Z .
Khorshidpour
The Use of Intelligent Hybrid Systems
Integration of Fuzzy Logic and Neural
Networks
Fuzzy Neurons
AND & OR Fuzzy Neuron
Kwan and
Cai’s
Fuzzy Neuron
Type of Fuzzy Neural Network
Regular Fuzzy Neural Network
Hybrid Fuzzy Neural Network
Intelligent
hybrid
systems
Process
control
Medical
diagnosis
Engineering
design
Cognitive
simulation
Credit
evaluation
Many complex domains
have many different component
problems, each of which may
require
different types of processing
While
fuzzy
logic
provides
an
inference
mechanism
under
cognitive
uncertainty
,
computational
neural
networks
offer
exciting
advantages,
such
as
learning,
adaptation
,
fault

tolerance
,
parallelism
and
generalization
.
To enable a system to deal with cognitive
uncertainties in a manner more like humans,
one may incorporate the concept of fuzzy
logic into the neural networks.
In
theory,
neural
networks,
and
fuzzy
systems
are
equivalent
in
that
they
are
convertible,
yet
in
practice
each
has
its
own
advantages
and
disadvantages
.
For
neural
networks,
the
knowledge
is
automatically
acquired
by
the
back
propagation
algorithm
,
but
the
learning
process
is
relatively
slow
and
analysis
of
the
trained
network
is
difficult
(
black
box
)
.
Fuzzy
systems
are
more
favorable
in
that
their
behavior
can
be
explained
based
on
fuzzy
rules
and
thus
their
performance
can
be
adjusted
by
tuning
the
rules
.
The signal x
i
interacts with the weight
w
i
Transfer function f, which could be a
sigmoidal
function
If we employ other operations like a
T

norm
,
or a
S

norm
, to combine the incoming data
to a neuron we obtain what we call
a hybrid
neural net
.
Basic requirements:
Boundary: T(0, 0) = 0, T(a, 1) = T(1, a) = a
Monotonicity
: T(a, b) < T(c, d) if a < c and b < d
Commutativity
: T(a, b) = T(b, a)
Associativity
: T(a, T(b, c)) = T(T(a, b), c)
Four
examples:
Minimum: T
m
(a, b)
Algebraic product: T
a
(a, b)
Bounded product: T
b
(a, b)
Drastic product: T
d
(a, b)
Basic requirements:
Boundary: S(1, 1) = 1, S(a, 0) = S(0, a) = a
Monotonicity
: S(a, b) < S(c, d) if a < c and b < d
Commutativity
: S(a, b) = S(b, a)
Associativity
: S(a, S(b, c)) = S(S(a, b), c)
Four examples
:
Maximum:
S
m
(a, b)
Algebraic sum: S
a
(a, b)
Bounded sum:
S
b
(a, b)
Drastic sum:
S
d
(a, b)
The inputs (which are usually
membership
degrees of a fuzzy concept
) x1 ,x2 and the
weights w1 ,w2 over the unit interval
[0, 1].
If
T = min
and
S = max
then the AND
neuron realizes the min

max composition
The min

max composition of two fuzzy relations
R
1
(defined on
X
and
Y
) and
R
2
(defined on
Y
and
Z
) is
If T = min and S = max then the AND neuron
realizes the max

min composition
The signal xi interacts with the weight
wi
to
produce the product
The input information pi is aggregated by an
aggregation function h to produce the input of
the neuron
The state of the neuron is computed by
Where f is an activation function and
θ
is the activating
threshold.
The m outputs of the neuron are computed
Where
gj
, j = 1,...,m
are the m output functions of the neuron
which represent the membership functions of the input pattern
x1 ,x2 ,...,
xn
in all the m fuzzy sets.
It is well

known that regular nets are
universal
approximator
, i.e. they can
approximate any continuous function on a
compact set to arbitrary accuracy.
classification
problem of a
fuzzy input vector
to a crisp class
Used to
implement
fuzzy IF

THEN
rules
Outputs are always
real numbers
because both
inputs and weights
are real numbers.
the
fuzziﬁcation
of weights
is not necessary because
targets are real numbers
A regular fuzzy neural network is a neural
network with
fuzzy signals and/or fuzzy
weights
,
sigmoidal
transfer function and all
the operations are
deﬁned
by
Zadeh’s
extension principle
.
A is a fuzzy set on X :
The image of A under f( ) is a fuzzy set B:
where
y
i
= f(x
i
),
i
= 1 to n.
If f( ) is a many

to

one mapping, then
The signal Xi interacts with the weight
Wi
Transfer function f:
membership function of the output fuzzy set Y
is computed by the
extension principle
A hybrid fuzzy neural network is a neural
network with fuzzy signals and/or fuzzy
weights.
We can combine Xi and
Wi
using a t

norm, t

conorm
, or some other continuous operation
We can aggregate P1 and P2 with a t

norm, t

conorm
, or any other continuous function f can
be any function from input to output
Consider a fuzzy expert system with one
block of rules:
The discrete version of the system is to input
Obtain output
Construct a fuzzy relation
R
k
to model rule
One way:
Then we combine all the
R
k
into one R
One way:
intersect the
R
k
to get R.
The value of R at the pair (
x
i
,y
j
):
The method of computing
b’j
from
a’i
is
called the
compositional rule of inference
.
Where * is some method (usually a t

norm)
One takes the data
A
k
(xi) and
B
k
(
yj
) to obtain
R
k
(
xi,yj
) for
each
rule.One
way to do this is
“Neural Fuzzy Systems”, Robert Fuller, ISBN 951

650

624

0, 1995.
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