MACHINE LEARNING TECHNIQUES - LASA

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130 6.4.1 The Adaline The Adaline is a simple one-layer feed-forward neural network, composed of perceptron units. Figure 6-4: one-layer neural network p Let P input patterns X , p 1,..., P = be the patterns you want to train the network with. p real output of the network when presented with each X , and network. One can compute an error measure over all patterns: 1 E = E = z −y 2 p y is the p z the desired output for the P P p p p ∑ ∑ ( ) 2 (6.12) p= 1 p= 1 In order to minimize the error, one can determine the gradient of the error with respect to the weights and move the weights in opposite direction. E Δ wj =−γ ∂ (6.13) ∂ w j Figure 6-5: A schematic diagram showing the principle of error descent. If the gradient is positive, changing the weights in a positive direction would increase the error. Therefore, we change the weights in a negative direction. Conversely, if the gradient is negative, one must change the weight in a positive direction to decrease the error. © A.G.Billard 2004 – Last Update March 2011
131 y= ∑ w x ) of Because of the linearity of the activation function of the Perceptron neuron (i.e. j j j the Adaline network, we have: ∂E ∂y ∂y ∂w p p p j = x p j p p ( z y ) =− − (6.14) (6.15) ( ) p p p Δ pwj = γ z −y ⋅ x (6.16) This has proven to be a most powerful rule and is at the core of almost all current supervised learning methods for ANN. But, it should be emphasized that nothing we have written guarantees that the method will cause the weight to converge. It can be proved that the method will give an optimal (in a least square error sense) approximation of the function being modeled. However, the method does not ensure convergence to a global optimum. 6.4.2 The Backpropagation Network An example of multi-layered Perceptron, or feed-forward neural network, is shown in Figure 6-6 Activity in the network is propagated forwards via a first set of weights from the input layer to the hidden layer, and, then, via a second set of weights from hidden layer to output layer. The error is calculated by Equation (6.12), similarly to what was done for the Adaline network. Now two sets of weights must be calculated. Here, however, one does not have access to the desired output of the hidden units. This is referred to, as the Credit assignment problem – in that we must assign how much effect each weight in the first layer of weights has on the final output of the network. In order to compute the weight change, we need to propagate backwards, to backpropagate, the error across the two layers. The algorithm is quite general and applies to any number of hidden layers. An example of multilayered perceptron is shown in Figure 6-6: Multi-layered feed-forward NN © A.G.Billard 2004 – Last Update March 2011
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SCHOOL OF ENGINEERING MACHINE LEARN
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3 4. 4 Regression Techniques ......
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5 9.2.2 Probability Distributions,
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7 Journals: • Machine Learning
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9 Performance What would be an opti
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11 1.2.3 Key features for a good le
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13 1.3.2 Crossvalidation To ensure
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15 In particular, we will consider
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17 2.1 Principal Component Analysis
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19 ( ) Xʹ′ = W X − µ (2.6) i
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21 2.1.2.2 Reconstruction error min
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23 PCA is an example of PP approach
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25 Algorithm: If one further assume
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27 The CCA algorithm consists thus
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29 Figure 2-6: Mixture of variables
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31 2.3.2 Why Gaussian variables are
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33 • In our general definition of
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35 2.3.5 ICA Ambiguities We cannot
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37 Denote by g the derivative of th
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39 3 Clustering and Classification
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41 An agglomerative clustering star
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43 3.1.1.1 The CURE Clustering Algo
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45 Disadvantages of hierarchical cl
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47 Cases where K-means might be vie
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49 3.1.4 Clustering with Mixtures o
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51 k ( σ j ) 2 = k ∑ i α = r k
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53 Theα are the so-called mixing c
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55 Figure 3-16: Clustering with 3 G
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57 When the transformation A is lin
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59 C: X → Y ( ) C x K = arg max
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61 Figure 3-18: Linear combination
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63 Figure 3-19: Bayes classificatio
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65 ⎛⎛ min ⎜⎜ w ⎝⎝ N i=
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67 T ( yi − xi w) 2 M ⎛⎛ ⎞
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69 Figure 4-2: Illustration of the
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71 4.4.2 Multi-Gaussian Case It is
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73 5 Kernel Methods These lecture n
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75 The kernel k provides a metric o
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77 M 1 T v = ∑ x ( x ) v M λ i j
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181 Often the crossover operator an
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183 ( A λI) x 0 − = (8.5) where
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185 Joint probability: The joint pr
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187 The two most classical distribu
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189 9.2.7 Statistical Independence
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191 1 1 1 1 h x ∫ a b a b 0 a a
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193 9.4 Estimators 9.4.1 Gradient d
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195 9.4.2.1 Maximum Likelihood Mach
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197 10 References • Machine Learn
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MACHINE LEARNING TECHNIQUES - LASA

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