Rough set

The rough set theory has some overlap with other methods handling vagueness and uncertainty, such as the Dempster-Shafter theory, which uses the belief func- tion as its main tool, while the rough set theory uses sets of lower and upper approximations. One of the advantages provided by the rough set theory is that there is no need for any preliminary or additional information about the data. For example, the probability distribution in statistics and basic probability assign- ment in the Dempster-Shafer theory are not needed in the rough set approaches. In the rough set theory, an information table is used to represent the input data of a domain in the real world, such as medicine, finance or transport. An ex- ample of the information table is shown in Table 2.2. Each row in the information table is called an ‘example’ (‘objects’ or ‘entities’). Properties of examples are perceived by assigning values to some variables. There are two kinds of variables: ‘attributes’ (sometimes called ‘condition attributes’) and ‘decisions’ (sometimes called ‘decision attributes’). Only a single decision is required in the informa- tion table. For example, in a hospital, patients are examples, while symptoms and tests are attributes, and diseases are decisions. Thus, each patient can be characterised by the results of tests and symptoms, and classified by disease.

The main concept of the rough set theory is indiscernibility, which is normally associated with a set of attributes. For example, let the set consist of headache and muscle pain in the information table shown in Table 2.2. Then e1, e2 and e3 are characterised by having the same value of these attributes, so that e1, e2

2.9. Rough set and granule mining 41

Attributes Decisions

Headache Muscle Pain Temperature Flu

e1 yes yes normal no

e2 yes yes high yes

e3 yes yes very high yes

e4 no yes normal no

e5 no no high no

e6 no yes very high yes

Table 2.2: An information table

and e3 are indiscernible from each other. Hence, the indiscernibility relation is an equivalent relation. Sets that are indiscernible are called ‘elementary sets’. In Table 2.2, the set of attributes of ‘headache’ and ‘muscle pain’ defines three elementary sets: {e1, e2, e3}, {e4, e6} and {e5}. Moreover, any finite union of elementary sets is called a ‘definable set’. For instance, {e1, e2, e3, e5} is a definable set defined by the attributes ‘headache’ and ‘muscle pain’ by setting the values of both attributes to ‘yes’ or ‘no’.

With the indiscernibility relation, redundant or dispensable attributes can be easily defined. If an attribute set and its super set define the same indiscerni- bility relation, then the attributes in the super set and those not in that set are redundant. For example, let ‘headache’ and ‘temperature’ be the set, and all three attributes be the super set. The elementary sets defined by the attribute set are all singletons–the same as the sets defined by the super set. Therefore, the attribute ‘muscle pain’ is redundant. Meanwhile, the attribute set Headache, Temperature has no redundant attribute because elementary sets of each attribute are not singletons. Such an attribute set is called ‘minimal’.

Attributes Decisions

Headache Temperature Flu

e1 yes normal no

e2 yes high yes

e3 yes very high yes

e4 no normal no

e5 no high no

e6 no very high yes

Table 2.3: Reduced information table

sets define the same indiscernibility relation. Set Headache, Temperature is a reduct of the original attribute set.

The elementary sets can also be defined by the decisions, and are called ‘con- cepts’. As shown in Table 2.3, there are two concepts defined by the decision ‘flu’, which represents whether the patient has the flu or not. Decision ‘flu’ depends on the attributes ‘headache’ and ‘temperature’ because all elementary sets of in- discernibility relation associated with this attribute set are subsets of the concept ‘flu’ in Table 2.3. Thus, one can determine whether the patient is sick by the rules from the table shown in Table 2.3. For example, (Temperature, normal) → (Flu, no) or (Headache, no) and (Temperature, high) → (Flu, yes).

The data shown in Table 2.3 are consistent because all elementary sets belong to some concepts. However, the data in Table 2.4 are inconsistent because {e5, e7} and {e6, e8} are not subsets of any concepts. That means that decision ‘flu’ does not depend on the attributes ‘temperature’ and ‘headache’. To solve this issue, the rough set theory introduces the lower approximation, which is the least definable set contained in each concept, X, and the upper approximation, which is the greatest definable set contained in X.

2.9. Rough set and granule mining 43

Attributes Decisions

Headache Temperature Flu

e1 yes normal no

e2 yes high yes

e3 yes very high yes

e4 no normal no

e5 no high no

e6 no very high yes

e7 no high yes

e8 no very high no

Table 2.4: Inconsistent information table

In Table 2.4, for the concept {e2, e3, e6, e7}, the lower approximation is {e2, e3} and the upper approximation is {e2, e3, e5, e6, e7, e8}, as shown in Figure 2.10. Sets such as {e5, e6, e7, e8} that contain elements from the upper approximation that are not in the lower approximation are called a ‘boundary region’. Thus, rough sets can be defined as sets that have non-empty boundary regions. With these tools, rules can be classified as follows: rules from the lower approximation are certainly valid, and rules from the upper approximation of the concepts are possibly valid.

The quality of the upper and lower approximation is used as the measure of uncertainty in the rough set theory. Given a set of examples, X, the quality of lower approximation is the ratio of the number of all elements in the lower approximation of X to the total number of X. It is the same for the upper ap- proximation. For concept X = {e1, e4, e5, e8} in Table 2.4, the quality of the upper approximation is 0.75(6 elements out of 8). The quality of both approxi- mations can also be calculated as the ratio of a number of certain (for lower) or possible (for upper) examples to the total number of examples in X. Thus, the

Figure 2.10: Lower and upper approximations [79]

quality is a relative frequency and this measure of quality is an objective belief function.