method are described in the literature [5,151,152,155]. For example, if two variables, $a$ and $b$ are given by $\left[a_l,a_u\right]$ and $\left[b_l,b_u\right]$, where $a_l\le a_u$ and $b_l\le b_u$, simple arithmetic operations are given by the following:

$$\begin{array}{ccl} a + b & = & \left[a_l+b_l,a_u+b_u\right] \... ...frac{1}{b_l}\right] ; 0 \notin \left[b_l,b_u\right] \end{array}$$ (2.1)

Furthermore, a range of computational tools exist for performing interval arithmetic on arbitrary functions of variables that are specified by intervals [122,132]. Symbolic computation packages such as Mathematica [10] and Maple [94] support interval arithmetic. In addition, extensions to FORTRAN language with libraries for interval arithmetic are also available [121,120]. Therefore, in principle, the ranges of uncertainty in any analytical or numerical model (FORTRAN model) can be analyzed by existing tools.

Various applications of interval analysis in the literature include the treatment of uncertainty in the optimal design of chemical plants [69], in the cost benefit analysis of power distribution [184], and in decision evaluation [24]. Interval analysis has also been applied to estimate the uncertainty in displacements in structures due to the uncertainties in external loads [129]. Dong et al. presented a methodology for the propagation of uncertainties using intervals [52]. Hurme et al. presented a review on semi qualitative reasoning based on interval mathematics in relation to chemical and safety engineering [104]. Applications in transport-transformation modeling include uncertainty analysis of groundwater flow models [143,53].

The primary advantage of interval mathematics is that it can address problems of uncertainty analysis that cannot be studied through probabilistic analysis. It may be useful for cases in which the probability distributions of the inputs are not known. However, this method does not provide adequate information about the nature of output uncertainty, as all the uncertainties are forced into one arithmetic interval [134]. Especially when the probability structure of inputs is known, the application of interval analysis would in fact ignore the available information, and hence is not recommended.

2.3.2 Fuzzy Theory

Fuzzy theory is a method that facilitates uncertainty analysis of systems where uncertainty arises due to vagueness or ``fuzziness'' rather than due to randomness alone [65]. This is based on a superset of conventional (Boolean) logic that has been extended to handle the concept of partial truth - truth values between ``completely true'' and ``completely false''. It was introduced by Zadeh as a means to model the uncertainty of natural language [222]. Fuzzy theory uses the process of ``fuzzification'' as a methodology to generalize any specific theory from a crisp (discrete) to a continuous (fuzzy) form.

Classical set theory has a ``crisp'' definition as to whether an element is a member of a set or not. However, certain attributes of systems cannot be ascribed to one set or another. For example, an attribute of a system can be specified as either ``low'' or ``high''. In such a case, uncertainty arises out of vagueness involved in the definition of that attribute. Classical set theory allows for either one or the other value. On the other hand, fuzzy theory provides allows for a gradual degree of membership. This can be illustrated as follows:

In the classical set theory, the truth value of a statement can be given by the membership function $\mu_A(x)$, as

$$\mu_A(x)= \left\{ \begin{array}{rcl}1 & \mbox{iff\ } & x \in A \\ 0 & \mbox{iff\ } & x \notin A \end{array}\right.$$ (2.2)

On the other hand, fuzzy theory allows for a continuous value of $\mu_A$between 0 and 1, as

$$\mu_A(x) = \left\{ \begin{array}{rcl}1 & \mbox{iff\ } & x \in... ...if\ } & x \mbox{\ \ partially belongs to} A \end{array}\right.$$ (2.3)

In fuzzy theory, statements are described in terms of membership functions, that are continuous and have a range [0,1]. For example, given the measured value of a parameter, the membership function gives the ``degree of truth'' that the parameter is ``high'' or ``low''.

Further, fuzzy logic is defined by the following the set relationships:

$$\begin{array}{rcl} \mu_{A'}(x) & = & 1.0 - \mu_A(x) \\ \mu_{... ...\\ \mu_{A \cup B}(x) & = & \max(\mu_A(x),\mu_B(x)) \end{array}$$ (2.4)

Using fuzzy arithmetic, based on the grade of membership of a parameter of interest in a set, the grade of membership of a model output in another set can be calculated. Fuzzy theory can be considered to be a generalization of the classical set theory. It must be noted that if the membership grades are restricted to only 0 and 1, the fuzzy theory simplifies to classical set theory.

A vast amount of literature is available on fuzzy theory, including extensive introductions to the concepts involved by Bezdek and by Smithson [15,191]. Fuzzy arithmetic and its applications are described by Klir et al., by Kauffmann et al., and by Puri et al. [166,119,126]. Kraslawski et al. applied fuzzy set theory to study uncertainty associated with incomplete and subjective information in process engineering [130]. Ayyub et al. studied structural reliability assessment by using fuzzy theory to study the uncertainty associated with ambiguity [9]. Juang et al. demonstrated the applicability of fuzzy theory in the modeling and analysis of non-random uncertainties in geotechnical engineering [115]. Further literature on industrial applications of fuzzy theory is available for the interested reader [221,174]. Specifically, it has been applied to characterize uncertainty in engineering design calculations [219,131], in quality control [167], in sludge application land selection [45], and in solute transport modeling [53,54,143]. However, drawbacks in its applicability to uncertainty analysis has been noted by some researchers [193,212]. Fuzzy theory appears to be more suitable for qualitative reasoning, and classification of elements into a fuzzy set, than for quantitative estimation of uncertainty.

2.3.3 Probabilistic Analysis

In the probabilistic approach, uncertainties are characterized by the probabilities associated with events. The probability of an event can be interpreted in terms of the frequency of occurrence of that event. When a large number of samples or experiments are considered, the probability of an event is defined as the ratio of the number of times the event occurs to the total number of samples or experiments. For example, the statement that the probability that a pollutant concentration $c$ lies between $c_1$ and $c_2$ equals $p$ means that from a large number of independent measurements of the concentration $c$, under identical conditions, the number of times the value of $c$ lies between $c_1$ and $c_2$ is roughly equal to the fraction $p$ of the total number of samples.

Probabilistic analysis is the most widely used method for characterizing uncertainty in physical systems, especially when estimates of the probability distributions of uncertain parameters are available. This approach can describe uncertainty arising from stochastic disturbances, variability conditions, and risk considerations. In this approach, the uncertainties associated with model inputs are described by probability distributions, and the the objective is to estimate the output probability distributions. This process comprises of two stages:

• Probability encoding of inputs: This process involves the determination of the probabilistic distribution of the input parameters, and incorporation of random variations due to both natural variability (from, e.g., emissions, meteorology) and ``errors.'' This is accomplished by using either statistical estimation techniques or expert judgments. Statistical estimation techniques involve estimating probability distributions from available data or by collection of a large number or representative samples. Techniques for estimating probability distributions from data are presented in the literature [37,103]. In cases where limited data are available, an expert judgment provides the information about the input probability distribution. For example, a uniform distribution is selected if only a range of possible values for an input is available, but no information about which values are more likely to occur is available. Similarly, normal distribution is typically used to describe unbiased measurement errors. Table 2.2 presents a list of some commonly used probability distributions in the uncertainty analysis of transport-transformation models.
• Propagation of uncertainty through models: Figure 2.2 depicts schematically the concept of uncertainty propagation: each point of the response surface (i.e., each calculated output value) of the model to changes in inputs ``1'' and ``2'' will be characterized by a probability density function (pdf) that will depend on the pdfs of the inputs. The techniques for uncertainty propagation are discussed in the following. This thesis uses probabilistic analysis...]

  

Figure 2.2: A schematic depiction of the propagation of uncertainties in transport-transformation models

Appendix A presents some background information on probabilistic analysis and random variables. A number of excellent texts on probabilistic analysis are available in the literature for the interested reader [160,75,201].

 

Table 2.2: Selected probability density functions useful in representing uncertainties in environmental and biological model inputs

Distribution	Parameters & Conditions	Probability density function	Moments
Uniform	$a, b$	$$\frac{1}{b-a}$$	Mean = $$\frac{a+b}{2}$$ Var = $$\frac{(b-a)^2}{12}$$
Normal	$\mu, \sigma\ , \sigma > 0$	$$\frac{1}{\sigma\sqrt{2\pi}}\ e^{\displaystyle-\frac{(x-\mu)^2}{2\sigma^2}}$$	Mean = $\mu$ Var = $$\sigma^2$$ Mode = $\mu$
Lognormal	$\mu, \sigma\ , \sigma > 0$	$$\frac{1}{x\sigma\sqrt{2\pi}}\ e^{\displaystyle-\frac{(\log(x)-\mu)^2}{2\sigma^2}}$$	Mean = $e^{\displaystyle(\mu + \sigma^2/2)}$ Var = $$(e^{\sigma^2}-1)e^{\displaystyle(2\mu+\sigma^2)}$$ Mode = $$e^{\displaystyle(\mu - \sigma^2)}$$
Gamma	$a,b,$ $a>0, b>0$	$$\frac{1}{\Gamma(a)b^a}\ x^{a-1}e^{-\displaystyle\frac{x}{b}}\,\ x>0$$	Mean = $ab$ Var = $$ab^2$$ Mode = $(a-1)b$
Exponential	$\lambda, \lambda > 0$	$$\lambda e^{-\lambda x}\ ,\ x > 0$$	Mean = $$\frac{1}{\lambda}$$ Var = $$\frac{1}{\lambda^2}$$ Mode = 0
Weibull	$a$	$$ax^{\displaystyle a-1}e^{\displaystyle-x^a}\ \ (x \ge 0)$$	Mean = $$\Gamma(1+\frac{1}{a})$$ Var = $$\Gamma(1+\frac{2}{a})-\Gamma(1+\frac{1}{a})$$ Mode = $$(1-\frac{1}{a})^{\displaystyle\frac{1}{a}}, a\ge1$$
Extreme Value		$$e^{\displaystyle-x-e^{\displaystyle-x}}$$	Mean = 0 Var = 1 Mode = 0