1. Rough Sets Theory
Speaker：Kun Hsiang

2. Outline Introduction Basic concepts of the rough sets theory
Decision table
Main steps of decision table analysis
An illustrative example of application of the rough set approach
Discussion

3. Introduction Often, information on the surrounding world is
Imprecise
Incomplete
uncertain.
We should be able to process uncertain and/or incomplete information.

4. Introduction
When dealing with inexact, uncertain, or vague knowledge, are the fuzzy set and the rough set theories.

5. Introduction Fuzzy set theory Rough set theory
Introduced by Zadeh in 1965 [1]
Has demonstrated its usefulness in chemistry and in other disciplines [2-9]
Rough set theory
Introduced by Pawlak in 1985 [10,11]
Popular in many other disciplines [12]

6. Introduction
Fuzzy Sets

7. Introduction Rough Sets
In the rough set theory, membership is not the primary concept.
Rough sets represent a different mathematical approach to vagueness and uncertainty.

8. Introduction Rough Sets
The rough set methodology is based on the premise that lowering the degree of precision in the data makes the data pattern more visible.
Consider a simple example. Two acids with pKs of
respectively pK 4.12 and 4.53 will, in many contexts,
be perceived as so equally weak, that they are
indiscernible with respect to this attribute.
They are part of a rough set ‘weak acids’ as compared to ‘strong’ or ‘medium’ or whatever other category, relevant to the context of this classification.

9. Basic concepts of the rough sets theory
Information system
IS=(U,A)
U is the universe ( a finite set of objects, U={x1,x2,….., xm}
A is the set of attributes (features, variables)
Va is the set of values a, called the domain of attribute a.

10. Basic concepts of the rough sets theory
Consider a data set containing the results of three
measurements performed for 10 objects. The results
can be organized in a matrix10x3.
Measurements
Objects

11. Basic concepts of the rough sets theory
IS=(U,A)
U={x1,x2,x3,x4,x5,x6….., x10}
A={a1,a2,a3}
The domains of attributes are：
V1={1,2,3}
V2={1,2}
V3={1,2,3,4}

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If X is crisp with respect to B.
If X is rough with respect to B.

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If the set of attributes is dependent, one can be interested in finding all possible minimal subsets of attributes.
The concepts of core and reduct are two fundamental concepts of the rough sets theory.
Keep only those attributes that preserve the indiscernibility relation and, consequently, set approximation.
There are usually several such subsets of attributes and those which are minimal are called reducts.

25. Basic concepts of the rough sets theory
To compute reducts and core, the discernibility matrix is used.
The discernibility matrix has the dimension nxn.
where n denotes the number of elementary sets and its elements are defined as the set of all attributes which discern elementary sets [x]i and [x]j .

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Discerns set 1 from both set 2 and set 3
Discerns set 1 from sets 2,3,4 and set 5

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The discernibility function has the following form:

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Classification：

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Decision Table：

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51. An illustrative example of application of the rough set approach

52. An illustrative example of application of the rough set approach

53. An illustrative example of application of the rough set approach

54. An illustrative example of application of the rough set approach

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56. An illustrative example of application of the rough set approach

57. Discussion