8 UNCERTAINTY MODELING
8.1 Application-oriented Modeling of Uncertainty
Until the 1960s probability theories and statistics were the only methods to model uncertainty which has always been considered by scientists as a rather dis- turbing feature of some scientific statements, of systems, phenomena or even in philosophy. Since the 1960s additional theories have been suggested as tools to model uncertainty. Some of these theories or their supporters even claim to be the only proper tool for modeling uncertainty, even though the notion of uncertainty has never been defined uniquely.
It has been defined in specific contexts-mainly formal theories-but then the semantic interpretation is generally restricted to this field. In decision logic , for instance, "decisions under uncertainty" are defined as acts of choice for which the state of the nature that will occur is unknown. Unluckily, as Schneider already observed in 1979, those situations occur in practice very seldomly, if at all [Schneider 1979].
One would expect to find an appropriate definition of uncertainty either in lexica or in scholarly books on "uncertainty" modeling [Goodman and Nguyen 1985, Klir and Folger 1988, Klir 1987]. Surprisingly enough I have not been successful to find any general definition for it.
The first question one should probably ask is whether uncertainty is a phenomenon, a feature of real world systems, a state of mind or a label for a situation in which a human being wants to make statements about phenomena (i.e, reality, models, theories) . One can also ask whether "uncertainty" is an objective fact or just a subjective impression which is closely related to individual persons.
Whether uncertainty is an objective feature of physical real systems seems to be a philosophical question. In the following we shall not consider these "objec- tive uncertainties" if they exist, but we shall focus on the human-related, sub- jective interpretation of "uncertainty" which depends on the quantity and quality of information which is available to a human being about a system or its behavior that the human being wants to describe, predict or prescribe.
In this respect it shall not matter whether the information is inadequate due to the specific individuum or whether it is due to the present state of knowledge, i.e. whether the information is not available at present to anybody. Figure 8-1 depicts our view of uncertainty used in this chapter.
In this figure the "system" denotes the phenomenon about which judgments are to be made . This can be parts of the physical reality, socio-economic systems, man-made systems or any other type of phenomena. Information or data emitted by the system might be impul ses, visible or measurable properties (noise, tem- perature etc.). Theses data or information are, however, very often not consid- ered directly by the "observer". They are rather the input to an uncertainty theory (e.g. probability theory), which processes this information in specified ways and supplies the observer with certain "measures of uncertainty" (e.g. mean values,
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Data Informatior
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Phenomenon Perception Figure 8-1 . Uncertainty as situational property.
variances etc.) or descriptions of uncertainty (e.g. probability distributions etc.).
Hence, the observer does not perceive the information about the phenomenon directly but only after it has been "filtered" by the uncertainty theory used.
The most important aspects of this view are:
1. "Causes" of uncertainty influence the information flow between the observed system and the uncertainty model (paradigm chosen by the observer).
2. A selected uncertainty model or theory has to be appropriate to the available quantity and quality of input information.
3. A chosen uncertainty theory also determines the type of information processing applied to available data or information.
4. For pragmatic reasons the information offered to the observer (human or other) by the uncertainty model should be in an adequate language.
5. Hence, the choice of an appropriate "uncertainty" calculus may depend on
• the causes of uncertainty,
• quantity and quality of information available,
• type of information processing required by the respective "uncertainty"
calculus and
• language required by the final observer.
Even this notion of uncertainty is rather vague, has many different appearances and many different causes. Itis, therefore, difficult to define it properly and in sufficient generality. Any definition of uncertainty is in a way arbitrary and subjective.Itcan be more or less extreme with respect to the situation. Here we chose a rather broad definition for uncertainty in order to include a large number of possible situations which can be considered "uncertain".
Definition8-1: A proposed definition of uncertainty
Uncertainty implies that in a certain situation a person does not dispose about information which quantitatively and qualitatively is appropriate to describe, prescribe or predict deterministically and numerically a system, its behavior or other characteristica.
"Situation" in the context of this definition includes features of the system as well as expectations or needs of the observer. The need to describe a phenome- non numerically was included because most of the known measures of uncer- tainty require a numerical description. In some situations a symbolic description of the phenomenon may be sufficient for the human observer to judge the situa- tion (e.g. the color of the traffic lights at a road intersection). But in this case he knows in addition to the color the meaning of the color and he will not be in a position to make statements about the traffic behavior at an intersection without involving numbers.
It seems that a lot of misunderstandings have been caused by confusing the "type of uncertainty" with the "cause of uncertainty" or with the theory which is used to model uncertainty. I shall, therefore, attempt to describe in the following these three aspects of uncertainty separately in order to arrive at a certain taxonomy of uncertainty, the classes of which may neither be disjunct nor exhaustive.
8. 1. 1 Causes of Uncertainty
Lack of Information. Lack of information is probably the most frequent cause for uncertainty. In decision logic, for instance, one calls "decisions under uncer- tainty" the situation in which a decision maker does not have any information about which of the possible states of nature will occur. This would obviously be a quantitative lack of information. With "decision making under risk" one nor- mally describes a situation in which the decision maker knows the probabilities for the occurrence of various states. This could be called a qualitative lack of information . Since information about the occurrence is available, it can also be considered complete in the sense of the availability of a complete probability function. But the kind of the available information is not sufficient to describe the situation deterministically. Another situation characterized by a lack of infor- mation might be called "approximation". Here one does not have or one does not want to gather sufficient information to make an exact description, even though this might be possible . In some cases the description of the system is explicitly called an "approximation", in other situations this is hidden and probably not
visible to the normal observer. Examples for the latter case can be found in math- ematics where symbols are used rather than real numbers because a description by real numbers is not feasible (for instance the "number" 1t, sin and cosine functions, or any complex or transcendental numbers). In this context the scale level on which numerical information is available also has to be considered. The situation of "certainty" normally assumes an absolute or at least a cardinal scale level of the information available. If only information on a ratio, ordinal or nominal scale level is available, this would also be called a "qualitative lack of information" in our view.
A transition from a situation of uncertainty caused by a lack of information to a situation of certainty can obviously only be achieved by gathering more or better information . Whether this is possible or desirable obviously depends on the situation and the goal of modeling.
Abundance of Information (Complexity). This type of uncertainty is due to the limited ability of human beings to perceive and process simultaneously large amounts of data [Newell and Simon 1972]. This situation is exemplified by real world situations in which more data is objectively available to human beings than they can "digest" or by situations in which human beings communicate about phenomena which are defined or described by a large number of features or prop- erties. What people do in these situations is normally, that they transform the available data into perceivable information by using a coarser grid or a rougher
"granularity" or by focusing their attention on those features which seem to them most important and neglecting all other information or data. If such a situation occurs in scientific activities, very often some kind of "scaling" is used to the same end.Itis obvious that in these situations a transfer to "certainty" cannot be achieved by gathering even more data, but rather by transforming available data to appropriate information.
Conflicting Evidence. Uncertainty might also be due to conflicting evidence, i.e. there might be considerable information available pointing to a certain behav- ior of a system and additionally there might also be information available point- ing to another behavior of the system.Ifthe two classes of available information are conflicting, then an increase of information might not reduce uncertainty at all, but rather increase the conflict. The reason for this conflict of evidence can certainly be different.Itcan be due to the fact that some of the information avail- able is wrong (but not identifiable as wrong information by the system), it can also be that information of non-relevant features of the system is being used, it might be that the model which the observer has of the system is wrong etc.
In this case a transition to a situation of certainty might call for checking the
available information again with respect to the correctness rather than gathering more information or putting the information on a rougher grid. In some cases, however, deleting some pieces of information might reduce the conflict and move the situation closer in the direction of certainty.
Ambiguity. By ambiguity we mean a situation in which certain linguistic information, for instance, has entirely different meanings or in which-mathe- matically speaking-we have a one-to-many mapping. All languages contain certain words which for several reasons have different meanings in different contexts. A human observer can normally easily interpret the word correctly semantically if he knows the context of the word. In so far this type of uncer- tainty could also be classified under "lack of information" because in this case adding more information about the context to the word may move us from uncertainty to certainty.
Measurement. The term "measurement" also has very different interpreta- tions in different areas [Zimmermann and Zysno 1980]. In the context of this chapter we mean "measurement" in the sense of "engineering measurement", i.e.
of measuring devices to measure physical features, such as weight, temperature, length etc.
The quality of our measuring technology has increased with time and the further this technology improves, the more exactly it can determine properties of physical systems. As long, however, as an "imagined" exact property cannot yet be measured perfectly, we have some uncertainty about the real measure and we only know the indicated measure. This is certainly also some type of uncertainty which could also be considered as a "lack of information".It is only considered to be a separate class in this paper due to the particular importance of this type of uncertainty to engineering.
Belief. Eventually, we would like to mention as cause of uncertainty situations in which all information available to the observer is subjective as a kind of belief in a certain situation. This situation is probably most disputable and it could also be considered as "lack of information" in the objective sense.
A possible interpretation of this situation is, however, also that a human being develops on the basis of available (objective) data and in a way which is unknown to us (subjective) beliefs which he afterwards considers as information about a system that he wants to describe or prescribe. The distinction of this class from the classes mentioned above is actually that, so far, we always have considered
"objective" information and now we are moving to "subjective" information.
Whether this distinction can and should be upheld at all is a matter for further discussion.
8.1.2 Type of Available Information
So far we have discussed causes of uncertainty which in most cases depend on the quality or quantity of available information. As already mentioned, however, we will have to consider the type of available information in a situation which we want to judge with respect to uncertainty in more detail: the information which is available for a system under consideration can, roughly speaking, be numerical, linguistic, interval-valued or symbolic.
Numerical Information. In our definition of certainty we requested that a system can be described numerically. This normally requires that the information about the system is also available numerically. Since this numerical information can come from quite a variety of sources, it is not sufficient to require just that the information is given in numbers, but we also have to determine the scale level on which this information is provided [Sneath and Sokal 1973]. This determines the type of information processing (mathematical operation) which we can apply to this information legitimately without pretending information which is not available. There is quite a number of taxonomies for scale levels, such as, for instance, distinguishing between nominal scale level, ordinal scale level, ratio scale level, interval scale level and absolute scale level. For our purposes we refer the reader to table 16-1 .
Roughly speaking, a nominal scale level indicates that the information pro- vided (even though in numerical form) only has the function of a name (such as the number on the back of a football player or a license plate of a car), that numer- ical information on an ordinal scale level provides information of an ordering type and information on a cardinal scale level also indicates information about the differences between the ordered quantities , i.e. contains a metric.
Interval-Information. In this case information is available, but not as precise in the sense of a real-valued number as above. If we want to process this infor- mation properly, we will have to use interval arithmetic and the outcome will again be interval-valued information.Itshould be clear, however, that this infor- mation is also "exact" or "dichotomous" in the sense that the boundaries of the intervals, no matter how they have been determined, are "crisp", "dichotomous", or "exact".
Linguistic Information. By linguistic information we mean that the informa- tion provided is given in a natural language and not in a formal language [Bellman and Zadeh 1970]. The properties of the type of information obviously differ from those of either numerical information or of information in a formal language.
Natural languages develop over time, they depend on cultural backgrounds, they
depend on educational backgrounds of the persons using this language and on many other things. One also has to distinguish between a word as a label and the meaning of a word. Very often there is neither a one-to -one relationship between these two nor are the meanings of words defined in a crisp and a context- independent way. By contrast to numerical information there are also hardly any measures of quality of information for natural languages (e.g. there are no defined scale levels for linguistic information). Linguistic information has developed as a means of communication between human beings and the "inference engines"
are the minds of people about which is still much too little known .
Symbolic Information. Very often information is provided in the form of symbols . This is obvious when numbers, letters or pictures are being used as symbols. This is often not as obvious if words are being used as symbols because sometimes it seems to be suggested or assumed that words have natural meanings while symbols do not. Hence, if symbolic information is provided, the information is as valuable as the definitions of the symbols are and the type of information processing also has to be symbolic and neither numerical nor linguistic .
8.1.3 Uncertainty Methods
As depicted in figure 8-1, information of the uncertain phenomenon is filtered by an uncertainty method before it is offered to the observer. By "uncertainty methods" we mean any of the probability theories, fuzzy set theory, rough set theory, evidence theory etc. These theories build on certain axioms with respect to the uncertainty to be modeled and they propose generally a mathematical framework to arrive at measures of uncertainty [Dubois and Prade 1989]. The mathematical models or methods suggested require a certain scale level of numer- ical information. Hence, a specific uncertainty method should not be used if its mathematical operations require a higher scale level than that on which the available information is provided. This is very often neglected when applying those theories. Rather one assumes, without checking, that numerical informa- tion is available on a cardinal or absolute scale level for which all mathematical operations would be legitimate.
To an increasing degree, moreover, uncertain information or information about
"uncertainties" is also processed in knowledge-based systems [Zimmermann 1988, Kandel and Langholz 1992, Klein and Methlie 1995, Turban 1988] which can either be systems which essentially perform symbol processing (classical expert system technology) or they perform meaning preserving inference . Obviously, for these systems different requirements exist and different types of
information are offered at the end. Eventually, information can be processed heuristically, i.e. according to well-defined procedures which can also require other types of languages.
To model, i.e. describe, prescribe or predict, a system or the behavior of a system normally serves a certain purpose. Itcould serve a human observer, it could be the input to another mechanical or electronic system, it could be used for other mathematical algorithms etc. In figure 8-1 a human observer was con- sidered as the recipient of the information . In this case the information does not only have to be "readable" by the recipient, but it may have to meet additional requirements, depending on what it is intended for. If the observer wants to rec- ognize certain patterns, a nominal scale level of the received information might already be sufficient. If he wants to evaluate or order phenomena, information will have to be at least on an ordinal scale level, etc. Hence, the information about the uncertain system will have to be provided in a suitable language, i.e.
either numerical, in the form of intervals, linguistically or symbolically, and on an appropriate scale level.
8. 1.4 Uncertainty Theories as Transformers of Information
Sections 8.1.1 to 8.1.3 of this chapter focused on informational features of the uncertain phenomenon. The uncertainty calculus, theory or method used to describe this phenomenon should obviously be compatible with the features of the phenomenon, i.e. not require information on a higher level than provided, not make any axiomatic assumptions about the cause of uncertainty etc. which are not satisfied by the real situation.
This certainly contradicts views that, for instance, any uncertainty can be modeled by probabilities, or by fuzzy sets, or by possibilities, or by any other single method. We do not believe that there exists any single method which is able to model all types of uncertainty equally well.
Most of the established theories and methods for uncertainty modeling are focused either on specific "types of uncertainty" defined by their causes or they at least imply certain causes and they also require specific types or qualities of information depending on the type of information processing they use. One could consider these uncertainty methods and their paradigms as glasses through which we consider uncertain situations or with other words: there is no "probabilistic uncertainty" as distinct from "possibilistic uncertainty". One is rather looking at an uncertain situation with the properties that were specified before and one tries to model this uncertain situation by means of probability theory or by means of possibility theory. Hence, the theory which is appropriate to model a specific