Artificial Neural Networks (1)

Dr. Hala Farouk

Biological Neurons of the Brain

History of Neural Networks

In 1943, Warren McCulloch and Walter Pitts introduced one of the first artificial neurons.

The main feature of their neuron model is that a weighted sum of input signals is compared to a threshold to determine the neuron output.

When the sum is greater than or equal to the threshold, the output is 0.

They went on to show that networks of these neurons could, in principle, compute any arithmetic or logical function.

Unlike biological networks, the parameters of their networks had to be designed, as no training method was available.

In the late 1950s, Frank Rosenblatt introduces a learning rule for training perceptron networks to solve pattern recognition problems. Perceptron learn from their mistakes even if they were initialized with random weights.

In the 1980s, improved multi-layer perceptron networks were introduced.

Analogy

Single-Input Neuron

The output depends on the transfer function f chosen by the designerW and b are parameters adjustable by some learning rule

Weight w

Bias b

Learning Rules

By learning we mean a procedure for modifying the weights and biases of a network

The training algorithm falls into three broad categories Supervised Learning

Here we need a set of example for the NN. {p1,t1}, {p2,t2},…, {pQ,tQ}

Where pq is the input and tq is the corresponding correct output.

Reinforcement Learning Similar to supervised but instead of the given correct output, only a

grade (score) is given. This is currently much less common than supervised.

Unsupervised Learning Here the weights and biases are modified in response to network

inputs only. Most of these algorithms perform clustering operation.

Typical Transfer Functions

Typical Transfer Functions

The log-sigmoid transfer function is commonly used in multilayer networks that are trained using the back-propagation algorithm, in part because this function is differentiable.

Example on Single-Input Neuron

The input to a single input neuron is 2.0, its weights is 2.3 and its bias is -3. What is the net input to the transfer function?

n=w p + b = (2.3) (2) + (-3) = 1.6

What is the neuron output, if it has the following transfer functions? Hard limit Linear Log-sigmoid

a= hardlim (1.6) =1.0

a= purelin (1.6) =1.6

a= logsig (1.6) = 1/ (1+e-

1.6)=0.8320

Multi-Input Neuron

n= w1,1 p1 + w1,2 p2 + w1,3 p3 + … +w1,R pR + b1

n=W p + b

First Index for the Neuron

Second Index for the Input

Multi-Input Neuron

The number of input is set by the external specifications of the problem.

If you want to design a neural network that is to predict kite-flying conditions and the inputs are air temperature, wind velocity, and humidity, then there would be three inputs to the network (R=3).

Example on Multi-Input Neuron

Given a two-input neuron with the following parameters: b=1.2, W=[3 2] and p=[-5 6]T, calculate the neuron output for the following transfer functions: A symmetrical hard limit transfer fn. A saturating linear transfer fn. A hyperbolic tangent sigmoid transfer fn.

n= W p + b = [ 3 2 ] * [ -5 6]T + (1.2)= -1.8 a= hardlims (-1.8)=-1 a= satlin(-1.8)=0 a= transig(-1.8)=0.9468

S Neurons with R Inputs each

S Neurons with R Inputs each

Example on Single-Neuron Perceptron

Ai= hardlim (ni) = hardlim(iwT p + bi) Given w1,1=1, w1,2=1, b= - 1

The decision boundary (line at which n=0) n=1wT p + b1=w1,1p1+w1,2p2+b =p1+p2-1 = 0

On one side of this boundary, the output will be 0; on the line and on the other side the output is 1

Test with any point to know the direction The boundary is always orthogonal to 1w

Adjusting the Bias

If the decision boundary line is given then we can adjust the bias according to the following equation:

1wT p + b1= 0

Design Problem

Lets design an AND gate using NN

1) Draw the input spaceBlack dots --> output=1

White dots --> output=0

Design Problem cont.

2. Select a decision boundarythere are infinite number of lines

that separates the black dots from

the white dots.

BUT it’s reasonable to choose a line halfway

3. Choose weight vector that is orthogonal

to the decision boundary

the weight vector can be of any length

so there are infinite possibilities

One choice is 1wT=[2 2]

Design Problem cont.

4. Find the biasPick any point on the decision boundary

line and substitute in

1wT p + b1= 0

For example, p=[1.5 0]T

1wT p + b1= [2 2]*[1.5 0]T+b = 3 + b =0

b= -3

5. Test the network

Multiple-Neuron Perceptron

There will be one decision boundary for EACH neuron.

The decision boundaries will be defined by

iwT p + bi= 0

Example on Supervision Learning

The problem has two inputs and one output

Therefore two inputs and one neuron

Example on Supervision Learning cont.

Lets start without any bias, so only two parameters w1,1 and w1,2

Removing the bias, then decision boundary passes through the origin

Example on Supervision Learning cont.

We want a learning rule that will find a weight vector that points in one of these directions (length of vector is not important)

So lets start with random weights

1wT=[1.0 -0.8]

Then present it to the network with p1. a=hardlim( [1.0 -0.8]*[1 2]T

a=hardlim( -0.6 ) = 0 The NN has made a mistake!

Example on Supervision Learning cont.

Why did it make a mistake?

If we adjust the weights but adding p1 to it

this would make 1w point more in the direction

of p1 so that it would hopefully not

be classified falsely into the wrong zone.

21

3 1w

SO If ( t=1 and a=0 ), then 1wnew = 1wold + p

Example on Supervision Learning cont.

Test the next input p2 with new weights a=hardlim( [2.0 1.2]*[-1 2]T

a=hardlim( 0.4 ) = 1

Again a mistake.

This time we want w to move away from p2

21

3 1w

1wnew = 1wold + p = [ 1.0 -0.8 ]T + [ 1 2 ]T= [ 2.0 1.2 ]T

SO If ( t=0 and a=1 ), then 1wnew = 1wold - p

Example on Supervision Learning cont.

Test the next input p3 with new weights a=hardlim( [3.0 -0.8 ]*[0 -1]T

a=hardlim( 0.8 ) = 1

Again a mistake.

We want w to move away from p3

21

3 1w

1wnew = 1wold + p = [2.0 1.2]T - [- 1 2 ]T= [ 3.0 -0.8 ]T

SO If ( t=0 and a=1 ), then 1wnew = 1wold - p

Example on Supervision Learning cont.

Test the next input p1 again with new weights a=hardlim( [ 3.0 0.2 ]*[1 2]T

a=hardlim( 3.4 ) = 1

Now correct.

Repeat for all other inputs

21

3 1w

1wnew = 1wold + p = [3.0 -0.8 ]T - [0 -1 ]T= [ 3.0 0.2 ]T

SO If ( t=a1) then wnew = 1wold

Multilayer Network

Two Hidden Layers One Output Layer

Recurrent Network

It is a network with feedback.

Some of its outputs is connected to its inputs

a(1)=satlins ( W a(0) + b )

a(2)=satlins ( W a(1) + b ), …