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Function Approx2009
Function Approx2009
Function Approx2009
Function Approx2009
Function Approx2009
Function Approx2009
Function Approx2009
Function Approx2009
Function Approx2009
Function Approx2009
Function Approx2009
Function Approx2009
Function Approx2009
Function Approx2009
Function Approx2009
Function Approx2009
Function Approx2009
Function Approx2009
Function Approx2009
Function Approx2009
Function Approx2009
Function Approx2009
Function Approx2009
Function Approx2009
Function Approx2009
Function Approx2009
Function Approx2009
Function Approx2009
Function Approx2009
Function Approx2009
Function Approx2009
Function Approx2009
Function Approx2009
Function Approx2009
Function Approx2009
Function Approx2009
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Function Approx2009

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Neural network viewed as a tool for Function approximation

Neural network viewed as a tool for Function approximation

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Transcript

  • 1.
    • Function Approximation
    • And
    • Pattern Recognition
    • Imthias Ahamed T. P.
    • Dept. of Electrical Engineering,
    • T.K.M.College of Engineering,
    • Kollam – 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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