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Neural Networks
• Outline
–
–
–
–
–
–
–
Introduction
From biological to artificial neurons
Self organizing maps
Backpropagation network
Radial basis functions
Associative memories Hopfield networks
Other applications of neural networks
Computer Science
Dept, UNC Charlotte
Copyright 2002 Kayvan Najarian
1
Introduction
• Why neural networks?
– Algorithms developed over centuries do not fit the complexity of
real world problem
– The human brain: most sophisticated computer suitable for solving
extremely complex problems
• Historical knowledge on human brain
– Greeks thought that the brain was where the blood is cooled off!
– Even till late 19th century not much was known about the brain and
it was assumed to be a continuum of non-structured cells
– Phineas Gage’s Story
• In a rail accident, a metal bar was shot
through the head of Mr. Phineas P.
Gage at Cavendish, Vermont, Sept 14, 1848
– Iron bar was 3 feet 7 inches long and weighed 13 1/2 pounds. It was 1 1/4
inches in diameter at one end
Computer Science
Dept, UNC Charlotte
Copyright 2002 Kayvan Najarian
2
Introduction (cont’d)
• He survived the accident!
– Originally he seemed to have fully recovered with no clear effect(s)
• After a few weeks, Phineas exhibited profound personality changes
– This is the first time, researchers have a clear evidence that the
brain is not a continuum of cell mass and rather each region has
relatively independent task
Computer Science
Dept, UNC Charlotte
Copyright 2002 Kayvan Najarian
3
Introduction (cont’d)
• Biological neural networks
– 1011 neurons (neural cells)
– Only a small portion of these
cells are used
– Main features
• distributed nature,
parallel processing
• each region of the brain
controls specialized task(s)
• no cell contains too
much information: simple
and small processors
• information is saved mainly in the connections among neurons
Computer Science
Dept, UNC Charlotte
Copyright 2002 Kayvan Najarian
4
Introduction (Continued)
• learning and generalization through examples
• simple building block: neuron
– Dendrites: collecting
signals from other neurons
– Soma (cell body): spatial
summation and processing
– Axon: transmitting signals to
dendrites of other cells
Computer Science
Dept, UNC Charlotte
Copyright 2002 Kayvan Najarian
5
Introduction (Continued)
• biological neural networks:
formation of neurons with different connection strengths
Computer Science
Dept, UNC Charlotte
Copyright 2002 Kayvan Najarian
6
From biological to artificial neural nets
• Biological vs. artificial neurons
– From biological neuron to schematic structure of artificial
neuron
• biological:
– Inputs
– Summation
–
–
of inputs
Processing
unit
Output
• artificial:
x1
xN
Computer Science
Dept, UNC Charlotte
w1
y  f (w1 x1  ...  wn xn )
wN
Copyright 2002 Kayvan Najarian
7
From biological to artificial neural net (continued)
– Artificial neural nets:
• Formation of artificial neurons
neuron 1
w11
x1
neuron 2
w1i
xN
y2
neuron i
yi
wNi
wNM
neuron M-1
yM 1
neuron M
Computer Science
Dept, UNC Charlotte
y1
Copyright 2002 Kayvan Najarian
yM
8
From biological to artificial neural nets (continued)
– Multi-layer neural nets:
• Serial connection of single layers:
y1
w11
x1
y2
w1i
xN
yi
wNi
wNM
yM 1
– Training: finding the best values of weights wij
yM
• Training happens iteratively and through exposing the network to
examples:
Computer Science
Dept, UNC Charlotte
wij (new )  wij (old )  wij
Copyright 2002 Kayvan Najarian
9
From biological to artificial neural nets (continued)
– Activation functions:
• Hard limiter (binary step):

1

f ( x)  0
 1

if x  
if    x  
if x  
f (x )
1
1

x
– Role of threshold
– Biologically supported
– Non-differentiable
Computer Science
Dept, UNC Charlotte
Copyright 2002 Kayvan Najarian
10
From biological to artificial neural nets (continued)
• Binary sigmoid (exponential sigmoid)
– Differentiable
– Biologically supported
– Saturation curve is controlled
by 
– In limit when   , hard limiter is achieved
• Bipolar sigmoid (atan)
f ( x)  tan 1 ( x)
– As popular as binary sigmoid
Computer Science
Dept, UNC Charlotte
Copyright 2002 Kayvan Najarian
11
From biological to artificial neural nets (continued)
• Supervised Vs Unsupervised Learning
– Supervised learning (classification)
•
•
Training data are labeled, i.e the output class of all training data are
given
Example: recognition of birds and insects
–
Training set:
–
Classification: sparrow  ?
eagle  bird , bee insect , ant insect , owl  bird 
– Unsupervised learning (clustering)
•
•
•
Training data are not labeled
Output classes must be generated during training
Similarity between features of training example creates different
classes
Computer Science
Dept, UNC Charlotte
Copyright 2002 Kayvan Najarian
12
From biological to artificial neural nets (continued)
•
Example : types of companies
–
Features: Number
of employees &
rate of growth
–
Training data
create natural
clusters
–
From graph:
100
500
1000
Number of Employees
Class #1: small size companies with small rate of growth
Class #2: small size companies with large rate of growth
Class #3: medium size companies with medium rate of growth
Class #4: large size companies with small rate of growth
–
Classification: a company with NOE=600 & ROG=12%
is mapped to Class #3
Computer Science
Dept, UNC Charlotte
Copyright 2002 Kayvan Najarian
13
From biological to artificial neural nets (continued)
– Artificial neural networks as classification tools:
•
•
•
If training data are labeled, supervised neural nets are used
Supervised leaning normally results to better performance
Most successful types of supervised ANNs:
–
–
–
Artificial neural networks as clustering tools:
•
•
If training data are not labeled, unsupervised neural nets are used
Most successful types of supervised ANNs:
–
–
Multi-layer perceptrons
Radial basis function networks
Kohonen network
Some networks can be trained both in supervised and
unsupervised modes
Computer Science
Dept, UNC Charlotte
Copyright 2002 Kayvan Najarian
14
Perceptron
• A more advanced version of
simple neuron
• Structure (architecture):
– Very similar to simple neuron
– The only difference:
activation function is
bipolar hard limiter
x1
xi
xn
y
n
y _ in  b j   xi wi
i 1
1

y  0
 1

Computer Science
Dept, UNC Charlotte
if y _ in  
if    y _ in  
if y _ in  

Copyright 2002 Kayvan Najarian
1
1

yin
15
Competitive Self-Organizing Networks
• Biological procedure:
– Each neuron (group of neurons)
stores a pattern
– Neurons in a neighborhood store
similar patterns
– During classification the similarity
of new pattern with all the patterns
in all neurons is calculated
– The neurons (or neighborhoods)
with highest similarity are the winners
– The new pattern is attributed to the
class of the winner neighborhood
Computer Science
Dept, UNC Charlotte
Rocking Chair
Study Chair
Trees
Forest
Books
Desk
Dining Table
Dishes
Food
New pattern: Conference Table
Rocking Chair
Study Chair
Trees
Forest
Books
Desk
Dining Table
Dishes
Food
Winner: Desk (and its neighborhood)
Copyright 2002 Kayvan Najarian
16
Kohonen Self-Organizing Maps
• Main idea: placing similar objects close to each other
– Example: object classification using one-dimensional array
Inputs
weight (0 to 1)
pencil
texture
(wood=0, metal =1)
size (0 to 1)
flies (1) or not (0)
airplane
chair
book
wooden
house
stapler
metal
desk
airplane
truck
truck
wooden
house
kite
metal
desk
chair
stapler
book
pencil
kite
– New pattern: hovercraft
• is mapped to the nationhood of truck or airplane
Computer Science
Dept, UNC Charlotte
Copyright 2002 Kayvan Najarian
17
Kohonen Self-Organizing Maps (continued)
• Two-dimensional arrays
– Proximity exists on two dimensions
– Better chance of positioning similar
objects in the same vicinity
– The most popular
type of Kohonen network
– How to define neighborhoods
Rectangular
Hexagonal
• Radius of neighborhood
(R)
• R = 0 means that each
neighborhood has only
one member
Computer Science
Dept, UNC Charlotte
Copyright 2002 Kayvan Najarian
18
Kohonen Self-Organizing Maps (continued)
• Example: two-dimensional array
–
Classification of objects
•
•
•
•
•
Objective: clustering of
a number of objects
10 input features: # of lines,
thickness of lines, angles, ...
Competitive layer is
a 2D-grid of neurons
Trained network clusters
objects rather successfully
Proximity of some objects
is not optimal
Computer Science
Dept, UNC Charlotte
Copyright 2002 Kayvan Najarian
19
Backpropagation neural networks
•
•
Idea: not as biologically-supported as Kohonen
Architecture:
z
v
–
–
•
Number
of layers &
number of
x1
neurons in
xN
each layer
Most popular structure
Activation functions:
–
j
jk
y1
w11
y2
w1i
yi
wNi
wNL
yL1
yM
sigmoid
Computer Science
Dept, UNC Charlotte
Copyright 2002 Kayvan Najarian
20
Backpropagation neural networks (continued)
•
Updating weights:
–
–
Based on Delta Rule
Function to be minimized:
E  0.5 t k  yk 
2
k
–
–
–
–
Best updating of
wJK is toward
the gradient
Error of output layer is propagated back towards the input layer
Error is calculated at output layer and propagated back towards
the input layer
As layers receive the backpropagated error, they adjust their
weights according to the negative direction of gradient
Computer Science
Dept, UNC Charlotte
Copyright 2002 Kayvan Najarian
21
Backpropagation neural networks (continued)
–
Define:
–
Then:
and:
–
 K  tK  yK  f ( y _ inK )
E
wJK
  K z J
w jk  
E
 k z j
w jk
Now:
E
vIJ
Computer Science
Dept, UNC Charlotte
   k wJk f  z _ in J xI
k
Copyright 2002 Kayvan Najarian
22
Backpropagation neural networks (continued)
•
Applications:
–
Time-series analysis and prediction
•
Problem statement (simplest case):
–
•
•
The future value of a signal depends on the previous values of
the same signal, i.e. y(t )  g  y(t 1), y(t  2), ..., y(t  p)
Objective: to use a neural net to estimate function “g”
Procedure:
–
Form a number of training points as:  y (i  1), y (i  2), ... , y (i  p)   y (i )
i  p, p  1, ...., n  p  1
–
•
Train a backpropagation net to learn the input-output relation
Advanced cases:
y(t )  g  y(t 1), y(t  2), ... , y(t  p), x(t 1), x(t  2), ... , x(t  r )
–
Computer Science
Dept, UNC Charlotte
Procedure is similar to the simple case
Copyright 2002 Kayvan Najarian
23
Backpropagation neural networks (continued)
•
Feedforward neural time series models are used in many
fields including:
–
–
–
–
–
–
•
Stock market prediction
Weather prediction
Control
System identification
Signal and image processing
Signal and image compression
Classification:
–
When a hard limiter is added to the output neurons
(only in classification and not during the training
phase), backpropagation network is used to classify
complex data sets
Computer Science
Dept, UNC Charlotte
Copyright 2002 Kayvan Najarian
24
Radial Basis Function networks (RBFN’s)
x1
Architecture:
–
–
Weights of the input
layer are all “1”, i.e.:
x  x1 , x2 , ..., xn 
2 (x)
x2
1
2

…..
–
1 (x)
…...
•
all 1
Based on the exact definition

 m (x)
xn
of radial basis functions i (.) ,
m
many different families of
y    ii x 
i 1
RBFN’s are defined
Main properties of all “radial” basis functions: “radial”
m
•
1
Example:  ( x) 
x  xC
is the same for
all points with equal
distance from point C.
Computer Science
Dept, UNC Charlotte
2
is a radial function because the value of 
Copyright 2002 Kayvan Najarian
C
25
y
RBFN’s (continued)
•
Basis functions
–
Gaussian basis functions:
•
•
–
•
 x x
Ci

i (x)  exp  
 i2


Coordinates of center: x C i
2





Width parameter:  i
Reciprocal Multi-Quadratic (RMQ) functions:
1
i (x) 
2
1  bi x  x Ci
•
Width parameter: bi
Training: Least Mean Square Techniques
Computer Science
Dept, UNC Charlotte
Copyright 2002 Kayvan Najarian
26
RBFN’s (continued)
•
Example:
–
–
Trying to approximate a no-linear function using RMQ-RBFN’s
Function to be estimated:
RBFNs (batch method)
g ( x)  sin( x). ln 1  x 
2.5
2
–
–
Actual and Estimated Outout
1.5
1
Red: Actual
0.5
0
135 Training points
19 Basis functions are
uniformly centered
between -10 and 10
All basis functions have: b = 0.5
Training method: batch
Estimation is rather successful
-0.5
-1
-1.5
-2
-2.5
-10
–
–
–
Computer Science
Dept, UNC Charlotte
Blue: Estimated
-8
-6
Copyright 2002 Kayvan Najarian
-4
-2
0
Input
2
4
6
8
27
10
Associative Memories
• Concept:
– Object or pattern A (input) reminds the network of object or pattern
B (output)
• Heteroassociative Vs. autoassociative memories
– If A and B are different, the system is called heteroassociative net
• Example: you see a large lake (A) and that reminds you of the Pacific
(B) ocean you visited last year
– If A and B are the same, the system is called autoassociative net
• Example: you see the Pacific ocean for the second time (A) and that
reminds you of the Pacific (B) ocean you visited last year
Computer Science
Dept, UNC Charlotte
Copyright 2002 Kayvan Najarian
28
Associative Memories (continued)
• Recognizing new or incomplete patterns
– Recognizing patterns that are similar to one of the patterns stored in
memory (generalization)
• Example: recognizing a football player you haven’t seen before from
his clothes
– Recognizing incomplete or noisy patterns whose complete (correct)
forms were previously stored in memory
• Example: recognizing somebody’s face from a picture that is partially
torn
• Unidirectional Vs. bidirectional memories
– Unidirectional: A reminds you of B
– Bidirectional: A reminds you of B and B reminds you of A
• Many biological neural nets are associative memories
Computer Science
Dept, UNC Charlotte
Copyright 2002 Kayvan Najarian
29
Hopfield Network
• Concept:
– A more advance type of autoassociative memory
– Is almost fully connected
• Architecture
– Symmetric weights
wij  w ji
– No feedback from a
cell to itself wii  0
– Notice the “feedback” in
the network structure
Computer Science
Dept, UNC Charlotte
Copyright 2002 Kayvan Najarian
30
Bidirectional Associative Memories (BAM)
• Concept:
– Bidirectional memory: pattern A reminds you of pattern B and
pattern B reminds you of pattern A
– Is almost fully connected
• Architecture
Y1
– Symmetric weights
wij  w ji
– No feedback from a
cell to itself
wii  0
– Notice the “feedback” in
the network structure
Computer Science
Dept, UNC Charlotte
…
Yj
…
Ym
wij
wnm
w11
X1
…
Copyright 2002 Kayvan Najarian
Xi
…
Xn
31
Other Applications of NNs
• Control
– Structure:
Desired
behavior
Neurocontroller
Control
decision
System
Actual
behavior
– Example: Robotic manipulation
Computer Science
Dept, UNC Charlotte
Copyright 2002 Kayvan Najarian
32
Applications of NNs (continued)
• Finance and Marketing
–
–
–
–
–
Stock market prediction
Fraud detection
Loan approval
Product bundling
Strategic planning
• Signal and image processing
–
–
–
–
Signal prediction (e.g. weather prediction)
Adaptive noise cancellation
Satellite image analysis
Multimedia processing
Computer Science
Dept, UNC Charlotte
Copyright 2002 Kayvan Najarian
33
Applications of NNs (continued)
• Bioinformatics
– Functional classification of protein
– Functional classification of genes
– Clustering of genes based on their expression (using DNA
microarray data)
• Astronomy
– Classification of objects (into stars and galaxies ad so on)
– Compression of astronomical data
• Function estimation
Computer Science
Dept, UNC Charlotte
Copyright 2002 Kayvan Najarian
34
Applications of NNs (continued)
• Biomedical engineering
– Modeling and control of complex biological system (e.g. modeling
of human respiratory system)
– Automated drug-delivery
– Biomedical image processing and diagnostics
– Treatment planning
• Clustering, classification, and recognition
– Handwriting recognition
– Speech recognition
– Face and gesture recognition
Computer Science
Dept, UNC Charlotte
Copyright 2002 Kayvan Najarian
35