MACHINE LEARNING TECHNIQUES - LASA

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188 Figure 9-2: How the Joint, Marginal, and Conditional distributions are related in a bi-dimensional Gaussian T − −( z−µ ) ∑ 1 ( z−µ ) 1 p( z = x, y | µ , ∑ ) = e N 1 2 ∑ 2 distribution { } ( 2π ) ( ) 9.2.5 Likelihood ( µσ x) = p( x µσ) ( ) For a parametrized density p x| µσ , , with parameters µσ , , the Likelihood Function is the likelihood of the data given the parameters: L , | | , . If The likelihood function (short – likelihood) of the model parameters is given by: 9.2.6 Probabilistic Independence Two events A and B are statistically independent iff the conditional probability of A given B ( | ) = P( A) P A B in which case the probability of A and B is just P( A∩ B) = P( A) ⋅ P( B) (8.24) © A.G.Billard 2004 – Last Update March 2011
189 9.2.7 Statistical Independence If y 1 and y 2 are two variables, generated by a random distribution p i (y i ) with zero-mean, then y 1 and y 2 are said to be mutually independent, if their joint density p(y 1 , y 2 ) is equal to the product of their density functions: pyy ( ) = py ( ) py ( ) (8.25) 1, 2 1 2 9.2.8 Uncorrelatedness y 1 and y 2 are said to be uncorrelated, if their covariance is zero ( y y ) cov , 0 1 2 = . If the variables are generated by a distribution with zero means, then uncorrelatedness is equal to product of their separate expectations: ( ) ( ) ( ) cov( yy) = E yy − E y E y = 0 (8.26) 1, 2 1, 2 1 2 E( y y ) E( y ) E( y ) = (8.27) 1, 2 1 2 Independence is in general a much stronger requirement than uncorrelatedness. Indeed, if the y i are independent, one has E( f y f y ) = E( f y ) E( f y ) (8.28) ( ) ( ) ( ) ( ) 1 2 1 2 for any functions f 1 and f 2 . Independence and uncorrelatedness are equivalent when y 1 and y 2 have a joint Gaussian distribution. The notion of independence depends, thus, on the definition of the probability distribution of any given variables. 9.3 Information Theory The mathematical formalism for measuring the information content, I, of an arbitrary measurement I is due largely to a seminal 1948 paper by Claude E. Shannon. Within the context of sending and receiving messages in a communication system, Shannon was interested in finding a measure of the information content of a received message. Shannon’s measure of information aimed at conveying the impression that an unlikely event contains more information than a likely event. The information content of an outcome x is defined to be: where P(x) is the probability of x. ( ) I x 1 = log2 =− log2 P( x) (8.29) P x ( ) © 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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