Fuzzy systems
Fuzzy systems
Inputs can be any value from 0 to 1.
The basic fuzzy principle is similar to Boolean logic.
Max and min operators are used instead of AND and OR.
The NOT operator also becomes 1 - #.
NEURAL NETWORKS
NEURAL NETWORKS
(ELEC 524
Introduction
NEURAL NETWORKS
nonlinear
terms
x1
How to handle nonlinear systems?
y1
x(t )dt
x2
nonlinear
terms
xn
Hardware Implementations
nonlinear
terms
x(t )dt
x2
yn
xn
nonlinear
terms
Implementations of Dynamic Linear Systems
easy
part
x(t )dt
y1 f
Principle Component Analysis
How to find proper architecture?
wij cxi y j
Hebbs rule
NEURAL NETWORKS
Predict theoretically
Reduce number of inputs using PCA
Use evolutionary computation technique
Try and error approach
Start with large system and prune it
Genetic Algorithms
NEURAL NETWORKS
Genetic Algorithms 2
The genetic algorithms follow the evolution process in the nature
to find the better solutions of some complicated problems.
Foundations of genetic algorithms are given in Holland (1975) and
Goldberg
Forming clusters as needed using
minimum distance concept
distance concept
Clustering problem
Find number of clusters and its location in 4-dim. space
NEURAL NETWORKS
4
4
2
4
-4
9
2
3
5
7
-5
9
3
3
4
-5
3
6
-5
(ELEC 5240 and ELEC 6240)
(ELEC 5240 and ELEC
Swarm Intelligence
Swarm Intelligence
Swarm intelligence (SI) is an artificial intelligence
technique based around the study of collective behavior in
decentralized, self-organized systems. The expression
"swarm intelligence" was introduced by Beni & Wang
Levenberg-Marquardt Algorithm (LM)
introduction
NEURAL NETWORKS
Steepest descent method
(ELEC 5240 and ELEC 6240)
(ELEC 5240 and ELEC 6240)
Bodgan M. Wilamowski
g 2 F2 w1 , w2 , w3 ,.wn
1
k
w k 1 w k A g
where g is gradient vector
E
w1
E
gradient g w2
E
Examples
Computational Intelligence
NEURAL NETWORKS
hidden
layer #1
Is this new technological revolution?
(ELEC 5240 and ELEC 6240)
Introduction
hidden
layer #2
Weather prediction
LP PE CVD
Loan Approval
output
layer
100+ years ago man power was replaced
Analytic Geometry in Euclidean Space with Cartesian Coordinates
Analytic Geometry in Euclidean Space with Cartesian Coordinates
gradient form
intercept form
NEURAL NETWORKS
for 3d:
y
(ELEC 5240 and ELEC 6240)
(ELEC 5240 and ELEC 6240)
y kx y0
analytical m
Area with 4 partitions. Inputoutput mapping
NEURAL NETWORKS
How to design?
Assuming linear
activation function for
activation function for
the output neuron
(ELEC 5240 and ELEC 6240)
(ELEC 5240 and ELEC 6240)
single neuron training
What is given?
is given
EBP Error Back Propagation algorithm
F1cfw_z
NEURAL NETWORKS
xp
(ELEC 5240 and ELEC 6240)
(ELEC 5240 and ELEC 6240)
f ( x)
modifications of EBP
EBP Error Back Propagation algorithm
The case with multiple outputs
z p F2cfw_z
op
Fnocfw_z
+1
The weight incre
EBP Error Back Propagation algorithm
EBP Error Back Propagation algorithm
The case with one output
The case with one output
NEURAL NETWORKS
xp
(ELEC 5240 and ELEC 6240)
zp
f ( x)
op
Fcfw_z
xp
np
Total _ Error d p o p
2
1
2
EBP Error Back Propagation algor
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