FUZZY MODEL BUILDING USING PROBABILISTIC RULES
Manish Agarwal
1
, K. K. Biswas
1
and Madasu Hanmandlu
2
1
Department of Computer Science, Indian Institute of Technology, New Delhi, India
2
Department of Electrical Engineering, Indian Institute of Technology, New Delhi, India
Keywords: Probabilistic fuzzy rules, Probability, Possibility, Decision making, Modelling.
Abstract: Uncertainty in the attributes and uncertainty in frequency of their occurrences are inherent to the real world
problems and an attempt is made here to tackle them together. The possible connections between the two
facets of uncertainty are explored and discussed. This paper also looks at the role of possibility and
probability in the context of decision making and in the process utilizes the existing fuzzy models by
incorporating the multiple probabilistic outputs in the associated fuzzy rules. This is needed to obtain the net
conditional possibility from the probabilistic fuzzy rules where the probabilistic information of the outputs
is given. A novel approach is devised to compute net conditional possibility from the given probabilistic
rules. The basis for extending the existing fuzzy models is also presented using the computed net
conditional possibility. The enhanced fuzzy models accruing from the addition of the probabilistic
information would usher in better decision making. The proposed approach is demonstrated through two
case-studies.
1 INTRODUCTION
Zadeh (1978) first coined the term possibility to
represent the imprecision in information. This
imprecision is quite different from the frequentist
uncertainty represented by well developed
probabilistic approach. But if we could appreciate
the real world around us, there is a constant interplay
between probability and possibility–even though the
two represent different aspects of uncertainty.
Hence, if not all, in many a situation, the two are
intricately interwoven in the linguistic representation
of a situation or an event by a human brain. And
often, it is possible to infer probabilistic information
from possibilistic one and vice versa. Even though
they are dissymmetrical and treated differently in
literature, there is a need to make an effort towards
exploring a unifying framework for their integration.
We feel that these two different, yet complimentary
formalisms can better represent practical situations,
going hand in hand.
Besides the vast potential of this study in more
closely representing the real world, we are also
motivated by its roots in philosophy. Non-
determinism is almost a constant feature in nature,
and together probability and possibility can go
farther in representing the real world situations.
Even though, probability and possibility represent
two different forms of uncertainty and are not
symmetrical, but still both are closely related, and
often needs to be transformed into one other, to
achieve computational simplicity and efficiency.
This transformation would pave the way for simpler
methods for the computation of net possibility. The
intelligent controllers utilizing these transformations
would represent the requirements and situations of
the real world more truly and accurately. They
would also be more computationally efficient in
terms of speed, storage and accuracy in processing
of the uncertain information.
Such transformations bridge two different facets
of uncertainty – statistical/probabilistic and
imprecision (on account of vagueness or lack of
knowledge). (Dubois et al., 1992; 1993) analyzed
the transformations between the two and judged the
consistency in the two representations.
This paper is concerned with devising a novel
approach for application of some of the research
results to the field of fuzzy theory under
probabilistic setting, and using the same to enhance
the existing fuzzy models to better infer the value of
possibility in the light of probabilistic information
available. It also relooks at the relevant results along
with their interpretations in the context of
361
Agarwal M., K. Biswas K. and Hanmandlu M..
FUZZY MODEL BUILDING USING PROBABILISTIC RULES.
DOI: 10.5220/0003616003610369
In Proceedings of the International Conference on Evolutionary Computation Theory and Applications (FCTA-2011), pages 361-369
ISBN: 978-989-8425-83-6
Copyright
c
 2011 SCITEPRESS (Science and Technology Publications, Lda.)

probabilistic fuzzy theory. This paper basically
addresses the following issues:
1. To amalgamate the field of fuzzy theory with the
probability theory and to discover the possible
linkages or connections between these two facets of
uncertainty.
2. To apply the probabilistic framework on the
existing fuzzy models for imparting the practical
utility to them.
3. To devise an approach to calculate the output of
the probabilistic fuzzy models.
4. To study the effect of probabilistic information
on the defuzzified outputs of fuzzy rules.
The paper is organized as follows: In Section 2,
relationship between probability and possibility is
explored by identifying the body of work in this
field and giving it a new look. This section also
gives the preliminaries needed for the paper. In
Section 3, a few mathematical relations are
presented in order to calculate the output of
probabilistic fuzzy rules (PFRs). The utility and
advantages of (PFR) are also discussed. Section 4
discusses an algorithm to compute net conditional
possibility from probabilistic fuzzy rules. In sections
5 and 6, two case studies are taken up to illustrate
the algorithm. Finally, Section 7 gives the
conclusions and the scope of further research in the
area.
2 PROBABILITY AND
POSSIBILITY: A RELOOK
The possible links between the two facets of
uncertainty: probability and possibility are explored
on the basis of the key contributions in the area.
The celebrated example of Zadeh (1978) “Hans
ate X eggs for Breakfast” illustrates the differences
and relationships between probability and possibility
in one go. The possibility of Hans eating 3 eggs for
breakfast is 1 whereas the probability that he may do
so might be quite small, e.g. 0.1. Thus, a high degree
of possibility does not imply a high degree of
probability; though if an event is impossible it is
bound to be improbable. This heuristic connection
between possibility and probability may be called
the possibility/probability consistency principle,
stated as: If a variable x takes values u
1
, u
2,
..., u
n
with respective possibilities ∏ = (π
1
, π
2,
..., π
n
) and
probabilities P= (p
1
, p
2
,.., p
n
) then the degree of
consistency of the probability distribution P with the
possibility distribution II is expressed by the
arithmetic sum as
γ = π
1
p
1
+ π
2
p
2
+... + π
n
p
n
Note that the above principle is not a precise law or
a relationship that is intrinsic to the concepts of
possibility and probability; rather it is an
approximate formalization of the heuristic
observation that a lessening of the possibility of an
event tends to lessen its probability, not vice-versa.
In this sense, the principle is applicable to situations
in which we know the possibility of a variable x
rather than its probability distribution. This principle
forms the most conceptual foundation of all the
works in the direction of probability/possibility
transformations having wide practical applications
Roisenberg (2009).
Having deliberated on the consistency principle,
we will look into: (i) Basic difference between
possibility and probability, (ii) Inter-relation
between possibility and probability and vice-versa,
(iii) Infer probability from possibility and vice-versa,
and (iv) Transformation of probability to possibility
and vice-versa, with a view to tackle real life
problems involving both probabilistic and
possibilistic information.
2.1 Basic Difference between
Possibility and Probability
In the perspective of example given by Zadeh,
possibility is the degree of ease with which Hans
may eat u eggs whereas probability is the chances of
actual reality; there may be significant difference
between the two. This difference is now elucidated
by noting that the possibility represents ‘likelihood’
of a physical reality with respect to some reference
whereas the probability represents the occurrences
of the same. To put it mathematically,

(

)
≜



()
(1)
where
A is a non fuzzy subset of U
II is possibility distribution of x
π (A) denotes the possibility measure of A in [0,1]
π
x
(u) is the possibility distribution function of ∏
x
.
Let A and B be arbitrary fuzzy subsets of U. In view
of (1), we can write that

(


) =

(

) 

()
(2)
The corresponding relation for probability is written
as
(


)

(

) +  ()
(3)
FCTA 2011 - International Conference on Fuzzy Computation Theory and Applications
362

2.2 Inter-relation between Possibility
and Probability
Any pair of dual necessity/possibility functions (Ν,
∏ ) can be interpreted as the upper and lower
probabilities induced from specific convex sets of
probability functions.
Let π be a possibility distribution inducing a pair
of functions [N, ∏]. Then we define

(

)
=

,∀ ,
(

)
≤()

=

,∀ ,
(

)
≤Π()

The family, 
(

)
, is entirely determined by the
probability intervals it generates. Any probability
measure  ∈ 
(

)
is said to be consistent with the
possibility distribution, π (Dubois, 1992); (De
Cooman, 1999). That is
sup
 ∈ 
(

)

(

)
= Π
(

)
(4)
A relevant work in this direction was carried out in
Walley (1999). It is shown that the imprecise
probability setting is capable of capturing fuzzy sets
representing linguistic information.
2.3 Inference of Probability from
Possibility and Vice-versa
In Zadeh (1978), Dubois (1982, 1992,1993), degrees
of possibility can be interpreted as the numbers that
generally stand for the upper probability bounds.
The probabilistic view is to prepare interpretive
settings for possibility measures. This enables us to
deduce a strong interrelation between the two. This
principle basically implies the following inferences:
High Probability → High Possibility
Low Probability ↛Low Possibility
Zero Possibility → Zero Probability
Zero Probability ↛ Zero Possibility
High Possibility → High Probability
Low Possibility ↛ Low Probability
(5)
From Klir (2000) and from the above properties of
possibility and necessity measures, we know that
maximizing the degree of consistency brings about
two strong restrictive conditions having a strong
coherence: cloudiness is directly pointing at more
probability of rain.
2.4 Transformation from Probability
to Possibility
Any transformation from probability to possibility
must comply with the following three basic
principles as in (Dubois, 1993).
1.Possibility-probability consistency:
γ
= π
1
p
1
+ π
2
p
2
+... +
π
n
p
n
2.Ordinal faithfulness: π (u) > π (u′) iff p (u) > p (u′)
3.Informativity: Maximization of information content of π
(6)
If P is a probability measure on a finite set U,
statistical in nature then, for a subset, E of U, its
possibility distribution on U, π
E
(u) is given by
(Dubois, 1982):


(
u
)
=
1   ∈,
1−
(

)
ℎ,
(7)
Also ∏
E
(A) ≥ P (A), ∀A ⊆ U
In other words, π
E
= x ∈ E with the confidence at
least P (E). In order to have a meaningful possibility
distribution, π
E
, care must be taken to balance the
nature of complimentary ingredients in (7), i.e. E
must be narrow and P (E) must be high.
There are quite a few ways, in which one can do
it. The one used in Dubois (1982) chooses a
confidence threshold α so as to minimize the
cardinality of E such that P (E) ≥ α. Conversely,
cardinality of E can be fixed and P (E) maximized.
This way, a probability distribution P can be
transformed into a possibility distribution π
P
(Dubois, 1982). Take p
i
as the probability
distribution on U and X = {x
1
, x
2
,.., x
n
} such that p
i
= P ({x
i
}). Similarly possibility distribution π
i
= ∏
({x
i
}) and p
1
≥ p
2
≥ ... ≥ p
n
, then we have
π


(u)= 



∀i = 1,n
(8)
For a continuous case, if the probability density
function so obtained is continuous unimodal having
bounded support [a, b], say p, then p is increasing
on [a, x
0
] and decreasing on [x
0
, b], where x
0
is the
modal value of p. This set is denoted as D in Dubois
(1982).
Let p be the probability density function (pdf) in
D such that a function f: [a, x
0
]  [x
0
, b] is defined
as f(x) = max {y| p(y) >= p(x)}. Then the most
specific possibility distribution  (minimizing the
integral of  on [a, b]) that dominates p is defined by

(

)
=
(

)
= 
(

)
 +  
(

)



(

)


(9)
3 PROBABILISTIC FUZZY
MODELING
A probabilistic fuzzy rule (PFR), first devised by
(Meghdadi, 2001), is an appropriate tool to represent
a real world situation possessing both the features of
uncertainty. In such cases, we often observe that for
FUZZY MODEL BUILDING USING PROBABILISTIC RULES
363

a set of inputs, there may be more than one possible
output. The probability of occurrence of the outputs
may be context dependent. In a fuzzy rule, there
being only a single output, we are unable to
accommodate this feature of the real world –
multiple outputs with different probabilities. This
ability is afforded with PFR. The PFR with multiple
outputs and their probabilities is defined as:
Rule R
q
:
If x is Aq then y is O
1
with probability P
1
& ...
& y is O
j
with probability P
j
& ...
& y is O
q
with probability P
n
Ρ = [P
1
, P
2
, P
1
, P
3
, P
4
, .., P
n
],
with P
1
+ P
2
+ P
1
+ P
3
+ P
4
+...... + P
n
= 1
(10)
Given the occurrence of the antecedent (an event) in
(10), one of the consequents (output) would occur
with the respective probability of occurrence, Ρ.
Therefore, y is associated with both qualitative (in
terms of membership function, O) and quantitative
(in terms of probability of occurrence, P)
information. Therefore y is both a stochastic and
fuzzy variable at the same time. The real outcome is
a function of the probability, while the quality of an
outcome is a function of the respective membership
function. The probability of an event is having a
larger role to play since it is the one that determines
the occurrence of the very event. More the
probability of an output event, more are the chances
of its certainty which in turn gives rise to the
respective possibility of the event (in terms of
membership function) determining the quality of the
outcome.
The above example illustrates the fact that both
these measures of uncertainty (probability and
possibility) are indispensable in fuzzy modelling of
real world multi-criteria decision making, and may
lead to incomplete and misleading result if one of
them is ignored. So the original fuzzy set theory, if
backed by probability theory could go miles in better
representing the decision making problems and
deriving realistic solutions.
Here, one question that naturally arises is: how
about treating probabilities in the antecedents? This
aspect is taken into account by having more than one
fuzzy rule and probabilistic outcome in the
consequent which is sufficient to handle the
frequentist uncertainty in the probabilistic fuzzy
event. For example in (10), the antecedent could be:
If x
1
is µ
1
and x
2
is µ
2.
Now, the range of probable values of occurrence of
inputs is either Input
1
or Input
n
etc. Thus for each
occurrence of an antecedent condition, there is a
corresponding probabilistic fuzzy consequent event
in (9).
As per the scope of this paper, we would be
considering similar PFRs with the same structure for
a probabilistic fuzzy system under consideration.
That is, any two PFRs would have the same order of
probabilistic outputs.
 ,,  ∶ 


= 



= 

where,
q and q′ represent two PFRs 

and 





is j
th
output in q
th
rule;




is j
th
output in q′
th
rule


is j
th
output that remains the same in any PFR
The mathematical framework follows from (Van den
Berg, 2002).Assuming two sample spaces, say X
and Y, in forming the fuzzy events A
i
and O
j
respectively, the following equations hold good,
∀: 


(

)
=1,∀:


(

)
=1
(11)
If the above conditions are satisfied then X and Y
are said to be well defined.
3.1 Input Conditional Probabilities of
Fuzzy Antecedents
Given a set of S samples (x
s
, y
s
), s = 1,.., S from two
well-defined sample spaces X, Y, the probability of
A
i
can be calculated as

(


)
=



=




=
1



(




)= ̂

(12)
where,
A
i:
is the antecedent fuzzy event, which leads to one
of the consequent events O
1, ..,
O
n
to occur.


Ai
: Relative Frequency of fuzzy sample values μ
i
(x
s
) for the fuzzy event A
i
ƒ
Ai
: Absolute Frequency of fuzzy sample values μ
i
(x
s
) for the fuzzy event A
i
The fuzzy conditional probability is given by,




)=
(

∩

)

(


)
≈
∑


(


)


(

)

∑


(


)

(13)
The density function, p
j
(y) can be approximated
using the fuzzy histogram [11] as follows:


() =




()



()


(14)
where denominator




j
(y)dy is a scaling factor.
FCTA 2011 - International Conference on Fuzzy Computation Theory and Applications
364

3.2 Input Conditional Probabilities of
Fuzzy Arbitrary Inputs
A input vector x, activates the firing of multiple
fuzzy rules, q, with multiple firing rates μ
q
(x), such
that 
q
μ
q
(x) = 1. In case this condition is true for a
single rule, only one of the consequents O
q
will
occur with the conditional probability P(O
j
| x).
In the light of (13) and (14) we obtain,
P

 ) = 


(

)
P

 

)



(

)




(15)
Extending the conditional probability P(O
j
|x) to
estimate the overall conditional probability density
function p (y | x), using (14), we get

(

|
) =
P



()



(

)



(16)
where, probabilities P(O
j
|x) is calculated using (15).
In view of (4) and (8) we obtain,

(

|
) = 
Pr



()



(

)




(17)
This value for conditional possibility can be used in
the expression for finding the defuzzified output of
fuzzy models
3.3 Obtaining Defuzzified Output
The existing fuzzy models can be used to obtain the
defuzzified output by replacing the conditional
possibility obtained.
3.3.1 Mamdani-larsen Model
Consider a rule of this model as:
Rule q: If x is Aq then y is Bq.
Here, fuzzy implication operator maps fuzzy subsets
from the input space A
q
to the output space B
q
(with
membership function ∅
(

)
) and generates the fuzzy
output B
q
with the fuzzy membership
Rule q: 
(

)
= (x)  ∅
(

)
The output fuzzy membership is:
φ
o
(y) = φ
1
(y) ∨ φ
2
(y) ∨ φ
3
(y) ∨ ..... ∨ φ
k
(y)
(18)
In Mamdani-Larsen (ML) model, the output of rule
q is represented by B
q
(b
q
, v
q
), with centroid b
q
and
the index of fuzziness v
q
given by


= (
y
)

(19)


=


(

)




(

)


(20)
where

(
y
)
is output membership function for rule q.
Now in the probabilistic fuzzy setting, the above
expressions (19) and (20) need to be modified.
Replacing the value of the output membership
function from (8) into (19) and (20) we get


=
P



()



(

)






(21)


=


∑
P



()



(

)







∑
P



()



(

)






(22)
where, v
q
is index of fuzziness and b
q
is Centroid.
The defuzzified output can be calculated in the
ML model by applying the weighted average gravity
method for the defuzzification. The defuzzified
output value of y
o
is given by


=



(

)




(

)


(23)
where 
(

)
is the output membership function
calculated using (18).
Also, the defuzzified output 

can be written as:


=


(

)
.

∑

 
(

)
.
 


 


.

(24)
where,


=


∑
P



()



(

)







∑
Pr



()



(

)








=
P



(

)



(

)






(25)
3.3.2 Generalized Fuzzy Model
The Generalized fuzzy model (GFM) model Azeem
(2000) generalizes both the ML model and the TS
(Takagi- Sugeno) model. The output in GFM model
has the properties of fuzziness (ML) around varying
centroid (TS) of the consequent part of a rule. Let us
consider a rule of the form
R
k
: if x
k
is A
k
then y is B
k
(f
k
(x
k
), 

).
where B
k
is the output fuzzy set,
v
k
is the index of fuzziness \
f
k
is the output function.
Using (23), we can obtain the defuzzified output y
0
as
FUZZY MODEL BUILDING USING PROBABILISTIC RULES
365



=


(

)
.

∑

 
(

)


 
.
 


.

(

)
(26)
where


(

)
is a varying singleton. It may be linear or non-
linear. The linear form is:


(

)
= = b


+ b


x

+ ...+ b


x

Replacing the value of b
q
from (22) into (26) we get


=


(

)
.

∑
Pr



()



(

)






∑

 
(

)


 
.

∑
Pr



()



(

)








.

(

)
(27)
4 COMPUTATION OF
PROBABILISTIC POSSIBILITY
FROM PROBABILITY FUZZY
RULES
How to compute the probabilistic possibility from a
probabilistic fuzzy rule is presented as an algorithm
here.
Step 1: Determine the fuzzy rules that are
applicable for the given test input, x.
Step 2: Evaluate the membership values of the
input fuzzy sets.
Step 3: Determine the membership values of the
output fuzzy sets.
Step 4: Calculate the conditional probability of
each probabilistic output using (15).
Step 6: Find the net conditional possibility of the
output using (17).
Step 7: This step is an optional step. The relations
for finding the defuzzified output for the fuzzy
models as in (25) and (27) may be used in case all
the values of parameters are available besides the
possibility term (as computed in Step 6).
5 CASE-STUDY 1
Let us contemplate the functioning of a fuzzy air
conditioner example in Kosko (1993) described by
five input linguistic terms/in the form of fuzzy sets
on X, along with five output linguistic terms
represented by fuzzy sets on Y:
• The input fuzzy sets on X are: Cold, Cool, Just
Right, Warm, and Hot
• The output fuzzy sets on Y are: Stop, Slow,
Medium, Fast, and Blast
The following fuzzy rules are framed from an
expert’s knowledge.
1. If temperature is cold, motor speed is stop
2. If temperature is cool, motor speed is slow
3. If temperature is just right, motor speed is
medium
4. If temperature is warm, motor speed is fast
5. If temperature is hot, motor speed is blast
A realistic representation of the above in the garb of
PFR when the probabilities are associated with the
outputs is the main concern now. The corresponding
PFR of Rule 1 is as follows:
If temperature is cold then
motor speed is stop with probability 70%
& motor speed is slow with probability 20%
& motor speed is medium with probability 8%
& motor speed is fast with probability 2%
Similarly, other PFRs can also be constructed. The
first column in Table 1 gives the antecedent value
for each rule. The remaining columns give the
values of the possible outputs for each rule. The
conditional possibility of the output, is calculated
when the inputs are 63

F and 68

F.
Table 1: The Probabilistic Fuzzy Rule-set.
# Temp(X) P
Sto
p
P
Slo
w
P
Medium
P
Fast
P
Blas
t
1 Cold 0.7 0.2 0.08 0.02 0.0
2 Cool 0.1 0.7 0.1 0.08 0.02
3 Jt Right 0.05 0.1 0.7 0.1 0.05
4 Warm 0.02 0.08 0.1 0.7 0.1
5 Hot 0.0 0.02 0.08 0.2 0.7
5.1 Case: Input 

F
It may be noted that the output fuzzy set with the
highest probability is only opted followed by the
others in the line. The farther a fuzzy set is from this
output, the lesser is its probability. We will elaborate
on the steps using the above example. The input and
output fuzzy sets for this example are shown in Fig.
1 and Fig. 2 respectively. The corresponding
applicable PFRs are as follows:
If temperature is just right
then motor speed is stop with probability 5%
& motor Speed is slow with probability 10%
& motor speed is medium with probability 70%
& motor speed is fast with probability 10%
& motor speed is blast with probability 5%
If temperature is cool
then motor speed is stop with probability 10%
& motor speed is slow with probability 70%
& motor speed is medium with probability 10%
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366

& motor speed is fast with probability 8%
& motor speed is blast with probability 2%
The fuzzy membership values for input fuzzy sets,
µ
0
(x) as noted from Fig. 1 and Fig. 2 are as follows:
µ
0
(Just Right): 80% (0.8) µ
0
(Cool): 15% (0.15)
The fuzzy membership values for output fuzzy sets,
µ
1
(x) as noted from Fig. 1 and Fig. 2 are as follows:
µ
1
(Slow): 15% (0.15) µ
1
(Medium): 80% (0.8)
Figure 1: Input fuzzy sets and their membership.
Figure 2: Output fuzzy sets and their membership values
when input temperature is 63oF.
We apply (15) to calculate the conditional
probability for each probabilistic output of a fuzzy
rule that is applicable, given the input temperature is
63

F.
In the light of (15), we have
P

 ) =  

(

)
P

 

)

P(O
Stop
|x) = (0.8 * 0.05) + (0.15 * 0.10) = 0.055
P (O
slow
| x) = [(0.8 * 0.1) + (0.15 * 0.7) = 0.185
P (O
Medium
| x) = [(0.8 * 0.7) + (0.15 * 0.1) = 0.575
P (O
fast
| x) = [(0.8 * 0.1) + (0.15 * 0.08) = 0.092
P (O
blast
| x) = [(0.8 * 0.05) + (0.15 * 0.02) = 0.043
The net conditional possibility for the output is
calculated using (17) as
π (y | x) = (0 + (0.185 * 0.15) + (0.575 * 0.8) + 0 +
0) = 0.48775
Thus having got the value of the conditional
probability, the same can be substituted along with
other values in the relations for ML and GFM
models as per (25) and (27) to obtain the defuzzified
output.
Comparison of the Output with Basic Fuzzy
Rules. We now use the above algorithm to estimate
the effect of the probabilistic output on the net
output conditional possibility. The fuzzy rules of
interest are as follows:
1. If temperature is cold then motor speed is stop
2. If temperature is cool then motor speed is slow
3. If temperature is just right then motor speed is
medium
4. If temperature is warm then motor speed is fast
5. If temperature is hot then motor speed is blast
The input and output fuzzy sets and their
corresponding membership values are the same as
above. The fuzzy sets for the given test input are
shown in Fig.2 and the valid fuzzy rules are:
If temperature is just right then motor speed is
medium.
If temperature is cool then motor speed is slow.
The conditional probability is computed using
(15) as
P

 ) =  

(

)
P

 

)

The conditional probabilities are evaluated as:
P (O
Stop
| x) = 0
P (O
slow
| x) = (0.15 * 1) = 0.15
P (O
Medium
| x) = (0.8 * 1) = 0.8
P (O
fast
| x) = 0 P (O
blast
| x) = 0
The net conditional possibility is found using (17) as
π (y | x) = 0 + (1 * 0.15) + (1 * 0.8) + 0 + 0 = 0.95
5.2 Case: Input 

F
The fuzzy input and output membership values are:
µ
0
(Warm): 0.2 µ
0
(Just Right): 0.55
µ
1
(Medium):0.55 µ
1
(Fast): 0.2
Applying (15) and taking Table 1 into account, the
conditional probability can be computed as in 5.1.
P(O
Stop
| x) = 0.0315 P (O
slow
| x) = 0.071
P (O
Medium
| x) = 0.525 P (O
fast
| x) = 0.075
P (O
blast
| x) = 0.0475
The net conditional possibility is found using (17) as
above in 5.1.
π (y | x) = (0.55 * 0.525) + (0.2 * 0.075) = 0.303
Comparison of the Output with Basic Fuzzy
Rules when Input is 68
0
F. The conditional
probabilities in the case of basic fuzzy rules can be
computed as
FUZZY MODEL BUILDING USING PROBABILISTIC RULES
367

P(O
Stop
| x) = 0 P (O
slow
| x) = 0
P (O
Medium
| x) = 0.55 P (O
fast
| x) = 0.2
P (O
blast
| x) = 0
The net conditional possibility is found using (17) as
π (y | x) = 0 + (1 * 0.55) + (1 * 0.2) + 0 + 0 = 0.75
It is pertinent to note that what we have here is the
possibility in the probabilistic framework. So, in this
example, the overall conditional possibility would
converge to the sum of the individual possibilities,
whereas in the case of probabilistic fuzzy rules, the
conditional possibility is a factor of probabilities as
well as possibilities.
6 CASE-STUDY 2
Consider designing a fuzzy controller for the control
of liquid level in a tank by varying its valve position
Meghdadi(2001). The simple fuzzy controller
employs Δh and dh/dt as inputs and dα/dt (rate of
change of valve position α , α∈[0,1]) as the output,
where h is the actual liquid level, h
d
is desired value
of the level, and Δh=h
d
- h is the error in the desired
level.
Three Gaussian membership functions for three
input fuzzy sets (negative, zero, positive) are
applicable on the input variables Δh and dh/dt. The
output fuzzy sets (close-fast, close-slow, no-change,
open-slow, open-fast) have triangular membership
functions. The following fuzzy rules are selected
using a human expert’s knowledge.
R1. If ∆h is zero then dα/dt is no-change
R2. If ∆h is positive then dα/dt is open-fast
R3. If ∆h is negative then dα/dt is close-fast
R4. If ∆h is zero and dh/dt is positive then dα/dt is
close-slow
R5. If ∆h is zero and dh/dt is negative then dα/dt is
open-slow
In order to model the existing scepticism of humans’
opinion in defining the optimal rule set, we may
substitute each conventional rule with a probabilistic
fuzzy rule with the output probability vector P
defined such that the only output sets of the
conventional fuzzy rules are the most probable from
the probabilistic fuzzy rules. Also the neighbouring
fuzzy sets in the PFR have smaller probabilities and
the other fuzzy sets have zero probabilities. For
example rule RI in the above rule set may be
modified as follows:
RI. If ∆h is zero then
dα/dt is no-change with probability 80%
& dα
/dt is close-slow with probability 10%
& dα/dt is open-slow with probability 10%
The consequent part of the PFR can be thus
expressed in a compact form using the output
probabilities vector P. The sample probabilistic
fuzzy rule set is given in Table 2.
Table 2: Probabilistic Fuzzy Rule-set for the Liquid Level
Fuzzy Controller.
# Q
1
V
1
Q
2
V
2
P
c-
f
P
c-s
P
n-c
P
o-s
P
o-
f
1
Δh
0 0 0.1 0.8 0.1 0
2
Δh
+ 0 0 0 0.2 0.8
3
Δh
- 0.8 0.2 0 0 0
4
Δh
0
ℎ

+ 0.1 0.8 0.1 0 0
5
Δh
0
ℎ

- 0 0 0.1 0.8 0.1
Let Input: Δh = 0.
The PFRs for the given input are as follows:
R1. If ∆h is zero then dα/dt is no-change with
probability 80%
& dα/dt is close-slow with probability 10%
& dα/dt is open-slow with probability 10%
R4. If ∆h is zero and dh/dt is positive then dα/dt is
no-change with probability 10%
& dα/dt is close-slow with probability 80%
& dα/dt is close-fast with probability 10%
R5. If ∆h is zero and dh/dt is negative then dα/dt is
no-change with probability 10%
& dα/dt is open-slow with probability 80%
& dα/dt is open-fast with probability 10%
The membership values, µ
zero
(x), µ
Positive
(x) and
µ
Negative
(x) for the given input are given as follows:
µ
Zero
(Δh): 1 µ
positive
(


): 1 µ
Negative
(


): 0
The membership grades for the output fuzzy sets are
given as follows:
µ
NoChange
(


): 1 µ
Slow
(


): 0.15 µ
Fast
(


): 0.15
The conditional probability is calculated using (15)
for each probabilistic output in each fuzzy rule that
is applicable, given the input value.
P(O
no-change
|x) = [(1 * 0.8) + (1 * 0.1) + (0 * 0.1)]/2 =
0.45.
Note:- The probability values are normalized by taking the
number of the input fuzzy sets as denominator. Similarly,
P (O
close-slow
| x) = 0.45 P (O
close-fast
| x) = 0.1
P (O
open-slow
| x) = 0.1 P (O
open-fast
| x) = 0
We arrive at the net consolidated conditional
FCTA 2011 - International Conference on Fuzzy Computation Theory and Applications
368

possibility for the output using (17) as
π (y|x) = (0.45 * 1) + (0.45 * 0.15) + (0.1 * 0.15) +
(0.1 * 0.15) + (0 * 0.15) = 0.5475
Thus having obtained the value of net membership,
the same can be substituted in the ML and GFM
models to obtain (v
q
, b
q
). It can also be noted that for
the basic fuzzy rules the net conditional possibility
for a given input is the sum of the memberships of
the various output fuzzy sets that are applicable.
7 CONCLUSIONS
It is shown how a probabilistic fuzzy framework is
more flexible and convenient than the conventional
methodology. As a consequence of this the
probabilistic possibility is derived from the
applicable probabilistic fuzzy rules which constitute
the probabilistic fuzzy system with the help of the
fuzzy modelling. The utility of probabilistic fuzzy
systems in representing real world situations is also
highlighted. Its ability to represent fuzzy nature of
situations along with corresponding probabilistic
information brings it much closer to real-world.
Two examples dealing with the practical
applications of an air-conditioner and a liquid level
controller are taken up to demonstrate a probabilistic
fuzzy system. It is noticed from this study how the
probability of the output affects the net possibility
for a particular test input.
It is observed that in the case of probabilistic
fuzzy rules, the conditional output probabilistic
possibility of an output fuzzy set for a given input
spans over the applicable output fuzzy sets. A basic
fuzzy rule is a special case of probabilistic fuzzy rule
in which there is only one output for a fuzzy rule
that translates into 100% probability for that
particular output. The methodology proposed for
calculating conditional probabilistic possibility for
PFRs fits well with basic fuzzy rules and leads to the
intuitively acceptable result. The proposed work
provides functionality to process the probabilistic
fuzzy rules that are better equipped to represent the
real-world situations.
Another feature of probabilistic fuzzy rules is the
enhanced adaptability in view of the outputs with
varying probabilities. This is borne out of the fact
that the outputs in the fuzzy rules are context-
dependent hence vary accordingly.
The proposed approach to calculate the
possibility from probability can be tailored to a
specific application depending upon the output
membership functions and their probabilities. This
can also be extended to represent probabilistic rough
fuzzy sets and other types of fuzzy sets so as to
increase its utility in capturing the higher forms of
uncertainty from probability since the probabilistic
information along with possibility aids the decision
making in the solution of the real-world problems.
The proposed framework has the capability to
address the uncertainty arising from fuzziness and
vagueness in the wake of their random occurrences.
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