Function Approx2009

Jun 2, 2011Download as PPT, PDF0 likes563 views

The document discusses function approximation and pattern recognition using neural networks. It introduces concepts like the perceptron, multi-layer perceptrons, backpropagation algorithm, supervised and unsupervised learning. It provides examples of using neural networks for function approximation and pattern recognition problems. Matlab code is also presented to illustrate training a neural network on sample datasets.

Function Approximation And Pattern Recognition Imthias Ahamed T. P. Dept. of Electrical Engineering, T.K.M.College of Engineering, Kollam �C 691005, [email_address]

Function Approximation Problem x = [0 1 2 3 4 5 6 7 8 9 10]; d = [0 1 2 3 4 3 2 1 2 3 4]; Find f such that

A matlab program clf clear x= [0 1 2 3 4 5 6 7 8 9 10]; d = [0 1 2 3 4 3 2 1 2 3 4]; plot(x,d,'x') pause

Optimization Technique: Steepest Descent The steepest descent algorithm: w (n+1)= w (n)- ? g (n)

Least-Mean-Square (LMS) Algorithm e(n) is the error signal measured at time n.

Model of a Simple Perceptron and Let b k =w k0 and x 0 =+1

Activation Functions Threshold Function Sigmoid Function

Multi Layer Perceptron Multi Layer Perceptron or Feedforward Network Consists of Input Layer One or more Hidden Layer Output Layer

net = newff([0 10],[5 1],{'tansig' 'purelin'}); ybeforetrain = sim(net,x) plot(x,d,'x',x,ybeforetrain,'o') legend('desired','actual') pause

net.trainParam.epochs = 50; net = train(net,x,d); Y = sim(net,x); plot(x,d,'x',x,Y,'o') legend('desired','actual') pause

xtest=0:.5:10; ytest = sim(net,xtest); plot(x,d,'x',xtest,ytest,'o') legend('desired','actual')

An Example x=[-0.5 -.5 .3 .1 .6 .7; -.5 .5 -.5 1 .8 1 ] Y=[ 1 1 0 0 0 0]

Summary Perceptron Weights Activation Function Error minimization Gradient descent Learning Rule

Training data Testing data Linearly separable Linearly non separable

Back-propagation algorithm . Notations: i, j and k refer to different neurons; with signals propagating through the network from left to right, neuron j lies in a layer to the right of neuron i. w ji (n): The synaptic weight connecting the output of neuron i to the input of neuron j at iteration n.

Back-propagation Algorithm Contd... Local Gradient

Case 2: Neuron j is a Hidden Node (Contd��)

Delta Rule If neuron j is an output node, If neuron j is an hidden node,

Back-propagation Algorithm: Summary Initialization. Pick all of the w ji from a uniform distribution. Presentations of Training Examples. Forward Computation. Backward Computation. Iteration.

Back-propagation Algorithm: Summary 1. Initialiaze 2. Forward Computation: 3. Backwar Computration: For all hidden layers l do

Learning with a Teacher (Supervised Learning)

Learning without a Teacher Reinforcement Learning

Learning Tasks Function Approximation d = f ( x ) x : input vector d : output vector f ( ? ) is assumed to be unknown Given a set of labeled examples: Requirement: Design a neural network to approximate this unknown function f( ? ) such that F( ? ). || F ( x )- f ( x )||< ? for all x , where ? is a small positive number

Learning Tasks Pattern Recognition Def: A received pattern/signal is assigned to one of a prescribed number of classes. Input pattern x Unsupervised network for feature extraction Feature vector y Supervised network for classification 1 2 r ��

This document discusses different machine learning paradigms including supervised learning, unsupervised learning, and learning rules. It then describes techniques such as function approximation, system identification, and inverse modeling. Function approximation involves using a neural network to approximate an unknown function based on examples. System identification uses a neural network model to learn the input-output mapping of an unknown system. Inverse modeling constructs a neural network that produces the input given the output to learn the inverse of the unknown system.

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Function Approx2009

1. Function Approximation And Pattern Recognition Imthias Ahamed T. P. Dept. of Electrical Engineering, T.K.M.College of Engineering, Kollam �C 691005, [email_address]

2. Function Approximation Problem x = [0 1 2 3 4 5 6 7 8 9 10]; d = [0 1 2 3 4 3 2 1 2 3 4]; Find f such that

3. A matlab program clf clear x= [0 1 2 3 4 5 6 7 8 9 10]; d = [0 1 2 3 4 3 2 1 2 3 4]; plot(x,d,'x') pause

4. Learning Problem

5. Optimization Technique: Steepest Descent The steepest descent algorithm: w (n+1)= w (n)- ? g (n)

6. Least-Mean-Square (LMS) Algorithm e(n) is the error signal measured at time n.

7. Model of a Simple Perceptron and Let b k =w k0 and x 0 =+1

8. Activation Functions Threshold Function Sigmoid Function

9. Multi Layer Perceptron Multi Layer Perceptron or Feedforward Network Consists of Input Layer One or more Hidden Layer Output Layer

10. Multi Layer Perceptron

11. A matlab program clf clear x= [0 1 2 3 4 5 6 7 8 9 10]; d = [0 1 2 3 4 3 2 1 2 3 4]; plot(x,d,'x') pause

12. net = newff([0 10],[5 1],{'tansig' 'purelin'}); ybeforetrain = sim(net,x) plot(x,d,'x',x,ybeforetrain,'o') legend('desired','actual') pause

13. net.trainParam.epochs = 50; net = train(net,x,d); Y = sim(net,x); plot(x,d,'x',x,Y,'o') legend('desired','actual') pause

14. xtest=0:.5:10; ytest = sim(net,xtest); plot(x,d,'x',xtest,ytest,'o') legend('desired','actual')

15. Pattern Recognition Problem ?

16. An Example x=[-0.5 -.5 .3 .1 .6 .7; -.5 .5 -.5 1 .8 1 ] Y=[ 1 1 0 0 0 0]

17. ?

18. ?

19. Linearly Non Separable data

20. Summary Perceptron Weights Activation Function Error minimization Gradient descent Learning Rule

21. Training data Testing data Linearly separable Linearly non separable

22. Back-propagation algorithm . Notations: i, j and k refer to different neurons; with signals propagating through the network from left to right, neuron j lies in a layer to the right of neuron i. w ji (n): The synaptic weight connecting the output of neuron i to the input of neuron j at iteration n.

23. Back-propagation Algorithm

24. Back-propagation Algorithm Contd... Local Gradient

25. Case 1: Neuron j is an Output Node

26. Case 2: Neuron j is a Hidden Node

27. Case 2: Neuron j is a Hidden Node (Contd��)

28. Delta Rule If neuron j is an output node, If neuron j is an hidden node,

29. Back-propagation Algorithm: Summary Initialization. Pick all of the w ji from a uniform distribution. Presentations of Training Examples. Forward Computation. Backward Computation. Iteration.

30. Back-propagation Algorithm: Summary 1. Initialiaze 2. Forward Computation: 3. Backwar Computration: For all hidden layers l do

31. 4. Update weights

32. Learning with a Teacher (Supervised Learning)

33. Learning without a Teacher Reinforcement Learning

34. Learning Tasks Function Approximation d = f ( x ) x : input vector d : output vector f ( ? ) is assumed to be unknown Given a set of labeled examples: Requirement: Design a neural network to approximate this unknown function f( ? ) such that F( ? ). || F ( x )- f ( x )||< ? for all x , where ? is a small positive number

35. Learning Tasks Pattern Recognition Def: A received pattern/signal is assigned to one of a prescribed number of classes. Input pattern x Unsupervised network for feature extraction Feature vector y Supervised network for classification 1 2 r ��

36. Thank You

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Function Approx2009

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