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Ihler, Hutchins and Smyth (2007). Learning to Detect Events with Markov-Modulated Poisson Processes. Outline. Problem: Finding unusual activity ( events ) in rhythms of natural human activity Method: Unsupervised learning
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Ihler, Hutchins and Smyth (2007) Learning to Detect Events with Markov-Modulated Poisson Processes
Outline • Problem: Finding unusual activity (events) in rhythms of natural human activity • Method: • Unsupervised learning • Time-varying Poisson process modulated by a hidden Markov process (events) • Bayesian framework for parameter learning
Why is it hard? • Chicken-and-egg problem • Where do we start? • Previous approaches: baseline • Simple threshold model • Has severe limitations • Need to quantify the notion of an unusual activity • How unusual is a measurement • How persistent is a deviating measurement
The Data Sets • 2 data sets used • Building data • Counts of people entering and exiting a building • 15 weeks of data • 30 minute time bins • 29 known events in the 15 weeks • Freeway Traffic data • Vehicle counts on a freeway on-ramp • 6 months of data • 5 minute time bins • 78 known events in the 6 months
Building Data • Example day
Example week Building Data
Example day Freeway Traffic Data
Example week Freeway Traffic Data
A naïve Poisson model • Is the data actually Poisson? • In a Poisson distribution the mean = the variance • Is this the case in out data?
A Baseline Model • Use a simple threshold approach • We say there is an event if • P(N;λ) < ε
Problems with this Approach • Hard to detect sustained small variation • Hard to capture event duration • Chicken and egg problem
The model (1) Assuming the processes are additive ...which is a fair assumption
A = Rainy B = Sunny What is a Markov Process? 0.1 0.5
Modelling Events with a Markov Process • We define a three state Markov chain • z(t) is the state at time t, the 3 possible states are • 0 if there is no event • +1 if there is a positive event • -1 if there is a negative even • With transition matrix
Details of the Markov Process • We give each row in the transition matrix a Dirichlet prior: • Given z(t), we can model NE(t) as a Poisson with rate γ(t). We give this a Gamma prior Γ(γ;aE,bE), which is independent of t • We can then marginalize out over γ(t):
Learning the parameters • If we are given the hidden variables N0(t), NE(t) and z(t), we can: • compute MAP estimates • draw posterior samples of the parameters λ(t) and Mz • So, we can use MCMC; iterate between sampling from the hidden variables (given the parameters), and the parameters (given the variables)
Sampling the hidden variables, given the parameters Rough outline: • First, use forward-backward algorithm [Baum et al. 1970] to sample z(t) • Then given z(t), determine N0(t) and NE(t) by sampling
Sampling the parameters, given the hidden variables • The conjugate prior distributions give us a straightforward way to compute the posteriors • Use the sufficient statistics of the data as (updating) parameters for the posterior:
Prior distributions of zij and γ(t) • Markov-modulated Poisson processes are sensitive to selection of priors for zij and γ(t) • For the domains of these models, we often have strong ideas on e.g. what constitutes a “rare” event • Use these ideas to build strong priors in the model in order to avoid overfitting, and to adjust threshold levels of event detection
Calculating Results • We are looking to detect unusual events, we can use our model to do this do this by calculating the posterior: • We can then compare our predictions with the known event occurrences
Other Possible Inferences • The model can be modified to test the degree of heterogeneity of the time process. We can ask questions like • are all week days essentially the same? • are all afternoons essentially the same? • We can estimate event attendance
Conclusion • Model much more affective than threshold approach • Good detection rate • Difficult to access false positive rate • Possibility for extension