soft computing

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Back Propagation Network : Soft Computing Course Lecture 15 – 20, notes, slides
www.myreaders.info/ , RC Chakraborty, e-mail rcchak@gmail.com , Aug. 10, 2010
http://www.myreaders.info/html/soft_computing.html
Back Propagation Network
Soft Computing
Back-Propagation Network, topics : Background, what is back-prop
network ? learning AND function, simple learning machines - Error
measure , Perceptron learning rule, Hidden Layer, XOR problem.
Back-Propagation Learning : learning by example, multi-layer
feed-forward back-propagation network, computation in input,
hidden and output layers, error calculation. Back-propagation
algorithm for training network - basic loop structure, step-by-step
procedure, numerical example.
www.myreaders.info

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Back Propagation Network
Soft Computing
Topics
(Lectures 15, 16, 17, 18, 19, 20 6 hours) Slides
1. Back-Propagation Network - Background
What is back-prop network ?; Learning : AND function; Simple
learning machines - Error measure , Perceptron learning rule ; Hidden
Layer , XOR problem.
03-11
2. Back-Propagation Learning - learning by example
Multi-layer Feed-forward Back-propagation network; Computation of
Input, Hidden and Output layers ; Calculation of Error.
12-16
3. Back-Propagation Algorithm
Algorithm for training Network - Basic loop structure, Step-by-step
procedure; Example: Training Back-prop network, Numerical example.
17-32
4. References 33
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Back-Propagation Network
What is BPN ?
• A single-layer neural network has many restrictions. This network
can accomplish very limited classes of tasks.
Minsky and Papert (1969) showed that a two layer feed-forward
network can overcome many restrictions, but they did not present
a solution to the problem as "how to adjust the weights from input
to hidden layer" ?
• An answer to this question was presented by Rumelhart, Hinton
and Williams in 1986. The central idea behind this solution is that
the errors for the units of the hidden layer are determined by
back-propagating the errors of the units of the output layer.
This method is often called the Back-propagation learning rule.
Back-propagation can also be considered as a generalization of the
delta rule for non-linear activation functions and multi-layer networks.
• Back-propagation is a systematic method of training multi-layer
artificial neural networks.
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SC - NN - BPN – Background
1. Back-Propagation Network – Background
Real world is faced with a situations where data is incomplete or noisy.
To make reasonable predictions about what is missing from the information
available is a difficult task when there is no a good theory available
that may to help reconstruct the missing data. It is in such situations the
Back-propagation (Back-Prop) networks may provide some answers.
• A BackProp network consists of at least three layers of units :
- an input layer,
- at least one intermediate hidden layer, and
- an output layer.
• Typically, units are connected in a feed-forward fashion with input
units fully connected to units in the hidden layer and hidden units
fully connected to units in the output layer.
• When a BackProp network is cycled, an input pattern is propagated
forward to the output units through the intervening input-to-hidden
and hidden-to-output weights.
• The output of a BackProp network is interpreted as a classification
decision.
[Continued in next slide]
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SC - NN - BPN –Background
[Continued from previous slide]
• With BackProp networks, learning occurs during a training phase.
The steps followed during learning are :
− each input pattern in a training set is applied to the input units and
then propagated forward.
− the pattern of activation arriving at the output layer is compared
with the correct (associated) output pattern to calculate an error signal.
− the error signal for each such target output pattern is then
back-propagated from the outputs to the inputs in order to
appropriately adjust the weights in each layer of the network.
− after a BackProp network has learned the correct classification for
a set of inputs, it can be tested on a second set of inputs to see
how well it classifies untrained patterns.
• An important consideration in applying BackProp learning is how
well the network generalizes.
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1.1 Learning :
AND function
Implementation of AND function in the neural network.
AND
X1 X2 Y
0 0 0
0 1 0
1 0 0
1 1 1
W1
Input I1
Output O
W2
Input I2
AND function implementation
− there are 4 inequalities in the AND function and they must be
satisfied.
w10 + w2 0 < θ , w1 0 + w2 1 < θ ,
w11 + w2 0 < θ , w1 1 + w2 1 > θ
− one possible solution :
if both weights are set to 1 and the threshold is set to 1.5, then
(1)(0) + (1)(0) < 1.5 assign 0 , (1)(0) + (1)(1) < 1.5 assign 0
(1)(1) + (1)(0) < 1.5 assign 0 , (1)(1) + (1)(1) > 1.5 assign 1
Although it is straightforward to explicitly calculate a solution to the
AND function problem, but the question is "how the network can
learn such a solution". That is, given random values for the weights
can we define an incremental procedure which will cover a set of
weights which implements AND function.
06
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C
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• Example 1
AND Problem
Consider a simple neural network made up of two inputs connected
to a single output unit.
AND
X1 X2 Y
0 0 0
0 1 0
1 0 0
1 1 1
W1
Input I1
Output O
W2
Input I2
Fig A simple two-layer network applied to the AND problem
− the output of the network is determined by calculating a weighted
sum of its two inputs and comparing this value with a threshold θ.
− if the net input (net) is greater than the threshold, then the output
is 1, else it is 0.
− mathematically, the computation performed by the output unit is
net = w1 I1 + w2 I2 if net > θ then O = 1, otherwise O = 0.
• Example 2
Marital status and occupation
In the above example 1
− the input characteristics may be : marital Status (single or married)
and their occupation (pusher or bookie).
− this information is presented to the network as a 2-D binary input vector
where 1st element indicates marital status (single = 0, married = 1)
and 2nd element indicates occupation ( pusher = 0, bookie = 1 ).
− the output, comprise "class 0" and "class 1".
− by applying the AND operator to the inputs, we classify an
individual as a member of the "class 0" only if they are both
married and a bookie; that is the output is 1 only when both of the
inputs are 1.
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1.2 Simple Learning Machines
Rosenblatt (late 1950's) proposed learning networks called Perceptron.
The task was to discover a set of connection weights which correctly
classified a set of binary input vectors. The basic architecture of the
perceptron is similar to the simple AND network in the previous example.
A perceptron consists of a set of input units and a single output unit.
As in the AND network, the output of the perceptron is calculated
by comparing the net input net = wi Ii and a threshold θ.
If the net input is greater than the threshold θ , then the output unit is
turned on , otherwise it is turned off.
To address the learning question, Rosenblatt solved two problems.
− first, defined a cost function which measured error.
− second, defined a procedure or a rule which reduced that error by
appropriately adjusting each of the weights in the network.
However, the procedure (or learning rule) required to assesses the
relative contribution of each weight to the total error.
The learning rule that Roseblatt developed, is based on determining
the difference between the actual output of the network with the
target output (0 or 1), called "error measure" which is explained
in the next slide.
08
Σ
i=1
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• Error Measure ( learning rule )
Mentioned in the previous slide, the error measure is the difference
between actual output of the network with the target output (0 or 1).
― If the input vector is correctly classified (i.e., zero error),
then the weights are left unchanged, and
the next input vector is presented.
― If the input vector is incorrectly classified (i.e., not zero error),
then there are two cases to consider :
Case 1 : If the output unit is 1 but need to be 0 then
◊ the threshold is incremented by 1 (to make it less likely that the
output unit would be turned on if the same input vector was
presented again).
◊ If the input Ii is 0, then the corresponding weight Wi is left
unchanged.
◊ If the input Ii is 1, then the corresponding weight Wi is
decreased by 1.
Case 2 : If output unit is 0 but need to be 1 then the opposite
changes are made.
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SC - NN – BPN – Background
• Perceptron Learning Rule : Equations
The perceptron learning rules are govern by two equations,
− one that defines the change in the threshold and
− the other that defines change in the weights,
The change in the threshold is given by
∆ θ = - (tp - op) = - dp
where p specifies the presented input pattern,
op actual output of the input pattern Ipi
tp specifies the correct classification of the input pattern ie target,
dp is the difference between the target and actual outputs.
The change in the weights are given by
∆ wi = (tp - op) Ipi = - dp Ipi
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1.3 Hidden Layer
Back-propagation is simply a way to determine the error values in
hidden layers. This needs be done in order to update the weights.
The best example to explain where back-propagation can be used is
the XOR problem.
Consider a simple graph shown below.
− all points on the right side of the line are +ve, therefore the output of
the neuron should be +ve.
− all points on the left side of the line are –ve, therefore the output of
the neuron should be –ve.
With this graph, one can make a simple table of
inputs and outputs as shown below.
AND
X1 X2 Y
1 1 1
1 0 0
0 1 0
0 0 0
Training a network to operate
as an AND switch can be
done easily through only one
neuron (see previous slides)
But a XOR problem can't be solved using only one neuron.
If we want to train an XOR, we need 3 neurons, fully-connected in a
feed-forward network as shown below.
XOR
X1 X2 Y
1 1 0
1 0 1
0 1 1
0 0 0
X1
X2
Y
X2
X1
11
X2
+ +
― + + +
― ― + + +
― ― + +
―
― X1
― ― ― ― ―
― ― ― ―
―
B
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SC - NN – Back Propagation Network
2. Back Propagation Network
Learning By Example
Consider the Multi-layer feed-forward back-propagation network below.
The subscripts I, H, O denotes input, hidden and output neurons.
The weight of the arc between i th
input neuron to j th
hidden layer is Vij .
The weight of the arc between i th hidden neuron to j th out layer is Wij
Fig Multi-layer feed-forward back-propagation network
The table below indicates an 'nset' of input and out put data.
It shows ℓ inputs and the corresponding n output data.
Table : 'nset' of input and output data
No Input Ouput
I1 I2 . . . . Iℓ O1 O2 . . . . On
1 0.3 0.4 . . . . 0.8 0.1 0.56 . . . . 0.82
2
:
nset
In this section, over a three layer network the computation in the input,
hidden and output layers are explained while the step-by-step
implementation of the BPN algorithm by solving an example is illustrated
in the next section.
12
Input Layer
i - nodes
Hidden Layer
m- nodes
Output Layer
n - nodes
11 1
2 2 2
ℓ nm
IO1OI1II1 V11 IH1
OH1 OO1W11
IO2OI2II2 IH2
OH2 OO2
IOnOIℓ
OHmIHmIIℓ OOn
Wm1Vl1
V21 W21
WijVij

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2.1 Computation of Input, Hidden and Output Layers
(Ref.Previous slide, Fig. Multi-layer feed-forward back-propagation network)
• Input Layer Computation
Consider linear activation function.
If the output of the input layer is the input of the input layer and
the transfer function is 1, then
{ O }I = { I }I
ℓ x 1 ℓ x 1 (denotes matrix row, column size)
The hidden neurons are connected by synapses to the input neurons.
- Let Vij be the weight of the arc between i
th
input neuron to
j
th
hidden layer.
- The input to the hidden neuron is the weighted sum of the outputs
of the input neurons. Thus the equation
IHp = V1p OI1 + V2p OI2 + . . . . + V1p OIℓ where (p =1, 2, 3 . . , m)
denotes weight matrix or connectivity matrix between input neurons
and a hidden neurons as [ V ].
we can get an input to the hidden neuron as ℓ x m
{ I }H = [ V ]
T
{ O }I
m x 1 m x ℓ ℓ x 1 (denotes matrix row, column size)
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• Hidden Layer Computation
Shown below the pth
neuron of the hidden layer. It has input from the
output of the input neurons layers. If we consider transfer function as
sigmoidal function then the output of the pth
hidden neuron is given by
where OHp is the output of the pth
hidden neuron,
IHp is the input of the pth
hidden neuron, and
θHP is the threshold of the pth
neuron;
Note : a non zero threshold neuron, is computationally equivalent to an input
that is always held at -1 and the non-zero threshold becomes the connecting
weight value as shown in Fig. below.
Fig. Example of Treating threshold
in hidden layer
Note : the threshold is not treated as
shown in the Fig (left); the outputs of
the hidden neuron are given by the
above equation.
Treating each component of the input of the hidden neuron separately,
we get the outputs of the hidden neuron as given by above equation .
The input to the output neuron is the weighted sum of the outputs of
the hidden neurons. Accordingly, Ioq the input to the qth
output neuron
is given by the equation
Ioq = W1q OH1 + W2q OH2 + . . . . + Wmq OHm , where (q =1, 2, 3 . . , n)
It denotes weight matrix or connectivity matrix between hidden neurons
and output neurons as [ W ], we can get input to output neuron as
{ I }O = [ W] T
{ O }H
n x 1 n x m m x 1 (denotes matrix row, column size)
14
1
OHp =
( 1 + e
-λ (IHP – θHP)
)
1
{ O }H =
( 1 + e
-λ (IHP – θHP)
)
–
–
–
–
θHP
V3p
V2p
Vℓp
V1p
p
OI1II1
1
OI2II2
2
OI3II3
3
OIℓIIℓ
ℓ
OIO = -1IIO = -1
O

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• Output Layer Computation
Shown below the qth
neuron of the output layer. It has input from
the output of the hidden neurons layers.
If we consider transfer function as sigmoidal function then the output
of the qth
output neuron is given by
where OOq is the output of the qth
output neuron,
IOq is the input to the qth
output neuron, and
θOq is the threshold of the qth
neuron;
Note : A non zero threshold neuron, is computationally equivalent to
an input that is always held at -1 and the non-zero threshold becomes
the connecting weight value as shown in Fig. below.
Note : Here again the threshold may be tackled by considering
extra Oth
neuron in the hidden layer with output of -1 and the threshold
value θOq becomes the connecting weight value as shown in Fig. below.
Fig. Example of Treating threshold
in output layer
Note : here again the threshold is not
treated as shown in the Fig (left); the
Outputs of the output neurons given by
the above equation.
15
1
OOq =
( 1 + e
-λ (IOq – θOq)
)
1
{ O }O =
( 1 + e
-λ (IOq – θOq)
)
–
–
–
–
q
θOq
W3q
w2q
Wmq
W1q
OOq
OH1IH1
1
OH2IH2
2
OH3IH3
3
OHmIHm
m
OHO = -1IHO = -1
O

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2.2 Calculation of Error
(refer the earlier slides - Fig. "Multi-layer feed-forward back-propagation network"
and a table indicating an 'nset' of input and out put data for the purpose of
training)
Consider any r
th
output neuron. For the target out value T, mentioned
in the table- 'nset' of input and output data" for the purpose of
training, calculate output O .
The error norm in output for the r th
output neuron is
E1
r = (1/2) e2
r = (1/2) (T –O)2
where E1
r is 1/2 of the second norm of the error er in the r
th
neuron
for the given training pattern.
e2
r is the square of the error, considered to make it independent
of sign +ve or –ve , ie consider only the absolute value.
The Euclidean norm of error E1
for the first training pattern is given by
E1
= (1/2) (Tor - Oor )2
This error function is for one training pattern. If we use the same
technique for all the training pattern, we get
E (V, W) = E j
(V, W, I)
where E is error function depends on m ( 1 + n) weights of [W] and [V].
All that is stated is an optimization problem solving, where the
objective or cost function is usually defined to be maximized or
minimized with respect to a set of parameters. In this case, the
network parameters that optimize the error function E over the 'nset'
of pattern sets [I
nset
, t nset
] are synaptic weight values [ V ] and
[ W ] whose sizes are
[ V ] and [ W ]
ℓ x m m x n
16
Σ
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n
Σ
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SC - NN - BPN – Algorithm
3. Back-Propagation Algorithm
The benefits of hidden layer neurons have been explained. The hidden layer
allows ANN to develop its own internal representation of input-output
mapping. The complex internal representation capability allows the
hierarchical network to learn any mapping and not just the linearly
separable ones.
The step-by-step algorithm for the training of Back-propagation network
is presented in next few slides. The network is the same , illustrated before,
has a three layer. The input layer is with ℓ nodes, the hidden layer with m
nodes and the output layer with n nodes. An example for training a
BPN with five training set have been shown for better understanding.
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3.1 Algorithm for Training Network
The basic algorithm loop structure, and the step by step procedure of
Back- propagation algorithm are illustrated in next few slides.
• Basic algorithm loop structure
Initialize the weights
Repeat
For each training pattern
"Train on that pattern"
End
Until the error is acceptably low.
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• Back-Propagation Algorithm - Step-by-step procedure
■ Step 1 :
Normalize the I/P and O/P with respect to their maximum values.
For each training pair, assume that in normalized form there are
ℓ inputs given by { I }I and
ℓ x 1
n outputs given by { O}O
n x 1
■ Step 2 :
Assume that the number of neurons in the hidden layers lie
between 1 < m < 21
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■ Step 3 :
Let [ V ] represents the weights of synapses connecting input
neuron and hidden neuron
Let [ W ] represents the weights of synapses connecting hidden
neuron and output neuron
Initialize the weights to small random values usually from -1 to +1;
[ V ] 0
= [ random weights ]
[ W ] 0
= [ random weights ]
[ ∆ V ] 0
= [ ∆ W ] 0
= [ 0 ]
For general problems λ can be assumed as 1 and threshold
value as 0.
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■ Step 4 :
For training data, we need to present one set of inputs and outputs.
Present the pattern as inputs to the input layer { I }I .
then by using linear activation function, the output of the input layer
may be evaluated as
{ O }I = { I }I
ℓ x 1 ℓ x 1
■ Step 5 :
Compute the inputs to the hidden layers by multiplying corresponding
weights of synapses as
{ I }H = [ V]
T
{ O }I
m x 1 m x ℓ ℓ x 1
■ Step 6 :
Let the hidden layer units, evaluate the output using the
sigmoidal function as
m x 1
21
1
{ O }H =
( 1 + e
- (IHi)
)
–
–
–
–

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■ Step 7 :
Compute the inputs to the output layers by multiplying corresponding
weights of synapses as
{ I }O = [ W]
T
{ O }H
n x 1 n x m m x 1
■ Step 8 :
Let the output layer units, evaluate the output using sigmoidal
function as
Note : This output is the network output
22
1
{ O }O =
( 1 + e - (IOj)
)
–
–
–
–

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■ Step 9 :
Calculate the error using the difference between the network output
and the desired output as for the j
th
training set as
EP
=
■ Step 10 :
Find a term { d } as
n x 1
23
√ ∑ (Tj - Ooj )2
n
{ d } = (Tk – OOk) OOk (1 – OOk )
–
–
–
–

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■ Step 11 :
Find [ Y ] matrix as
[ Y ] = { O }H 〈 d 〉
m x n m x 1 1 x n
■ Step 12 :
Find [ ∆ W ]
t +1
= α [ ∆ W ]
t
+ η [ Y ]
m x n m x n m x n
■ Step 13 :
Find { e } = [ W ] { d }
m x 1 m x n n x 1
m x 1 m x 1
Find [ X ] matrix as
[ X ] = { O }I 〈 d* 〉 = { I }I 〈 d* 〉
1 x m ℓ x 1 1 x m ℓ x 1 1 x m
24
(OHi) (1 – OHi )
{ d* } =
–
–
–
–
ei

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■ Step 14 :
Find [ ∆ V ]
t +1
= α [ ∆ V ]
t
+ η [ X ]
1 x m 1 x m 1 x m
■ Step 15 :
Find [ V ]
t +1
= [V ]
t
+ [ ∆ V ]
t +1
[ W ]
t +1
= [W ]
t
+ [ ∆ W ]
t +1
■ Step 16 :
Find error rate as
∑ Ep
error rate =
nset
■ Step 17 :
Repeat steps 4 to 16 until the convergence in the error rate is less
than the tolerance value
■ End of Algorithm
Note : The implementation of this algorithm, step-by-step 1 to 17,
assuming one example for training BackProp Network is illustrated in
the next section.
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3.2 Example : Training Back-Prop Network
• Problem :
Consider a typical problem where there are 5 training sets.
Table : Training sets
S. No. Input Output
I1 I2 O
1 0.4 -0.7 0.1
2 0.3 -0.5 0.05
3 0.6 0.1 0.3
4 0.2 0.4 0.25
5 0.1 -0.2 0.12
In this problem,
- there are two inputs and one output.
- the values lie between -1 and +1 i.e., no need to normalize the values.
- assume two neurons in the hidden layers.
- the NN architecture is shown in the Fig. below.
Fig. Multi layer feed forward neural network (MFNN) architecture
with data of the first training set
The solution to problem are stated step-by-step in the subsequent
slides.
26
0.2
-0.20.4 TO = 0.1
0.10.4
0.2
-0.7 -0.5
Input
layer
Hidden
layer
Output
layer

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■ Step 1 : Input the first training set data (ref eq. of step 1)
from training set s.no 1
■ Step 2 : Initialize the weights as (ref eq. of step 3 & Fig)
;
from fig initialization from fig initialization
■ Step 3 : Find { I }H = [ V] T
{ O }I as (ref eq. of step 5)
{ I }H
Values from step 1 & 2
27
{ O }I =
ℓ x 1
{ I }I =
ℓ x 1
0.4
-0.7
2 x 1
[ V ] 0
=
0.1 0.4
-0.2 0.2
2x2
[ W ] 0
=
0.2
-0.5
2 x1
0.1 -0.2
-0.4 0.2
0.4
-0.7
=
0.18
0.02
=

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■ Step 4 : (ref eq. of step 6)
Values from step 3 values
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1
( 1 + e
- (0.18)
)
1
( 1 + e - (0.02)
)
{ O }H =
0.5448
0.505
=

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{ I }O = [ W] T
{ O }H = ( 0.2 - 0.5 ) = - 0.14354
Values from step 2 , from step 4
Values from step 5
Error = (TO – OO1 )2
= (0.1 – 0.4642)2
= 0.13264
table first training set o/p from step 6
29
0.5448
0.505
1
( 1 + e - (0.14354)
)
{ O }O = = 0.4642

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d = (TO – OO1 ) ( OO1 ) (1 – OO1 )
= (0.1 – 0.4642) (0.4642) ( 0.5358) = – 0.09058
Training o/p all from step 6
(ref eq. of step 11)
[ Y ] = { O }H (d ) = (– 0.09058) =
from values at step 4 from values at step 8 above
[ ∆ W ]
1
= α [ ∆ W ]
0
+ η [ Y ] assume η =0.6
=
from values at step 2 & step 8 above
{ e } = [ W ] { d } = (– 0.09058) =
from values at step 8 above
from values at step 2
30
0.5448
0.505
–0.0493
–0.0457
–0.02958
–0.02742
0.2
-0.5
–0.018116
–0.04529

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{ d* } = =
from values at step 10 at step 4 at step 8
[ X ] = { O }I ( d* ) = ( – 0.00449 0.01132)
from values at step 1 from values at step 11 above
=
[ ∆ V ]
1
= α [ ∆ V ]
0
+ η [ X ] =
from values at step 2 & step 8 above
31
(–0.018116) (0.5448) (1- 0.5448)
(0.04529) (0.505) ( 1 – 0.505)
–0.00449
–0.01132
0.4
-0.7
– 0.001796 0.004528
0.003143 –0.007924
– 0.001077 0.002716
0.001885 –0.004754

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[ V ]
1
= +
from values at step 2 from values at step 13
=
[ W ]
1
= + =
from values at step 2, from values at step 9
■ Step 15 :
With the updated weights [ V ] and [ W ] , error is calculated again
and next training set is taken and the error will then get adjusted.
■ Step 16 :
Iterations are carried out till we get the error less than the tolerance.
■ Step 17 :
Once the weights are adjusted the network is ready for
inferencing new objects .
32
–0.02958
–0.02742
0.1 0.4
-0.2 0.2
– 0.001077 0.002716
0.001885 –0.004754
– 0.0989 0.04027
0.1981 –0.19524
0.2
-0.5
0.17042
–0.52742

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SC - NN - BPN – References
4. References : Textbooks
1. "Neural Network, Fuzzy Logic, and Genetic Algorithms - Synthesis and
Applications", by S. Rajasekaran and G.A. Vijayalaksmi Pai, (2005), Prentice Hall,
Chapter 3, page 34-86.
2. "Soft Computing and Intelligent Systems Design - Theory, Tools and Applications",
by Fakhreddine karray and Clarence de Silva (2004), Addison Wesley, chapter 5,
page 249-293.
3. "Elements of Artificial Neural Networks", by Kishan Mehrotra, Chilukuri K. Mohan
and Sanjay Ranka, (1996), MIT Press, Chapter 3, page 65-106.
4. "Fundamentals of Neural Networks: Architecture, Algorithms and Applications", by
Laurene V. Fausett, (1993), Prentice Hall, Chapter 6, page 289-332.
5. "Neural Network Design", by Martin T. Hagan, Howard B. Demuth and Mark
Hudson Beale, ( 1996) , PWS Publ. Company, Chapter 11-12, page 11-1 to 12-50.
6. Related documents from open source, mainly internet. An exhaustive list is
being prepared for inclusion at a later date.
33

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