MACHINE LEARNING TECHNIQUES - LASA

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118 Non-linear case The non-linear case of classification using Gaussian Process proceeds similarly to Gaussian Process Regression. Instead of putting a prior on the weight as in the linear case, we put a prior f x , such that we have on the latent function ( ) 1 p( y =+ 1| x) = (5.94) f ( x) 1 + e − This has for effect to ensure that the output is bounded between 0 and +1 and can hence be interpreted as a probability (as in the linear case), see Figure Figure 5-20 Figure 5-20: Example of an arbitrary prior function f(x) (here composed of the superposition of two Gaussian). Applyinhg the sigmoid function on f(x) flattens the function, while normalizing between 0 and +1. One now can build an estimate of the class label * y for a query point * x by computing the posterior distribution of the function f ( x ) applied on our query point. If we make this a distribution that is a function of the training datapoints, we have ( ) posterior distribution we want to compute is given by: ( ) ( ) ( ) ( ( ) ) p y * | x * , X, Y sigmoid f x * p f x * | x * , X, Y df * ( | * , , ) p f x x X Y and the = ∫ (5.95) The integral on the righthandside compute all values on our prior on f ( ) x . While in GPR there was an analytical solution, in the classification case, the integral is usually analytically intractable. To solve this, one must use either an analytic approximations of integrals, or solutions based on Monte Carlo sampling. © A.G.Billard 2004 – Last Update March 2011
119 Figure 5-21: Example of successful classification with Gaussian Process classification, adapted from C. E. Rasmussen & C. K. I. Williams, Gaussian Processes for Machine Learning, the MIT Press, 2006.Panel a shows the location of the data points in the two-dimensional space [0, 1]. The two classes are labelled as open circles (+1) and closed circles (- 1). Panels (b)-(d) show contour plots of the predictive probability Eq[_(x_)|y] for signal variance f = 9 and length-scales ` of 0.1, 0.2 and 0.3 respectively. The decision boundaries between the two classes are shown by the thicker black lines. The maximum value attained is 0.84, and the minimum is 0.19. © 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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Page 73 and 74: 73 5 Kernel Methods These lecture n
Page 75 and 76: 75 The kernel k provides a metric o
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Page 81 and 82: 81 5.4 Kernel CCA The linear versio
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Page 85 and 86: 85 Figure 5-3: TOP: Marginal (left)
Page 87 and 88: 87 statistical independence. We def
Page 89 and 90: 89 J j ( µ 1,...., µ K) = ∑∑
Page 91 and 92: 91 A simple pattern recognition alg
Page 93 and 94: 93 ( ) ( , ) f x = sign w x + b (5.
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Page 113 and 114: 113 5.9.2 Equivalence of Gaussian P
Page 115 and 116: 115 5.9.3 Curse of dimensionality,
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Page 125 and 126: 125 The sigmoid f x ( x) ( ) = tanh
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Page 133 and 134: 133 6.5 Willshaw net David Willshaw
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Page 137 and 138: 137 Figure 6-11: The weight vector
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169 7.2.5 Further Readings Rabiner,
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171 7.3.1 Principle In reinforcemen
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173 general, acting to maximize imm
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175 of reinforcement learning makes
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177 8 Genetic Algorithms We conclud
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179 However, you must define geneti
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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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