Journal of Software Engineering and Applications
Vol.11 No.06(2018), Article ID:85233,14 pages
10.4236/jsea.2018.116018
Generalized Fuzzy Data Mining for Incomplete Information
Poli Venkata Subba Reddy
Department of Computer Science and Engineering, Sri Venkateswara University, Tirupati, India
Copyright © 2018 by author and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: March 1, 2018; Accepted: June 10, 2018; Published: June 13, 2018
ABSTRACT
Defining data with fuzziness made the knowledge discovery process easy and secure to data in data mining. The fuzzy data bases may have linguistic variables. In this paper, fuzzy conditional inference and reasoning are studied for generalized fuzzy data mining. Generalized fuzzy data mining and reasoning is studied with two membership functions “Belief” and “Disbelief”. The fuzzy logic with two membership functions will give more evidence than single membership function. The fuzzy certainty factor is studied as difference between these functions and made it as single membership function. The fuzzy data mining methods are studied. The generalized data mining is studied with different fuzzy conditional inferences. The business intelligence is given as an example.
Keywords:
Fuzzy Logic, Generalized Fuzzy Logic, Fuzzy Certainty Factor, Business Intelligence
1. Introduction
Zadeh [1] defined fuzzy set with single membership function. Zadeh [2] , Mamdani [3] and TSK [4] proposed fuzzy conditional inference. The main objective of fuzzy data mining is knowledge discovery process. The reasoning may be considered as one of the data mining technique during knowledge discovery process. The data mining with fuzzy databases will reduce the time and make easy to access for Big Data analysis. The fuzzy data mining may be dealt with linguistic variables. The generalized fuzzy data mining with two membership function will give more evidence. The fuzzy data mining and fuzzy reasoning made the knowledge discovery process easy through the overall observation and reasoning. The two membership functions shall be made as single fuzzy membership function with fuzzy certainty factor. The fuzzy certainty factor will give single membership as difference between two membership functions.
In the following, fuzzy conditional inference and reasoning are studied. Generalized fuzzy logic is discussed. The fuzzy certainty factor is studied as single membership function. The generalized fuzzy data mining and reasoning are studied.
2. Fuzzy Logic
Various theories are studied to deal with imprecise, inconsistent and inexact information and these theories deal with likelihood (probability) where as fuzzy logic with mind (commonsense). Zadeh [1] has introduced fuzzy set as a model to deal with incomplete information as single membership functions. The fuzzy set is a class of objects with a continuum of grades of membership. The set A of X is characterized by its membership function µA(x) and ranging values in the unit interval [0, 1]
For example, the fuzzy proposition “x is demand”
For instance “Item 1 has demand” and the fuzziness of “demand” is 0.8.
The Graphical representation of “demand” and “not demand” is shown in Figure 1.
The fuzzy logic is defined as combination of fuzzy sets using logical operators [1] . Some of the logical operations are given below.
Let A, B and C be fuzzy sets. The operations on fuzzy sets are given bellow.
Figure 1. Fuzzy membership function.
Negation
x is not A
Conjunction
x is A and y is B → (x, y) is AΛB
Disjunction
x is A and x is B→ (x, x) is AVB
Composition
The fuzzy propositions may contain quantifiers like “very”, “more or Less”. These fuzzy quantifiers may be eliminated as
Concentration
x is very A
Diffusion
x is very A
The fuzzy reasoning [2] is a drawing conclusion from fuzzy propositions using fuzzy inference rules.
Some of the fuzzy reasoning rules are given below.
R1: x is A
x and y are B
y is AΛB
R2: x is A
x or y are B
y is AVB
R3: x and y are A
y and z are B
y and z are A o B
R4: x is A1
if x is A then y is B
y is A1 o (A à B)
3. Fuzzy Conditional Inference
Zadeh [2] fuzzy conditional inference is given by
if x is A then y is B
if x is A and x is B then x is C
Mamdani [3] fuzzy conditional inference is given by
if x is A and x is B then x is C
TSK [4] fuzzy conditional inference is given by
if x is A then y= f(x) is B
if x1 is A1 and x2 is A2 and … and xn is An then y is B
where
The proposed fuzzy conditional inference using TSK is given by
The additive mapping f: R à R is called derivation if
t-norm is used in several fuzzy classification system
Substitute fuzzy sets A1 and A2 instead of x and y
The fuzzy conditional inference is given by
if x1 is A1 and x2 is A2 and … and xn is An then
where
,
(3.1)
Here is the “Consequent part” is given from “Precedent part” of the fuzzy rule.
Using Mamdani fuzzy conditional inference, the proposed fuzzy conditional inference is given by
if x1 is A1 and x2 is A2 ….. and xn is An then y is B
(3.2)
where .
Proposed fuzzy conditional inference give by
if x is A then y is B
Here is the fuzzy conditional inference is given for fuzzy rule.
The Mamdani [3] nested fuzzy conditional inference “if x is A then if y is B then z is C” is given by
if x is A then if y is B then z is C is equivalent to
if x is A and y is B then z is C
The proposed nested fuzzy conditional inference “if x is A then if y is B then z is C” is given by
The advantages of proposed fuzzy conditional inferences are:
It gives inference for consequent part;
It gives different fuzzy conational inference for fuzzy rule;
It gives different fuzzy conditional inference for nested fuzzy rule.
4. Fuzzy Certainty Factor
Zadeh [1] defined fuzzy set with single membership function. The generalized fuzzy logic is defending by two fold fuzzy set [5] . The two fold fuzzy set is a fuzzy set with two membership functions “belief” and “disbelief”.
The generalized fuzzy set simply as two fold fuzzy set and is defined by
In MYCIN [6] , the CF[h,e] is defined with MB[h,e] and MD[h,e],
where “e” is evidence and “h” is hypothesis and CF, MB and MD are probabilities.
The fuzzy certainty factor (FCF) is defined with fuzziness instead of probability.
The above are interpreted as redundant, insufficient and sufficient information respectively.
The FCF is a single membership function. The fuzzy logic and reasoning of FCF is applicable similar to the fuzzy logic with single membership function.
For instance
The graphical representation of FCF is shown in Figure 2.
Application to Fuzzy Conditional Inference
The business intelligence is needed to deal with incomplete information. Fuzzy logic deals with incomplete information. The proposed fuzzy conditional inference [7] is discussed for business intelligence.
The business intelligence needs commonsense. The fuzzy logic deals incomplete information with commonsense.
Consider Business fuzzy rule
If x is demand of the product then x is Price
Let x1, x2, x3, x4, x5 be the Items.
Consider Generalized fuzzy set
Figure 2. Fuzzy certainty factor.
Zadeh [1] [2] inference is given by
Mamdani [3] inference is given by
Proposed inference is given by
Zadeh [2] fuzzy reasoning is given by
Mamdani [3] fuzzy reasoning is given by
Proposed fuzzy reasoning is given by
Similarly the fuzzy quantifiers may be given as
Zadeh [2] fuzzy reasoning is given by
Momdani [3] fuzzy reasoning is given by
Proposed fuzzy reasoning is given by
5. Generalized Fuzzy Data Mining
The relational database is a Cartesian product of attributes and is represented as
or
The fuzzy relational database in Table 1 may be defined for Attributes
Where “+” is union, D is domain and ti are tupls.
The fuzzy quantifiers “very” and “more” are given by
Table 1. Fuzzy relational database.
It is shown in Table 2.
It is shown in Table 3.
1) Negation in Table 4.
2) Union in Table 5.
3) Intersection in Table 6.
4) Fuzzy Implication in Table 7.
Table 2. Fuzzy sales database.
Table 3. Fuzzy Price database.
Table 4. The negation of price.
Table 5. The union of sales and price.
Table 6. The intersection of Sales or Price.
Table 7. Fuzzy Implication sales ® price.
Table 8. Customers who purchased > 0.5.
5) Fuzzy frequency in Table 8.
Fuzzy frequency in Table 9 may be defined as
6) Fuzzy Association
The fuzzy functional dependency [8] FFD; X à Y or Y is depending on X is defined by
if then
if then
7) Fuzzy association in Table 10.
8) Fuzzy Clustering in Table 11.
Fuzzy sales database and Fuzzy Price database are shown in Table 12 and Table 13.
Table 9. Fuzzy frequency.
Table 10. Customers the items together purchased.
Table 11. Clustering of items purchased > 0.9.
Table 12. Fuzzy sales database.
Table 13. Fuzzy Price database.
6. Fuzzy Reasoning
The fuzzy reasoning is drawing conclusions.
Consider the fuzzy reasoning:
If x is A then y is B
x is more A
y is more A o (A ® B)
If x is sales then y is price
is more sales
y is more sales o (sales ® price)
It is shown in Tables 14-17.
Table 14. Fuzzy sales.
Table 15. Fuzzy price.
Table 16. More sales.
Table 17. Sales ® price.
Table 18. Fuzzy reasoning for price.
Zadeh fuzzy reasoning is given by
y is more sales o (sales ® price)
=min{more sales, min(1, 1-sales + price)}
Mamdani fuzzy reasoning is given by
y is more sales o (sales ® price)
=min{ more sales, min(sales, price)}
Proposed fuzzy reasoning is given by
yis more sales o (sales ® price)
=min{more sales,, sales)}
It is shown in Table 18.
Consider the nested fuzzy conditional inference for business intelligence:
If Demand then if Supply then increase price.
which is equivalent to:
If Demand and Supply then increase price.
The nested conditional fuzzy inference may be applied in fuzzy data mining similarly.
Acknowledgements
The author would like thank Editor-in-Chief, JSEA for accepting this paper.
Cite this paper
Reddy, P.V.S. (2018) Generalized Fuzzy Data Mining for Incomplete Information. Journal of Software Engineering and Applications, 11, 285-298. https://doi.org/10.4236/jsea.2018.116018
References
- 1. Zadeh, L.A. (1965) Fuzzy Sets. In Control, 8, 338-353. https://doi.org/10.1016/S0019-9958(65)90241-X
- 2. Zadeh, L.A. (1975) Calculus of Fuzzy Restrictions. In: Zadeh, L.A., King-Sun, F.U., Tanaka, K. and Shimura, M., Eds., Fuzzy Sets and Their Applications to Cognitive and Decision Processes, Academic Press, New York, 1-40. https://doi.org/10.1016/B978-0-12-775260-0.50006-2
- 3. Mamdani, E.H. and Assilian, S. (1975) An Experiment in Linguistic Synthesis with a Fuzzy Logic Control. International Journal of Man-Machine Studies, 7, 1-13. https://doi.org/10.1016/S0020-7373(75)80002-2
- 4. Takagi, T. and Sugeno, M. (1985) Fuzzy Identification of Systems and Its Application to Modelling and Control. Systems, Man and Cybernetics, 1, 194-207.
- 5. Reddy, P.V.S. (2015) Fuzzy Data Mining and Web Intelligence. International Conference on iFuuzy 2015, Taiwan, 18-20 November 2015, 74-79.
- 6. Buchanan, B.G. and Shortliffe, E.H. (1984) Rule Based Expert Systems: The MYCIN Experiments of the Stanford Heuristic Programming Project. Addison-Wesley, Reading, MA.
- 7. Reddy, P.V.S. (1993) Fuzzy Conditional Inference for Medical Diagnosis. Proceedings of Second International Conference on Fuzzy Theory and Technology, Summary FT&T 1993, 193-195.
- 8. Reddy, P.V.S. (2016) Fuzzy Map Reducing Algorithm for BIG DATA. iFuzzy 2016, IEEE Explore, 4-8.