Artificial Neural Networks

Brian TaleckiCSC 8520

Villanova University

ANN - Artificial Neural Network • A set of algebraic equations and functions

which determine the best output given a set of inputs.

• An artificial neural network is modeled on a very simplified version of the a human neuron which make up the human nervous system.

• Although the brain operates at 1 millionth the speed of modern computers, it functions faster than computers because of the parallel processing structure of the nervous system.

Human Nerve Cell

•picture from: G5AIAI Introduction to AI by Graham Kendall

•www.cs.nott.ac.uk/~gxk/courses/g5aiai

• At the synapse – the nerve cell releases a chemical compounds called neurotransmitters, which excite or inhibit a chemical / electrical discharge in the neighboring nerve cells.

• The summation of the responses of the adjacent neurons will elicit the appropriate response in the neuron.

Brief History of ANN• McCulloch and Pitts (1943) designed the first neural network • Hebb (1949) who developed the first learning rule. If two

neurons were active at the same time then the strength between them should be increased.

• Rosenblatt (1958) – introduced the concept of a perceptron which performed pattern recognition.

• Widrow and Hoff (1960) introduced the concept of the ADALINE (ADAptive Linear Element) . The training rule was based on the idea of Least-Mean-Squares learning rule which minimizing the error between the computed output and the desired output.

• Minsky and Papert (1969) stated that the perceptron was limited in its ability to recognize features that were separated by linear boundaries. “Neural Net Winter”

• Kohonen and Anderson – independently developed neural networks that acted like memories.

• Webros(1974) – developed the concept of back propagation of an error to train the weights of the neural network.

• McCelland and Rumelhart (1986) published the paper on back propagation algorithm. “Rebirth of neural networks”.

• Today - they are everywhere a decision can be made. Source : G5AIAI - Introduction to Artificial Intelligence Graham Kendall:

Basic Neural Network

Inputs – normally a vector of measured parametersBias – may/may not be addedf() – transfer or activation function

Outputs = f(∑ W p + b)

f()W ∑ Outputs

Inputs

b - Bias ∑ Wp +b

T

Activation Functions

Source: Supervised Neural Network Introduction CISC 873. Data Mining Yabin Meng

Log Sigmoidal Function

Source: Artificial Neural Networks Colin P. Faheyhttp://www.colinfahey.com/2003apr20_neuron/index.htm

Hard Limit Function

1.0

-1.0

x

y

Log Sigmoid and Derivative

Source : The Scientist and Engineer’s Guide to Digital Signal Processing by Steven Smith

Derivative of the Log Sigmoidal Functions(x) = (1 + e )

s’(x) = -(1+e ) * (-e )

= e * (1+ e )

= ( e ) * ( 1 )

(1+ e ) ( 1 + e )

= (1 + e – 1) * ( 1 )

( 1+ e ) ( 1 + e )

= (1 - ( 1 ) ) * ( 1 )

(1+ e ) (1 + e )

s’(x) = (1-s(x)) * s(x)

-1-x

-2-x -x

-x -2-x

-x

-x -x

-x

-x -x

-x -x

Derivative is important for the back error propagation algorithm used to train multilayer neural networks.

Example : Single Neuron Given : W = 1.3, p = 2.0, b = 3.0

Wp + b = 1.3(2.0) + 3.0 = 5.6

Linear:

f(5.6) = 5.6

Hard limit

f(5.6) = 1.0

Log Sigmoidal

f(5.6) = 1/(1+exp(-5.6)

= 1/(1+0.0037)

= .9963

Simple Neural NetworkOne neuron with a linear activation function => Straight Line

Recall the equation of a straight Line : y = mx +b

m is the slope (weight), b is the y-intercept (bias).

Bad

Good

Decision Boundary

p2

p1

Mp1 + b >= p2

Mp1 + b < p2

Perceptron Learning Extend our simple perceptron to two inputs and hard limit

activation function

F()W

bias

OutputW1

W2

o = f (∑ W p + b)

W is the weight matrix

p is the input vector

o is our scalar output

p1

p2Hard limit function

∑

T

Rules of Matrix MathAddition/Subtraction

1 2 3 9 8 7 10 10 10

4 5 6 +/- 6 5 4 = 10 10 10

7 8 9 3 2 1 10 10 10

Multiplication by a scalar Transpose

a 1 2 = a 2a 1 = 1 2

3 4 3a 4a 2

Matrix Multiplication

2 4 5 = 18 , 5 2 4 = 10 20

2 2 4 8

T

Data Points for the AND Function

q1 = 0 , o1 = 0

0

q2 = 1 , o2 = 0

0

q3 = 0 , o3 = 0

1

q4 = 1 , o4 = 1

1

Truth Table

P1 P2 O 0 0 0

0 1 0

1 0 0

1 1 1

Weight Vector and the Decision Boundary W = 1.0

1.0Magnitude and Direction

Decision Boundary is the line where

W p = b or W p – b = 0T T

W p < b

W p > b

T

TAs we adjust the weights and biases of the neural network,

we change the magnitude and direction of the weight vector or the slope and intercept of the

decision boundary

Perceptron Learning Rule• Adjusting the weights of the Perceptron

• Perceptron Error : Difference between the desired and derived

outputs. e = Desired – Derived When e = 1 W new = Wold + p

When e = -1 W new = Wold - p

When e = 0 W new = Wold

Simplifing W new = Wold + λ * ep

b new = bold + e λ is the learning rate ( = 1 for the perceptron).

AND Function ExampleStart with W1 = 1, W2 = 1, and b = -1 W p + b => t - a = e

1 1 0 + -1 => 0 - 0 = 0 N/C 0

1 1 0 + -1 => 0 - 1 = -1 1

1 0 1 + -2 => 0 - 0 = 0 N/C 0 1 0 1 + -2 => 1 - 0 = 1 1

T

W p + b => t - a = e

2 1 0 + -1 => 0 - 0 = 0 N/C 0

2 1 0 + -1 => 0 - 1 = -1 1

2 0 1 + -2 => 0 - 1 = -1 0

1 0 1 + -3 => 1 - 0 = 1 1

T

W p + b => t - a = e

2 1 0 + -2 => 0 - 0 = 0 N/C 0

2 1 0 + -2 => 0 - 0 = 0 N/C 1

2 1 1 + -2 => 0 - 1 = -1 0

1 1 1 + -3 => 1 - 0 = 1 1

T

W p + b => t - a = e

2 2 0 + -2 => 0 - 0 = 0 N/C 0

2 2 0 + -2 => 0 - 1 = -1 1

2 1 1 + -3 => 0 - 0 = 0 N/C 0

2 1 1 + -3 => 1 - 1 = 0 N/C 1

T

W p + b => t - a = e

2 1 0 + -3 => 0 - 0 = 0 N/C

0

2 1 0 + -3 => 0 - 0 = 0 N/C

1 Done !

T

2f()

1 Hardlim()

p1

p2

Σ

-3

XOR FunctionTruth Table X Y Z = (X and not Y) or (not X and Y)

0 0 0

0 1 1

1 0 1

1 1 0

1

0

No single decision boundary can separate the favorable and unfavorable outcomes.

z

xy

We will need a more complicated neural net to realize this function

Circuit Diagram

XOR Function – Multilayer Perceptron

W5

W6

W1

W3

W2W4

f1()

f1()

f()

z

b2b11

b12

Σ

Σ

x

y

Z = f (W5*f1(W1*x + W4*y+b11) +W6*f1(W2*x + W3*y+b12)+b2)

Weights of the neural net are independent of each other, so that we can compute the partial derivatives of z with respect to the

weights of the network.

i.e. δz / δW1, δz / δW2, δz / δW3,

δz / δW4, δz / δW5, δz / δW6

Back Propagation Diagram

Neural Networks and Logistic Regression by Lucila Ohno-MachadoDecision Systems Group, Brigham and Women’s Hospital, Department of Radiology

Input units

Output units

H iddenunits

what we gotwhat we wanted-error

∆ rule

∆ rule

Back Propagation Algorithm• This algorithm to train Artificial Neural Networks

(ANN) depends to two basic concepts: a) Reduced the Sum Squared Error, SSE, to an

acceptable value.

b) Reliable data to train your network under

your supervision.

Simple case : Single input no bias neural net.zx W1

n1

f1W2

a1 n2

T = desired output

f2

BP Equationsn1 = W1 * xa1 = f1(n1) = f1(W1 * x)n2 = W2 * a1 = W2 * f1(n1) = W2 * f1(W1 * x)z = f2(n2) = f2(W2 * f1(W1 * x))SSE = ½ (z – T) Lets now take the partial derivatives δSSE/ δW2 = (z - T) * δ(z - T)/ δW2 = (z – T) * δz/ δW2 = (z - T) * δf2(n2)/δW2 Chain Ruleδf2(n2)/δW2 = (δf2(n2)/δn2)* (δn2/δW2) = (δf2(n2)/δn2)* a1δSSE/ δW2 = (z - T) * (δf2(n2)/δn2)* a1

Define λ to our learning rate (0 < λ < 1, typical λ = 0.2)Compute our new weight:

W2(k+1) = W2(k) - λ (δSSE/ δW2)

= W2(k) - λ ((z - T) * (δf2(n2)/δn2)* a1)

2

Sigmoid function: δf2(n2)/δn2 = f2(n2)(1 – f2(n2)) = z(1 – z)Therefore: W2(k+1) = W2(k) - λ ((z - T) * ( z(1 –z) )* a1)

Analysis for W1 n1 = W1 * x a1 = f1(W1*x) n2 = W2 * f1(n1) = W2 * f1(W1 * x) δSSE/ δW1 = (z - T) * δ(z -T )/ δW1 = (z - T) * δz/ δW1 = (z - T) * δf2(n2)/δW1 δf2(n2)/δW1 = (δf2(n2)/δn2)* (δn2/δW1) -> Chain Rule δn2/δW1 = W2 * (δf1(n1)/δW1) = W2 * (δf1(n1)/δn1) * (n1/δW1) -> Chain Rule = W2 * (δf1(n1)/δn1) * x δSSE/ δW1 = (z - T ) * (δf2(n2)/δn2)* W2 * (δf1(n1)/δn1) * x

W1(k+1) = W1(k) - λ ((z - T ) * (δf2(n2)/δn2)* W2 * (δf1(n1)/δn1) * x) δf2(n2)/δn2 = z (1 – z) and δf1(n1)/δn1 = a1 ( 1 – a1)

Gradient Descent

Local minimum

Global minimum

Error

Training timeNeural Networks and Logistic Regression by Lucila Ohno-Machado

Decision Systems Group, Brigham and Women’s Hospital, Department of Radiology

2-D Diagram of Gradient Descent

Source : Back Propagation algorithm by Olena Lobunetswww.essex.ac.uk/ccfea/Courses/ workshops03-04/Workshop4/Workshop%204.ppt

Learning by Example• Training Algorithm: backpropagation

of errors using gradient descent training.

• Colors:– Red: Current weights– Orange: Updated weights

– Black boxes: Inputs and outputs to a neuron

– Blue: Sensitivities at each layer

Source : A Brief Overview of Neural Networks

Rohit Dua, Samuel A. Mulder, Steve E. Watkins, and Donald C. Wunsch

campus.umr.edu/smartengineering/ EducationalResources/Neural_Net.ppt

First Pass

0.5

0.5

0.5

0.50.5

0.5

0.5

0.51

0.5

0.5 0.6225

0.62250.6225

0.6225

0.6508

0.6508

0.6508

0.6508

Error=1-0.6508=0.3492

G3=(1)(0.3492)=0.3492

G2= (0.6508)(1-0.6508)(0.3492)(0.5)=0.0397

G1= (0.6225)(1-0.6225)(0.0397)(0.5)(2)=0.0093

Gradient of the neuron= G =slope of the transfer function×[Σ{(weight of the neuron to the next neuron) × (output of the neuron)}]

Gradient of the output neuron = slope of the transfer function × error

Weight Update 1

New Weight=Old Weight + {(learning rate)(gradient)(prior output)}

0.5+(0.5)(0.3492)(0.6508)

0.6136

0.5124 0.5124

0.51240.6136

0.5124

0.5047

0.5047

0.5+(0.5)(0.0397)(0.6225)0.5+(0.5)(0.0093)(1)

Source : A Brief Overview of Neural NetworksRohit Dua, Samuel A. Mulder, Steve E. Watkins, and Donald C. Wunsch

campus.umr.edu/smartengineering/ EducationalResources/Neural_Net.ppt

Second Pass

0.5047

0.5124

0.6136

0.61360.5047

0.5124

0.5124

0.51241

0.5047

0.5047

0.6391

0.63910.6236

0.6236

0.8033

0.6545

0.6545

0.8033

Error=1-0.8033=0.1967

G3=(1)(0.1967)=0.1967

G2= (0.6545)(1-0.6545)(0.1967)(0.6136)=0.0273

G1= (0.6236)(1-0.6236)(0.5124)(0.0273)(2)=0.0066

Source : A Brief Overview of Neural NetworksRohit Dua, Samuel A. Mulder, Steve E. Watkins, and Donald C. Wunsch

campus.umr.edu/smartengineering/ EducationalResources/Neural_Net.ppt

Weight Update 2

New Weight=Old Weight + {(learning rate)(gradient)(prior output)}

0.6136+(0.5)(0.1967)(0.6545)

0.6779

0.5209 0.5209

0.52090.6779

0.5209

0.508

0.508

0.5124+(0.5)(0.0273)(0.6236)0.5047+(0.5)(0.0066)(1)

Source : A Brief Overview of Neural NetworksRohit Dua, Samuel A. Mulder, Steve E. Watkins, and Donald C. Wunsch

campus.umr.edu/smartengineering/ EducationalResources/Neural_Net.ppt

Third Pass

0.508

0.5209

0.6779

0.67790.508

0.5209

0.5209

0.52091

0.508

0.508

0.6504

0.65040.6243

0.6243

0.8909

0.6571

0.6571

0.8909

Source : A Brief Overview of Neural NetworksRohit Dua, Samuel A. Mulder, Steve E. Watkins, and Donald C. Wunsch

campus.umr.edu/smartengineering/ EducationalResources/Neural_Net.ppt

Weight Update Summary

Output Expected OutputErrorw1 w2 w3

Initial conditions 0.5 0.5 0.5 0.6508 1 0.3492Pass 1 Update 0.5047 0.5124 0.6136 0.8033 1 0.1967Pass 2 Update 0.508 0.5209 0.6779 0.8909 1 0.1091

Weights

W1: Weights from the input to the input layerW2: Weights from the input layer to the hidden layerW3: Weights from the hidden layer to the output layer

Source : A Brief Overview of Neural NetworksRohit Dua, Samuel A. Mulder, Steve E. Watkins, and Donald C. Wunsch

campus.umr.edu/smartengineering/ EducationalResources/Neural_Net.ppt

ECG Interpretation

R-R interval

S-T elevation

P-R interval

QRS duration

AVF lead

QRS amplitude

SV tachycardia

Ventricular tachycardia

LV hypertrophy

RV hypertrophy

Myocardial infarction

Neural Networks and Logistic Regression by Lucila Ohno-MachadoDecision Systems Group, Brigham and Women’s Hospital,

Department of Radiology

Other Applications of ANNLip Reading Using Artificial Neural Network

Ahmad Khoshnevis, Sridhar Lavu, Bahar Sadeghi

and Yolanda Tsang ELEC502 Course Project

www-dsp.rice.edu/~lavu/research/doc/502lavu.ps

AI Techniques in Power Electronics and DrivesDr. Marcelo G. Simões Colorado School of Mines

egweb.mines.edu/msimoes/tutorial

Car Classification with Neural Networks

Koichi Sato & Sangho Park

hercules.ece.utexas.edu/course/ ee380l/1999sp/present/carclass.ppt

Face Detection and Neural Networks

Todd Wittman

www.ima.umn.edu/~whitman/faces/face_detection2.ppt

A Neural Network for Detecting and Diagnosing Tornadic Circulations

V Lakshmanan, Gregory Stumpf, Arthur Witt www.cimms.ou.edu/~lakshman/Papers/mdann_talk.ppt

BibliographyA Brief Overview of Neural Networks Rohit Dua, Samuel A. Mulder, Steve E. Watkins, and Donald C. Wunsch campus.umr.edu/smartengineering/ EducationalResources/Neural_Net.ppt

Neural Networks and Logistic Regression Lucila Ohno-Machado Decision Systems Group, Brigham and Women’s Hospital,Department of Radiology dsg.harvard.edu/courses/hst951/ppt/hst951_0320.ppt

G5AIAI Introduction to AI by Graham Kendall Schooll of Computer Science and IT , University of Nottingham www.cs.nott.ac.uk/~gxk/courses/g5aiai The Scientist and Engineer's Guide to Digital Signal Processing Steven W. Smith, Ph.D.

California Technical Publishing www.dspguide.com

Neural Network Design Martin Hagen, Howard B. Demuth, and Mark Beale Campus Publishing Services, Boulder Colorado 80309-0036

ECE 8412 lectures notes by Dr. Anthony Zygmont Department of Electrical Engineering Villanova University January 2003

Supervised Neural Network Introduction CISC 873. Data Mining Yabin Meng meng@cs.queensu.ca