introduction to neural networks (under graduate course) lecture 2 of 9

Neural Networks

Dr. Randa Elanwar

Lecture 2

Lecture Content

• Neural network concepts:

– Basic definition.

– Connections.

– Processing elements.

2Neural Networks Dr. Randa Elanwar

Artificial Neural Network: Structure

• ANN posses a large number of processing elements called nodes/neurons which operate in parallel.

• Neurons are connected with others by connection link.

• Each link is associated with weights which contain information about the input signal.

• Each neuron has an internal state of its own which is a function of the inputs that neuron receives- Activation level

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Artificial Neural Network: Neuron Model

(dendrite) (axon)

(soma)

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f()Y

Wa

Wb

Wc

Connection weights

Summing function

Computation

(Activation Function)

X1

X3

X2

Input units

How are neural networks being used in solving problems

• From experience: examples / training data

• Strength of connection between the neurons is stored as a weight-value for the specific connection.

• Learning the solution to a problem = changing the connection weights

5Neural Networks Dr. Randa Elanwar

How are neural networks being used in solving problems

• The problem variables are mainly: inputs, weights and outputs

• Examples (training data) represent a solved problem. i.e. Both the inputs and outputs are known

• Thus, by certain learning algorithm we can adapt/adjust the NN weights using the known inputs and outputs of training data

• For a new problem, we now have the inputs and the weights, therefore, we can easily get the outputs.

6Neural Networks Dr. Randa Elanwar

How NN learns a task: Issues to be discussed

- Initializing the weights.

- Use of a learning algorithm.

- Set of training examples.

- Encode the examples as inputs.

-Convert output into meaningful results.

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Linear Problems

• The simplest type of problems are the linear problems.

• Why ‘linear’? Because we can model the problem by a straight line equation (ax+by+c=z)

• or

• Example: logic linear problems And, OR, NOT problems. We know the truth tables thus we have examples and we can model the operation using a neuron

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iii inw

1

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outbinwinwinw ......332211

bXWOUT .

Linear Problems

• Example: AND (x1,x2), f(net) = 1 if net>1 and 0 otherwise

• Check the truth table: y = f(x1+x2)

9Neural Networks Dr. Randa Elanwar

x1 x2 y

0 0 0

0 1 0

1 0 0

1 1 1

x1

x2

y

1

1

Linear Problems

• Example: OR(x1,x2), f(net) = 1 if net>1 and 0 otherwise

• Check the truth table: y = f(2.x1+2.x2)

10Neural Networks Dr. Randa Elanwar

x1 x2 y

0 0 0

0 1 1

1 0 1

1 1 1

x1

x2

y

2

2

Linear Problems

• Example: NOT(x1), f(net) = 1 if net>1 and 0 otherwise

• Check the truth table: y = f(-1.x1+2)

11Neural Networks Dr. Randa Elanwar

x1 y

0 1

1 0

x1

y

-1

2

bias

1

Linear Problems

• Example: AND (x1,NOT(x2)), f(net) = 1 if net>1 and 0 otherwise

• Check the truth table: y = f(2.x1-x2)

12Neural Networks Dr. Randa Elanwar

x1 x2 y

0 0 0

0 1 0

1 0 1

1 1 0

x1

x2

y

2

-1

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The McCulloch-Pitts Neuron

• This vastly simplified model of real neurons is also known as a Threshold Logic Unit – A set of connections brings in activations from other neurons.– A processing unit sums the inputs, and then applies a non-linear activation function (i.e.

squashing/transfer/threshold function).– An output line transmits the result to other neurons.

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McCulloch-Pitts Neuron Model

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Features of McCulloch-Pitts model

• Allows binary 0,1 states only

• Operates under a discrete-time assumption

• Weights and the neurons’ thresholds are fixed in the model and no interaction among network neurons

• Just a primitive model

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McCulloch-Pitts Neuron Model

• When T = 1 and w = 1• The input passes as is• Thus if input is =1 then o = 1• Thus if input is =0 then o = 0 (buffer)• Works as ‘1’ detector

• When T = 1 and w = -1• The input is inverted• Thus if input is =0 then o = 0• Thus if input is =1 then o = 0 • useless

16Neural Networks Dr. Randa Elanwar

McCulloch-Pitts Neuron Model

• When T = 0 and w = 1• The input passes as is• Thus if input is =0 then o = 1• Thus if input is =1 then o = 1• useless

• When T = 0 and w = -1• The input is inverted• Thus if input is =1 then o = 0• Thus if input is =0 then o = 1 (inverter)• Works as Null detector

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McCulloch-Pitts NOR

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•Can be implemented using an OR gate design followed by inverter

•We need ‘1’ detector, thus first layer is (T=1) node preceded by +1 weights

Zeros stay 0 and Ones stay 1

•We need inverter in the second layer, (T=0) node preceded by -1 weights

•Check the truth table

McCulloch-Pitts NAND

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•Can be implemented using an inverter design followed by OR gate

•We need inverter in the first layer is (T=0) node preceded by -1 weightsZeros will be 1 and Ones will be zeros

•We need ‘1’ detector, thus first layer is (T=1) node preceded by +1 weights

Zeros stay 0 and Ones stay 1

General symbol of neuron consisting of processing node and synaptic connections

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Neuron Modeling for ANN

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Is referred to activation function. Domain is set of activation values net. (Not a single value fixed threshold)

Scalar product of weight and input vector

Neuron as a processing node performs the operation of summation of its weighted input.