Rough Data Set Based Applications in Framing Decision Rules

Rough set theory is used for dealing with uncertainty in the hidden pattern of data. This paper outlines concepts of the rough set theory for lower and upper approximations, reduction of attributes and decision rules. It assumes that the information about the real world is given in the form of an information table which represents input data, gathered from any domain, such as, medicine, finance or the military. In this study we have acquired the data from the real estate domain and framed some possible and certain rules to make the decision related to the price of the house. nonempty sets called the universe, and the set of attributes, respectively. Set A will contain two disjoint sets of attributes called condition and decision attributes and the system is donated by S = (U, C, D) where C is called condition attribute and D is called decision attribute. With every attribute aє A we associate a set Va, of its values, called the domain of a. Now we define two approximations ( ) P X and ( ) P X called the P-lower and the P-upper approximation of X respectively where ( ) P X = x U ∈  { } ( ) : ( ) P x P x X ⊆ and ( ) P X = x U ∈  { } ( ) : ( ) . P x P x X ∩ ≠ φ Lower approximation will consist of all the members which surely belongs to the set and Upper approximation consist of all the members which possibly belongs to the set. The boundary region is given by the set difference ( ) P X ( ) P X consists of those objects that can neither be ruled in nor ruled out as members of the target set X. If the boundary region is empty i.e ( ) P X = ( ) P X then the set is crisp otherwise the set is rough. Information Table in Rough Set In this article we will assume that the information about the real world is given in the form of an information table. Thus, the information table represents input data, gathered from any domain, such as medicine, finance or the military. An example of such an information table is given in Table 1. Rows of a table, labeled e1, e2, e3, e4, e5 and e6 in Table 1, are called examples (objects, entities). Properties International Journal of Computer Science & Communication (IJCSC) 56 of examples are perceived through assigning values to some variables. We will distinguish between two kinds of variables: attributes (sometimes called condition attributes) and decisions (sometimes called decision attributes). For example, if the information table describes a hospital, the examples may be patients; the attributes, symptoms and tests and the decisions, diseases. Each patient is characterized by the results of tests and symptoms and is classified by the physicians (experts) as being on some level of disease severity. Here we are considering the data from the real estate. The attributes are the various parameters of a house and the decision is Price of house. Table 1 Information Table Condition Attributes Location Fireplace Basement e1 bad yes yes e2 good yes yes e3 v_good yes yes e4 bad yes no e5 good no no e6 v_good yes no Indiscernibility Relation The main concept of rough set theory is an indiscernibility relation, normally associated with a set of attributes for example the set consisting of attributes Basement and fireplace from Table1. Examples e1 and e2 are characterized by the same values of both attributes, for the attribute Basement the value is yes for e1 and e2 and for the attribute fireplace the value is yes for both e1 and e2. Moreover, example e3 is indiscernible from e1 and e2. Examples e4 and e6 are also indiscernible from each other. Obviously, the indiscernibility relation is an equivalence relation. Sets that are indiscernible are called elementary sets. Thus, the set of attributes Basement and fireplace defines the following elementary sets: {e1, e2, e3}, {e4,e6}, and {e5}. Any finite union of elementary sets is called a definable set. In our case, set {e1, e2, e3, e5} is definable by the attributes basement and fireplace, since we may define this set by saying that any member of it is characterized by the attribute Basement equal to yes and the attribute fireplace equal to yes or by the attribute Basement equal to no and the attribute fireplace equal to no. With the help of indescernibility relation, we can define redundant or dispensable attributes. Table2 is called Decision Table because this table is having a Decision attribute. Usually decision is a single attribute. The decisions are the prices of houses according to the values of various condition attributes. Decision Table 2 Location Fireplace Basement Decision Price e1 bad yes yes Low e2 good yes yes High e3 v_good yes yes High e4 bad yes no Low e5 good no no Low e6 v_good yes no High If a set of attributes and its superset define the same indiscernibility relation (i.e. if elementary sets of both relations are identical), then any attribute that belongs to the superset and not to the set is redundant. In the example from Table 2, let the set of attributes be the set {Location, Basement} and its superset be the set of all three attributes, i.e. the set { Location, Fireplace, Basement }. Elementary sets of the indiscernibility relation defined by the set { Location, Basement } are singletons, i.e., sets {e1}, {e2}, {e3}, {e4}, {e5}, and {e6}, and elementary sets of the indiscernibility relation defined by the superset { Location, Fireplace, Basement } are also {e1}, {e2}, {e3}, {e4}, {e5}, and {e6} . Here the attribute Fireplace belongs to the superset and does’t belong to the set. Thus, the attribute Fireplace is redundant. On the other hand, the set {Location, Basement} does not contain any redundant attribute, since elementary sets for attribute sets {Location} and {Basement} are not singletons. Such a set of attributes, with no redundant attribute, is called minimal (or independent).