1. Artificial Neural Networks (1)
Dr. Hala Farouk

2. Biological Neurons of the Brain

3. 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.

4. Analogy

5. Single-Input Neuron Weight w Bias b
The output depends on the transfer function f chosen by the designer
W and b are parameters adjustable by some learning rule

6. 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.

7. Typical Transfer Functions

8. 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.

9. 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

10. Multi-Input Neuron Second Index for the Input
First Index for the Neuron
n= w1,1 p1 + w1,2 p2 + w1,3 p3 + … +w1,R pR + b1
n=W p + b

11. 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).

12. 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

13. S Neurons with R Inputs each

14. S Neurons with R Inputs each

15. 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

16. 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

17. Design Problem Lets design an AND gate using NN Draw the input space
Black dots --> output=1
White dots --> output=0

18. Design Problem cont. Select a decision boundary
there are infinite number of lines
that separates the black dots from
the white dots.
BUT it’s reasonable to choose a line halfway
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]

19. Design Problem cont. Test the network Find the bias
Pick 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
Test the network

20. Multiple-Neuron Perceptron
There will be one decision boundary for EACH neuron.
The decision boundaries will be defined by
iwT p + bi= 0

21. Example on Supervision Learning
The problem has two inputs and one output
Therefore two inputs and one neuron

22. 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

23. 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=[ ]
Then present it to the network with p1.
a=hardlim( [ ]*[1 2]T
a=hardlim( -0.6 ) = 0
The NN has made a mistake!

24. Example on Supervision Learning cont.2 1 3
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. 1w
SO If ( t=1 and a=0 ), then 1wnew = 1wold + p

25. Example on Supervision Learning cont.2 1 3
1wnew = 1wold + p = [ ]T + [ ]T= [ ]T
Test the next input p2 with new weights
a=hardlim( [ ]*[-1 2]T
a=hardlim( 0.4 ) = 1 1w
Again a mistake.
This time we want w to move away from p2
SO If ( t=0 and a=1 ), then 1wnew = 1wold - p

26. Example on Supervision Learning cont.2 1 3
1wnew = 1wold + p = [ ]T - [ ]T= [ ]T
Test the next input p3 with new weights
a=hardlim( [ ]*[0 -1]T
a=hardlim( 0.8 ) = 1 1w
Again a mistake.
We want w to move away from p3
SO If ( t=0 and a=1 ), then 1wnew = 1wold - p

27. Example on Supervision Learning cont.2 1 3
1wnew = 1wold + p = [ ]T - [0 -1 ]T= [ ]T
Test the next input p1 again with new weights
a=hardlim( [ ]*[1 2]T
a=hardlim( 3.4 ) = 1 1w
Now correct.
Repeat for all other inputs
SO If ( t=a1) then wnew = 1wold

28. Multilayer Network
Two Hidden Layers
One Output Layer

29. 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 ), …