MACHINE LEARNING TECHNIQUES - LASA

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196 EM is a description of a class of related algorithm, not of a particular algorithm. EM is a recipe or meta-algorithm, which is used to devise particular algorithms. The Baum-Welch algorithm is an example of an EM algorithm applied to Hidden Markov Models. Another example is the K-means clustering algorithm. It can be shown that an EM iteration does not decrease the observed data likelihood function, and that the only stationary points of the iteration are the stationary points of the observed data likelihood function. In practice, this means that an EM algorithm will converge to a local maximum of the observed data likelihood function. EM proceeds thus iteratively and is particularly suited for parameter estimation under incomplete data or missing data situations. By using the EM procedure, the so-called marginal (or incomplete-data) likelihood is obtained by computing the average or expectation of the complete-data likelihood with respect to the missing data using the current parameter estimates (E-step), then the new parameter estimates are obtained by maximizing the marginal likelihood (M-step). © A.G.Billard 2004 – Last Update March 2011
197 10 References • Machine Learning, Tom Mitchell, McGraw Hill, 1997. • Pattern Classification Richard O. Duda, Peter E. Hart, David G. Stork, • Information Theory, Inference and Learning Algorithms, David J.C Mackay, Cambridge University Press, 2003. • Independent Component Analysis, A. Hyvarinen, J. Karhunen and E. Oja, Wiley Inter- Sciences. 2001. • Artificial Neural Networks and Information Theory, Colin Fyfe, Tech. Report, Dept. of Computing and Infortion Science, The University of Paisley, 2000. • Neural Networks, Simon Haykin, Prentice Hall International Editions, 1994. • Self-Organizing Maps, Teuvo Kohonen, Springer Series in Information Sciences, 30, Springer. 2001. • Learning with Kernels, B. Scholkopf and A. Smola, MIT Press 2002 • Reinforcement Learning: An Introduction,. R. Sutton & A. Barto, A Bradford Book. MIT Press, 1998. • Neural Networks for Pattern Recognition, Bishop, C.M. 1996. New York: Oxford University Press. 10.1 Other Recommended Textbooks • Elements of Machine Learning, Pat Langley, Morgan Kaufmann, 1996. • Applied Nonlinear Control, J-J. Slotine, Slotine, J.J.E., and Li, W., Prentice-Hall, 1991. • Neural Networks, Simon Haykin, Prentice Hall International Editions, 1994. • Cluster Analysis, Copyright StatSoft, Inc., 1984-2004 © 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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79 1 M The solutions to the dual ei
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81 5.4 Kernel CCA The linear versio
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83 additional ridge parameter induc
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85 Figure 5-3: TOP: Marginal (left)
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87 statistical independence. We def
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89 J j ( µ 1,...., µ K) = ∑∑
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91 A simple pattern recognition alg
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93 ( ) ( , ) f x = sign w x + b (5.
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95 Figure 5-6: A binary classificat
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97 where N is the number of support
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99 5.8 Support Vector Regression In
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101 The optimization problem given
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103 Note that since we never have t
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105 Figure 5-13: Effect of the kern
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107 To better understand the effect
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109 5.9 Gaussian Process Regression
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111 One can then use the above expr
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113 5.9.2 Equivalence of Gaussian P
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115 5.9.3 Curse of dimensionality,
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117 The weight w determines the slo
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119 Figure 5-21: Example of success
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121 • its performance tends to de
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123 neurons. Furthermore, they lear
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125 The sigmoid f x ( x) ( ) = tanh
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127 6.3.2 Information Theory and th
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129 ( R) ⎛⎛det ⎞⎞ I( x, y)
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131 y= ∑ w x ) of Because of the
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133 6.5 Willshaw net David Willshaw
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135 6.6.1 Weights bounds One of the
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137 Figure 6-11: The weight vector
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139 6.6.4 Oja’s one Neuron Model
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141 If y i and y are highly correla
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143 Foldiak’s second model allows
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