Nonparametric Bayesian Learning of Switching Dynamical Processes

1
Nonparametric Bayesian Learning of Switching
Dynamical Processes

• Emily Fox, Erik Sudderth, Michael Jordan, and
  Alan Willsky
• Nonparametric Bayes Workshop 2008
• Helsinki, Finland

2
Applications
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Priors on Modes

• Switching linear dynamical processes useful for
  describing nonlinear phenomena
• Goal allow uncertainty in number of dynamical
  modes
• Utilize hierarchical Dirichlet process (HDP)
  prior
• Cluster based on dynamics

Switching Dynamical Processes
4
Outline

• Background
• Switching dynamical processes SLDS, VAR
• Prior on dynamic parameters
• Sticky HDP-HMM
• HDP-AR-HMM and HDP-SLDS
• Sampling Techniques
• Results
• Synthetic
• IBOVESPA stock index
• Dancing honey bee

5
Linear Dynamical Systems

• State space LTI model

6
Linear Dynamical Systems

• State space LTI model

• Vector autoregressive (VAR) process

7
Switching Dynamical Systems

• Switching linear dynamical system (SLDS)

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Prior on Dynamic Parameters
Rewrite VAR process in matrix form
Results in K decoupled linear regression problems
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Sticky HDP-HMM
Infinite HMM Beal, et.al., NIPS 2002HDP-HMM
Teh, et. al., JASA 2006 Sticky HDP-HMM Fox,
et.al., ICML 2008
Time

• Dirichlet process (DP)
• Mode space of unbounded size
• Model complexity adapts to observations
• Hierarchical
• Ties mode transition distributions
• Shared sparsity
• Sticky self-transition bias parameter

Mode
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Sticky HDP-HMM

• Global transition distribution

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HDP-AR-HMM and HDP-SLDS
HDP-AR-HMM
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Blocked Gibbs Sampler

• Sample parameters
• Approximate HDP
• Truncate stick-breaking
• Weak limit approximation
• Sample transition distributions
• Sample dynamic parameters using state sequence as
  VAR(1) pseudo-observations

Fox, et.al., ICML 2008
13
Blocked Gibbs Sampler

• Sample mode sequence
• Use state sequence as pseudo-observations of an
  HMM
• Compute backwards messages
• Block sample as

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Blocked Gibbs Sampler

• Sample state sequence
• Equivalent to LDS with time-varying dynamic
  parameters
• Compute backwards messages (backwards information
  filter)
• Block sample as

All Gaussian distributions
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Hyperparameters

• Place priors on hyperparameters and learn them
  from data
• Weakly informative priors
• All results use the same settings

hyperparameters
can be set using the data
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Results Synthetic VAR(1)
5-mode VAR(1) data
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Results Synthetic AR(2)
3-mode AR(2) data
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Results Synthetic SLDS
3-mode SLDS data
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Results IBOVESPA
Daily Returns

• Data Sao Paolo stock index
• Goal detect changes in volatility
• Compare inferred change-points to 10 cited world
  events

Carvalho and Lopes, Comp. Stat. Data Anal., 2006
20
Results Dancing Honey Bee
Sequence 1
Sequence 2
Sequence 3
Sequence 4
Sequence 5
Sequence 6

• 6 bee dance sequences with expert labeled dances
• Turn right (green)
• Waggle (red)
• Turn left (blue)

Time
Oh et. al., IJCV, 2007
21
Movie Sequence 6
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Results Dancing Honey Bee

• Nonparametric approach
• Model HDP-VAR(1)-HMM
• Set hyperparameters
• Unsupervised training from each sequence
• Infer
• Number of modes
• Dynamic parameters
• Mode sequence

• Supervised Approach Oh07
• Model SLDS
• Set number of modes to 3
• Leave one out training fixed label sequences on
  5 of 6 sequences
• Data-driven MCMC
• Use learned cues (e.g., head angle) to propose
  mode sequences

Oh et. al., IJCV, 2007
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Results Dancing Honey Bee
Sequence 4
Sequence 5
Sequence 6
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Results Dancing Honey Bee
Sequence 1
Sequence 2
Sequence 3
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Conclusion

• Examined HDP as a prior for nonparametric
  Bayesian learning of SLDS and switching VAR
  processes.
• Presented efficient Gibbs sampler
• Demonstrated utility on simulated and real
  datasets