# Z . Khorshidpour

AI and Robotics

Oct 20, 2013 (4 years and 6 months ago)

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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

such

as

learning,

,

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

and

.

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

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

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.