Feedback Particle Filter and its Applications to Neuroscience

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3rd IFAC Workshop on Distributed Estimation and Control in Networked Systems …

3rd IFAC Workshop on Distributed Estimation and Control in Networked Systems
Santa Barbara, Sep 14-15, 2012

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  • 1. Feedback Particle Filter and its Applications to Neuroscience 3rd IFAC Workshop on Distributed Estimation and Control in Networked Systems Santa Barbara, Sep 14-15, 2012 Prashant G. Mehta Department of Mechanical Science and Engineering and the Coordinated Science Laboratory University of Illinois at Urbana-Champaign Research supported by NSF and AFOSR
  • 2. Background Bayesian Inference/Filtering Mathematics of prediction: Bayes’ rule Signal (hidden): X X ∼ P(X), (prior, known) Solution Bayes’ rule: P(X|Y ) Posterior ∝ P(Y |X)P(X) Prior This talk is about implementing Bayes’ rule in dynamic, nonlinear, non-Gaussian settings! 2
  • 3. Background Bayesian Inference/Filtering Mathematics of prediction: Bayes’ rule Signal (hidden): X X ∼ P(X), (prior, known) Observation: Y (known) Solution Bayes’ rule: P(X|Y ) Posterior ∝ P(Y |X)P(X) Prior This talk is about implementing Bayes’ rule in dynamic, nonlinear, non-Gaussian settings! 2
  • 4. Background Bayesian Inference/Filtering Mathematics of prediction: Bayes’ rule Signal (hidden): X X ∼ P(X), (prior, known) Observation: Y (known) Observation model: P(Y |X) (known) Solution Bayes’ rule: P(X|Y ) Posterior ∝ P(Y |X)P(X) Prior This talk is about implementing Bayes’ rule in dynamic, nonlinear, non-Gaussian settings! 2
  • 5. Background Bayesian Inference/Filtering Mathematics of prediction: Bayes’ rule Signal (hidden): X X ∼ P(X), (prior, known) Observation: Y (known) Observation model: P(Y |X) (known) Problem: What is X ? Solution Bayes’ rule: P(X|Y ) Posterior ∝ P(Y |X)P(X) Prior This talk is about implementing Bayes’ rule in dynamic, nonlinear, non-Gaussian settings! 2
  • 6. Background Bayesian Inference/Filtering Mathematics of prediction: Bayes’ rule Signal (hidden): X X ∼ P(X), (prior, known) Observation: Y (known) Observation model: P(Y |X) (known) Problem: What is X ? Solution Bayes’ rule: P(X|Y ) Posterior ∝ P(Y |X)P(X) Prior This talk is about implementing Bayes’ rule in dynamic, nonlinear, non-Gaussian settings! 2
  • 7. Background Bayesian Inference/Filtering Mathematics of prediction: Bayes’ rule Signal (hidden): X X ∼ P(X), (prior, known) Observation: Y (known) Observation model: P(Y |X) (known) Problem: What is X ? Solution Bayes’ rule: P(X|Y ) Posterior ∝ P(Y |X)P(X) Prior This talk is about implementing Bayes’ rule in dynamic, nonlinear, non-Gaussian settings! 2
  • 8. Background Applications Engineering applications Filtering is important to: Air moving target indicator (AMTI) systems, Space situational awareness Remote sensing and surveillance: Air traffic management, weather surveillance, geophysical surveys Autonomous navigation & robotics: Simultaneous localization and map building (SLAM) 3
  • 9. Background Applications Engineering applications Filtering is important to: Air moving target indicator (AMTI) systems, Space situational awareness Remote sensing and surveillance: Air traffic management, weather surveillance, geophysical surveys Autonomous navigation & robotics: Simultaneous localization and map building (SLAM) 3
  • 10. Background Applications Engineering applications Filtering is important to: Air moving target indicator (AMTI) systems, Space situational awareness Remote sensing and surveillance: Air traffic management, weather surveillance, geophysical surveys Autonomous navigation & robotics: Simultaneous localization and map building (SLAM) 3
  • 11. Background Applications Engineering applications Filtering is important to: Air moving target indicator (AMTI) systems, Space situational awareness Remote sensing and surveillance: Air traffic management, weather surveillance, geophysical surveys Autonomous navigation & robotics: Simultaneous localization and map building (SLAM) 3
  • 12. Background Applications in Biology Bayesian model of sensory signal processing 4
  • 13. Background Applications in Biology Bayesian model of sensory signal processing 4
  • 14. Part I Theory: Nonlinear Filtering
  • 15. Nonlinear Filtering Nonlinear Filtering Mathematical Problem Signal model: dXt = a(Xt)dt + dBt, X0 ∼ p∗ 0(·) 6
  • 16. Nonlinear Filtering Nonlinear Filtering Mathematical Problem Signal model: dXt = a(Xt)dt + dBt, X0 ∼ p∗ 0(·) Observation model: dZt = h(Xt)dt + dWt 6
  • 17. Nonlinear Filtering Nonlinear Filtering Mathematical Problem Signal model: dXt = a(Xt)dt + dBt, X0 ∼ p∗ 0(·) Observation model: dZt = h(Xt)dt + dWt Problem: What is Xt ? given obs. till time t =: Zt 6
  • 18. Nonlinear Filtering Nonlinear Filtering Mathematical Problem Signal model: dXt = a(Xt)dt + dBt, X0 ∼ p∗ 0(·) Observation model: dZt = h(Xt)dt + dWt Problem: What is Xt ? given obs. till time t =: Zt Answer in terms of posterior: P(Xt|Zt) =: p∗ (x,t). 6
  • 19. Nonlinear Filtering Nonlinear Filtering Mathematical Problem Signal model: dXt = a(Xt)dt + dBt, X0 ∼ p∗ 0(·) Observation model: dZt = h(Xt)dt + dWt Problem: What is Xt ? given obs. till time t =: Zt Answer in terms of posterior: P(Xt|Zt) =: p∗ (x,t). 6
  • 20. Nonlinear Filtering Nonlinear Filtering Mathematical Problem Signal model: dXt = a(Xt)dt + dBt, X0 ∼ p∗ 0(·) Observation model: dZt = h(Xt)dt + dWt Problem: What is Xt ? given obs. till time t =: Zt Answer in terms of posterior: P(Xt|Zt) =: p∗ (x,t). Posterior is an information state P(Xt ∈ A|Zt) = A p∗ (x,t)dx E(Xt|Zt) = R xp∗ (x,t)dx 6
  • 21. Nonlinear Filtering Pretty Formulae in Mathematics More often than not, these are simply stated Euler’s identity eiπ = −1 Euler’s formula v −e +f = 2 Pythagoras theorem x2 +y2 = z2 Kenneth Chang “What Makes an Equation Beautiful” in The New York Times on October 24, 2004 7
  • 22. Nonlinear Filtering Kalman filter Solution in linear Gaussian settings dXt = αXt dt + dBt (1) dZt = γXt dt + dWt (2) [?] R. E. Kalman, Trans. ASME, Ser. D: J. Basic Eng.,1961 8
  • 23. Nonlinear Filtering Kalman filter Solution in linear Gaussian settings dXt = αXt dt + dBt (1) dZt = γXt dt + dWt (2) Kalman filter: p∗ = N( ˆXt,Σt) d ˆXt = α ˆXt dt + K(dZt −γ ˆXt dt) Update [?] R. E. Kalman, Trans. ASME, Ser. D: J. Basic Eng.,1961 8
  • 24. Nonlinear Filtering Kalman filter Solution in linear Gaussian settings dXt = αXt dt + dBt (1) dZt = γXt dt + dWt (2) Kalman filter: p∗ = N( ˆXt,Σt) d ˆXt = α ˆXt dt + K(dZt −γ ˆXt dt) Update Kalman Filter - + [?] R. E. Kalman, Trans. ASME, Ser. D: J. Basic Eng.,1961 8
  • 25. Nonlinear Filtering Kalman filter Solution in linear Gaussian settings dXt = αXt dt + dBt (1) dZt = γXt dt + dWt (2) Kalman filter: p∗ = N( ˆXt,Σt) d ˆXt = α ˆXt dt + K(dZt −γ ˆXt dt) Update Kalman Filter - + Kalman Filter Observation: dZt = γXt dt + dWt [?] R. E. Kalman, Trans. ASME, Ser. D: J. Basic Eng.,1961 8
  • 26. Nonlinear Filtering Kalman filter Solution in linear Gaussian settings dXt = αXt dt + dBt (1) dZt = γXt dt + dWt (2) Kalman filter: p∗ = N( ˆXt,Σt) d ˆXt = α ˆXt dt + K(dZt −γ ˆXt dt) Update Kalman Filter - + Kalman Filter Observation: dZt = γXt dt + dWt Prediction: dˆZt = γ ˆXt dt [?] R. E. Kalman, Trans. ASME, Ser. D: J. Basic Eng.,1961 8
  • 27. Nonlinear Filtering Kalman filter Solution in linear Gaussian settings dXt = αXt dt + dBt (1) dZt = γXt dt + dWt (2) Kalman filter: p∗ = N( ˆXt,Σt) d ˆXt = α ˆXt dt + K(dZt −γ ˆXt dt) Update Kalman Filter - + Kalman Filter Observation: dZt = γXt dt + dWt Prediction: dˆZt = γ ˆXt dt Innov. error: dIt = dZt − dˆZt = dZt −γ ˆXt dt [?] R. E. Kalman, Trans. ASME, Ser. D: J. Basic Eng.,1961 8
  • 28. Nonlinear Filtering Kalman filter Solution in linear Gaussian settings dXt = αXt dt + dBt (1) dZt = γXt dt + dWt (2) Kalman filter: p∗ = N( ˆXt,Σt) d ˆXt = α ˆXt dt + K(dZt −γ ˆXt dt) Update Kalman Filter - + Kalman Filter Observation: dZt = γXt dt + dWt Prediction: dˆZt = γ ˆXt dt Innov. error: dIt = dZt − dˆZt = dZt −γ ˆXt dt Control: dUt = K dIt [?] R. E. Kalman, Trans. ASME, Ser. D: J. Basic Eng.,1961 8
  • 29. Nonlinear Filtering Kalman filter Solution in linear Gaussian settings dXt = αXt dt + dBt (1) dZt = γXt dt + dWt (2) Kalman filter: p∗ = N( ˆXt,Σt) d ˆXt = α ˆXt dt + K(dZt −γ ˆXt dt) Update Kalman Filter - + Kalman Filter Observation: dZt = γXt dt + dWt Prediction: dˆZt = γ ˆXt dt Innov. error: dIt = dZt − dˆZt = dZt −γ ˆXt dt Control: dUt = K dIt Gain: Kalman gain [?] R. E. Kalman, Trans. ASME, Ser. D: J. Basic Eng.,1961 8
  • 30. Nonlinear Filtering Kalman filter d ˆXt = α ˆXt dt Prediction + K(dZt −γ ˆXt dt) Update This illustrates the key features of feedback control: 1 Use error to obtain control (dUt = K dIt) 2 Negative gain feedback serves to reduce error (K = γ σ2 W SNR Σt) Simple enough to be included in the first undergraduate course on control 9
  • 31. Nonlinear Filtering Kalman filter d ˆXt = α ˆXt dt Prediction + K(dZt −γ ˆXt dt) Update This illustrates the key features of feedback control: 1 Use error to obtain control (dUt = K dIt) 2 Negative gain feedback serves to reduce error (K = γ σ2 W SNR Σt) Simple enough to be included in the first undergraduate course on control 9
  • 32. Nonlinear Filtering Kalman filter d ˆXt = α ˆXt dt Prediction + K(dZt −γ ˆXt dt) Update Kalman Filter - + This illustrates the key features of feedback control: 1 Use error to obtain control (dUt = K dIt) 2 Negative gain feedback serves to reduce error (K = γ σ2 W SNR Σt) Simple enough to be included in the first undergraduate course on control 9
  • 33. Nonlinear Filtering Filtering Problem Nonlinear Model: Kushner-Stratonovich PDE Signal & Observations dXt = a(Xt)dt +σB dBt, (1) dZt = h(Xt)dt +σW dWt (2) Posterior distribution p∗ is a solution of a stochastic PDE: dp∗ = L † (p∗ )dt + 1 σ2 W (h − ˆh)(dZt − ˆhdt)p∗ where ˆh = E[h(Xt)|Zt] = h(x)p∗ (x,t)dx L † (p∗ ) = − ∂(p∗ ·a(x)) ∂x + 1 2 σ2 B ∂2p∗ ∂x2 No closed-form solution in general. Closure problem. [?] R. L. Stratonovich, SIAM Theory Probab. Appl., 1960. [?] H. J. Kushner, SIAM J. Control, 1964 10
  • 34. Nonlinear Filtering Filtering Problem Nonlinear Model: Kushner-Stratonovich PDE Signal & Observations dXt = a(Xt)dt +σB dBt, (1) dZt = h(Xt)dt +σW dWt (2) Posterior distribution p∗ is a solution of a stochastic PDE: dp∗ = L † (p∗ )dt + 1 σ2 W (h − ˆh)(dZt − ˆhdt)p∗ where ˆh = E[h(Xt)|Zt] = h(x)p∗ (x,t)dx L † (p∗ ) = − ∂(p∗ ·a(x)) ∂x + 1 2 σ2 B ∂2p∗ ∂x2 No closed-form solution in general. Closure problem. [?] R. L. Stratonovich, SIAM Theory Probab. Appl., 1960. [?] H. J. Kushner, SIAM J. Control, 1964 10
  • 35. Nonlinear Filtering Filtering Problem Nonlinear Model: Kushner-Stratonovich PDE Signal & Observations dXt = a(Xt)dt +σB dBt, (1) dZt = h(Xt)dt +σW dWt (2) Posterior distribution p∗ is a solution of a stochastic PDE: dp∗ = L † (p∗ )dt + 1 σ2 W (h − ˆh)(dZt − ˆhdt)p∗ where ˆh = E[h(Xt)|Zt] = h(x)p∗ (x,t)dx L † (p∗ ) = − ∂(p∗ ·a(x)) ∂x + 1 2 σ2 B ∂2p∗ ∂x2 No closed-form solution in general. Closure problem. [?] R. L. Stratonovich, SIAM Theory Probab. Appl., 1960. [?] H. J. Kushner, SIAM J. Control, 1964 10
  • 36. Nonlinear Filtering Particle Filter An algorithm to solve nonlinear filtering problem Approximate posterior in terms of particles p∗ (x,t) = 1 N N ∑ i=1 δXi t (x) Algorithm outline 1 Initialization at time 0: Xi 0 ∼ p∗ 0(·) 2 At each discrete time step: Importance sampling (Bayes update step) Resampling (for variance reduction) 11
  • 37. Nonlinear Filtering Particle Filter An algorithm to solve nonlinear filtering problem Approximate posterior in terms of particles p∗ (x,t) = 1 N N ∑ i=1 δXi t (x) Algorithm outline 1 Initialization at time 0: Xi 0 ∼ p∗ 0(·) 2 At each discrete time step: Importance sampling (Bayes update step) Resampling (for variance reduction) 11
  • 38. Nonlinear Filtering Particle Filter An algorithm to solve nonlinear filtering problem Approximate posterior in terms of particles p∗ (x,t) = 1 N N ∑ i=1 δXi t (x) Algorithm outline 1 Initialization at time 0: Xi 0 ∼ p∗ 0(·) 2 At each discrete time step: Importance sampling (Bayes update step) Resampling (for variance reduction) 11
  • 39. Nonlinear Filtering Particle Filter An algorithm to solve nonlinear filtering problem Approximate posterior in terms of particles p∗ (x,t) = 1 N N ∑ i=1 δXi t (x) Algorithm outline 1 Initialization at time 0: Xi 0 ∼ p∗ 0(·) 2 At each discrete time step: Importance sampling (Bayes update step) Resampling (for variance reduction) e.g. dZt = Xt dt + small noise 11
  • 40. Nonlinear Filtering Particle Filter An algorithm to solve nonlinear filtering problem Approximate posterior in terms of particles p∗ (x,t) = 1 N N ∑ i=1 δXi t (x) Algorithm outline 1 Initialization at time 0: Xi 0 ∼ p∗ 0(·) 2 At each discrete time step: Importance sampling (Bayes update step) Resampling (for variance reduction) e.g. dZt = Xt dt + small noise 11
  • 41. Nonlinear Filtering Particle Filter An algorithm to solve nonlinear filtering problem Approximate posterior in terms of particles p∗ (x,t) = 1 N N ∑ i=1 δXi t (x) Algorithm outline 1 Initialization at time 0: Xi 0 ∼ p∗ 0(·) 2 At each discrete time step: Importance sampling (Bayes update step) Resampling (for variance reduction) Innovation error, feedback?And most importantly, is this pretty? 11
  • 42. Control-Oriented Approach to Particle Filtering Research goal: Bringing pretty back! 10 2 10 3 10 −3 10 −2 10 −1 N (number of particles) Bootstrap (BPF) Feedback (FPF) MSE Control-Oriented Approach to Particle Filtering 12
  • 43. Control-Oriented Approach to Particle Filtering Feedback Particle Filter Signal & Observations dXt = a(Xt)dt +σB dBt (1) dZt = h(Xt)dt +σW dWt (2) Controlled system (N particles): dXi t = a(Xi t )dt +σB dBi t + dUi t mean field control , i = 1,...,N (3) {Bi t}N i=1 are ind. standard white noises. Objective: Choose control Ui t, as a function of history {Zs,Xi s : 0 ≤ s ≤ t}, such that the two posteriors coincide: x∈A p∗ (x,t) dx = P{Xt ∈ A | Zt} x∈A p(x,t) dx = P{Xi t ∈ A | Zt} Motivation: Work of Huang, Caines and Malhame on Mean-field games (IEEE TAC 2007) 13
  • 44. Control-Oriented Approach to Particle Filtering Feedback Particle Filter Signal & Observations dXt = a(Xt)dt +σB dBt (1) dZt = h(Xt)dt +σW dWt (2) Controlled system (N particles): dXi t = a(Xi t )dt +σB dBi t + dUi t mean field control , i = 1,...,N (3) {Bi t}N i=1 are ind. standard white noises. Objective: Choose control Ui t, as a function of history {Zs,Xi s : 0 ≤ s ≤ t}, such that the two posteriors coincide: x∈A p∗ (x,t) dx = P{Xt ∈ A | Zt} x∈A p(x,t) dx = P{Xi t ∈ A | Zt} Motivation: Work of Huang, Caines and Malhame on Mean-field games (IEEE TAC 2007) 13
  • 45. Control-Oriented Approach to Particle Filtering Feedback Particle Filter Signal & Observations dXt = a(Xt)dt +σB dBt (1) dZt = h(Xt)dt +σW dWt (2) Controlled system (N particles): dXi t = a(Xi t )dt +σB dBi t + dUi t mean field control , i = 1,...,N (3) {Bi t}N i=1 are ind. standard white noises. Objective: Choose control Ui t, as a function of history {Zs,Xi s : 0 ≤ s ≤ t}, such that the two posteriors coincide: x∈A p∗ (x,t) dx = P{Xt ∈ A | Zt} x∈A p(x,t) dx = P{Xi t ∈ A | Zt} Motivation: Work of Huang, Caines and Malhame on Mean-field games (IEEE TAC 2007) 13
  • 46. Control-Oriented Approach to Particle Filtering FPF Solution Linear model Controlled system: for i = 1,...N: dXi t = αXi t dt +σB dBi t Prediction + K dZt −γ Xi t + µt 2 dt Update (via mean field control) (3) Feedback Particle Filter - + 14
  • 47. Control-Oriented Approach to Particle Filtering FPF Solution Linear model Controlled system: for i = 1,...N: dXi t = αXi t dt +σB dBi t Prediction + K dZt −γ Xi t + µt 2 dt Update (via mean field control) (3) Feedback Particle Filter - + 14
  • 48. Control-Oriented Approach to Particle Filtering FPF Update Steps Linear model Feedback particle filter Kalman filter Observation: dZt = γXt dt +σW dWt dZt = γXt dt +σW dWt 15
  • 49. Control-Oriented Approach to Particle Filtering FPF Update Steps Linear model Feedback particle filter Kalman filter Observation: dZt = γXt dt +σW dWt dZt = γXt dt +σW dWt Prediction: dˆZi t = 1 2 γXi t +γµt dt dˆZt = γ ˆXt dt 15
  • 50. Control-Oriented Approach to Particle Filtering FPF Update Steps Linear model Feedback particle filter Kalman filter Observation: dZt = γXt dt +σW dWt dZt = γXt dt +σW dWt Prediction: dˆZi t = 1 2 γXi t +γµt dt dˆZt = γ ˆXt dt Innovation error: dIi t = dZt − dˆZi t dIt = dZt − dˆZt = dZt −γ Xi t +µt 2 dt = dZt −γ ˆXt dt 15
  • 51. Control-Oriented Approach to Particle Filtering FPF Update Steps Linear model Feedback particle filter Kalman filter Observation: dZt = γXt dt +σW dWt dZt = γXt dt +σW dWt Prediction: dˆZi t = 1 2 γXi t +γµt dt dˆZt = γ ˆXt dt Innovation error: dIi t = dZt − dˆZi t dIt = dZt − dˆZt = dZt −γ Xi t +µt 2 dt = dZt −γ ˆXt dt Control: dUi t = K dIi t dUt = K dIt 15
  • 52. Control-Oriented Approach to Particle Filtering FPF Update Steps Linear model Feedback particle filter Kalman filter Observation: dZt = γXt dt +σW dWt dZt = γXt dt +σW dWt Prediction: dˆZi t = 1 2 γXi t +γµt dt dˆZt = γ ˆXt dt Innovation error: dIi t = dZt − dˆZi t dIt = dZt − dˆZt = dZt −γ Xi t +µt 2 dt = dZt −γ ˆXt dt Control: dUi t = K dIi t dUt = K dIt Gain: K is the Kalman gain 15
  • 53. Control-Oriented Approach to Particle Filtering Linear Feedback Particle Filter Mean field model is the Kalman filter Feedback particle filter: dXi t = αXi t dt +σB dBi t +K dZt − γ 2 Xi t + 1 N N ∑ j=1 Xj t dt (3) Xi 0 ∼ p∗ (x,0) = N(µ(0),Σ(0)) Mean-field model: Kalman filter! Let p denote cond. dist. of Xi t given Zt. Then p = N(µt,Σt) where dµt = αµt dt + γΣt σ2 W (dZt −γµt dt) dΣt = 2αΣt +σ2 B − γ2Σ2 t σ2 W dt As N → ∞, the empirical distribution approximates the posterior p∗ 16
  • 54. Control-Oriented Approach to Particle Filtering Linear Feedback Particle Filter Mean field model is the Kalman filter Feedback particle filter: dXi t = αXi t dt +σB dBi t +K dZt − γ 2 Xi t + 1 N N ∑ j=1 Xj t dt (3) Xi 0 ∼ p∗ (x,0) = N(µ(0),Σ(0)) Mean-field model: Kalman filter! Let p denote cond. dist. of Xi t given Zt. Then p = N(µt,Σt) where dµt = αµt dt + γΣt σ2 W (dZt −γµt dt) dΣt = 2αΣt +σ2 B − γ2Σ2 t σ2 W dt As N → ∞, the empirical distribution approximates the posterior p∗ 16
  • 55. Control-Oriented Approach to Particle Filtering Linear Feedback Particle Filter Mean field model is the Kalman filter Feedback particle filter: dXi t = αXi t dt +σB dBi t +K dZt − γ 2 Xi t + 1 N N ∑ j=1 Xj t dt (3) Xi 0 ∼ p∗ (x,0) = N(µ(0),Σ(0)) Mean-field model: Kalman filter! Let p denote cond. dist. of Xi t given Zt. Then p = N(µt,Σt) where dµt = αµt dt + γΣt σ2 W (dZt −γµt dt) dΣt = 2αΣt +σ2 B − γ2Σ2 t σ2 W dt As N → ∞, the empirical distribution approximates the posterior p∗ 16
  • 56. Control-Oriented Approach to Particle Filtering Variance Reduction Filtering for simple linear model. Mean-square error: 1 T T 0 Σ (N) t −Σt Σt 2 dt 10 2 10 3 10 −3 10 −2 10 −1 N (number of particles) Bootstrap (BPF) Feedback (FPF) MSE 17
  • 57. Feedback Particle Filter Methodology: Variational Formulation How do we derive the feedback particle filter? Time-stepping procedure: Signal, observ. process: dXt = a(Xt)dt +σB dBt Ztn = h(Xtn )+Wtn Feedback Particle filter Filter: dXi t = a(Xi t )dt +σB dBi t Control: Xi tn = Xi t− n +u(Xi t− n ) control Conditional distributions: p∗ n(·): cond. pdf of Xt|Zt pn(·;u): cond. pdf of Xi t |Zt Variational problem: min u D (pn(u) p∗ n) As ∆t → 0: Optimal control, u = u◦ , yields the feedback particle filter, Nonlinear filter is the gradient flow and u◦ is the optimal transport. 18
  • 58. Feedback Particle Filter Methodology: Variational Formulation How do we derive the feedback particle filter? Time-stepping procedure: Signal, observ. process: dXt = a(Xt)dt +σB dBt Ztn = h(Xtn )+Wtn Feedback Particle filter Filter: dXi t = a(Xi t )dt +σB dBi t Control: Xi tn = Xi t− n +u(Xi t− n ) control Conditional distributions: p∗ n(·): cond. pdf of Xt|Zt pn(·;u): cond. pdf of Xi t |Zt Variational problem: min u D (pn(u) p∗ n) As ∆t → 0: Optimal control, u = u◦ , yields the feedback particle filter, Nonlinear filter is the gradient flow and u◦ is the optimal transport. 18
  • 59. Feedback Particle Filter Methodology: Variational Formulation How do we derive the feedback particle filter? Time-stepping procedure: Signal, observ. process: dXt = a(Xt)dt +σB dBt Ztn = h(Xtn )+Wtn Feedback Particle filter Filter: dXi t = a(Xi t )dt +σB dBi t Control: Xi tn = Xi t− n +u(Xi t− n ) control Conditional distributions: p∗ n(·): cond. pdf of Xt|Zt pn(·;u): cond. pdf of Xi t |Zt Variational problem: min u D (pn(u) p∗ n) As ∆t → 0: Optimal control, u = u◦ , yields the feedback particle filter, Nonlinear filter is the gradient flow and u◦ is the optimal transport. 18
  • 60. Feedback Particle Filter Methodology: Variational Formulation How do we derive the feedback particle filter? Time-stepping procedure: Signal, observ. process: dXt = a(Xt)dt +σB dBt Ztn = h(Xtn )+Wtn Feedback Particle filter Filter: dXi t = a(Xi t )dt +σB dBi t Control: Xi tn = Xi t− n +u(Xi t− n ) control Conditional distributions: p∗ n(·): cond. pdf of Xt|Zt pn(·;u): cond. pdf of Xi t |Zt Variational problem: min u D (pn(u) p∗ n) As ∆t → 0: Optimal control, u = u◦ , yields the feedback particle filter, Nonlinear filter is the gradient flow and u◦ is the optimal transport. 18
  • 61. Feedback Particle Filter Feedback Particle Filter Filtering in nonlinear non-Gaussian settings Signal model: dXt = a(Xt)dt + dBt, X0 ∼ p∗ 0(·) Observation model: dZt = h(Xt)dt + dWt FPF: dXi t = a(Xi t )dt + dBi t +K(Xi t )◦ dIi t Update Innovations: dIi t =:dZt − 1 2 (h(Xi t )+ ˆh)dt, with cond. mean ˆh = p,h . 19
  • 62. Feedback Particle Filter Feedback Particle Filter Filtering in nonlinear non-Gaussian settings Signal model: dXt = a(Xt)dt + dBt, X0 ∼ p∗ 0(·) Observation model: dZt = h(Xt)dt + dWt FPF: dXi t = a(Xi t )dt + dBi t +K(Xi t )◦ dIi t Update Innovations: dIi t =:dZt − 1 2 (h(Xi t )+ ˆh)dt, with cond. mean ˆh = p,h . 19
  • 63. Feedback Particle Filter Update Step How does feedback particle filter implement Bayes’ rule? Feedback particle filter Linear case Observation: dZt = h(Xt)dt + dWt dZt = γXt dt + dWt 20
  • 64. Feedback Particle Filter Update Step How does feedback particle filter implement Bayes’ rule? Feedback particle filter Linear case Observation: dZt = h(Xt)dt + dWt dZt = γXt dt + dWt Prediction: dˆZi t = h(Xi t )+ˆh 2 dt dˆZi t = γXi t +γµt 2 dt ˆh = 1 N ∑N i=1 h(Xi t ) 20
  • 65. Feedback Particle Filter Update Step How does feedback particle filter implement Bayes’ rule? Feedback particle filter Linear case Observation: dZt = h(Xt)dt + dWt dZt = γXt dt + dWt Prediction: dˆZi t = h(Xi t )+ˆh 2 dt dˆZi t = γXi t +γµt 2 dt ˆh = 1 N ∑N i=1 h(Xi t ) Innov. error: dIi t = dZt − dˆZi t dIi t = dZt − dˆZi t = dZt − h(Xi t )+ˆh 2 dt = dZt −γ Xi t +µt 2 dt 20
  • 66. Feedback Particle Filter Update Step How does feedback particle filter implement Bayes’ rule? Feedback particle filter Linear case Observation: dZt = h(Xt)dt + dWt dZt = γXt dt + dWt Prediction: dˆZi t = h(Xi t )+ˆh 2 dt dˆZi t = γXi t +γµt 2 dt ˆh = 1 N ∑N i=1 h(Xi t ) Innov. error: dIi t = dZt − dˆZi t dIi t = dZt − dˆZi t = dZt − h(Xi t )+ˆh 2 dt = dZt −γ Xi t +µt 2 dt Control: dUi t = K(Xi t )◦ dIi t dUi t = K(Xi t )◦ dIi t 20
  • 67. Feedback Particle Filter Update Step How does feedback particle filter implement Bayes’ rule? Feedback particle filter Linear case Observation: dZt = h(Xt)dt + dWt dZt = γXt dt + dWt Prediction: dˆZi t = h(Xi t )+ˆh 2 dt dˆZi t = γXi t +γµt 2 dt ˆh = 1 N ∑N i=1 h(Xi t ) Innov. error: dIi t = dZt − dˆZi t dIi t = dZt − dˆZi t = dZt − h(Xi t )+ˆh 2 dt = dZt −γ Xi t +µt 2 dt Control: dUi t = K(Xi t )◦ dIi t dUi t = K(Xi t )◦ dIi t Gain: K is a solution of a linear BVP K is the Kalman gain 20
  • 68. Feedback Particle Filter Boundary Value Problem Euler-Lagrange equation for the variational problem Multi-dimensional boundary value problem ∇·(Kp) = −(h − ˆh)p solved at each time-step. Linear case: Nonlinear case: 21
  • 69. Feedback Particle Filter Boundary Value Problem Euler-Lagrange equation for the variational problem Multi-dimensional boundary value problem ∇·(Kp) = −(h − ˆh)p solved at each time-step. Linear case: Nonlinear case: 21
  • 70. Feedback Particle Filter Boundary Value Problem Euler-Lagrange equation for the variational problem Multi-dimensional boundary value problem ∇·(Kp) = −(h − ˆh)p solved at each time-step. Linear case: Nonlinear case: 21
  • 71. Feedback Particle Filter Boundary Value Problem Euler-Lagrange equation for the variational problem Multi-dimensional boundary value problem ∇·(Kp) = −(h − ˆh)p solved at each time-step. Linear case: Nonlinear case: 21
  • 72. Feedback Particle Filter Boundary Value Problem Euler-Lagrange equation for the variational problem Multi-dimensional boundary value problem ∇·(Kp) = −(h − ˆh)p solved at each time-step. Linear case: Nonlinear case: 21
  • 73. Feedback Particle Filter Boundary Value Problem Euler-Lagrange equation for the variational problem Multi-dimensional boundary value problem ∇·(Kp) = −(h − ˆh)p solved at each time-step. Linear case: Nonlinear case: 21
  • 74. Feedback Particle Filter Boundary Value Problem Euler-Lagrange equation for the variational problem Multi-dimensional boundary value problem ∇·(Kp) = −(h − ˆh)p solved at each time-step. Linear case: Nonlinear case: 21
  • 75. Feedback Particle Filter Boundary Value Problem Euler-Lagrange equation for the variational problem Multi-dimensional boundary value problem ∇·(Kp) = −(h − ˆh)p solved at each time-step. Linear case: Nonlinear case: 21
  • 76. Feedback Particle Filter Boundary Value Problem Euler-Lagrange equation for the variational problem Multi-dimensional boundary value problem ∇·(Kp) = −(h − ˆh)p solved at each time-step. Linear case: Nonlinear case: 21
  • 77. Feedback Particle Filter Consistency Feedback particle filter is exact p∗ : conditional pdf of Xt given Zt, dp∗ = L † (p∗ )dt +(h − ˆh)(σ2 W )−1 (dZt − ˆhdt)p∗ p : conditional pdf of Xi t given Zt, dp = L † (p)dt − ∂ ∂x (Kp) dZt − ∂ ∂x (up) dt + σ2 W 2 ∂2 ∂x2 pK2 dt Consistency Theorem Consider the two evolution equations for p and p∗ . Provided the FPF is initialized with p(x,0) = p∗ (x,0), then p(x,t) = p∗ (x,t) for all t ≥ 0 22
  • 78. Feedback Particle Filter Consistency Feedback particle filter is exact p∗ : conditional pdf of Xt given Zt, dp∗ = L † (p∗ )dt +(h − ˆh)(σ2 W )−1 (dZt − ˆhdt)p∗ p : conditional pdf of Xi t given Zt, dp = L † (p)dt − ∂ ∂x (Kp) dZt − ∂ ∂x (up) dt + σ2 W 2 ∂2 ∂x2 pK2 dt Consistency Theorem Consider the two evolution equations for p and p∗ . Provided the FPF is initialized with p(x,0) = p∗ (x,0), then p(x,t) = p∗ (x,t) for all t ≥ 0 22
  • 79. Feedback Particle Filter Kalman Filter Kalman Filter - + Innovation Error: dIt = dZt −h( ˆX)dt Gain Function: K = Kalman Gain Feedback Particle Filter Feedback Particle Filter - + Innovation Error: dIi t = dZt − 1 2 h(Xi t )+ ˆht dt Gain Function: K is solution of a linear BVP. 23
  • 80. Feedback Particle Filter Kalman Filter Kalman Filter - + Innovation Error: dIt = dZt −h( ˆX)dt Gain Function: K = Kalman Gain Feedback Particle Filter Feedback Particle Filter - + Innovation Error: dIi t = dZt − 1 2 h(Xi t )+ ˆht dt Gain Function: K is solution of a linear BVP. 23
  • 81. Feedback Particle Filter Kalman Filter Kalman Filter - + Innovation Error: dIt = dZt −h( ˆX)dt Gain Function: K = Kalman Gain Feedback Particle Filter Feedback Particle Filter - + Innovation Error: dIi t = dZt − 1 2 h(Xi t )+ ˆht dt Gain Function: K is solution of a linear BVP. 23
  • 82. Feedback Particle Filter Kalman Filter Kalman Filter - + Innovation Error: dIt = dZt −h( ˆX)dt Gain Function: K = Kalman Gain Feedback Particle Filter Feedback Particle Filter - + Innovation Error: dIi t = dZt − 1 2 h(Xi t )+ ˆht dt Gain Function: K is solution of a linear BVP. 23
  • 83. Feedback Particle Filter Kalman Filter Kalman Filter - + Innovation Error: dIt = dZt −h( ˆX)dt Gain Function: K = Kalman Gain Feedback Particle Filter Feedback Particle Filter - + Innovation Error: dIi t = dZt − 1 2 h(Xi t )+ ˆht dt Gain Function: K is solution of a linear BVP. 23
  • 84. Feedback Particle Filter Kalman Filter Kalman Filter - + Innovation Error: dIt = dZt −h( ˆX)dt Gain Function: K = Kalman Gain Feedback Particle Filter Feedback Particle Filter - + Innovation Error: dIi t = dZt − 1 2 h(Xi t )+ ˆht dt Gain Function: K is solution of a linear BVP. 23
  • 85. Part II Neural Rhythms, Bayesian Inference
  • 86. Oscillators in Biology Normal Form Reduction Derivation of oscillator model C dV dt = −gT ·m2 ∞(V )·h ·(V −ET ) −gh ·r ·(V −Eh)−...... dh dt = h∞(V )−h τh(V ) dr dt = r∞(V )−r τr (V ) [?] J. Guckenheimer, J. Math. Biol., 1975; [?] J. Moehlis et al., Neural Computation, 2004 25
  • 87. Oscillators in Biology Normal Form Reduction Derivation of oscillator model C dV dt = −gT ·m2 ∞(V )·h ·(V −ET ) −gh ·r ·(V −Eh)−...... dh dt = h∞(V )−h τh(V ) dr dt = r∞(V )−r τr (V ) [?] J. Guckenheimer, J. Math. Biol., 1975; [?] J. Moehlis et al., Neural Computation, 2004 25
  • 88. Oscillators in Biology Normal Form Reduction Derivation of oscillator model C dV dt = −gT ·m2 ∞(V )·h ·(V −ET ) −gh ·r ·(V −Eh)−...... dh dt = h∞(V )−h τh(V ) dr dt = r∞(V )−r τr (V ) Normal form reduction −−−−−−−−−−−−→ dθi (t) = ωi dt +ui (t)·Φ(θi (t))dt [?] J. Guckenheimer, J. Math. Biol., 1975; [?] J. Moehlis et al., Neural Computation, 2004 25
  • 89. Oscillators in Biology Collective Dynamics of a Large Number of Oscillators Synchrony, Neural rhythms 26
  • 90. Oscillators in Biology Functional Role of Neural Rhythms Is synchronization useful? Does it have a functional role? Books/review papers: Buzsaki, Destexhe, Ermentrout, Izhikevich, Kopell, Trout and Whittington (2009), Llinas and Ribary (2001), Pareti and Palma (2004), Sejnowski and Paulsen (2006), Singer (1993)... Computations: Computing with intrinsic network states Destexhe and Contreras (2006); Izhikevich (2006); Zhang and Ballard (2001). Synaptic plasticity: Neurons that fire together wire together And several other hypotheses: Communication and information flow (Laughlin and Sejnowski); Binding by synchrony (Singer); Memory formation (Jutras and Fries); Probabilistic decision making (Wang); Stimulus competition and attention selection (Kopell); Sleep/wakefulness/disease (Steriade) 27
  • 91. Oscillators in Biology Prediction Brain as a reality emulator “[Prediction] is the primary function of the neocortex, and the foundation of intelligence. If we want to understand how your brain works, and how to build intelligent machines, we must understand the nature of these predictions and how the cortex makes them.” “The capacity to predict the outcome of future events – critical to successful movement – is, most likely, the ultimate and most common of all brain functions.” 28
  • 92. Oscillators in Biology Prediction Brain as a reality emulator “[Prediction] is the primary function of the neocortex, and the foundation of intelligence. If we want to understand how your brain works, and how to build intelligent machines, we must understand the nature of these predictions and how the cortex makes them.” “The capacity to predict the outcome of future events – critical to successful movement – is, most likely, the ultimate and most common of all brain functions.” 28
  • 93. Oscillators in Biology Prediction Brain as a reality emulator “[Prediction] is the primary function of the neocortex, and the foundation of intelligence. If we want to understand how your brain works, and how to build intelligent machines, we must understand the nature of these predictions and how the cortex makes them.” “The capacity to predict the outcome of future events – critical to successful movement – is, most likely, the ultimate and most common of all brain functions.” 28
  • 94. Oscillators in Biology Prediction Brain as a reality emulator “[Prediction] is the primary function of the neocortex, and the foundation of intelligence. If we want to understand how your brain works, and how to build intelligent machines, we must understand the nature of these predictions and how the cortex makes them.” “The capacity to predict the outcome of future events – critical to successful movement – is, most likely, the ultimate and most common of all brain functions.” 28
  • 95. Oscillators in Biology Filtering in Brain? Bayesian model of sensory signal processing Theory: Lee and Mumford, Hierarchical Bayesian inference Framework (2003) Rao; Rao and Ballard; Rao and Sejnowski. Predictive coding framework (2002) Dayan, Hinton, Neal and Zemel. The Helmholtz machine (1995) Ma, Beck, Latham and Pouget. Probabilistic population codes (2006) Kording and Wolpert. Bayesian decision theory (2006) And others: See Doya, Ishii, Pouget and Rao, Bayesian Brain, MIT Press (2007) Rao, Olshausen & Lewicki, Probabilistic Models of Brain, MIT Press (2002) 29
  • 96. Oscillators in Biology Filtering in Brain? Bayesian model of sensory signal processing Theory: Lee and Mumford, Hierarchical Bayesian inference Framework (2003) Rao; Rao and Ballard; Rao and Sejnowski. Predictive coding framework (2002) Dayan, Hinton, Neal and Zemel. The Helmholtz machine (1995) Ma, Beck, Latham and Pouget. Probabilistic population codes (2006) Kording and Wolpert. Bayesian decision theory (2006) And others: See Doya, Ishii, Pouget and Rao, Bayesian Brain, MIT Press (2007) Rao, Olshausen & Lewicki, Probabilistic Models of Brain, MIT Press (2002) 29
  • 97. Oscillators in Biology Filtering in Brain? Bayesian model of sensory signal processing Experiments (see reviews): Gold & Shadlen, The neural basis of decision making, Ann. Rev. of Neurosci. (2007) R. T. Knight, Neural networks debunk phrenology, Science (2007) Such theories naturally feed into computer vision & more generally on how to make computer “intelligent” 30
  • 98. Oscillators in Biology Filtering in Brain? Bayesian model of sensory signal processing Experiments (see reviews): Gold & Shadlen, The neural basis of decision making, Ann. Rev. of Neurosci. (2007) R. T. Knight, Neural networks debunk phrenology, Science (2007) Such theories naturally feed into computer vision & more generally on how to make computer “intelligent” 30
  • 99. Oscillators in Biology Bayesian Inference in Neuroscience Lee and Mumford’s hierarchical Bayesian inference framework . . . Bayes’rule Bayes’rule Bayes’rule Similar ideas also appear in: 1 Dayan, Hinton, Neal and Zemel. The Helmholtz machine (1995) 2 Lewicki and Sejnowski. Bayesian unsupervised learning (1995) 3 Rao and Ballard; Rao and Sejnowski. Predictive coding framework (1999;2002) 31
  • 100. Oscillators in Biology Bayesian Inference in Neuroscience Lee and Mumford’s hierarchical Bayesian inference framework . . . Bayes’rule Bayes’rule Bayes’rule . . . Part. Filter Part. Filter Part. Filter Similar ideas also appear in: 1 Dayan, Hinton, Neal and Zemel. The Helmholtz machine (1995) 2 Lewicki and Sejnowski. Bayesian unsupervised learning (1995) 3 Rao and Ballard; Rao and Sejnowski. Predictive coding framework (1999;2002) 31
  • 101. Part III Application: Filtering with Rhythms
  • 102. Gait Cycle Biological Rhythm 33
  • 103. Gait Cycle Biological Rhythm 33
  • 104. Gait Cycle Biological Rhythm 33
  • 105. Gait Cycle Biological Rhythm 33
  • 106. Gait Cycle Biological Rhythm 33
  • 107. Gait Cycle Biological Rhythm 33
  • 108. Gait Cycle Biological Rhythm 33
  • 109. Application: Ankle-foot Orthoses Estimation of gait cycle using sensor measurements Ankle-foot orthoses (AFOs) : For lower-limb neuromuscular impairments. Provides dorsiflexor (toe lift) and plantarflexor (toe push) torque assistance Sensors: heel, toe, and ankle joint Compressed CO2 Actuator Solenoid valves: control the flow of CO2 to the actuator AFO system components: Power supply, Valves, Actuator, Sensors. Professor Liz Hsiao-Wecksler Acknowledgement: Professor Liz Hsiao-Wecksler for sharing the AFO device picture and sensor data. 34
  • 110. Application: Ankle-foot Orthoses Estimation of gait cycle using sensor measurements Ankle-foot orthoses (AFOs) : For lower-limb neuromuscular impairments. Provides dorsiflexor (toe lift) and plantarflexor (toe push) torque assistance Sensors: heel, toe, and ankle joint Compressed CO2 Actuator Solenoid valves: control the flow of CO2 to the actuator AFO system components: Power supply, Valves, Actuator, Sensors. Professor Liz Hsiao-Wecksler Acknowledgement: Professor Liz Hsiao-Wecksler for sharing the AFO device picture and sensor data. 34
  • 111. Gait Cycle Signal model Stance phase Swing phase Model (Noisy oscillator) dθt = ω0 dt natural frequency + noise 35
  • 112. Gait Cycle Signal model Stance phase Swing phase Model (Noisy oscillator) dθt = ω0 dt natural frequency + noise 35
  • 113. Gait Cycle Signal model Stance phase Swing phase Model (Noisy oscillator) dθt = ω0 dt natural frequency + noise 35
  • 114. Gait Cycle Signal model Stance phase Swing phase Model (Noisy oscillator) dθt = ω0 dt natural frequency + noise 35
  • 115. Gait Cycle Signal model Stance phase Swing phase Model (Noisy oscillator) dθt = ω0 dt natural frequency + noise 35
  • 116. Gait Cycle Signal model Stance phase Swing phase Model (Noisy oscillator) dθt = ω0 dt natural frequency + noise 35
  • 117. Gait Cycle Signal model Stance phase Swing phase Model (Noisy oscillator) dθt = ω0 dt natural frequency + noise 35
  • 118. Gait Cycle Signal model Stance phase Swing phase Model (Noisy oscillator) dθt = ω0 dt natural frequency + noise 35
  • 119. Problem: Estimate Gait Cycle θt Sensor model Observation model: dZt = h(θt)dt+ noise Problem: What is θt given noisy observations? 36
  • 120. Problem: Estimate Gait Cycle θt Sensor model Observation model: dZt = h(θt)dt+ noise Problem: What is θt given noisy observations? 36
  • 121. Problem: Estimate Gait Cycle θt Sensor model Observation model: dZt = h(θt)dt+ noise Problem: What is θt given noisy observations? 36
  • 122. Problem: Estimate Gait Cycle θt Sensor model Observation model: dZt = h(θt)dt+ noise Problem: What is θt given noisy observations? 36
  • 123. Problem: Estimate Gait Cycle θt Sensor model Observation model: dZt = h(θt)dt+ noise Problem: What is θt given noisy observations? 36
  • 124. Solution: Particle Filter Algorithm to approximate posterior distribution “Large number of oscillators” Posterior distribution: P(φ1 < θt < φ2|Sensor readings) = Fraction of θi t in interval (φ1,φ2) Circuit: dθi t = ωi dt natural freq. of ith oscillator + noisei + dUi t mean-field control , i = 1,...,N Feedback Particle Filter: Design control law Ui t 37
  • 125. Solution: Particle Filter Algorithm to approximate posterior distribution “Large number of oscillators” Posterior distribution: P(φ1 < θt < φ2|Sensor readings) = Fraction of θi t in interval (φ1,φ2) Circuit: dθi t = ωi dt natural freq. of ith oscillator + noisei + dUi t mean-field control , i = 1,...,N Feedback Particle Filter: Design control law Ui t 37
  • 126. Solution: Particle Filter Algorithm to approximate posterior distribution “Large number of oscillators” Posterior distribution: P(φ1 < θt < φ2|Sensor readings) = Fraction of θi t in interval (φ1,φ2) Circuit: dθi t = ωi dt natural freq. of ith oscillator + noisei + dUi t mean-field control , i = 1,...,N Feedback Particle Filter: Design control law Ui t 37
  • 127. Solution: Particle Filter Algorithm to approximate posterior distribution “Large number of oscillators” Posterior distribution: P(φ1 < θt < φ2|Sensor readings) = Fraction of θi t in interval (φ1,φ2) Circuit: dθi t = ωi dt natural freq. of ith oscillator + noisei + dUi t mean-field control , i = 1,...,N Feedback Particle Filter: Design control law Ui t 37
  • 128. Solution: Particle Filter Algorithm to approximate posterior distribution “Large number of oscillators” Posterior distribution: P(φ1 < θt < φ2|Sensor readings) = Fraction of θi t in interval (φ1,φ2) Circuit: dθi t = ωi dt natural freq. of ith oscillator + noisei + dUi t mean-field control , i = 1,...,N Feedback Particle Filter: Design control law Ui t 37
  • 129. Solution: Particle Filter Algorithm to approximate posterior distribution “Large number of oscillators” Posterior distribution: P(φ1 < θt < φ2|Sensor readings) = Fraction of θi t in interval (φ1,φ2) Circuit: dθi t = ωi dt natural freq. of ith oscillator + noisei + dUi t mean-field control , i = 1,...,N Feedback Particle Filter: Design control law Ui t 37
  • 130. Filtering for Oscillators Signal & Observations dθt = ω dt + dBt mod 2π dZt = h(θt)dt + dWt − π 0 π Particle evolution, dθi t = ωi dt + dBi t +K(θi t )◦[dZt − 1 2 (h(θi t )+ ˆh)dt] mod 2π, i = 1,...,N. where ωi is sampled from a distribution. 38
  • 131. Filtering for Oscillators Signal & Observations dθt = ω dt + dBt mod 2π dZt = h(θt)dt + dWt − π 0 π Particle evolution, dθi t = ωi dt + dBi t +K(θi t )◦[dZt − 1 2 (h(θi t )+ ˆh)dt] mod 2π, i = 1,...,N. where ωi is sampled from a distribution. 38
  • 132. Filtering for Oscillators Signal & Observations dθt = ω dt + dBt mod 2π dZt = h(θt)dt + dWt − π 0 π Particle evolution, dθi t = ωi dt + dBi t +K(θi t )◦[dZt − 1 2 (h(θi t )+ ˆh)dt] mod 2π, i = 1,...,N. where ωi is sampled from a distribution. Feedback Particle Filter - + 38
  • 133. Simulation Results Solution of the Estimation of Gait Cycle Problem [Click to play the movie] 39
  • 134. Filtering of Biological Rhythms with Brain Rhythms Connection to Lee and Mumford’s hierarchical Bayesian inference framework . . . Part. Filter Part. Filter Part. Filter Prior Noisy input . . . Part. Filter Part. Filter 40
  • 135. Filtering of Biological Rhythms with Brain Rhythms Connection to Lee and Mumford’s hierarchical Bayesian inference framework . . . Part. Filter Part. Filter Part. Filter Prior Noisy input . . . Part. Filter Part. Filter Noisy measurements Rhythmic movement Prior Mumford’s box with neurons Normal form reduction Normal form reduction Estimate Mumford’s box with oscillators 40
  • 136. Filtering of Biological Rhythms with Brain Rhythms Connection to Lee and Mumford’s hierarchical Bayesian inference framework . . . Part. Filter Part. Filter Part. Filter Prior Noisy input . . . Part. Filter Part. Filter Noisy measurements Rhythmic movement Prior Mumford’s box with neurons Normal form reduction Normal form reduction Estimate Mumford’s box with oscillators 40
  • 137. Filtering of Biological Rhythms with Brain Rhythms Connection to Lee and Mumford’s hierarchical Bayesian inference framework . . . Part. Filter Part. Filter Part. Filter Prior Noisy input . . . Part. Filter Part. Filter Noisy measurements Rhythmic movement Prior Mumford’s box with neurons Normal form reduction Normal form reduction Estimate Mumford’s box with oscillators 40
  • 138. Acknowledgement Adam Tilton Tao Yang Huibing Yin Liz Hsiao-Wecksler Sean Meyn 1 T. Yang, P. G. Mehta, and S. P. Meyn. Feedback particle filter with mean-field coupling. In Procs. of IEEE Conf. on Decision and Control, December 2011. 2 T. Yang, P. G. Mehta, and S. P. Meyn. A mean-field control-oriented approach to particle filtering. In Procs. of American Control Conference, June 2011. 3 A. Tilton, E. Hsiao-Wecksler, P. G. Mehta. Filtering with rhythms: Application to estimation of gait cycle. In Procs. of American Control Conference, 2012. 4 T. Yang, G. Huang and P. G. Mehta. Joint probabilistic data association-feedback particle filter with applications to multiple target tracking. In Procs. of American Control Conference, 2012. 5 A. Tilton, T. Yang, H. Yin and P. G. Mehta. Feedback particle filter-based multiple target tracking using bearing-only measurements. In Procs. of Information Fusion, 2012. 6 T. Yang, R. Laugesen, P. G. Mehta, and S. P. Meyn. Multivariable feedback particle filter. To appear in IEEE Conf. on Decision and Control, 2012. 7 T. Yang, P. G. Mehta, and S. P. Meyn. Feedback particle filter. Conditionally accepted to IEEE Transactions on Automatic Control.