MACHINE LEARNING TECHNIQUES - LASA

More documents

Recommendations

Info

44 The CURE algorithm has a number of features of general significance. It takes special care of outliers. It also uses two devices to achieve scalability. A major feature of CURE is that it represents a cluster by a fixed number c of points scattered around it. The distance between two clusters used in the agglomerative process is equal to the minimum of distances between two scattered representatives. Therefore, CURE takes a middleground approach between the graph (all-points) methods and the geometric (one centroid) methods. Single and average link closeness is replaced by representatives. Selecting representatives scattered around a cluster makes it possible to cover non-spherical shapes. As before, agglomeration continues until the requested number k of clusters is achieved. CURE employs one additional device: originally selected scattered points are shrunk to the geometric centroid of the cluster by user-specified factor α. Shrinkage suppresses the affect of the outliers since outliers happen to be located further from the cluster centroid than the other scattered representatives. CURE is capable of finding clusters of different shapes and sizes, and it is insensitive to outliers. Since CURE uses sampling, estimation of its complexity is not straightforward. Figure 3-7: Agglomeration with CURE. Three clusters, each with three representatives, are shown before and after the merge and shrinkage. Two closest representatives are connected by arrow. Summary: Advantages of hierarchical clustering include: • Embedded flexibility regarding the level of granularity • Ease of handling of any forms of similarity or distance • Consequently, applicability to any attribute types © A.G.Billard 2004 – Last Update March 2011
45 Disadvantages of hierarchical clustering are related to: • Vagueness of termination criteria • The fact that most hierarchical algorithms do not revisit once constructed (intermediate) clusters with the purpose of their improvement Exercise: Determine the optimal distance metric for two examples, in which the data are either not linearly separable or linearly separable. 3.1.2 K-means clustering K-Means clustering generates a number K of disjoint, flat (non-hierarchical) k clusters k = K so as to minimize the sum-of-squares criterion C , 1,... , J ( ,... ) µ µ = x −µ 1 K K i k k = 1 k i∈C 2 ∑∑ (3.2) where x i is a vector representing the i th data point and µ is the geometric centroid of the data k points associated to cluster C . K-means is well suited to generating globular clusters. The K- Means method is numerical, unsupervised, non-deterministic and iterative. In general, the algorithm is not guaranteed to converge to a global minimum of J. k K-Means Algorithm: 1 Initialization: Pick K arbitrary centroids and set their geometric means µ ,..., 1 µ to K random values. 2 Calculate the distance dx ( , µ ) from each data point i to each centroid k . 3 Assignment Step: Assign the responsibility i k i r k of each data point i to its “closest” centroid k i (E-step). If a tie happens (i.e. two centroids are equidistant to a data point, one assigns the data point to the smallest winning centroid). { ( )} k arg min d x , µ = (3.3) i i k k i k ⎧⎧ 1 if k = k ri = ⎨⎨ ⎩⎩ 0 otherwise 4 Update Step: Adjust the centroids to be the means of all data points assigned to them (M-step) µ = k ∑ i ∑ i rx k i r k i i (3.4) © A.G.Billard 2004 – Last Update March 2011
Page 1 and 2: SCHOOL OF ENGINEERING MACHINE LEARN
Page 3 and 4: 3 4. 4 Regression Techniques ......
Page 5 and 6: 5 9.2.2 Probability Distributions,
Page 7 and 8: 7 Journals: • Machine Learning
Page 9 and 10: 9 Performance What would be an opti
Page 11 and 12: 11 1.2.3 Key features for a good le
Page 13 and 14: 13 1.3.2 Crossvalidation To ensure
Page 15 and 16: 15 In particular, we will consider
Page 17 and 18: 17 2.1 Principal Component Analysis
Page 19 and 20: 19 ( ) Xʹ′ = W X − µ (2.6) i
Page 21 and 22: 21 2.1.2.2 Reconstruction error min
Page 23 and 24: 23 PCA is an example of PP approach
Page 25 and 26: 25 Algorithm: If one further assume
Page 27 and 28: 27 The CCA algorithm consists thus
Page 29 and 30: 29 Figure 2-6: Mixture of variables
Page 31 and 32: 31 2.3.2 Why Gaussian variables are
Page 33 and 34: 33 • In our general definition of
Page 35 and 36: 35 2.3.5 ICA Ambiguities We cannot
Page 37 and 38: 37 Denote by g the derivative of th
Page 39 and 40: 39 3 Clustering and Classification
Page 41 and 42: 41 An agglomerative clustering star
Page 43: 43 3.1.1.1 The CURE Clustering Algo
Page 47 and 48: 47 Cases where K-means might be vie
Page 49 and 50: 49 3.1.4 Clustering with Mixtures o
Page 51 and 52: 51 k ( σ j ) 2 = k ∑ i α = r k
Page 53 and 54: 53 Theα are the so-called mixing c
Page 55 and 56: 55 Figure 3-16: Clustering with 3 G
Page 57 and 58: 57 When the transformation A is lin
Page 59 and 60: 59 C: X → Y ( ) C x K = arg max
Page 61 and 62: 61 Figure 3-18: Linear combination
Page 63 and 64: 63 Figure 3-19: Bayes classificatio
Page 65 and 66: 65 ⎛⎛ min ⎜⎜ w ⎝⎝ N i=
Page 67 and 68: 67 T ( yi − xi w) 2 M ⎛⎛ ⎞
Page 69 and 70: 69 Figure 4-2: Illustration of the
Page 71 and 72: 71 4.4.2 Multi-Gaussian Case It is
Page 73 and 74: 73 5 Kernel Methods These lecture n
Page 75 and 76: 75 The kernel k provides a metric o
Page 77 and 78: 77 M 1 T v = ∑ x ( x ) v M λ i j
Page 79 and 80: 79 1 M The solutions to the dual ei
Page 81 and 82: 81 5.4 Kernel CCA The linear versio
Page 83 and 84: 83 additional ridge parameter induc
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.
Page 95 and 96:
95 Figure 5-6: A binary classificat
Page 97 and 98:
97 where N is the number of support
Page 99 and 100:
99 5.8 Support Vector Regression In
Page 101 and 102:
101 The optimization problem given
Page 103 and 104:
103 Note that since we never have t
Page 105 and 106:
105 Figure 5-13: Effect of the kern
Page 107 and 108:
107 To better understand the effect
Page 109 and 110:
109 5.9 Gaussian Process Regression
Page 111 and 112:
111 One can then use the above expr
Page 113 and 114:
113 5.9.2 Equivalence of Gaussian P
Page 115 and 116:
115 5.9.3 Curse of dimensionality,
Page 117 and 118:
117 The weight w determines the slo
Page 119 and 120:
119 Figure 5-21: Example of success
Page 121 and 122:
121 • its performance tends to de
Page 123 and 124:
123 neurons. Furthermore, they lear
Page 125 and 126:
125 The sigmoid f x ( x) ( ) = tanh
Page 127 and 128:
127 6.3.2 Information Theory and th
Page 129 and 130:
129 ( R) ⎛⎛det ⎞⎞ I( x, y)
Page 131 and 132:
131 y= ∑ w x ) of Because of the
Page 133 and 134:
133 6.5 Willshaw net David Willshaw
Page 135 and 136:
135 6.6.1 Weights bounds One of the
Page 137 and 138:
137 Figure 6-11: The weight vector
Page 139 and 140:
139 6.6.4 Oja’s one Neuron Model
Page 141 and 142:
141 If y i and y are highly correla
Page 143 and 144:
143 Foldiak’s second model allows
Page 145 and 146:
145 ∂ ∂ J 1 = fy − 1 2 λ 1yf
Page 147 and 148:
147 6.8 The Self-Organizing Map (SO
Page 149 and 150:
149 6. Decrease the size of the nei
Page 151 and 152:
151 the resulting distribution is a
Page 153 and 154:
153 To simplify the description of
Page 155 and 156:
155 C µν −1 where ( ) is the µ
Page 157 and 158:
157 The continuous time Hopfield ne
Page 159 and 160:
159 ∂f If the slope is negative,
Page 161 and 162:
161 7.2 Hidden Markov Models Hidden
Page 163 and 164:
163 Figure 7-2: Schematic illustrat
Page 165 and 166:
165 these two quantities to compute
Page 167 and 168:
167 7.2.4 Decoding an HMM There are
Page 169 and 170:
169 7.2.5 Further Readings Rabiner,
Page 171 and 172:
171 7.3.1 Principle In reinforcemen
Page 173 and 174:
173 general, acting to maximize imm
Page 175 and 176:
175 of reinforcement learning makes
Page 177 and 178:
177 8 Genetic Algorithms We conclud
Page 179 and 180:
179 However, you must define geneti
Page 181 and 182:
181 Often the crossover operator an
Page 183 and 184:
183 ( A λI) x 0 − = (8.5) where
Page 185 and 186:
185 Joint probability: The joint pr
Page 187 and 188:
187 The two most classical distribu
Page 189 and 190:
189 9.2.7 Statistical Independence
Page 191 and 192:
191 1 1 1 1 h x ∫ a b a b 0 a a
Page 193 and 194:
193 9.4 Estimators 9.4.1 Gradient d
Page 195 and 196:
195 9.4.2.1 Maximum Likelihood Mach
Page 197 and 198:
197 10 References • Machine Learn
show all

MACHINE LEARNING TECHNIQUES - LASA

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?