Problem Formulation Continuous-Discrete SIR Example Problem Conclusion On Sequential Monte Carlo Sampling of Discretely Observed Stochastic Differential Equations Simo Särkkä <simo.sarkka@hut.fi> Laboratory of Computational Engineering Helsinki University of Technology Finland September 15, 2006 Simo Särkkä <simo.sarkka@hut.fi>
Problem Formulation Continuous-Discrete SIR Example Problem Conclusion Contents Problem Formulation 1 Continuous-Discrete SIR 2 3 Example Problem Conclusion 4 Simo Särkkä <simo.sarkka@hut.fi>
Problem Formulation Continuous-Discrete SIR Example Problem Conclusion Contents Problem Formulation 1 Continuous-Discrete SIR 2 3 Example Problem Conclusion 4 Simo Särkkä <simo.sarkka@hut.fi>
Problem Formulation Continuous-Discrete SIR Example Problem Conclusion Contents Problem Formulation 1 Continuous-Discrete SIR 2 3 Example Problem Conclusion 4 Simo Särkkä <simo.sarkka@hut.fi>
Problem Formulation Continuous-Discrete SIR Example Problem Conclusion Contents Problem Formulation 1 Continuous-Discrete SIR 2 3 Example Problem Conclusion 4 Simo Särkkä <simo.sarkka@hut.fi>
Problem Formulation Continuous-Discrete SIR Example Problem Conclusion Continuous-Discrete Filtering Problem Estimate the unobserved continuous-time signal from noisy discrete-time measurements Signal 0.25 Signal derivative Measurement 0.2 0.15 0.1 0.05 0 −0.05 −0.1 −0.15 −0.2 −0.25 0 2 4 6 8 10 12 14 16 Time Simo Särkkä <simo.sarkka@hut.fi>
Problem Formulation Continuous-Discrete SIR Example Problem Conclusion Mathematical Problem Formulation The dynamics of state x ( t ) modeled as a stochastic differential equation (diffusion process) d x = f ( x , t ) d t + L d β ( t ) . Measurements y k are obtained at discrete times y k ∼ p ( y k | x ( t k )) . Formal solution: Compute the posterior distribution(s) p ( x ( t ) | y 1 , . . . , y k ) , t ≥ t k . Simo Särkkä <simo.sarkka@hut.fi>
Problem Formulation Continuous-Discrete SIR Example Problem Conclusion Mathematical Problem Formulation The dynamics of state x ( t ) modeled as a stochastic differential equation (diffusion process) d x = f ( x , t ) d t + L d β ( t ) . Measurements y k are obtained at discrete times y k ∼ p ( y k | x ( t k )) . Formal solution: Compute the posterior distribution(s) p ( x ( t ) | y 1 , . . . , y k ) , t ≥ t k . Simo Särkkä <simo.sarkka@hut.fi>
Problem Formulation Continuous-Discrete SIR Example Problem Conclusion Mathematical Problem Formulation The dynamics of state x ( t ) modeled as a stochastic differential equation (diffusion process) d x = f ( x , t ) d t + L d β ( t ) . Measurements y k are obtained at discrete times y k ∼ p ( y k | x ( t k )) . Formal solution: Compute the posterior distribution(s) p ( x ( t ) | y 1 , . . . , y k ) , t ≥ t k . Simo Särkkä <simo.sarkka@hut.fi>
Problem Formulation Continuous-Discrete SIR Example Problem Conclusion Formal solution Optimal filter Prediction step: Solve the Kolmogorov-forward 1 (Fokker-Planck) partial differential equation. ∂ 2 ∂ p ∂ ( f i ( x , t ) p ) + 1 � � � � [ L Q L T ] ij p ∂ t = − ∂ x i 2 ∂ x i ∂ x j i ij Update step: Apply the Bayes’ rule. 2 p ( y k | x ( t k )) p ( x ( t k ) | y 1 : k − 1 ) p ( x ( t k ) | y 1 : k ) = � p ( y k | x ( t k )) p ( x ( t k ) | y 1 : k − 1 ) d x ( t k ) Simo Särkkä <simo.sarkka@hut.fi>
Problem Formulation Continuous-Discrete SIR Example Problem Conclusion Formal solution Optimal filter Prediction step: Solve the Kolmogorov-forward 1 (Fokker-Planck) partial differential equation. ∂ 2 ∂ p ∂ ( f i ( x , t ) p ) + 1 � � � � [ L Q L T ] ij p ∂ t = − ∂ x i 2 ∂ x i ∂ x j i ij Update step: Apply the Bayes’ rule. 2 p ( y k | x ( t k )) p ( x ( t k ) | y 1 : k − 1 ) p ( x ( t k ) | y 1 : k ) = � p ( y k | x ( t k )) p ( x ( t k ) | y 1 : k − 1 ) d x ( t k ) Simo Särkkä <simo.sarkka@hut.fi>
Problem Formulation Continuous-Discrete SIR Example Problem Conclusion Numerical Approximations Gaussian models and approximations, extended Kalman filters and unscented Kalman filters. FEM and finite difference based approximations to the Kolmogorov forward PDE. Bootstrap filter: Simulates trajectories from the SDE. Sequential importance resampling: Not applicable! Simo Särkkä <simo.sarkka@hut.fi>
Problem Formulation Continuous-Discrete SIR Example Problem Conclusion Numerical Approximations Gaussian models and approximations, extended Kalman filters and unscented Kalman filters. FEM and finite difference based approximations to the Kolmogorov forward PDE. Bootstrap filter: Simulates trajectories from the SDE. Sequential importance resampling: Not applicable! Simo Särkkä <simo.sarkka@hut.fi>
Problem Formulation Continuous-Discrete SIR Example Problem Conclusion Numerical Approximations Gaussian models and approximations, extended Kalman filters and unscented Kalman filters. FEM and finite difference based approximations to the Kolmogorov forward PDE. Bootstrap filter: Simulates trajectories from the SDE. Sequential importance resampling: Not applicable! Simo Särkkä <simo.sarkka@hut.fi>
Problem Formulation Continuous-Discrete SIR Example Problem Conclusion Numerical Approximations Gaussian models and approximations, extended Kalman filters and unscented Kalman filters. FEM and finite difference based approximations to the Kolmogorov forward PDE. Bootstrap filter: Simulates trajectories from the SDE. Sequential importance resampling: Not applicable! Simo Särkkä <simo.sarkka@hut.fi>
Problem Formulation Continuous-Discrete SIR Example Problem Conclusion Sequential Importance Resampling Sequential Importance Resampling Draw a random sample from the importance distribution 1 x ( i ) ( t k ) ∼ q ( x ( i ) ( t k ) | x ( i ) ( t k − 1 )) (1) Evaluate the importance weight 2 ∝ p ( y k | x ( i ) ( t k )) p ( x ( i ) ( t k ) | x ( i ) ( t k − 1 )) w ( i ) k q ( x ( i ) ( t k ) | x ( i ) ( t k − 1 )) Do resampling if needed. 3 Simo Särkkä <simo.sarkka@hut.fi>
Problem Formulation Continuous-Discrete SIR Example Problem Conclusion Sequential Importance Resampling Sequential Importance Resampling Draw a random sample from the importance distribution 1 x ( i ) ( t k ) ∼ q ( x ( i ) ( t k ) | x ( i ) ( t k − 1 )) (1) Evaluate the importance weight 2 ∝ p ( y k | x ( i ) ( t k )) p ( x ( i ) ( t k ) | x ( i ) ( t k − 1 )) w ( i ) k q ( x ( i ) ( t k ) | x ( i ) ( t k − 1 )) Do resampling if needed. 3 Simo Särkkä <simo.sarkka@hut.fi>
Problem Formulation Continuous-Discrete SIR Example Problem Conclusion Sequential Importance Resampling Sequential Importance Resampling Draw a random sample from the importance distribution 1 x ( i ) ( t k ) ∼ q ( x ( i ) ( t k ) | x ( i ) ( t k − 1 )) (1) Evaluate the importance weight 2 ∝ p ( y k | x ( i ) ( t k )) p ( x ( i ) ( t k ) | x ( i ) ( t k − 1 )) w ( i ) k q ( x ( i ) ( t k ) | x ( i ) ( t k − 1 )) Do resampling if needed. 3 Simo Särkkä <simo.sarkka@hut.fi>
Problem Formulation Continuous-Discrete SIR Example Problem Conclusion The Problem of SIR Weight Evaluation The weight evaluation of SIR is of the form ∝ p ( y k | x ( i ) ( t k )) p ( x ( i ) ( t k ) | x ( i ) ( t k − 1 )) w ( i ) k q ( x ( i ) ( t k ) | x ( i ) ( t k − 1 )) But p ( x ( t k ) | x ( t k − 1 )) is the solution of an arbitrary second order partial differential equation and cannot be solved. Actually we only need the likelihood ratio p ( x ( t k ) | x ( t k − 1 )) q ( x ( t k ) | x ( t k − 1 )) This can be computed with the Girsanov theorem without solving the PDE. Simo Särkkä <simo.sarkka@hut.fi>
Problem Formulation Continuous-Discrete SIR Example Problem Conclusion The Problem of SIR Weight Evaluation The weight evaluation of SIR is of the form ∝ p ( y k | x ( i ) ( t k )) p ( x ( i ) ( t k ) | x ( i ) ( t k − 1 )) w ( i ) k q ( x ( i ) ( t k ) | x ( i ) ( t k − 1 )) But p ( x ( t k ) | x ( t k − 1 )) is the solution of an arbitrary second order partial differential equation and cannot be solved. Actually we only need the likelihood ratio p ( x ( t k ) | x ( t k − 1 )) q ( x ( t k ) | x ( t k − 1 )) This can be computed with the Girsanov theorem without solving the PDE. Simo Särkkä <simo.sarkka@hut.fi>
Problem Formulation Continuous-Discrete SIR Example Problem Conclusion The Problem of SIR Weight Evaluation The weight evaluation of SIR is of the form ∝ p ( y k | x ( i ) ( t k )) p ( x ( i ) ( t k ) | x ( i ) ( t k − 1 )) w ( i ) k q ( x ( i ) ( t k ) | x ( i ) ( t k − 1 )) But p ( x ( t k ) | x ( t k − 1 )) is the solution of an arbitrary second order partial differential equation and cannot be solved. Actually we only need the likelihood ratio p ( x ( t k ) | x ( t k − 1 )) q ( x ( t k ) | x ( t k − 1 )) This can be computed with the Girsanov theorem without solving the PDE. Simo Särkkä <simo.sarkka@hut.fi>
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