Neural Network

Contents

Diagram of a Neuron The Simple Perceptron Multilayer Neural Network What is Hidden Layer? Why do we Need a Hidden Layer? How do Multilayer Neural Networks Learn?

Diagram of a Neuron

Neuron

w1

w2

wn

Y

.

.

.

x1

x2

x3

Input signalsWeight Output Signals

Example of NN: The Perceptron

w1

w2

Σ

Th

Threshold

Linear Combiner

　 Hard 　　Limiter

Y-output

x1

x2

10

n

i iix w Th

Step & sign activation function called hard limit functions.

Single neuron with adjustable synaptic weight and a hard limiter.

Multilayer Neural Network

A multilayer Perceptron is a feedforward network with one

or more hidden layers

The network consists of:

an input layer of source neurons,

at least one middle or hidden layer of computation

neurons

An output layer of computation neurons

The input signals are propagated in a forward direction on a

layer-by-layer basis

Multilayer Perceptron with two Hidden Layers

What is Hidden Layer?

A hidden layer hides its desired output

Neurons in the hidden layer cannot be observed through the

input/output behavior of the network.

There is no obvious way to know what the desired output of the

hidden layer should be.

Why do we Need a Hidden Layer?

The input layer accepts input signals from the outside world

and redistributes these signals to all neurons in the hidden

layer.

Neuron in the hidden layer detect the features; the weights of

the neurons represent the features hidden in the input

patterns.

The output layer accepts output signal from the hidden layer

and establishes the output pattern of the entire network.

How Do Multilayer Neural Networks Learn?

Most popular method of learning is back-propagation.

Learning in a multi-layer network proceeds the same way as for

a Perceptron

A training set of input patterns is presented to the network

The network computes the output pattern.

If there is an error, the weight are adjusted to reduce this error.

In multilayer network, there are many weights, each of which

contributes to more than one output.

Back Propagation Neural Network (1/2)

A back-propagation network is a multilayer network that has

three or four layers.

The layers are fully connected, i.e, every neuron in each layer

is connected to every other neuron in the adjacent forward

layer

A neuron determines its output in a manner similar to

Rosenblatt’s Perceptron.

1

n

i ii

X x w

Back Propagation Neural Network (2/2)

The net weighted input value is passed through the activation

function.

Unlike a Perceptron, neuron in the back propagation network

use a sigmoid activation function:

1

1sigmoid

XY

e

Three-layer Back Propagation Neural Network

Learning Law Used in Back- Propagation Network

In three layer network, i,j and k refer to neurons in the input, hidden and

output layers.

Input signal x1, x2, …….. xn are propagated through the network from left

to right

Error signals e1, e2, en from right to left.

The symbol Wij denotes the weight for the connection between neuron i

in the input layer and neuron j in the hidden layer

The symbol Wjk denotes the weight for the connection between neuron j

in the hidden layer and neuron k in the output layer

Learning Law Used in Back- Propagation Network

The error signal at the output of neuron k at iteration p is

defined by,

,( ) ( ) ( )k d k ke p y p y p

The updated weight at the output layer is defined by,

( 1) ( ) ( ) ( )jk jk j kW p W p y p p

Learning Law Used in Back- Propagation Network

( )( ) ( )

( )k

k kk

y pp e p

x p

The error gradient is determined as the derivative of the activation function multiplied by the error at the neuron output,

Where yk(p) is the output of neuron k at iteration p and xk(p) is the net weighted input to neuron k,

( ) ( )[1 ( )] ( )k k k kp y p y p e p

( )

1( )

1 kk X py p

e

Learning Law Used in Back- Propagation Network

The weight correction for the hidden layer,

( 1) ( ) ( ) ( )ij ij i jW p W p x p p

1( ) ( )[1 ( )] ( ) ( )

l

j j j k jkk

p y p y p p W p

1( ) ( ) ( )

n

j i ij ji

X p x p W p

( )

1( )

1 jj X py p

e

Back Propagation Training Algorithm

Initialization : Set all the weights and threshold levels of the network to random numbers uniformly distributed inside a small range (Haykin 1994):(-2.4/Fi, +2.4/Fi), Where Fi is the total number of inputs of neuron i in the network.

Activation: Calculate the actual outputs of the neurons in the hidden layer Calculate the actual outputs of the neurons in the output layer

Weight Training: Update the weights in the back-propagation network propagating backward the errors associated with output neurons.

Iteration: Increase iteration p by one, go back to step 2 and repeat the process until the selected error criterion is satisfied.

Back-propagation: Activation

(A) Calculate the actual outputs of the neurons in the hidden layer

(B) Calculate the actual outputs of the neurons in the output layer

1( ) [ ( ) ( ) ]

n

j i ij ji

y p sigmoid x p W p

1( ) [ ( ) ( )

m

k jk jk kj

y p sigmoid x p W p

Back-propagation: Weight Training

(A) Calculate the error gradient for the neurons in the output layer.

( ) ( )[1 ( )] ( )k k k kp y p y p e p

,( ) ( ) ( )k d k ke p y p y p

( 1) ( ) ( ) ( )jk jk j kW p W p y p p

Back-propagation: Weight Training

(B) Calculate the error gradient for the neurons in the hidden layer.

1( ) ( )[1 ( )] ( ) ( )

l

j j j k jkk

p y p y p p W p

( 1) ( ) ( ) ( )ij ij i jW p W p x p p

Recommended Textbooks

[Negnevitsky, 2001] M. Negnevitsky “ Artificial Intelligence: A guide to Intelligent Systems”, Pearson Education Limited, England, 2002.

[Russel, 2003] S. Russell and P. Norvig Artificial Intelligence: A Modern Approach Prentice Hall, 2003, Second Edition

[Patterson, 1990] D. W. Patterson, “Introduction to Artificial Intelligence and Expert Systems”, Prentice-Hall Inc., Englewood Cliffs, N.J, USA, 1990.

[Minsky, 1974] M. Minsky “A Framework for Representing Knowledge”, MIT-AI Laboratory Memo 306, 1974.