Lecture 9
Gaussian Membership Functions
Another fuzzy membership function that is often used to represent vague, linguistic terms is
the Gaussian which is given by:
(ci − x)2
),
µAi (x) = exp(−
2
2σi
(1)
where ci and σi are the centre and width of the
ith fuzzy set Ai, respectively.
Ci
Gaussian
1
0.8
0.6
σi
0.4
0.2
0
0
0.5
1
1.5
2
2.5
3
3.5
4
Centres
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Lecture 9
• local although not strictly compact
• the output is very smooth
• not probability
• Multivariate Gaussian functions can be formed
from the product of the univariate sets.
• Gaussian fuzzy membership functions are
quite popular in the fuzzy logic literature,
as they are the basis for the connection between fuzzy systems and radial basis function (RBF) neural networks.
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Lecture 9
Fuzzy operators
Fuzzy Intersection: AND
The fuzzy intersection of two sets A and B
refers to linguistic statement
(x is A) AN D (y is B),
where x and y could potentially refer to the
same variable.
For fuzzy intersection, the family of operators
is known as the set of triangular or T-norms
and the new membership function is generated
by:
µA∩B (x, y) = µA(x)ˆ
∗µB (y),
(2)
where ˆ
∗ is the T-operator. For ease of notation, let a, b, c, d ∈ [0, 1] denote the value of the
fuzzy membership functions.
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Lecture 9
Definition(T-norm) T-norms is the class of functions obeying the following relationships:
• aˆ
∗b = bˆ
∗a
• (aˆ
∗b)ˆ
∗c = aˆ
∗(bˆ
∗c)
• If a ≤ c and b ≤ d then aˆ
∗b ≤ cˆ
∗d
• aˆ
∗1 = a
• If a T-norm is Archimedean then aˆ
∗a <
a, ∀a ∈ (0, 1).
e.g., algebraic product and min functions:
µA∩B (x, y) = µA(x) ∗ µB (y),
(3)
µA∩B (x, y) = min{µA(x), µB (y)},
(4)
for which the former is an Archimedian Tnorm.
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Lecture 9
algebraic product / min operator
• the ‘min’ operator acts as a truncation operator losing information contained in the
membership functions {µA(x), µB (y)}
• the product operator allows error information to be back propagated through the
network as the first derivative is well defined.
• generally gives a smoother output surface
when univariate B-spline and Gaussian fuzzy
membership functions are used.
• ‘min’ operator forms an upper bound on
the space of fuzzy intersection operators.
For any T-norm, we have
µA(x)ˆ
∗µB (x) ≤ min{µA(x), µB (x)},
(5)
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Lecture 9
In a fuzzy algorithm, the antecedent of a fuzzy
production rule is formed from the fuzzy intersection of n-univariate fuzzy sets:
(x1 is Ai1) AN D ... AN D (xn is Ain),
which produces a new multivariate membership function
µAi ∩...∩Ai (x1, ..., xn) or µAi (x),
n
1
defined on the original n-dimensional input space
whose output is given by
µAi (x) =
where
Q̂
Ŷ
(µAi (x1), ..., µAi (xn)),
1
(6)
n
is the multivariate T-norm operator.
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Lecture 9
Fuzzy Union: OR
The fuzzy union of two sets A and B refers to
a linguistic statement of the form:
(x is A) OR (y is B),
where x and y could potentially refer to the
same variable. A new fuzzy membership function is generated by this operation defined on
the X × Y space, and is denoted by µA∪B (x, y).
This family of operators is known as S-norms
and new membership functions are obtained
from:
µA S B (x, y) = µA(x)+̂µB (y),
(7)
where +̂ is the binary S-norm operator.
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Lecture 9
Definition 4.9 (S-norm) The set of triangular
w-norms or S-norms is the class of functions
which satisfy the following:
• a+̂b = b+̂a,
• (a+̂b)+̂c = a+̂(b+̂c),
• If a ≤ c, b ≤ d, then a+̂b ≤ c+̂d,
• a+̂0 = a.
e.g., algebraic sum and max functions:
µA S B (x, y) = µA(x) + µB (y),
(8)
µA S B (x, y) = max{µA(x), µB (y)}.
(9)
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Lecture 9
If the fuzzy membership functions do not form
a partition of unity, the sum operator is sometimes replaced by the bounded sum:
µA S B (x, y) = µA(x) + µB (y) − µA(x) ∗ µB (y),
(10)
which always has a membership value that lies
in the unit interval. The max operator can be
shown to be the most pessimistic S-norm as
max{µA(x), µB (y)} ≤ µA(x)+̂µB (y).
(11)
Unlike T -norms, when the arguments of an
S-norm are unimodal, the resulting membership function is unlikely to retain the unimodal
property.
µ (y)
µ (y)
1
1
B1
B2
B1
B2
y
(a)
y
(b)
Comparison of max (a) and sum (b) operators
where the shaded area represents the membership function µB 1∪B 2 (·)
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Lecture 9
Fuzzy Implication: IF (.) THEN (.)
A general fuzzy algorithm is composed of a set
of production rules of the form:
rij :
IF (x1 is Ai1 AN D ... AN D xn is Ain)
T HEN (y is B j ) cij ,
(12)
where rij rule is the ijth fuzzy production rule
which relates the ith input fuzzy set Ai to the
jth output fuzzy set B j . The degree of confidence, with which the input fuzzy set Ai (which
is composed of fuzzy intersection (AND) of
several univariate fuzzy sets) is related to the
output fuzzy set B j , is given by a rule confidence cij ∈ [0, 1]. When cij is zero, the rule is
inactive and does not contribute to the output.
Otherwise, it partially fires whenever its antecedent is activated to a degree greater than
zero.
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Lecture 9
The rule base is characterized by a set of rule
confidence {cij } (i = 1, 2, ..., p; j = 1, 2, ..., q)
which can be stored in a rule confidence matrix
C = {cij }. Once fuzzy membership functions
have been defined, the rule confidences encapsulate the experts knowledge about a particular process and also form a convenient set
of parameters to train the fuzzy system from
training data.
An example:
Some cars have a separate antilock brake system using fuzzy control, with sensors that monitor the rotational speeds of wheels when brakes
are applied. When one of these wheels approaches lockup, the controller reduces brake
torque to that wheel (or set of wheels) just
enough to allow rotation again. Discuss how a
fuzzy intersection operator can be used to describe the condition of a wheel lockup, relating
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Lecture 9
to its rotational speed and vehicle speed, and
how this information can be used to control
the brake.
Sample answer: Intersection operator AND define two situations happen simultaneously. Wheel
lockup happens when the rotational speed approaches zero (small), but the speed of the bus
is still large . A fuzzy rule can be formed such
as IF (Wheel rotational speed is small) AND
(speed of the bus is not small)
The fuzzified value of the above rule can be
transformed into a control signal as the input
to antilock system, e.g. turn on/off the antilock that reduces the brake torque.