An information system is a type of knowledge representation, and attribute reduction is crucial in big data, machine learning, data mining, and intelligent systems. There are several ways for solving attribute reduction problems, but they all require a common categorization. The selection of features in most scientific studies is a challenge for the researcher. When working with huge datasets, selecting all available attributes is not an option because it frequently complicates the study and decreases performance. On the other side, neglecting some attributes might jeopardize data accuracy. In this case, rough set theory provides a useful approach for identifying superfluous attributes that may be ignored without sacrificing any significant information; nonetheless, investigating all available combinations of attributes will result in some problems. Furthermore, because attribute reduction is primarily a mathematical issue, technical progress in reduction is dependent on the advancement of mathematical models. Because the focus of this study is on the mathematical side of attribute reduction, we propose some methods to make a reduction for information systems according to classical rough set theory, the strength of rules and similarity matrix, we applied our proposed methods to several examples and calculate the reduction for each case. These methods expand the options of attribute reductions for researchers.

Pawlak proposed Rough set theory (RST) in 1982 [

In the early 1980s, RST was introduced by Pawlak as a new mathematical tool for dealing with uncertainty and vagueness [

where Universe U and Attributes A are finite nonempty sets.

Set A consists of two distinct sets of attributes called: Condition C and Decision D attributes.

For every

and

where

The boundary region

If the boundary region is nonempty, then the set is rough; otherwise, the set is crisp. The ratio of lower- and upper-approximation is used to compute the approximation accuracy of the set X from the elementary subsets. Most of these concepts illustrated in

where

which is determined as the union of the equivalence classes of the relation

For every set of condition attributes

Two objects

In the following car information table, see

U/A | V | N | I | C |
---|---|---|---|---|

Medium | Medium | Fair | 5 | |

Medium | Medium | Fair | 4 | |

Medium | Medium | Good | 5 | |

Low | Low | Fair | 5 | |

Low | Medium | Fair | 2 | |

Medium | Low | Excellent | 4 | |

Low | Low | Good | 4 | |

Low | Low | Good | 2 |

We can obtain the indiscernibility relation from

1) First, we find the equivalence classes of all attributes dented by U/A.

2) Second, we find the equivalence classes for each attribute alone.

3) Third, we find the equivalence classes of double attributes only.

4) Fourth, we find the equivalence classes of triple attributes only.

From the previous relationships, we conclude that.

Therefore, attribute V or N can be dispensed with.

where

which is used for the reduction of condition attributes relative to decision attributes.

In the following Car decision table

U/A | Condition attributes | Decision attribute | |||
---|---|---|---|---|---|

V | N | I | C | D | |

Medium | Medium | Fair | 5 | Low | |

Medium | Medium | Fair | 4 | Low | |

Medium | Medium | Good | 5 | Low | |

Low | Low | Fair | 5 | Medium | |

Low | Medium | Fair | 2 | Medium | |

Medium | Low | Excellent | 4 | High | |

Low | Low | Good | 4 | High | |

Low | Low | Good | 2 | High |

We obtain the relative discernibility matrix from

U/U | C |
C |
C |
C |
C |
C |
C |
C |
---|---|---|---|---|---|---|---|---|

– | – | – | N,V | C,V | C,I,N | C,I,N,V | C,I,N,V | |

– | – | N,V | C,V | I,N | I,N,V | C,I,N,V | ||

– | I,N,V | C,I,V | C,I,N | C,N,V | C,N,V | |||

– | – | C,I,V | C,I | C,I | ||||

– | C,I,N,V | C,I,N | I,N | |||||

– | – | – | ||||||

– | – | |||||||

– |

Then;

=

=

We reduce

U/A | V | N | D |
---|---|---|---|

_{1} |
Medium | Medium | Low |

_{2} |
Medium | Medium | Low |

_{3} |
Medium | Medium | Low |

_{4} |
Low | Low | Medium |

_{5} |
Low | Medium | Medium |

_{6} |
Medium | Low | High |

_{7} |
Low | Low | High |

_{8} |
Low | Low | High |

To find the core of each example, we proceed with

U/A | N | V | D |
---|---|---|---|

_{1} |
Medium | Medium | Low |

_{2} |
Low | Low | Medium |

_{3} |
Medium | Low | Medium |

_{4} |
Low | Medium | High |

_{5} |
Low | Low | High |

U/A | N | V | D |
---|---|---|---|

_{1} |
Medium | Medium | Low |

_{2} |
* | Low | Medium |

_{3} |
* | Low | Medium |

_{4} |
Low | * | High |

_{5} |
Low | * | High |

We delete the repeated rows, as in

U/A | N | V | D |
---|---|---|---|

_{1} |
Medium | Medium | Low |

_{2} |
* | Low | Medium |

_{3} |
Low | * | High |

1) If N(Medium) and V(Medium)

2) If V(Low)

3) If N(Low)

Core =

Let _{1} (x), …,c_{n} (x), d_{1}(x)……d_{m}((x), where

Then the strength of rules is

where

From

The rules strength for attribute C may be found as follows:

(C = 5)

(C = 4)

(C = 5)

(C = 2)

(C = 4)

(C = 2)

The average of the strength of rules for attribute C = 50%

The rules strength for attribute I may be found as follows:

(I = Fair)

(I = Good)

(I = Fair)

(I = Excellent)

(I = Good)

The average of the strength of rules for attribute I = 60 %

The rules strength for attribute N may be found as follows:

(N = Medium)

(N = Low)

(N = Medium)

(N = Low)

The average of the strength of rules for attribute N = 75%

The rules strength for attribute V may be found as follows:

(V = Medium)

(V = Low)

(V = Medium)

(V = Low)

The average of the strength of rules for attribute V = 75%

U/A | V | N | D |
---|---|---|---|

Medium | Medium | Low | |

Medium | Medium | Low | |

Medium | Medium | Low | |

Low | Low | Medium | |

Low | Medium | Medium | |

Medium | Low | High | |

Low | Low | High | |

Low | Low | High |

Reduce

U/A | N | V | D |
---|---|---|---|

Medium | Medium | Low | |

Low | Low | Medium | |

Medium | Low | Medium | |

Low | Medium | High | |

Low | Low | High |

We find the core of

U/A | N | V | D |
---|---|---|---|

Medium | Medium | Low | |

* | Low | Medium | |

* | Low | Medium | |

Low | * | High | |

Low | * | High |

By removing the same rows again, we obtain

U/A | N | V | D |
---|---|---|---|

Medium | Medium | Low | |

* | Low | Medium | |

Low | * | High |

1) If N(Medium) and V(Medium)

2) If V(Low)

3) If N(Low)

The similarity matrix is considered a novel method reducing condition attributes. It is easy to use and produces more accurate results, depending on the deletion or dispensing of the attribute that has the least influence on decision making under specific conditions.

The ratio of the similarity between two objects denoted by

From

U/U | C |
C |
C |
C |
C |
C |
C |
C |
---|---|---|---|---|---|---|---|---|

1 | 1 | 1 | ||||||

1 | 1 | |||||||

1 | ||||||||

1 | ||||||||

1 | 1 | |||||||

1 | 1 | |||||||

1 | 1 | |||||||

1 | 1 | 1 |

From

U/IND(D) =

From

we get;

Then the degree of dependency is

By eliminating attribute C from

U/U | C |
C |
C |
C |
C |
C |
C |
C |
---|---|---|---|---|---|---|---|---|

1 | 1 | 0 | 0 | |||||

1 | 1 | 0 | 0 | |||||

1 | 0 | |||||||

0 | 1 | |||||||

1 | 0 | |||||||

0 | 1 | |||||||

0 | 0 | 1 | 1 | |||||

0 | 0 | 1 | 1 |

From

Then The degree of dependency is

Also, by eliminating attributes {I}, {N}, {V, {C, N}, {I, V}, {C, I}, {C, V}, {I, N}, {N, V} from

From previous calculations, we have;

Therefore, the reduct(A) =

We emphasised in our research the need of decreasing the size of the dataset before beginning any research and how rough set theory provides an effective approach for determining the minimal dataset’s reduct. We also discussed how the rough set may be unable to discover the minimal reduct by itself since doing so may need computing all combinations of attributes, which is not achievable in huge datasets. We proposed two methods to find the reduct of a dataset. One of them is the strength of rules which calculate the strength of rules for all attributes, and the other is similarity matrix, which is considered a novel method reducing condition attributes, it is easy to use and produce more accurate results.