Hidden Markov Models: A Brief Introduction

1
Hidden Markov Models A Brief Introduction
2
Introduction

• Most of the methods weve looked at
• Have examined images and tried to compute a few
  pieces of information
• These give us observations that tell us about the
  world in some way

• Hidden Markov Models (HMMs)
• Provide a way of representing a model of what
  happens to cause these observations
• We can use this to explain a sequence of
  observations, such as those made from a sequence
  of images

3
Hidden, Markov, Model

• Hidden
• We cant observe the model directly, but we know
  that it is there, and we can observe things
  related to it
• Markov
• The state of the model at any time depends only
  on the previous state, and not on those before
  (is Markovian c.f. CONDENSATION)

4
Hidden Markov Models

• A HMM consists of two sets and three functions
• The sets
• A set of n states, Ss1, s2, , sn
• A set of m symbols that can be observed, Aa1,
  a2, , am

• The functions
• A start state function, I(si), giving the prob.
  of starting in state si
• A transition function, T(si, sj), giving the
  prob. of going from state si to state sj
• An observation function, O(ak , si) giving the
  prob. of seeing symbol ak in state si

5
Example

• Suppose we have a camera watching some people
• We segment the scene into individuals
• We track each person to class their speed into
  still, slow, or fast
• We use stereo to class their height as being low,
  or high

• We expect people to have several actions
• Sitting, standing, walking, or running
• Each of these is a state in our HMM for that
  person
• Each state has an expected observation (eg
  sitting people are still and appear low)

6
Eg Sets and Start States

• The sets
• The set of states, S, is sitting, standing,
• walking, running
• The observations, A, are
• (still, low),
• (still, high),
• (slow, low),

• The function I tells us how people enter the
  scene
• I(sitting) 0
• I(standing) 0
• I(walking) 0.9
• I(running) 0.1

7
Eg Transition Function

• Tells us how peoples actions change
• Given that we saw someone sitting in the last
  frame what are they likely to be doing now?
• Thell probably still be sitting, might have
  stood up or walked off, but wont have run

• Eg
• T(sit, sit) 0.7
• T(sit, stand) 0.2
• T(sit, walk) 0.1
• T(sit, run) 0
• We repeat this for all other pairs of actions.
  Note that T(x,x) is often high

8
Eg Transition Function
9
Eg Observation Function

• Tells us how each state looks
• Often a bit uncertain because of errors in
  measurements
• We assume there are a discrete set of
  measurements made (not necessary, but simpler)

• Eg A sitting person is almost certainly low
  and probably still but not fast
• O(sit, (low, still)) 0.6
• O(sit, (low, slow)) 0.3
• O(sit, (low, fast)) 0
• O(sit, (high, still)) 0.1
• O(sit, (high, slow)) 0
• O(sit, (high, fast)) 0

10
Three Problems

• Given a set of observations, what set of
  parameters is the model likely to have?
• Given the model, what is the chance of some event
  happening?
• Given the model and an observation, what is the
  most likely sequence of states to explain that
  observation?

11
The Viterbi Algorithm

• We can view the problem as a graph search problem
• Suppose our observation is x1x2xk
• We make a graph with k columns of vertices, each
  column has a vertex for each state in the HMM
• It makes it easier to add a start vertex S and a
  finish vertex F
• With the right weighting we can find the optimal
  sequence of states as the shortest path from S to
  F

12
The Viterbi Algorithm
S
F




x1
x2
x3
xk
13
Weighting the Arcs

• Given a (partial) path through the graph,
  S,y1,y2,,yj, the probability of taking this path
  and getting the observations x1x2xj is
• I(y1)O(x1,y1)T(y1,y2)O(x2,y3)T(yk-1,yk)O(xk,yk)
• But graph search algorithms use the sum of the
  weights through the graph, and this is a product
  we need to convert it into a sum

14
Weighting the Arcs

• We can convert the product into a sum using
  log(xy) log(x) log(y)
• We take the log of the weight to give
• log(I(y1)) log(O(x1,y1)) log(T(y1,y2))
  log(O(x2,y3)) log(O(xk,yk))
• This will be high if the probability is high, and
  algorithms find the shortest path, we multiply by
  -1 giving
• -log(I(y1)) - log(O(x1,y1)) - log(T(y1,y2)) -
  log(O(x2,y3)) - - log(O(xk,yk))

15
The Viterbi Algorithm
log(I(y1)
log(O(xi,yi) log(T(yi,yi1))
-log(O(xk,yk)
S
F




x1
x2
x3
xk
16
Graph Search

• We can now find the shortest path from S to F
  through the graph
• Since all the arcs point left to right, we can
  find the shortest path to all the vertices in the
  first column, then all the vertices in the second
  column, and so on
• At each stage we keep track of the total path
  length and the arcs that lead to these shortest
  paths

17
Gesture Recognition
The fish was this big..

• http//research.microsoft.com/awilson/papers/ijpr
  ai.pdf

18
For More Information

• HMMs have been used for a number of tasks, both
  in vision and in artificial intelligence (notably
  NLP)
• Dugad and Desai give a good tutorial, available
  online at
• http//uirvli.ai.uiuc.edu/dugad/hmm_tut.html