MACHINE LEARNING TECHNIQUES - LASA

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160 7 Markov-Based Models Except for ICA, recurrent neural networks and continuous Hopfield network, these lectures notes have covered primarily methods to encode static datasets, i.e. data that do not vary in time. Being able to predict patterns that vary in time is fundamental to many applications. Such patterns are often referred to as time-series. In this chapter, we cover two models to encode time-series based on the hypothesis of a firstorder Markov Process, which we describe next. 7.1 Markov Process A Markov Process is by definition discrete and assumes that the evolution of a system can be described by a finite set of states. If N particular values { s ,... 1 s N} , i.e. x { s ,... 1 sN} x is the state of the system at time, then x can take only t ∈ . A first-order markov process is a process by which the future state of the system can be determined solely by knowing the current state of the system. In other words, if one has made T consecutive observation of the state x of the system: { t} t 1 the system at the next time step T 1 current state, i.e.: T X x = = , the probability that the state of x + take any of the N possible value is determined solely by the ( t+ 1 j | t, t− 1,...., 0) ( t+ 1 j | t) p x = s x x x = p x = s x (7.1) When modeling time-series, to assume that a process description is first-order markov is advantageous in that it simplifies greatly computation. Instead of computing the complete x , x ,...., x conditional on all previously observed states , one must solely compute the t t− 1 0 conditional on the last state. We will see how this is exploited in two different techniques widely used in machine learning next. © A.G.Billard 2004 – Last Update March 2011
161 7.2 Hidden Markov Models Hidden Markov Models (HMMs) are used to model the temporal evolution of a complex problem of which one can have only a partial description. The goal is to use this model to predict the evolution of the system in the future. One may be provided solely with the current state of the system or with several of the previous states of the system. Take for example the problem of predicting traffic jams. This seems to be a very intricate problem, especially as one can usually only observe part of the problem. One may have video cameras and other type of sensors monitoring the traffic at every single junction. However, one cannot know what individual path each car will follow. While one may suddenly measure an increase in traffic at all points and one may hence infer that there are good chances that a traffic jam may be created. It is very difficult to determine when and on which particular road the traffic jam will happen, if any. The underlying process leading to traffic jam is stochastic and highly complex. HMM-s aim at encapsulating such stochasticity and complexity in a partially observable problem. HMM are used widely in speech recognition and gesture recognition. In speech recognition, they are used to model the temporal evolution of speech pattern and to recognize either complete words or parts of the words, such as phonemes. In gesture recognition, HMM are used to recognize patterns of motions (from either video data or from recording joint motion through e.g. exoskeleton or motion sensors). One usually builds one HMM per gesture and one uses a measure of the likelihood that a new observed gesture was generated by a particular HMM to recognize this newspecific gesture (e.g. stop motion, waving motion, pointing motion, etc). These motions can either be computed from searching for body features in an image or through the use of an exoskeleton to measure directly displacement of each body segment. 7.2.1 Formalism HMM extends the principle of first-order Markov Process and assumes that the observed time series was generated by an underlying hidden process. This hidden process is assumed to consist of stochastic finite state automata where the states sequence is not observed directly. Each state has an underlying probabilistic function describing the distribution of observable outputs. Two concurrent stochastic processes are involved, one modeling the sequential structure of the data, and one modeling the local properties of the data. A typical graphical representation of this process is given in Figure 7-1. T O o = = be the set of observations. These correspond to the set of values for all Let { t} t 1 parameters describing the system which one recorded over a time period T. Each observation is usually multidimensional, e.g. q N ot ∈ ° . The set S { s i } i= 1 = of N hidden states is finite. © 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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17 2.1 Principal Component Analysis
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19 ( ) Xʹ′ = W X − µ (2.6) i
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23 PCA is an example of PP approach
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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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35 2.3.5 ICA Ambiguities We cannot
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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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59 C: X → Y ( ) C x K = arg max
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65 ⎛⎛ min ⎜⎜ w ⎝⎝ N i=
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67 T ( yi − xi w) 2 M ⎛⎛ ⎞
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77 M 1 T v = ∑ x ( x ) v M λ i j
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87 statistical independence. We def
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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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