MACHINE LEARNING TECHNIQUES - LASA

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104 Examples: Figure 5-12 illustrates the application of SVR on the same dataset (these data were generated using MLdemos). When using a linear kernel with a large e, one can completely encapsulate the datapoints, while still yielding a very poor fit. Using a Gaussian kernel allows to fit better the nonlinearities. This is however very sensitive to the choice of penalty given to poor fit through parameter C. A high penalty will result in a smoother fit. Figure 5-12: Example of e-SVR on a two-dimensional datasets. Datapoints are shown in plain dot. The regression signal is in solid line. The boundaries around the e-insensitite tube are drawn in light grey lines. using a linear kernel (top) and a Gaussian kernel (bottom). TOP: effect of the increasing the e-tube with e=0.02 (left) and e=0.1 (right). BOTTOM: effect of increasing parameter C (from left to right, C=10, 100 and 1000) that penalies for poorly fit datapoints. SVR as all other regression techniques remain very sensitive also the choice of hyperparameter for the kernel. This is illustrated in Figure 5-13. Too small a kernel width will lead to overfitting of the data. Too large may lead to very poor performance. The latter effect can be compensated by choosing appropriate value for the hyperparameters of the optimization function, namely C and e, see Figure 5-14. The small kernel width that had led to a poor fit in Figure 5-13 is compensated by relaxing the constraints using a large e and smaller C. We also see that reducing the constraints and widening e decreases the number of support vectors. While the very tight fit in Figure 5-13 used almost all datapoints as support vector, the looser fit in Figure 5-14 used much less. © A.G.Billard 2004 – Last Update March 2011
105 Figure 5-13: Effect of the kernel width on the fit. Here fit using C=1000, e=0.01, kernel width=0.01 (top), 0.1 (bottom). The points encircled represent the support vectors. Figure 5-14: Reduction of the effect of the kernel width on the fit by choosing appropriate hyperparameters. Gaussian SVR fit using C=100, e=0.03, kernel width=0.1. © 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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Page 63 and 64: 63 Figure 3-19: Bayes classificatio
Page 65 and 66: 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 87 and 88: 87 statistical independence. We def
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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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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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