Unit 1

INTRODUCTION AND ARTIFICIAL NEURAL
NETWORKS
UNIT I
Mr. S. VINOD
ASSISTANT PROFESSOR
EEE DEPARTMENT

• computing techniques
• applications of soft computing
• Neuron
• Nerve structure and synapse-
• Artificial Neuron and its model
• Activation functions
• Neural network architecture
• single layer and multilayer feed forward networks
• McCullochPitts neuron model
• perceptron model- Adaline and Madaline
• multilayer perception model
• back propagation learning methods
• effect of learning rule coefficient
• back propagation algorithm
• factors affecting back propagation training applications.
2

Neural Networks
● Artificial neural network (ANN) is a machine learning
approach that models human brain and consists of a number
of artificial neurons.
● Neuron in ANNs tend to have fewer connections than
biological neurons.
● Each neuron in ANN receives a number of inputs.
● An activation function is applied to these inputs which results
in activation level of neuron (output value of the neuron).
● Knowledge about the learning task is given in the form of
examples called training examples.

Contd..
● An Artificial Neural Network is specified by:
− neuron model: the information processing unit of the NN,
− an architecture: a set of neurons and links connecting neurons.
Each link has a weight,
− a learning algorithm: used for training the NN by modifying the
weights in order to model a particular learning task correctly on
the training examples.
● The aim is to obtain a NN that is trained and generalizes
well.
● It should behaves correctly on new instances of the
learning task.

Neuron
● The neuron is the basic information processing unit of a
NN. It consists of:
1 A set of links, describing the neuron inputs, with weights W1, W2,
…, Wm
2 An adder function (linear combiner) for computing the weighted
sum of the inputs:
(real numbers)
3 Activation function for limiting the amplitude of the neuron
output. Here ‘b’ denotes bias.


m
1
jjxwu
j

)(uy b

The Neuron Diagram
Input
values
weights
Summing
function
Bias
b
Activation
functionInduced
Field
v
Output
y
x1
x2
xm
w2
wm
w1
 
 )(

Bias of a Neuron
● The bias b has the effect of applying a transformation to
the weighted sum u
v = u + b
● The bias is an external parameter of the neuron. It can be
modeled by adding an extra input.
● v is called induced field of the neuron
bw
xwv j
m
j
j

 
0
0

Neuron Models
● The choice of activation function determines the
neuron model.
Examples:
● step function:
● ramp function:
● sigmoid function with z,x,y parameters
● Gaussian function:














 

2
2
1
exp
2
1
)(




v
v
)exp(1
1
)(
yxv
zv











otherwise))/())(((
if
if
)(
cdabcva
dvb
cva
v






cvb
cva
v
if
if
)(

• The Gaussian function is the probability
function of the normal distribution. Sometimes
also called the frequency curve.

Network Architectures
● Three different classes of network architectures
− single-layer feed-forward
− multi-layer feed-forward
− recurrent
● The architecture of a neural network is linked
with the learning algorithm used to train

Single Layer Feed-forward
Input layer
of
source nodes
Output layer
of
neurons

Perceptron: Neuron Model
(Special form of single layer feed forward)
− The perceptron was first proposed by Rosenblatt (1958) is a simple
neuron that is used to classify its input into one of two categories.
− A perceptron uses a step function that returns +1 if weighted sum
of its input  0 and -1 otherwise
x1
x2
xn
w2
w1
wn
b (bias)
v y
(v)






0if1
0if1
)(
v
v
v

Perceptron for Classification
● The perceptron is used for binary classification.
● First train a perceptron for a classification task.
− Find suitable weights in such a way that the training examples are
correctly classified.
− Geometrically try to find a hyper-plane that separates the examples of the
two classes.
● The perceptron can only model linearly separable classes.
● When the two classes are not linearly separable, it may be
desirable to obtain a linear separator that minimizes the mean
squared error.
● Given training examples of classes C1, C2 train the perceptron in
such a way that :
− If the output of the perceptron is +1 then the input is assigned to class C1
− If the output is -1 then the input is assigned to C2

X1
1 true true
false true
0 1 X2
Boolean function OR – Linearly separable

Learning Process for Perceptron
● Initially assign random weights to inputs between -0.5 and
+0.5
● Training data is presented to perceptron and its output is
observed.
● If output is incorrect, the weights are adjusted accordingly
using following formula.
wi  wi + (a* xi *e), where ‘e’ is error produced
and ‘a’ (-1  a  1) is learning rate
− ‘a’ is defined as 0 if output is correct, it is +ve, if output is too low and
–ve, if output is too high.
− Once the modification to weights has taken place, the next piece of
training data is used in the same way.
− Once all the training data have been applied, the process starts again
until all the weights are correct and all errors are zero.
− Each iteration of this process is known as an epoch.

Example: Perceptron to learn OR
function
● Initially consider w1 = -0.2 and w2 = 0.4
● Training data say, x1 = 0 and x2 = 0, output is 0.
● Compute y = Step(w1*x1 + w2*x2) = 0. Output is correct so
weights are not changed.
● For training data x1=0 and x2 = 1, output is 1
● Compute y = Step(w1*x1 + w2*x2) = 0.4 = 1. Output is correct
so weights are not changed.
● Next training data x1=1 and x2 = 0 and output is 1
● Compute y = Step(w1*x1 + w2*x2) = - 0.2 = 0. Output is
incorrect, hence weights are to be changed.
● Assume a = 0.2 and error e=1
wi = wi + (a * xi * e) gives w1 = 0 and w2 =0.4
● With these weights, test the remaining test data.
● Repeat the process till we get stable result.

Perceptron: Limitations
● The perceptron can only model linearly separable
functions,
− those functions which can be drawn in 2-dim graph and single
straight line separates values in two part.
● Boolean functions given below are linearly separable:
− AND
− OR
− COMPLEMENT
● It cannot model XOR function as it is non linearly
separable.
− When the two classes are not linearly separable, it may be
desirable to obtain a linear separator that minimizes the mean
squared error.

XOR – Non linearly separable function
● A typical example of non-linearly separable function is the
XOR that computes the logical exclusive or..
● This function takes two input arguments with values in {0,1}
and returns one output in {0,1},
● Here 0 and 1 are encoding of the truth values false and
true,
● The output is true if and only if the two inputs have
different truth values.
● XOR is non linearly separable function which can not be
modeled by perceptron.
● For such functions we have to use multi layer feed-forward
network.

These two classes (true and false) cannot be separated using a
line. Hence XOR is non linearly separable.
Input Output
X1 X2 X1 XOR X2
0 0 0
0 1 1
1 0 1
1 1 0
X1
1 true false
false true
0 1 X2

Multi layer feed-forward NN (FFNN)
● FFNN is a more general network architecture, where there are
hidden layers between input and output layers.
● Hidden nodes do not directly receive inputs nor send outputs to
the external environment.
● FFNNs overcome the limitation of single-layer NN.
● They can handle non-linearly separable learning tasks.
Input
layer
Output
layer
Hidden Layer
3-4-2 Network

Inputs OutputofHiddenNodes Output
Node
X1XORX2
X1 X2 H1 H2
0 0 0 0 –0.50 0
0 1 –10 1 0.5 1 1
1 0 1 –10 0.5 1 1
1 1 0 0 –0.50 0
Since we are representing two states by 0 (false) and 1 (true), we
will map negative outputs (–1, –0.5) of hidden and output layers
to 0 and positive output (0.5) to 1.

FFNN for XOR
● The ANN for XOR has two hidden nodes that realizes this non-linear
separation and uses the sign (step) activation function.
● Arrows from input nodes to two hidden nodes indicate the directions of
the weight vectors (1,-1) and (-1,1).
● The output node is used to combine the outputs of the two hidden
nodes.
Input nodes Hidden layer Output layer Output
H1 –0.5
X1 1
–1 1
Y
–1 H2
X2 1 1

FFNN NEURON MODEL
● The classical learning algorithm of FFNN is based on the
gradient descent method.
● For this reason the activation function used in FFNN are
continuous functions of the weights, differentiable
everywhere.
● The activation function for node i may be defined as a
simple form of the sigmoid function in the following
manner:
where A > 0, Vi =  Wij * Yj , such that Wij is a weight of the link
from node i to node j and Yj is the output of node j.
)*(
1
1
)( ViA
e
Vi 



Training Algorithm: Back-propagation
● The Back propagation algorithm learns in the same way as
single perceptron.
● It searches for weight values that minimize the total error of
the network over the set of training examples (training set).
● Back propagation consists of the repeated application of the
following two passes:
− Forward pass: In this step, the network is activated on one example
and the error of (each neuron of) the output layer is computed.
− Backward pass: in this step the network error is used for updating
the weights. The error is propagated backwards from the output
layer through the network layer by layer. This is done by recursively
computing the local gradient of each neuron.

Feed-forward Network
Feed-forward networks often have one or more hidden layers of sigmoid neurons followed
by an output layer of linear neurons.
Multiple layers of neurons with nonlinear transfer functions allow the network to learn
nonlinear and linear relationships between input and output vectors.
The linear output layer lets the network produce values outside the range -1 to +1. On the
other hand, if you want to constrain the outputs of a network (such as between 0 and 1),
then the output layer should use a sigmoid transfer function (such as logsig).

Backpropagation Learning
Algorithm
The following slides describes teaching process of multi-layer neural network
employing back-propagation algorithm. To illustrate this process the three layer neural
network with two inputs and one output, which is shown in the picture below, is used:

Learning Algorithm
Backpropagation
Each neuron is composed of two units. First unit adds products of weights coefficients and
input signals. The second unit realise nonlinear function, called neuron transfer (activation)
function. Signal e is adder output signal, and y = f(e) is output signal of nonlinear element.
Signal y is also output signal of neuron.

Learning Algorithm:
Backpropagation
To teach the neural network we need training data set. The training data set consists of
input signals (x1 and x2 ) assigned with corresponding target (desired output) z.
The network training is an iterative process. In each iteration weights coefficients of nodes
are modified using new data from training data set. Modification is calculated using
algorithm described below:
Each teaching step starts with forcing both input signals from training set. After this stage
we can determine output signals values for each neuron in each network layer.

Learning Algorithm:
Backpropagation
Pictures below illustrate how signal is propagating through the network,
Symbols w(xm)n represent weights of connections between network input xm and
neuron n in input layer. Symbols yn represents output signal of neuron n.

Learning Algorithm:
Backpropagation

Learning Algorithm:
Backpropagation
Propagation of signals through the hidden layer. Symbols wmn represent weights
of connections between output of neuron m and input of neuron n in the next
layer.

Learning Algorithm:
Backpropagation
n

Learning Algorithm:
Backpropagation
Propagation of signals through the output layer.

Learning Algorithm:
Backpropagation
In the next algorithm step the output signal of the network y is
compared with the desired output value (the target), which is found in
training data set. The difference is called error signal d of output layer
neuron

Learning Algorithm:
Backpropagation
The idea is to propagate error signal d (computed in single teaching step)
back to all neurons, which output signals were input for discussed
neuron.

Learning Algorithm:
Backpropagation
The weights' coefficients wmn used to propagate errors back are equal to
this used during computing output value. Only the direction of data flow
is changed (signals are propagated from output to inputs one after the
other). This technique is used for all network layers. If propagated errors
came from few neurons they are added. The illustration is below:

Learning Algorithm:
Backpropagation
When the error signal for each neuron is computed, the weights
coefficients of each neuron input node may be modified. In formulas
below df(e)/de represents derivative of neuron activation function
(which weights are modified).

Weight Update Rule
● The Backprop weight update rule is based on the gradient
descent method:
− It takes a step in the direction yielding the maximum decrease of
the network error E.
− This direction is the opposite of the gradient of E.
● Iteration of the Backprop algorithm is usually terminated
when the sum of squares of errors of the output values for
all training data in an epoch is less than some threshold
such as 0.01
ijijij www 
ij
ij
w
-w



E


Back-prop learning algorithm
(incremental-mode)
n=1;
initialize weights randomly;
while (stopping criterion not satisfied or n <max_iterations)
for each example (x,d)
- run the network with input x and compute the output y
- update the weights in backward order starting from those of
the output layer:
with computed using the (generalized) Delta rule
end-for
n = n+1;
end-while;
jijiji www 
jiw

Stopping criterions
● Total mean squared error change:
− Back-prop is considered to have converged when the absolute
rate of change in the average squared error per epoch is
sufficiently small (in the range [0.1, 0.01]).
● Generalization based criterion:
− After each epoch, the NN is tested for generalization.
− If the generalization performance is adequate then stop.
− If this stopping criterion is used then the part of the training set
used for testing the network generalization will not used for
updating the weights.

● Data representation
● Network Topology
● Network Parameters
● Training
● Validation
NN DESIGN ISSUES

● Data representation depends on the problem.
● In general ANNs work on continuous (real valued) attributes.
Therefore symbolic attributes are encoded into continuous ones.
● Attributes of different types may have different ranges of values
which affect the training process.
● Normalization may be used, like the following one which scales
each attribute to assume values between 0 and 1.
for each value xi of ith attribute, mini and maxi are the minimum and
maximum value of that attribute over the training set.
Data Representation
i
i
minmax
min



i
i
i
x
x

● The number of layers and neurons depend on the specific
task.
● In practice this issue is solved by trial and error.
● Two types of adaptive algorithms can be used:
− start from a large network and successively remove some neurons
and links until network performance degrades.
− begin with a small network and introduce new neurons until
performance is satisfactory.
Network Topology

● How are the weights initialized?
● How is the learning rate chosen?
● How many hidden layers and how many
neurons?
● How many examples in the training set?
Network parameters

Initialization of weights
● In general, initial weights are randomly chosen, with
typical values between -1.0 and 1.0 or -0.5 and 0.5.
● If some inputs are much larger than others, random
initialization may bias the network to give much more
importance to larger inputs.
● In such a case, weights can be initialized as follows:


Ni
N
,...,1
|x|
1
2
1
ij i
w For weights from the input to the first layer
For weights from the first to the second layer


Ni
N
i
,...,1
)xw(
1
2
1
jk ij
w 

● The right value of  depends on the application.
● Values between 0.1 and 0.9 have been used in
many applications.
● Other heuristics is that adapt  during the
training as described in previous slides.
Choice of learning rate

Training
● Rule of thumb:
− the number of training examples should be at least five to ten
times the number of weights of the network.
● Other rule:
|W|= number of weights
a=expected accuracy on test seta)-(1
|W|
N 

Contd..
● The networks generated using these
weights and input vectors are stable, except
X2.
● X2 stabilizes to X1 (which is at hamming
distance 1).
● Finally, with the obtained weights and
stable states (X1 and X3), we can stabilize
any new (partial) pattern to one of those

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