1. Sharing Features among Dynamical Systems with Beta Processes
Emily B. Fox, Erik B. Sudderth, Michael I. Jordan, Alan S. Willsky
NIPS 2009
Some pictures and slides are adopted from Emily Fox’s paper and slides in NIPS 2009

2. Outline Introduction Sticky HDP-HMM
Beta process autoregressive HMM (BP-AR-HMM)
Experiment Results
Conclusion

3. Introduction
Problem: To discover and model common behaviors in related time series.
Jointly model multiple time series
Transfer knowledge between related time series
Uncover interesting relationships
Standard way to represent the set of behaviors time series exhibit via associated features:
Binary matrix
Time Series
Dynamic Behavior
If implies that time series i exhibit feature k

4. picture is adopted from Emily Fox’s paper in ICML 2008
Sticky HDP-HMM
HDP-HMM assumes all time series exhibit the same behaviors in same way
picture is adopted from Emily Fox’s paper in ICML 2008

5. BP-AR-HMM Model Beta process prior
masses
The beta process distributed measure is
represented by its masses and locations as
Beta process prior
Shared set of features or behaviors

6. BP-AR-HMM Model Features indicating behaviors of object i
Transition patterns for object i

7. BP-AR-HMM Model Latent dynamical mode for object i
Switching Vector Autoregressive (VAR) Process with order r

8. BP-AR-HMM Inference Metropolis-Hastings binary feature sampling
“Birth and dead” reversible jump MCMC (RJMCMC)
Proposal Distribution:
is defined similarly, but using
where

9. BP-AR-HMM Inference MH acceptance probability is where
MH sampling is also used for updating and

10. Synthetic Experiments
Five time series are generated from
with

11. Motion Capture (MoCap) Experiments
6 CMU MoCap exercise routines
Each routine includes some of the following 12 motions:
running in place, jumping jacks, arm circles, side twists, knee raises, squats, punching, up and down, two variants of toe touches, arch over, and a reach out stretch
Observation: 62-dimension
25 chains of the sampler for 15,000 burn-in and 5000 collections
Chosen MCMC sample minimizes an expected Hamming distance criterion

12. MoCap Results
Share motions: Jumping jacks, Side twists, Arm circles, Squats
Unique motion: Bend Over, Punching, Toe Touch 1, Toe Touch 2
Split motions: knee raises, up and down, running in place

13. Comparison with Other Models

14. Conclusion
Bayesian nonparametric framework for discovering dynamical features common to multiple time series
Novel exact sampling algorithm for non-conjugate beta process models