Model Based on Rough Set Theory

13.2 Model Based on Rough Set Theory

In 1982 Zdzisław Pawlak proposed an alternative, or rather complementary, approach [216] for representing vague notions. This approach, called rough set theory, has been developed considerably since then [218, 219, 220]. As we have mentioned at the beginning of this chapter, fuzzy set theory is used for solving the problem of ambiguity of notions, whereas in rough set theory vagueness of notions is considered

in the aspect of their degree of precision (detail, accuracy). 9 The degree of precision of characterization of a notion should be adequate for the problem considered. In rough set theory this adequacy is described as a feature of a system which allows it to distinguish between phenomena/objects which are considered to belong to different categories and not to distinguish between phenomena/objects which belong to the same category. Let us consider this feature with the help of the following example.

Let the domain containing objects to be considered, called the universe of dis- course U , consist of passenger cars. Let us assume that we analyze the problem of placing new car service stations for various car marques in various regions of Poland, such as the Warsaw region, the Cracow region, etc. For such an analysis we can define

a set of attributes describing objects 10 A = {marque _of_car, region_of_registration}. For every attribute we should define a set of possible values, i.e., the domain of the attribute . Then, for an object x belonging to the universe U an expression marque_of _car (x) =BMW denotes the fact that x is a BMW model. A partition of the universe

U into subareas determined by the given set of attributes 11 is shown in Fig. 13.6 a. We say that objects which belong to the same subarea (e.g., two BMW cars registered in the Warsaw region and marked with white dots in Fig. 13.6 a) are indiscernible objects

9 Rough set theory is discussed in this book only from the AI point of view. However, this theory is applied much more broadly and it is interpreted in a more general way in computer science.

10 Passenger cars are objects. 11 A set of attributes means a set of attributes together with their domains.

198 13 Defining Vague Notions in Knowledge-Based Systems

(a)

BMW Ford

Warsaw Cracow …

region region

(b)

sedan passenger

cars hatchback

pickup trucks

Poland

Germany

(c)

passenger cars

pickup trucks

Fig. 13.6 An example of controlling the level of generality of considerations with the help of a set of attributes

with respect to the given set of attributes. 12 A subarea of the universe determined in this way is called an elementary set. Elementary sets are called also knowledge

12 In fact, in rough set theory these two cars are not distinguishable. In order to distinguish between them one should introduce an additional attribute, e.g., the registration number. However, in the

context of our problem this is not necessary.

13.2 Model Based on Rough Set Theory 199 granules . They correspond to elementary notions, which can be used for defining

more complex notions, including vague notions. The set of attributes is used for controlling the degree of precision of the defi- nition of elementary notions. In other words, it is used for determining the degree of granularity of the universe of discourse. For example, let us consider a universe of cars and a manager responsible for sales. He/she wants to know the preferences of customers from the EU countries for different types of car (sedan, hatchback, etc.). Then, an adequate set of attributes should be defined as A = {type_of_car, country _of_registration}. In this case, a partition of the universe into elementary sets

is defined as shown in Fig. 13.6 b. As we can see, the degree of granularity of the universe is less than in the previous case, i.e., knowledge granules are bigger. This results from using more general attributes. Now, two BMW sedans from the Warsaw region are indiscernible from two Ford sedans from the Cracow region. Of course, we can go to a higher level of abstraction and consider types of cars with respect to

bigger markets, like the EU market or the US market, as shown in Fig. 13.6 c. Now, the granules are even bigger. This means that some cars which have been distinguishable previously are now indistinguishable.

In our example we have controlled the degree of precision of definitions of elemen- tary notions by determining more/less general attributes. In fact, we often control it by increasing/decreasing the number of attributes. The fewer attributes we use, the less the granularity. For example, if we remove the attribute marque_of_car in

Fig. 13.6 a, then the granularity of the universe is reduced. (Then, it is partitioned only into vertical granules, which contain all passenger cars in one region of Poland.) Now, we can consider the issue of defining vague notions in rough set theory. A vague notion X is defined by two crisp notions: a lower approximation BX and an upper approximation BX . A lower approximation BX is defined as a set B X which contains knowledge granules that necessarily are within the scope of a vague notion

X . An upper approximation BX is defined as a set B X which contains knowledge granules that possibly are within the scope of a vague notion X . Let us consider this way of representing a vague notion with the following exam- ple. Let us assume that we construct a system which selects ripe plums on the basis of two attributes: hardness and color. The vague notion ripe plum (R) is represented

by the set R shown in Fig. 13.7 a, which is determined with the help of ripe plums denoted by black dots. Unripe plums are denoted by white dots. The border of the set R is also marked in this figure. The area of the whole rectangle represents the universe of discourse U . After defining attributes together with their domains, the universe U

is divided into the knowledge granules shown in Fig. 13.7 b. The lower approximation BR is defined as the set B R, which contains knowledge granules marked with a grey color in Fig. 13.7 b. These granules contain only ripe plums. In Fig. 13.7 c the set B R, which corresponds to an upper approximation BR, is marked with a grey color. The set B R contains all ripe plums and also some unripe plums. However, using such a degree of granularity of the universe we are not able to distinguish between ripe and unripe plums for three granules. These three granules determine a boundary region,

which is marked with a grey color in Fig. 13.7 d. It contains objects which cannot be assigned to R in an unambiguous way.

200 13 Defining Vague Notions in Knowledge-Based Systems

(a)

color

(b)

violet

yellow green

(c)

(d)

yellow green soft

Fig. 13.7 An example of basic notions of rough set theory: a a universe of discourse with a set R marked, b a lower approximation of the set R, c an upper approximation of the set R, d the boundary region of the set R

Finally we can define a rough set. For a rough set a lower approximation is different from an upper approximation assuming a fixed granularity of the universe. In other words, the boundary region is not empty for a rough set. On the other hand, for a crisp (exact) set the lower approximation is equal to the upper approximation, i.e., its boundary region is the empty set.

Usually after defining a vague notion with the help of a lower and an upper approximation we would like to know how good our approximation is. There are several measurements of approximation quality in rough set theory. The coefficient of accuracy of approximation is one of them. It is calculated by dividing the number of objects belonging to the lower approximation by the number of objects belonging to the upper approximation. In our example it is equal to 4/10 = 0.4. Let us notice that this coefficient belongs to the interval [0, 1]. For a crisp set it equals 1.

The right choice of attributes is a very important issue in the application of rough set theory in AI. The number of attributes should be as small as possible (for reasons of computation efficiency), yet sufficient for a proper granulation of the universe. Methods for determining an optimum number of attributes have been developed in rough set theory.

13.2 Model Based on Rough Set Theory 201 Rough sets can be used, like fuzzy sets, for implementing rule-based systems.

Methods for automatic generation of rules on the basis of lower and upper approxi- mations have been defined as well [277].

Bibliographical Note Foundations of fuzzy set theory are presented in [63, 75, 257, 323]. Fuzzy logic and

fuzzy rule-based systems are discussed in [249].

A good introduction to rough set theory can be found in [71, 128, 217, 229, 257, 278].