Gibbs Sampling and Hidden Markov Models in the Event Detection Problem

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Gibbs Sampling and Hidden Markov Models in the
Event Detection Problem

• By Marc Sobel

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Event Detection Problems

• A process like e.g., traffic flow, crowd
  formation, or financial electronic transactions
  is unfolding in time. We can monitor and observe
  the flow frequencies at many fixed time points.
  Typically, there are many causes influencing
  changes in these frequencies.

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Causes for Change

• Possible causes for change include
• a) changes due to noise i.e., those
  best modeled by e.g., a Gaussian error
  distribution.
• b) periodic changes i.e., those
  expected to happen over periodic intervals.
• c) changes not due to either of the
  above these are usually the changes we would
  like to detect.

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Examples

• Examples include
• 1) Detecting Events, which are not
  pre-planned, involving large numbers of people at
  a particular location.
• 2) Detecting Fraudulent transactions.
• We observe a variety of electronic
  transactions at many time intervals. We would
  like to detect when the number of transactions is
  significantly different from what is expected.

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Model for Changes Due to Noise or Periodic
Changes or other

• We model changes in flow frequency due to
  all possible known causes. This is done using
  latent Poisson processes.
• 
  
• The frequency count N(t) at time t is observed.
  N0(t), NE(t) are independent latent Poisson
  processes.
• N0(t) denotes the frequency due to periodic and
  noise changes at time t. We use the
  terminology, ?(t) for the average rate for such
  changes. We write N0(t)P(N,?(t)) in this case.
• NE(t) denotes the frequency due to causes
  different from periodic and noise changes. It
  has a rate function ?(t). We write,
  NE(t)P(N,?(t)) in this case.
• The rate function ?(t) is regressed on a
  parametric function of the periodic effects as
  follows

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The Process N0(t)

• We focus on the first example given above and
  consider the problem of modeling the frequencies
  of people entering a building with the eventual
  purpose of modeling special events connected
  with these frequencies.
• We let
• a) d stand for day,
• b) hh for half hour time interval.
• c) b for base

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Rate Function Due to Periodic and Noise Changes.

• The rate function due to periodic and noise
  changes is

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Rate Function Explained

• This makes sense because we are thinking of the
  fact that, for a time t in day d and
  half-hour period h, we have (by Bayes rule)
• In the sequel, we assume time t has been broken
  up into half-hour periods without re-indexing.

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Example Work week

• Say you worked 21 hours on average per week. So
  your base work rate (per week) is ?b 21.
• Therefore your daily base work rate is 3.
• Your average work rate for Sunday relative to
  this base is ?Sunday (total Sunday rate)/3.
• The sum of your work rates for Sunday, Saturday
  is ?Sunday ?Saturday7

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Modeling Occasional changes in event flow and
Noise

• Where does the noise come in? How do we model
  occasional changes in the periodic rate
  parameters? The missing piece is (dramatic
  pause) ??????!!!!!!!!!

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Priors Come to the Rescue

• Priors serve the purpose of modeling noise
• and occasional changes in
• the values of parameters.
• 
  
  
  
  
• 
  
  
  
  Thus
  spake the prior
• The base parameter is given a gamma prior,
• ?base p(?)?a-1 da exp(-?d)/G(a)
• By flexibly assigning values to the
  hyperparameter a,d we can biuld a distribution
  which properly characterizes the base rate.

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Interpretation

• The ?day s, being conditional rates, satisfy,
• ??day ?(average day i totals)/?base7
• Similarly, summing over periods,
• ??jth period during day i
• ?(average jth period frequency in day
  i)/?day iD
• where D stands for the number of half hour
  intervals in a day.

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A Simple example illustrating the issue.

• What do these mean. Assume, for purposes of
  simplification, that there are a total of 2 days
  in a week Sunday and Monday. Daily rates are
  measured in events per day.
• The base rate is the average rate for
  sundays and mondays combined.
• A) The Sunday and Monday relative rates add up to
  2.
• B) If we observe 10 people (total) on sunday and
  30 on Monday, over a total of 10 weeks.

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(continued)

• C) maximum likelihood dictates estimating the
  base rate (per week) as (40/10)4 people per week
  (or 2 people per day), sundays relative rate is
• (10 (people)/210 (weeks)) .5 and Mondays
  relative rate is 1.5.
• D) But, this is incorrect because,
• (i) it turns out that one week out of
  10, the conditional Monday rate shoots up to
  1.90, while the Sunday rate decreases to 0.1.
• (ii) it turns out that usually, the
  conditional Sunday rate is 1 rather than .5.

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The Bayesian formulation wins out

• We can biuld a model with this new information by
  assuming a beta prior for half the Monday,Sunday
  relative rates
• (.5)?sunday ?(.66-1)(1-?)(.66-1)/ß(.66,.66)
• This prior has the required properties that
  the Sunday rate dips down to .10 about 10 percent
  of the time, but averages 1 over the entire
  interval.

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The Failure of classical theory

• The MLE of ?sunday is
• The Bayes estimator of ?sunday is
• But even more importantly, the posterior
  distribution of the parameter provides
  information useful to all other inference and
  prediction in the problem.
• Medicare for classical statistics?

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illustration

• Posterior distribution for the twice the Sunday
  frequency rate.
• 0 .5 .52
  1

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Actual Priors Used

• For our example, we have seven rather than 2 days
  in a week. We use scaled Dirichlet priors
  (extensions of beta priors) for this
• Smaller alphas indicate smaller apriori relative
  frequencies. Smaller sum of alphas indicate
  greater relative frequency variance for the ps.
• This provides a flexible way to model the daily
  rates.

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Events The Process NE

• Events signify times during which there are
  higher frequencies which are not due to periodic
  or noise causes. We can model this by assuming
  z(t)1 for such events and z(t)0 if not.
• P(z(t)1z(t-1)0) 1-z00
• P(z(t)0z(t-1)0) z00
• P(z(t)1z(t-1)1)z11
• P(z(t)0z(t-1)1)1-z11
• i.e., if there is no event at time t-1, the
  chance of an event at time t is 1-z00

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The need for a Bayesian treatment for events

• This gives a latent geometric distribution.
  Assume z00.8 z11.1. Then non-events tend to
  last an average of (1/.2)5 half-hours while
  events tend to last an average of (1/.9)1.11
  half-hours. Classical statistics would dictate
  direct estimation of the zs but note that this
  says nothing about the tendencies of events to
  exhibit non-average behavior. It doesnt provide
  information about prediction and estimation.

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Priors for event probabilities

• Beta distributions priors for the zs.
• z00 z00a0-1 (1-z00)b0-1 and
  z11 analogously.
• This characterizes the behavior of the
  underlying latent process. The hyperparameters
  a,b are designed to model that behavior.
• Recall that N0(t) (the non-event process)
  characterizes periodic and noise changes. The
  event process NE(t) characterizes other changes.
• NE(t) is 0 if z(t)0 and Poisson with rate
  ?(t) if z(t)1.
• So, if there is no event, NN0(t). If there
  is an event, the frequency due to periodic or
  noise changes is NN0(t)NE(t)
• The rate ?(t) is itself gamma with parameters
  aE and bE. Hence it is marginally negative
  binomial with p(bE/(1bE) and nN.

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Gibbs Sampling

• Gibbs sampling works by simulating each
  parameter/latent variable conditional on all the
  rest. The ?s are parameters and the zs,Ns are
  the latent variables. The resulting simulated
  values have an empirical distribution similar to
  the true posterior distribution. It works as a
  result of the fact that the joint distribution of
  parameters is determined by the set of all such
  conditional distributions.

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Gibbs Sampling

• Given z(t)0 and the remaining parameters, Put
  N0(t)N(t) and NE(t)0.
• If z(t)1, simulate NE(t) as negative binomial
  with parameters, N(t) and bE/(1bE). Put
  N0(t)N(t)-NE(t).
• To simulate z(t), define

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More of Gibbs Sampling

• Then, if the previous state was 0, we get

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Gibbs Sampling (Continued)

• Having simulated z(t), we can simulate the
  parameters as follows
• Where Nday denotes the number of day units in
  the data, Nhh denotes the number of hh
  periods in the data.

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Gibbs Sampling (conclusion)

• We can simulate from the remaining conditional
  posterior distributions using standard MCMC
  techniques.

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• END
• Thank You

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Polya Tree Priors

• A more general methodology for introducing
  multiple prior levels is through Polya Tree
  Priors. (see Michael Lavine). For these priors,
  we divide the time interval (e.g., a week) into
  parts with relative frequencies p1,,pk. p
  has a dirichlet distribution. Given p, we further
  divide up each of the time interval parts into
  parts with corresponding conditional dirichlet
  distributions. We can continue to subdivide
  until it is no longer useful.