MACHINE LEARNING TECHNIQUES - LASA

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62 quantities are comparable. Since a pdf is a positive, but unbounded (above), function, its likelihood may grow arbitrarily. It usually grows with the numer of parameters. A good precaution is then to make sure that the two classifiers have the same number of parameters and are trained on the same number of datapoints. The latter means that one must have at its disposal as many examples of the positive class as that of the negative class. If this can not be ensured, then one may apply some normalization term on the likelihood of each classifier or one may use a threshold on the likelihood of both models to determine when inference is warranted. Despite these caveats, Bayes classifiers remain quite popular, in part because of their extreme simplicity, but also because they yield very good performance in practice. Note that Bayes classification can easily be extended to multiclass classification. Assume that i you wish to classify a set of datapoints { } i= 1 M X = x into K classes i i y associated to each datapoint x , i = 1.... M, can then take any value compute the posterior probability of each class, using: 1 K C ,...., C . Each label 1 K C ,...., C . One can then i ( | ) p y = C x = i i p( y = C ) p( x| y = C ) K k k ∑ p( y = C ) p( x| y = C ) k = 1 (3.33) We see next how this can be used when using Gaussian Mixture Models. Note that we will revisit multi-class classification using kernel methods in Section 5.10. 3.4 Linear classification with Gaussian Mixture Models The Gaussian Mixture Model, introduced in Section 3.1.5, can be used in conjunction with the Bayes classifier to perform binary classification. For two knows classes of datapoints, one can train two separate GMM-s, one for each class of datapoints and use the Bayes classifier to determine the labeling of any given datapoint. In other word, each GMM yields a density p and p which can then be used to determine the label of a new training point using (3.32). + − To ensure that the two models are comparable, one should train two GMM with same number of Gaussians, using the same modeling (with full, diagonal or isotropic covariance matrix in each case), and using similar number of training points for each GMM. Figure 3-19 shows one example of such classification using two GMM-s with 8 Gaussians each. © A.G.Billard 2004 – Last Update March 2011
63 Figure 3-19: Bayes classification using a two GMMs with 8 Gaussians each and full covariance matrix. From left to right top to bottom: i) original datapoint (in dark, datapoints belonging to class +1, empty circles denotes datapoints belonging to class -1), ii) result of the classification; all datapoints are correctly classified in their respective class; iii) the 16 Gaussians superimposed on the datapoints; iv) the region associated to each class using Bayes classification. 3.5 Further Readings There exist numerous methods to cluster or classify data. This chapter dealt only with some of the major algorithms for clustering and classification from which most of state-of-the-art methods are derived. In this manuscript, we will also see two other techniques for classification as part of the Artificial Neural Networks chapter. These are the perceptron and the backpropagation algorithm. In part-II of this manuscript, we will also consider other techniques for classification, such as Support Vector Machine and Support Vector Clustering that exploit non-linear transformation of the data through kernel projection. © 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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Page 13 and 14: 13 1.3.2 Crossvalidation To ensure
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Page 17 and 18: 17 2.1 Principal Component Analysis
Page 19 and 20: 19 ( ) Xʹ′ = W X − µ (2.6) i
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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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145 ∂ ∂ J 1 = fy − 1 2 λ 1yf
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147 6.8 The Self-Organizing Map (SO
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149 6. Decrease the size of the nei
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151 the resulting distribution is a
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153 To simplify the description of
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155 C µν −1 where ( ) is the µ
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157 The continuous time Hopfield ne
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159 ∂f If the slope is negative,
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161 7.2 Hidden Markov Models Hidden
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163 Figure 7-2: Schematic illustrat
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165 these two quantities to compute
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167 7.2.4 Decoding an HMM There are
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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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