1. Particle Filters for Event Detection
By Marc Sobel

2. The Event detection problem
We observe the number of people entering a building like e.g., Tuttleman Hall in each half hour. Most of the time the numbers of people entering the building are reasonably uniform – but sometimes there are larger numbers of people entering the building. We’d like to identify the moments when there are substantially larger numbers of people ‘than usual’ entering the building.

3. Event detection problem
This is like a hidden markov model problem.
Let N0(t) represent the number of people entering the building ‘normally’. N1(t) is the number of people entering the building over and above the number N0(t). Let
Θt denote the occurrence (=1) or nonoccurrence of an event (=0).

4. Setup
At each timepoint t:

5. Prior Distributions
We have that:

6. Prior Distributions II
We also have,

7. Training the hyperparameters
We have that,

8. Training the hyperparameters
Train λ0 using:

9. Marginalizing over λ1
We find the marginal distribution of N1(t) in order to avoid simulation in this case.
This is the negative binomial distribution.

10. Posterior Distributions (I)
We have,

11. Posterior Distribution (II)
We also have,

12. Smyth strategy II
Estimate λ0 and generate the events by choosing the map estimator of Θ(t). Maximize over i=0,1
This comes to:

13. Particle Strategy I
Instead of employing MAP estimation. Create particles out of the lambda’s via
Simulate lambda zeros from the prior:
Then create theta’s at each step:

14. Particle strategy I
The theta’s are simulated aposteriori via the equation given above.

15. Residual Resampling
After θ(t) and N0(t) particles have been generated i.e.,

16. Calculate Weights
Calculate weights:

17. Residual Sampling
Determine the class of parameters for which the weights are >(1/k). Call these
the good weights. If a weight < (1/k), then
Resample it from among the good weights.

18. Comparison between process and predicted events

19. Improved model
Instead of λ0 define a sequence of parameters, λ0(t) --- together with an updating mechanism: