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.

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 µ_A(x) and ranging values in the unit interval [0, 1]

μ A ( x ) : X → [ 0 , 1 ] ,   x ∈ X   where   “ + ”   is   union

For example, the fuzzy proposition “x is demand”

demand = 0.4 / x 1 + 0.5 / x 2 + 0.6 / x 3 + 0.8 / x 4 + 0.9 / x 5

notdemand = 0.6 / x 1 + 0.5 / x 2 + 0.4 / x 3 + 0.2 / x 4 + 0.1 / x 5

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.

Negation

x is not A

A ′ = 1 − μ A ( x ) / x

Conjunction

x is A and y is B → (x, y) is AΛB

A Λ B = min { μ A ( x ) , μ B ( y ) } / ( x , x )

Disjunction

x is A and x is B→ (x, x) is AVB

A V B = max { μ A ( x ) , μ B ( y ) } / ( x , x )

Composition

A   o   R = min { μ A ( x ) , μ R ( x , y ) } / x

The fuzzy propositions may contain quantifiers like “very”, “more or Less”. These fuzzy quantifiers may be eliminated as

Concentration

x is very A

μ very A ( x ) = μ A ( x ) ²

Diffusion

x is very A

μ moreorless A ( x ) = μ A ( x ) 0.5

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)

Zadeh [2] fuzzy conditional inference is given by

if x is A then y is B

A → B = min { 1 , 1 − μ A ( x ) + μ B ( x ) } / x

if x is A and x is B then x is C

= min { 1 , 1 − ( μ A ( x ) + μ B ( x ) ) + μ C ( x ) } / x

Mamdani [3] fuzzy conditional inference is given by

A → B = min { μ A ( x ) , μ B ( x ) } / x

if x is A and x is B then x is C

= min { ( μ A ( x ) , μ B ( x ) ) , μ C ( x ) } / x

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 x_n is A_n then y is B

where y = f ( x 1 , x 2 , ⋯ , x n ) .

The proposed fuzzy conditional inference using TSK is given by

The additive mapping f: R à R is called derivation if

f ( x + y ) = f ( x ) + f (y)

t-norm is used in several fuzzy classification system

t ( x + y ) ≤ max ( t ( x ) , t (y))

t ( x ∗ y ) ≤ min ( t ( x ) , t (y))

Substitute fuzzy sets A₁ and A₂ instead of x and y

t ( A 1 + A 2 ) ≤ max { ( t ( A 1 ) , t ( A 2 ) }

t ( A 1 ∗ A 2 ) ≤ min { t ( A 1 ) , t ( A 2 ) }

The fuzzy conditional inference is given by

if x₁ is A₁ and x₂ is A₂ and … and x_n is A_n then B = t ( A 1 , A 2 ⋯ , A n )

where

A 1 + A 2 = A 1 V A 2 ,

A 1 ∗ A 2 = A 1 Λ A 2

B = t ( A 1 , A 2 , ⋯ , A n ) = min ( A 1 , A 2 , ⋯ , A n )

B = min ( A 1 , A 2 , ⋯ , A n ) (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 x_n is A_n then y is B

= min { min ( A 1 , A 2 , ⋯ , A n ) , B }

= min ( A 1 , A 2 , ⋯ , A n ) (3.2)

where B = min ( A 1 , A 2 , ⋯ , A n ) .

Proposed fuzzy conditional inference give by

if x is A then y is B

= min ( μ A ( x ) , μ B (y))

= min ( μ A ( x ) , μ A ( x ) ) = { μ A ( x ) }

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

A → ( B → C ) = min { μ A ( x ) , min ( μ B ( y ) , μ C ( z ) ) } = min { μ A ( x ) , min ( μ B ( y ) , μ C ( z ) ) }

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

A → ( B → C ) = min { μ A ( x ) , min ( μ B ( y ) ) } = min { μ A ( x ) , min ( μ B ( y ) ) } = μ A (x)

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.

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

A = { μ A belief ( x ) , μ A disbelief ( x ) }

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.

μ A FCF ( x ) = μ A belief ( x ) − μ A disbelief (x)

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

demand = { 0.4 / x 1 + 0.5 / x 2 + 0.6 / x 3 + 0.8 / x 4 + 0.9 / x 5 ,       0.05 / x 1 + 0.1 / x 2 + 0.15 / x 3 + 0.2 / x 4 + 0.25 / x 5 } = 0.35 / x 1 + 0.4 / x 2 + 0.45 / x 3 + 0.6 / x 4 + 0.65 / x 5

The graphical representation of FCF is shown in Figure 2.

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

demand = { 0.3 / x 1 + 0.4 / x 2 + 0.5 / x 3 + 0.7 / x 4 + 0.8 / x 5 ,       0 / x 1 + 0 / x 2 + 0.5 / x 3 + 1 / x 4 + 1 / x 5 }

μ demand FCF ( x ) = 0.3 / x 1 + 0.4 / x 2 + 0.45 / x 3 + 0.6 / x 4 + 0.7 / x 5

price = { 0.4 / x 1 + 0.5 / x 2 + 0.6 / x 3 + 0.8 / x 4 + 0.9 / x 5 ,                         0 / x 1 + 0 / x 2 + 0 / x 3 + 1 / x 4 + 1 / x 5 }

μ price FCF ( x ) = 0.4 / x 1 + 0.5 / x 2 + 6 / x 3 + 0.7 / x 4 + 0.8 / x 5

Zadeh [1] [2] inference is given by

A → B = min { 1 , 1 − μ A ( x ) + μ B ( x ) }

μ demand → Price FCF ( x ) = 1.0 / x 1 + 1.0 / x 2 + 1.0 / x 3 + 1.0 / x 4 + 1.0 / x 5

Mamdani [3] inference is given by

A → B = min { μ A ( x ) , μ B ( x ) }

μ demand → Price FCF ( x ) = 0.3 / x 1 + 0.4 / x 2 + 0.45 / x 3 + 0.6 / x 4 + 0.7 / x 5

Proposed inference is given by

A → B = { μ A ( x ) }

μ demand → Price FCF ( x ) = 0.3 / x 1 + 0.4 / x 2 + 0.45 / x 3 + 0.6 / x 4 + 0.7 / x 5

verysmalldemand = { 0.09 / x 1 + 0.16 / x 2 + 0.20 / x 3 + 0.36 / x 4 + 0.49 / x 5 }

Zadeh [2] fuzzy reasoning is given by

verysmalldemand o demand → price = { 0.09 / x 1 + 0.16 / x 2 + 0.20 / x 3 + 0.36 / x 4 + 0.49 / x 5 }       o   { 1.0 / x 1 + 1.0 / x 2 + 1.0 / x 3 + 1.0 / x 4 + 1.0 / x 5 } = { 0.09 / x 1 + 0.16 / x 2 + 0.20 / x 3 + 0.36 / x 4 + 0.49 / x 5 }

Mamdani [3] fuzzy reasoning is given by

verysmalldemand o demand → price = { 0.09 / x 1 + 0.16 / x 2 + 0.20 / x 3 + 0.36 / x 4 + 0.49 / x 5 }       o   { 0.3 / x 1 + 0.4 / x 2 + 0.45 / x 3 + 0.6 / x 4 + 0.7 / x 5 } = { 0.09 / x 1 + 0.16 / x 2 + 0.20 / x 3 + 0.36 / x 4 + 0.49 / x 5 }

Proposed fuzzy reasoning is given by

verysmalldemand o demand → price = { 0.09 / x 1 + 0.16 / x 2 + 0.20 / x 3 + 0.36 / x 4 + 0.49 / x 5 }       o   { 0.3 / x 1 + 0.4 / x 2 + 0.45 / x 3 + 0.6 / x 4 + 0.7 / x 5 } = { 0.09 / x 1 + 0.16 / x 2 + 0.20 / x 3 + 0.36 / x 4 + 0.49 / x 5 }

Similarly the fuzzy quantifiers may be given as

moredemand = { 0.55 / x 1 + 0.63 / x 2 + 0.67 / x 3 + 0.77 / x 4 + 0.84 / x 5 }

Zadeh [2] fuzzy reasoning is given by

verysmalldemand o demand → price = { 0.55 / x 1 + 0.63 / x 2 + 0.67 / x 3 + 0.77 / x 4 + 0.84 / x 5 }       o   { 1.0 / x 1 + 1.0 / x 2 + 1.0 / x 3 + 1.0 / x 4 + 1.0 / x 5 } = { 0.55 / x 1 + 0.63 / x 2 + 0.67 / x 3 + 0.77 / x 4 + 0.84 / x 5 }

Momdani [3] fuzzy reasoning is given by

verysmalldemand o demand → price = { 0.55 / x 1 + 0.63 / x 2 + 0.67 / x 3 + 0.77 / x 4 + 0.84 / x 5 }       o   { 0.3 / x 1 + 0.4 / x 2 + 0.45 / x 3 + 0.6 / x 4 + 0.7 / x 5 } = { 0.3 / x 1 + 0.4 / x 2 + 0.45 / x 3 + 0.6 / x 4 + 0.7 / x 5 }

Proposed fuzzy reasoning is given by

verysmalldemand o demand → price = { 0.55 / x 1 + 0.63 / x 2 + 0.67 / x 3 + 0.77 / x 4 + 0.84 / x 5 }       o   { 0.3 / x 1 + 0.4 / x 2 + 0.45 / x 3 + 0.6 / x 4 + 0.7 / x 5 } = { 0.3 / x 1 + 0.4 / x 2 + 0.45 / x 3 + 0.6 / x 4 + 0.7 / x 5 }

The relational database is a Cartesian product of attributes and is represented as

R = A 1 × A 2 × ⋯ × A n

or

t i = ( d i 1 , d i 2 , ⋯ , d i i n ) , d i j ∈ A i

R ( A 1 , A 2 , ⋯ , A n )

The fuzzy relational database in Table 1 may be defined for Attributes

R = { t , μ d FCF ( t ) }

μ d FCF ( x ) = μ d belief ( x ) − μ d disbelief (x)

μ D ( r ) = μ d ( t 1 ) + μ d ( t 2 ) + ⋯ + μ d (tn)

Where “+” is union, D is domain and t_i are tupls.

1 − C = 1 − μ C ( x )         Negation

C V D = max { μ C ( x ) ⋅ μ D ( x ) }       Disjunction

C Λ D = min { μ C ( x ) ⋅ μ D ( x ) }       Conjunction

C → D = min { 1 , 1 − μ C ( x ) + μ D ( x ) }       Implication

C 1   o   C → D = min { C 1 , C → D }       Composition

The fuzzy quantifiers “very” and “more” are given by

μ very d ( r ) = { μ very d ( r ) } 2

μ more d ( r ) = { μ more   d ( r ) } 0.5

sales = ( 0.5 − 0.1 ) = 0.4 / 40 + ( 0.6 − 0.1 ) = 0.5 / 50 + ( 0.7 − 0.1 ) = 0.6 / 60 + ( 0.9 − 0.1 ) = 0.8 / 80 + ( 1.0 − 0.1 ) = 0.9 / 100

It is shown in Table 2.

price = ( 0.5 − 0.1 ) = 0.4 / 40 + ( 0.6 − 0.1 ) = 0.5 / 50 + ( 0.7 − 0.1 ) = 0.6 / 60 + ( 0.9 − 0.1 ) = 0.8 / 80 + ( 1.0 − 0 ) = 1.0 / 100

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.

5) Fuzzy frequency in Table 8.

Fuzzy frequency in Table 9 may be defined as

Frequency = 0.2 / 1 + 0.2 / 3 + 0.35 / 3 + 0.45 / 4 + 0.45 / 5

6) Fuzzy Association

The fuzzy functional dependency [8] FFD; X à Y or Y is depending on X is defined by

if E Q ( t 1 ( X ) , t 2 ( X ) ) then E Q ( t 1 ( Y ) , t 2 (y))

if F A ( t 1 ( X ) , t 2 ( X ) ) then F A ( t 1 ( Y ) , t 2 ( Y ) ) = min ( t 1 ( Y ) , t 2 ( Y ) )

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.

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.

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.

The author would like thank Editor-in-Chief, JSEA for accepting 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