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# chapter4 - Chapter 4 Multilayer Perceptron Multilayer...

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Chapter 4 Multilayer Perceptron

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Chapter 4 --- Multilayer Perceptron 2 Multilayer Perceptron ring2 A generalization of the single-layer perceptron to enhance its computational power. ring2 Training method : error backpropagation algorithm which is based on the error- correction learning rule. ring2 Requirement : nonlinear neuronal function should be smooth (i.e., differentiable everywhere ).
Chapter 4 --- Multilayer Perceptron 3 Multilayer Perceptron

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Chapter 4 --- Multilayer Perceptron 4 Backpropagation Training Algorithm ring2 A systematic method for training multilayer ANN, error is back ward propagated to adjust the weights during training phase. Therefore it s called backpropagation training. ring2 Requires that the nonlinear neuronal function be differentiable everywhere . A good choice is the sigmoid function (or logistic / squashing function)
Chapter 4 --- Multilayer Perceptron 5 Backpropagation Training Algorithm ring2 Training Objective : to adjust the weights so that application of a set of inputs produces the desired set of outputs. ring2 Belongs to the category of Supervised Learning.

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Chapter 4 --- Multilayer Perceptron 6 Graphical Representation
Chapter 4 --- Multilayer Perceptron 7 Mathematical Analysis ring2 Consider neuron j , ring2 Define error signal e j ( n ) = d j ( n ) - y j ( n ) ring2 Define instantaneous squared error for output neuron j = ring2 Instantaneous sum of squared errors of the network where the summation is over all output neurons. v n w n y n and y n v n j ji i i j j ( ) ( ) ( ) ( ) ( ( )) = = ϕ 1 2 2 e n j ( ) E n e n j j ( ) ( ) = 1 2 2

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Chapter 4 --- Multilayer Perceptron 8 Mathematical Analysis Make use of the steepest gradient descent concept, where local gradient points to the required changes in synaptic weights. E n w n E n e n e n y n y n v n v n w n ji j j j j j j ji ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = = ⋅- ⋅ e n v n y n j j i ( ) ( ( )) ( ) ' 1 ϕ Δ w n E n w n n y n ji ji j i ( ) ( ) ( ) ( ) ( ) = - = η ηδ δ ϕ j j j n e n v n ( ) ( ) ( ( )) ' = ---- (4.1)
Chapter 4 --- Multilayer Perceptron 9 Mathematical Analysis If neuron j is an output neuron Easy, as we know the target value of output neurons. e j ( n ) = d j ( n ) - y j ( n ) ---- (4.2) ---- (4.3) = Δ ) ( ) ( ) ( n y j neuron of signal input n gradient local parameter rate learning n w correction Weight i j ji δ η δ ϕ j j j n e n v n ( ) ( ) ( ( )) ' =

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Chapter 4 --- Multilayer Perceptron 10 Mathematical Analysis If neuron j is a hidden neuron Difficult, as there s no target value for hidden neurons. We have to calculate δ ϕ j j j j j j n E n y n y n v n E n y n v n ( ) ( ) ( ) ( ) ( ) ( ) ( ) '( ( )) = - = -
Chapter 4 --- Multilayer Perceptron 11 Mathematical Analysis = - = = j j kj k k k k k k y w v and v d e n e n E ) ( , ) ( 2 1 ) ( 2 ϕ E n y n e n e n v n v n y n j k k k k k j ( ) ( ) ( ) ( ) ( ) ( ) ( ) = = ⋅- e n v n v n y n k k k k j ( ) ( ( )) ( ) ( ) ' ϕ = - δ k kj k n w n ( ) ( ) δ ϕ δ j j k kj k n v n n w n ( ) ( ( )) ( ) ( ) ' = --- (4.4)

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Chapter 4 --- Multilayer Perceptron 12 Mathematical Analysis Error backward
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