Table 3.1: Summary results from the outputs
System Mean STD Max Min
Case 1 0.5696 0.1690 0.8679 0.1334
Case 2 0.5822 0.1554 0.8679 0.2659
Case 3 0.5679 0.1688 0.8679 0.1334
Case 4 0.5808 0.1552 0.8679 0.2659
3.4
Discussion
This section has described the well-known properties normal, convex and distinct used in the vast majority of terms implemented in fuzzy systems in the literature. It has been argued that while these properties are undoubtedly useful in the context of fuzzy control, they restrict the more general shapes of terms that might be used within linguistic variables in fuzzy systems. Examples have been given in which potentially useful terms do not adhere to each of these three properties, and a case study is presented to demonstrate the use of such non-regular membership functions in a fuzzy expert system.
We repeat our assertion in our previous ideas [128] that we are not cognitive scientists and are not arguing that the unusual membership shapes described in this work are how such concepts are internally represented at a cognitive level. Whether concepts can be non-convex at a cognitive level has been discussed by, for example, G¨ardenfors [132], in which he asserts that:
“most properties expressed by simple words in natural language can be anal- ysed as convex regions of a domain in a conceptual space” (our italics)
However, while he supports this (rather hedged) assertion with some examples, it remains far from proven. Whatever the reality at the cognitive level, we merely assert that non- regular fuzzy sets may be useful to consider when modelling human reasoning in a fuzzy system.
3.4. Discussion 74 terms can be used in a fuzzy logic system and they can perform together with regular membership functions. From these illustrations, we firmly believe that non-convex mem- bership functions such as MealTime featured in the Time (of day) variable are plausible, reasonable membership functions in the sense originally intended by Zadeh.
We are particularly interested in the role of linguistic variables, and their associated terms as used in the fuzzy inferencing process. Within the general category of inferencing (rule-based) systems there are two broad aspects: control systems and expert systems (emulating human reasoning). Although human reasoning has been investigated since the inception of fuzzy logic (e.g. [4, 133]), by far the majority of published work has been concerned with fuzzy control. Indeed, both the two main methods of implementing fuzzy inferencing, namely the Mamdani method and the Takagi-Sugeno method, were introduced to solve control applications [25, 26].
This historical bias towards the control domain has, we believe, led to a relative ne- glect of aspects of inferencing in the context of human decision making. Thus, there has been a tendency to restrict membership functions to well-known forms. Triangular, left- shoulder, right-shoulder and trapezoidal, or more generally piecewise linear, functions are common. Also used are standard Gaussian or Sigmoid type curves.
In the case study to illustrate the use of non-convex fuzzy sets, the shapes of terms used in fuzzy systems have adopted several ‘conventions’. Terms are almost invariably normalised (having a maximum membership value of 1), convex (having a single maxi- mum or plateau maxima) and distinct (being restricted in their degree of overlap: often expressed as some variation on the concept that all membership values at any point in the universe of discourse sum to 1 across that universe). The shape of these terms are generated by certain accepted membership functions: piecewise linear functions (with restrictions), Gaussians or Sigmoids are almost exclusively used. As such these consti- tute only a small subset of the total set of possible shapes of terms. These conventions are largely empirical or are justified by arguments based on what might loosely be called ‘fuzzy control principles’. However, in many applications involving the modelling of hu- man decision making (expert systems), these traditional membership functions may not provide a wide enough choice for the system developer. They are therefore missing an opportunity to, potentially, produce simpler or better systems. This work extends pre-
3.5. Summary 75 vious work in which it was suggested that non-convex membership functions might be considered for use in the context of fuzzy expert systems. In particular, the merits of non- convex fuzzy sets are discussed and a case study is presented which investigates whether is is possible to build an expert system featuring usual Mamdani style fuzzy inference in which a time-related non-convex fuzzy set is used together with traditional fuzzy sets. It is shown that this is indeed possible and an examination is made of the resultant output surface generated by four different sub-classes of non-convex membership functions.
3.5
Summary
In this Chapter, the use of non-standard membership functions to better model reasoning in a variety of complex domains, including when modelling human reasoning, has been described. It has been shown that the use of such membership functions has been limited in practice, for no good reason. It is concluded that non-convex membership functions are useful and their further use is encouraged.
In next chapter, type-1 and type-2 fuzzy systems with a varying number of tunable parameters are investigated. Their performance, in their ability to predict the Mackey- Glass time series with various levels of added noise, was compared. The concept of non- deterministic fuzzy reasoning is also presented and how to implement non-deterministic fuzzy sets is described.
Chapter 4
Investigating the Performance of Type-1
and Type-2 Fuzzy Systems for
Time-Series Forecasting
4.1
Introduction
As we mentioned in Chapter 2, many decision-making and problem solving tasks are too complex to be understood quantitatively, but by using knowledge that is imprecise rather than precise [1] and [9] it is possible to overcome this. Fuzzy logic resembles human reasoning in its use of approximate information and uncertainty to generate decisions. It was specifically designed to represent uncertainty and vagueness and provide formalized tools for dealing with the imprecision in many real problems. Since knowledge can be expressed more naturally by using fuzzy sets, many complex decision problems can be significantly simplified. Although many applications have been found for type-1 fuzzy logic systems, it is its application to rule-based systems that has most significantly shown its importance as a powerful design methodology, but yet it is unable to model and mini- mize the effect of all uncertainties. To overcome this limitation, type-2 fuzzy systems can be introduced as they can model uncertainties better and minimize their effects. Type-2 fuzzy systems are characterized by IF-THEN rules, but their antecedent or consequent sets are type-1 or type-2. A type-2 fuzzy set can represent and handle uncertain informa- tion effectively. More details about type-2 fuzzy sets and fuzzy systems can be found in
4.2. Time-Series Forecasting Using Type-1 Fuzzy Systems 77