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**Cognitive Computer Vision**

Kingsley Sage and Hilary Buxton Prepared under ECVision Specific Action 8-3

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**Lecture 7 (Hidden) Markov Models What are they?**

What can you do with them?

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**So why are HMMs relevant to Cognitive CV?**

Provides a well-founded methodology for reasoning about temporal events One method that you can use as a basis for our model of expectation

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Markov Models Markov Models are used to represent and model temporal relationships, e.g.: Motion of objects in a tracker Gestures Interpreting sign language Speech recognition

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Markov Models So for MIT’s Smart Room, we can use Markov Models to represent cue gestures that control the behaviour of other agents …

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**Markov Models What is a Markov Model? Observable and hidden variables**

The Markov assumption Forward evaluation Order of a Markov Model Observable and hidden variables The Hidden Markov Model (HMM) Viterbi decoding Learning the HMM parameters

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**What is a Markov Model? The (first order) Markov assumption**

That the distribution P(XT ) depends solely on the distribution P(XT-1). The present (current state) can be predicted using local knowledge of the past (state at previous time step)

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**What is a Markov Model? Can represent as a state transition diagram**

State Transition Matrix Sunny Rain Wet state at time t Sunny Rain Wet 0.6 0.3 0.1 0.2 state at time t-1

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**What is a Markov Model? Formally a Markov Model = (, A)**

vector is the probability that you are in a state at time t=0 A is the State Transition Matrix You can use this information to calculate the probability for our weather example at any future t using Forward Evaluation …

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**Forward evaluation (1) t=0 t=1 1.0**

Sunny Rain Wet 0.6 0.3 0.1 0.2 t=0 t=1 Sunny 1.0 (1.0*0.6) + (0.0*0.1) + (0.0*0.2) = 0.6 Rain 0.0 (1.0*0.3) + (0.0*0.6) + (0.0*0.2) = 0.3 Wet (1.0*0.1) + (0.0*0.3) + (0.0*0.6) = 0.1

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**Forward evaluation (2) t=0 t=1 t=2 1.0 0.6 0.0 0.3 0.1 Sunny**

Rain Wet 0.6 0.3 0.1 0.2 t=0 t=1 t=2 Sunny 1.0 0.6 (0.6*0.6) + (0.3*0.1) + (0.1*0.2) = 0.41 Rain 0.0 0.3 (0.6*0.3) + (0.3*0.6) + (0.1*0.2) = 0.38 Wet 0.1 (0.6*0.1) + (0.3*0.3) + (0.1*0.6) = 0.21

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**The order of a Markov Model (1)**

The (N-th order) Markov assumption That the distribution P(XT ) depends solely on the joint distribution P(XT-1,XT-2, … XT-N) The present (current state) can be predicted using only a knowledge of the past (state of N previous time steps) Problem: number of parameters for state transition matrix increases as |S|*|S|N

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**Before we can discuss Hidden Markov Models (HMMs) …**

Observable and hidden variables: A variable is observable if its value can be directly measured, or given as an Observation sequence O A variable is hidden if its value cannot be measured directly, but we can infer its value indirectly So …

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**Before we can discuss Hidden Markov Models (HMMs) …**

Consider a hermit living in a cage. He cannot see the weather conditions, but he does have a magic crystal which reacts to environmental conditions. The crystal turns one of 3 colours (red, green or blue). The actual weather states (sunny, rainy, wet) are hidden to the hermit, but the crystal states (red, green and blue) are observable.

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**The Hidden Markov Model (HMM) - model structure**

Sunny Rain Wet Red Green Blue Observable variables Hidden

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**What is a Hidden Markov Model?**

Formally a Hidden Markov Model = (, A, B) vector and A matrix as before M observable states Red Green Blue Sunny 0.8 0.1 Rain Wet 0.2 0.6 The B (confusion) matrix Single N * M matrix iff M is discrete. If M is continuous, B is usually represented as set of Gaussian mixtures N hidden states

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**So what can you do with a HMM?**

Given and a sequence of observations O Calculate p(O| ) – forward evaluation Given and O, calculate the most likely sequence of hidden states (Viterbi decoding) Given O, find to maximise p(|O) – Baum Welch (model parameter) learning Use to generate new O (the HMM as a generative model) – stochastic sampling

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**Forward evaluation (1) Assume and O = {o1,o2, … ,oT} are given … Red**

M observable states Red Green Blue Sunny 0.8 0.1 Rain Wet 0.2 0.6 Sunny Rain Wet 0.6 0.3 0.1 0.2 N hidden states Assume and O = {o1,o2, … ,oT} are given …

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**Forward evaluation (2) t=1 Here O = {o1 = red} Sunny 1.0**

M observable states Sunny Rain Wet 0.6 0.3 0.1 0.2 Red Green Blue Sunny 0.8 0.1 Rain Wet 0.2 0.6 N hidden states t=1 Sunny 1.0 1.0 * 0.8 = 0.80 Rain 0.0 0.0 * 0.1 = 0.00 Wet 0.0 * 0.2 = 0.00 time t = 1 is a special case Here O = {o1 = red}

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**Forward evaluation (3) t=1 t=2 Here O = {o1 = red, o2 = green} Sunny**

Blue Sunny 0.8 0.1 Rain Wet 0.2 0.6 Sunny Rain Wet 0.6 0.3 0.1 0.2 N hidden states t=1 t=2 Sunny 1 0.80 {(0.80*0.6) + (0.00*0.1) + (0.00*0.2)} * 0.1 = 0.048 Rain 0.00 {(0.80*0.3) + (0.00*0.6) + (0.00*0.2)} * 0.8 = 0.192 Wet {(0.80*0.1) + (0.00*0.3) + (0.00*0.6)} * 0.2 = 0.016 Here O = {o1 = red, o2 = green}

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**Seminar Prior assumption**

Practical issues in computing the forward evaluation matrix Measure of likelihood per observation symbol Backwards evaluation Reference: “An Introduction to Hidden Markov Models”, L. R. Rabiner & B. H. Juang, IEEE ASSP Magazine, January 1986

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Further reading Try reading the Rabiner paper (it’s quite friendly really) … Mixture of Gaussians: many maths books will cover MOG. Non-trivial maths involved … Variable length Markov Models: “The Power of Amnesia”, D. Ron, Y. Singer and N. Tishby, In Advances in Neural Information Processing Systems (NIPS), vol 6, pp , 1994

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Summary An N-th order Markov model incorporates the assumption that the future depends only on the last N timesteps In Markov model reasoning over time, we use a state transition matrix A and a vector representing the probabilities at time step t=1 We use a matrix B which maps the observation O to the hidden states

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Next time … Gaussian mixtures and HMMs with continuous valued data

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