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PERCEPTRON
Chapter 3: The Basic Neuron
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The structure of the brain can be viewed as a highly interconnected
network of relatively simple processing elements/ neuron.
The brain has at least 1010 neurons, each connected to 104 others
We are not attempting to build computer brains – extremely simplified
versions of natural neural systems- rather we are aiming to discover
the properties of models.
The idea behind neural computing - by modeling the major features
of the brain- can produce computers that exhibit many of the useful
properties of the brain.
Whereas, we are concerned here with maybe a few hundred neurons
at most, connected to a few thousand input lines
The aim of a model is to produce a simplified version of a system.
Biological Neural Networks
Neuron - three components: its dendrites, soma, and axon
(Fig. 1.3).
 Dendrites receive signals from other neurons.
 The signals are electric impulses that are transmitted
across a synaptic gap.
 The soma/ cell body, sums the incoming signals.
 When sufficient input is received, the cell fires (transmits a
signal to other cells.)
 However, the frequency of firing varies a- either greater or
lesser magnitude.
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The human brain contains
about 10 billion nerve cells,
or neurons. On average,
each neuron is connected to
other neurons through about
10 000 synapses. (The actual
figures vary greatly,
depending on the local
neuroanatomy.) The brain's
network of neurons forms a
massively parallel
information processing
system. This contrasts with
conventional computers, in
which a single processor
executes a single series of
http://www.idsia.ch/NNcourse/brain.html
instructions.
human brain
…
the brain has quite remarkable capabilities:
1. its performance tends to degrade gracefully
under partial damage. In contrast, most
programs and engineered systems are brittle:
if…
you remove some arbitrary parts, very likely
the whole will cease to function.
2.
it can learn (reorganize itself) from experience.
this means that partial recovery from damage
is possible if healthy units can learn to take
over the functions previously carried out by
the damaged areas.
3.
it performs massively parallel computations
extremely efficiently. For example, complex
visual perception occurs within less than 100
ms, that is, 10 processing steps!
4.
it supports our intelligence and selfawareness. (Nobody knows yet how this
occurs.)
As a discipline of Artificial Intelligence,
Neural Networks attempt to bring computers
a little closer to the brain's capabilities by imitating
certain aspects of information processing in the
brain, in a highly simplified way.
MODELLING THE SINGLE NEURON
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The basic function of a biological neuron is to add up its inputs, and
to produce an output if this sum is greater than some value, known
as the threshold value.
The inputs to the neuron arrive along the dendrites, which are
connected to the outputs from other neurons by specialized junctions
called synapses.
The junctions pass a large signal across, whilst others are very poor.
The cell body receives all inputs, and fires if the total input exceeds
the threshold.
Our model of the neuron must capture these important features:
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The output from a neuron is either on or off.
The output depends only on the inputs. A certain number must be on (threshold
value) at any one time in order to make the neuron fire.
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The synapses can be modeled by having a multiplicative factor on the input.
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MODELLING THE SINGLE NEURON
Artificial Neural Network (ANN) is an information processing
system that have similar characteristic with biological neural
network.
It has been developed as a general representation for human
mathematical model.
MODELLING THE SINGLE NEURON
With assumption
1.
Information processing occurs within the simple element called
neuron.
2.
A neuron consist of a cell body, soma, fibres, dendrites and a
long fibre called axon.
3.
Signals are transferred within the neurons through connections.
4.
Each connection has its weight.
5.
Each neuron uses an activation function to produce output.
BRAIN ANALOGY AND NN
Biological Neural Network
Soma
Dendrite
Axon
Synapse
Artificial Neural Network
Neuron
Input
Output
Weight
Biological Neuron
LEARNING IN SIMPLE NEURONS
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We need a mechanism for achieving learning in our model neuron.
Connecting these neurons together then train them in order to do
useful task.
Example in Classification problem:
Figure 3.5 - Two groups - one group of several differently written A’s,
and the other of B’s, we may want our neuron to output a 1 when an
A is presented and a 0 when it sees a B.
The guiding principle is to allow the neuron to learn from its mistakes:
LEARNING IN SIMPLE NEURONS
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If it produces an incorrect output, we want to reduce the chances of
that happening again; if it comes up with correct output, then we
need do nothing.
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If the neuron produces a 0 when we show it an A, then increase the
weighted sum so that next time it will exceed the threshold and so
produces the correct output 1.
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If the neuron produces a 1 when we show it an B, then decrease the
weighted sum so that next time it will less than threshold and so
produces the correct output 0.
Learning strategy
increase the weights on the active inputs when we want the
output to be active,
 decrease them when we want the output to be inactive.
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To achieve - add the input values to the weights when we
the output to be on, and subtracting the input values from
the weights when we want the output to be off.
 This defines our learning rule.
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This learning rule is a variant on that proposed in 1949 by Donald
Hebb, and is therefore called Hebbian learning.
Since the learning is guided by knowing what we want to achieve,
it is known as supervised learning.
Learning strategy
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Our learning paradigm can be summarized as follows:
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Set the weights and thresholds randomly
Present an input
Calculate the actual output - thresholding the weighted sum
of the inputs. (0 or 1)
 Alter the weights to reinforce correct decisions – i.e, reduce
the error.
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The Perceptron
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The operation of Rosenblatt’s perceptron is based on the
McCulloch and Pitts neuron model. The model
consists of a linear combiner followed by a hard limiter.
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The weighted sum of the inputs is applied to the hard
limiter, which produces an output equal to +1 if its input is
positive and 1 if it is negative.
The Perceptron
Input Unit
1
X1
X2
Output Unit
b
w1
Y
w2
Single-layer net for pattern classification
The Perceptron
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Negative and Positive Response
The Algorithm
Step 0:
Step 1:
Initialize all weights and bias:
wi = 0 (i= 1 to n), b=0
Set learning rate  (0 <  ≤ 1)
=0
While stopping condition is false,
do steps 2-6.
Step2: For each training pair s:t, do
steps 3-5
Step 3. Set activations for input units:
xi = si
Step 4.Compute response of output
unit:
y_in = b +  xi wi ;
y=
1
if y_in > 
0
if -  ≤ y_in <
-1
if y_in < - 
Step 5. Update weights and bias if an
error occurred for this pattern
If y  t,
wi(new) = wi(old) +  txi
b(new) = b(old) +  t
else
wi(new) = wi(old)
b(new) = b(old)
Step 6. Test stopping condition:
If no weights changed in Step 2,
stop;
else, continue
The Algorithm
Step 0:
Initialize all weights and bias:
wi = 0 (i= 1 to n), b=0
Set learning rate  (0 <  ≤ 1)
=0
The Algorithm
Step 1: While stopping condition is false, do steps 2-6.
Step2: For each training pair s:t, do steps 3-5
Step 3. Set activations for input units:
xi = si
Step 4. Compute response of output unit:
NET = y_in = b +  xi wi ;
OUT= y =
1
if y_in > 
0
if -  ≤ y_in <
-1
if y_in < - 
The Algorithm
Step 5. Update weights and bias if an error occurred for this pattern
If y  t,
wi(new) = wi(old) +  txi (i = 1 to n).
b(new) = b(old) +  t
else
wi(new) = wi(old)
b(new) = b(old)
Step 6.
Test stopping condition:
If no weights changed in Step 2,
stop;
else, continue
Perceptron net for And function: binary inputs and bipolar targets
1st Epoch
Input
(x1
(1
(1
(0
(0
x2
1
0
1
0
Target
1)
1)
1)
1)
1)
Input
=1,  = 0.2
wi=0
b=0
1
-1
-1
-1
Net Out Target
(x1 x2 1)
(1
(1
(0
(0
1
0
1
0
1)
1)
1)
1)
0
2
1
-1
0
1
1
-1
1
-1
-1
-1
Weight Changes
Weights
( w1 w2 b)
(w1
(0
(1
(0
(0
(0
(1
(-1
(0
(0
1
0
-1
0
1)
-1)
-1)
-1)
w2 b)
0 0)
1 1)
1 0)
0 -1)
0 -2)
Separating lines for 1st training input
x2
-
-
Formula asas lukis graf
b +  xi wi > 
+
-
x1
1 + x1(1)+ x2(1)=0.2 and
1 + x1(1)+ x2(1)=-0.2
Separating lines for 2nd training input
x2
-
-
Formula asas lukis graf
b +  xi wi > 
+
-
x1
0 + x1(0)+ x2(1)= 0.2 and
0 + x1(0)+ x2(1)= -0.2
Separating lines for 3rd and 4th training input
For 3rd input the weight derived is –ve
For the 4th input – no weight changes
Decision boundary is still not correct for 1st input
We are not finished training
Perceptron net for And function: binary inputs and bipolar targets
2nd Epoch
Input
(x1
(1
(1
(0
(0
x2
1
0
1
0
Target
1)
1)
1)
1)
1)
Input
=1,  = 0.2
wi=0
b=0
1
-1
-1
-1
Net Out Target
(x1 x2 1)
(1
(1
(0
(0
1
0
1
0
1)
1)
1)
1)
-1
1
0
-2
-1
1
0
-1
1
-1
-1
-1
Weight Changes
Weights
( w1 w2 b)
(w1
(0
(1
(0
(0
(0
(1
(-1
(0
(0
1
0
-1
0
1)
-1)
-1)
0)
w2
0
1
1
0
0
b)
-2)
0)
-1)
-2)
-2)
Separating lines for 1st training input, 2nd epoch
x2
-
-
Formula asas lukis graf
b +  xi wi > 
+
-
x1
0 + x1(1)+ x2(1)= 0.2 and
0 + x1(1)+ x2(1)= -0.2
Separating lines for 2nd training input 2nd epoch
x2
-
-
Formula asas lukis graf
b +  xi wi > 
+
-
x1
-1 + x1(0)+ x2(1)= 0.2 and
-1 + x1(0)+ x2(1)= -0.2
Perceptron net for And function: binary inputs and bipolar targets
3rd Epoch
Input
(x1
(1
(1
(0
(0
x2
1
0
1
0
Target
1)
1)
1)
1)
1)
Input
 =1,  = 0.2
wi=0
b=0
1
-1
-1
-1
Net Out Target
(x1 x2 1)
(1
(1
(0
(0
1
0
1
0
1)
1)
1)
1)
-2
0
-1
-2
-1
0
-1
-1
1
-1
-1
-1
Weight Changes
Weights
( w1 w2 b)
(w1
(0
(1
(0
(0
(0
(1
(-1
(0
(0
1
0
0
0
1)
-1)
0)
0)
w2
0
1
1
1
1
b)
-2)
-1)
-2)
-2)
-2)
Perceptron net for And function: binary inputs and bipolar targets
10th Epoch
Input
(x1
(1
(1
(0
(0
x2
1
0
1
0
Target
1)
1)
1)
1)
1)
Input
=1,  = 0.2
wi=0
b=0
1
-1
-1
-1
Net Out Target
(x1 x2 1)
(1
(1
(0
(0
1
0
1
0
1)
1)
1)
1)
1
-2
-1
-4
1
-1
-1
-1
1
-1
-1
-1
Weight Changes
Weights
( w1 w2 b)
(w1 w2 b)
(0
(0
(0
(0
(2
(2
(2
(2
0
0
0
0
0)
0)
0)
0)
3
3
3
3
-4)
-4)
-4)
-4)
Separating lines for Final decision Boundaries
x2
-
-
Formula asas lukis graf
b +  xi wi > 
+
-
x1
-4 + 2x1+ 3x2> 0.2 and
-4 + 2x1+ 3x2< -0.2
Perceptron net for And function: bipolar inputs and bipolar targets
1st and 2nd epoch
=1,  = 0.2
wi=0
b=0
Input
(x1
(1
(1
(-1
(-1
x2
1
-1
1
-1
1)
1)
1)
1)
1)
Input
Net Out Target
Weight Changes
0
1
2
-3
( w1 w2 b)
(1
1 1)
(-1 1 -1)
(1
-1 -1)
(0
0 0)
(w1 w2 b)
(0 0 0 )
(1 1 1)
(0 2 0)
(1 1 -1)
(1 1 -1)
Weight Changes
Weights
( w1 w2 b)
(w1 w2 b)
(0
(0
(0
(0
(1
(1
(1
(1
0
1
1
-1
1
-1
-1
-1
Net Out Target
(x1 x2 1)
(1
(1
(-1
(-1
1
-1
1
-1
1)
1)
1)
1)
1
-1
-1
-3
1
-1
-1
-1
1
-1
-1
-1
0 0)
0 0)
0 0)
0 0)
Weights
1
1
1
1
-1)
-1)
-1)
-1)
LIMITATIONS OF PERCEPTRONS
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The perceptron is trying to find the straight line that separates
classes.
It can separate classes that lie on either side of a straight line easily
enough,
but there are many situations where the division between classes is
much more complex. Consider the case of the exclusive-or (XOR)
problem.
LIMITATIONS OF PERCEPTRONS
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The XOR logic function has two inputs and one output
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It produces an output as shown in table 3.1.
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Such patterns are known as linearly inseparable since no straight line
can divide them up.
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The single-layer perceptron has shown great success for such a
simple model.
Perceptron
Perceptron learning applet
 http://diwww.epfl.ch/mantra/tutorial/english/perceptron/html/
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