Continuous-Discrete Filtering using the Zakai Equation: Smooth Likelihood Surface Hermann Singer Department of Economics FernUniversität in Hagen Statistische Woche Trier September 2019 October 16, 2019 1/21
Continuous Time State Space Model dY ( t ) = f ( Y ( t ) , t ) dt + G ( Y ( t ) , t ) dW ( t ) dZ ( t ) = h ( Y ( t ) , t ) dt + dV ( t ) sampled measurements: = h ( Y ( t i ) , t i ) + ǫ i z i Goal: Optimal Filtering and Maximum Likelihood Estimation Wiener process W ( t ) Itô stochastic differential equations sampled measurements ˙ Z ( t i ) = z i , ρ ( t ) / dt = R ( t ) 2/21
(some) Solution Methods Kalman Filtering (sequential) Moment based methods Taylor expansion: EKF, SNF, HNF Numerical integration: UKF, GHF, Smolyak sparse grid PDE based methods: Stratonovich-Kushner and Duncan-Mortensen-Zakai (DMZ) equation Exact filters: Daum, Benes Particle Filters: Sequential Monte Carlo Markov Chain Monte Carlo (nonsequential) Simulated likelihood Bayesian approaches 3/21
Nonlinear bistable diffusion: Ginzburg-Landau model − [ α Y + β Y 3 ] dt + σ dW ( t ) dY = = −∇ Φ( Y ) + σ dW ( t ) z i = Y ( t i ) + ǫ i � � � � � � � � - � - � - � - � - � - � � ��� ���� ���� ���� � ��� ���� ���� ���� Figure: Simulated data (left) and extended Kalman filter (right). 4/21
Likelihood surface: Particle filter and Gauß-Hermite filter � � � � - � - � - � � ��� ���� ���� ���� Likelihood surface particle filter { sample size } = { 1000 } { ���� ���� ��������� ��������� ���� ���� ������������ ������ } ����� - �� ��� ��� - �� ��� - �� ��� - �� ��� - �� - ��� - �� - ��� - ���� ���� ���� ���� - ���� ���� ���� ���� Figure: SIR particle filter (mean and SD) and trajectories (top, right), likelihood and score for β (bottom). Increment d β = 0 . 0025. 5/21
Economic example: Equilibrium model: Herings (1996) 2 y 2 + β Potential Φ( y ) ∼ α 4 y 4 6/21
State estimation: Continuous-discrete Kalman Filter time update: t i ≤ t < t i + 1 (Fokker-Planck equation) ∂ t p ( y , t | Z i ) F ( y , t ) p ( y , t | Z i ) = measurement update: t = t i + 1 (Bayes formula) p ( z i + 1 , t i + 1 | y i + 1 , Z i ) p ( y i + 1 , t i + 1 | Z i ) p ( y i + 1 , t i + 1 | z i + 1 , Z i ) = p ( z i + 1 , t i + 1 | Z i ) filter density p ( y , t | Z t ) Fokker-Planck operator F ( y , t ) = − ∂ α f α + 1 2 ∂ α ∂ β Ω αβ 7/21
Recursive likelihood � p ( z i + 1 | Z i ; ψ ) p ( z i + 1 | y i + 1 , Z i ) p ( y i + 1 | Z i ) dy i + 1 = � u ( y i + 1 | z i + 1 , Z i ) dy i + 1 := � w l u ( y i + 1 , l | z i + 1 , Z i ) ≈ l unnormalized filter density u ( y , t | Z t ) numerical integration using quadrature formulas measurements up to time t i : Z i = { Z ( s ) | s ≤ t i } 8/21
Continuous time filtering: DMZ equation SPDE: Zakai (1969) ∂ t u ( y , t | Z t ) [ F + h ′ ρ − 1 ( ˙ Z − h / 2 )] ◦ u ( y , t | Z t ) = [ F ( y , t ) + M ( y , t )] ◦ u ( y , t | Z t ) = measurement precision ρ − 1 ( y , t ) = � i π ( t − t i )( R i dt ) − 1 measurement density p ( dZ ( t ) | y , Z t ) ∝ exp − 1 � 2 ( dZ − hdt ) ′ ( ρ dt ) − 1 ( dZ − hdt ) � dZ ◦ u : symmetrized product 9/21
Stochastic Representation: Feynman-Kac Formula � t � � t 0 M ( Y ( τ ) ,τ ) d τ δ ( y − Y ( t )) � Z t � u ( y , t | Z t ) = E e � use Lie -Trotter and Zassenhaus formula e ( F + M ) δ t δ ( y − y ′ ) ≈ e M δ t e F δ t δ ( y − y ′ ) = e M δ t p ( y , t + δ t | y ′ , t ) dY ( t ) = f ( Y , t ) dt + G ( Y , t ) dW ( t ) , Y ( t 0 ) ∼ p ( y , t 0 | Z t 0 ) � Dirac delta function lim n →∞ δ n ( x ) φ ( x ) dx = φ ( 0 ) 10/21
Importance Sampling: Backward DMZ Equation time reversal c ( x , s ) = u ( x , T − s ) , s ≤ T ∂ s c + Lc + ( M + v ) c = 0 terminal condition c ( x , T ) = h ( x ) = u ( x , 0 ) � T � � � X ( s ) = x , Z T − s � s ( M + v )( X ( τ ) ,τ ) d τ h ( X ( T )) c ( x , s ) = E e � dX ( τ ) = ˜ f ( X , T − τ ) dt + G ( X , T − τ ) dW ( τ ) , X ( s ) = x backward operator L = [ − f α + ( ∂ β Ω αβ )] ∂ α + 1 2 Ω αβ ∂ α ∂ β scalar potential v = − ( ∂ α f α ) + 1 2 ( ∂ α ∂ β Ω αβ ) 11/21
Simulation of Backward DMZ Equation Stochastic representation � T � � � s ( M + v )( X ( τ ) ,τ ) d τ h ( X ( T )) c ( x , s ) = � X ( s ) = x E e � ˜ dX ( τ ) = f ( X , T − τ ) dt + G ( X , T − τ ) dW ( τ ) X ( s ) = x Importance sampling: drift correction (Milstein; 1995) Ω( X , T − τ ) ∇ log u ( X , T − τ ) approximate filter solution ˆ u ( X , T − τ ) : EKF, GHF, UKF or particle filter 12/21
Ginzburg-Landau Model: Forward and backward simulation ������� ���������� � { ���� �������� ���� _ ����� = �������� � * ����� = ���� { ������ ���� }={ ���� ��� } � ������� = ����������� ���� _ ��� = ������� �� = ����� ��� } � ���� � � ���� - � ���� - � - � ���� � �� ��� ��� ��� - � - � - � � � � � �������� ���������� �������� ���������� � � � � � � � � - � - � - � - � - � - � � �� ��� ��� ��� � �� ��� ��� ��� Figure: Estimated filter density (top, right), backward simulation (bottom) and forward simulation (top, left) 13/21
Ginzburg-Landau Model: Forward and backward simulation ������� ���������� � { ���� �������� ���� _ ����� = �������� � * ����� = ���� { ������ ���� }={ ���� �� } � ������� = ����������� ���� _ ��� = �������� �� = ����� ��� } ��� � � ��� � ��� - � ��� - � - � ��� � ��� ��� ��� ��� - � - � - � � � � � �������� ���������� �������� ���������� � � � � � � � � - � - � - � - � - � - � � ��� ��� ��� ��� � ��� ��� ��� ��� Figure: Estimated filter density (top, right), backward simulation (bottom) and forward simulation (top, left) 14/21
Likelihood surface: Particle filter and Zakai Equation Likelihood surface particle filter { sample size } = { 1000 } { ���� ���� ��������� ��������� ���� ���� ������������ ������ } ����� - �� ��� ��� - �� ��� - �� ��� - �� ��� - �� - ��� - �� - ��� - ���� ���� ���� ���� - ���� ���� ���� ���� Likelihood surface Zakai filter { sample size,beta,alpha,dx } = { 20, 1.5, 1.5, { 0.25 }} { ���� ���� ��� } ����� ��� - �� ��� - �� ��� - �� ��� - �� ��� - �� - ��� - �� - ��� - ���� ���� ���� ���� - ���� ���� ���� ���� Figure: Likelihood for SIR particle filter (top) and ZKF (Riemann), GHF, TKF (Riemann). Increment d β = 0 . 0025. 15/21
Likelihood surface: Zakai Equation (UT) Likelihood surface Zakai filter { sample size,beta,alpha,method } = { 20, 1.5, 1.5, { UT, 2, _}} { ���� ���� ��� } ����� - �� ��� ��� - �� ��� - �� ��� - �� ��� - �� - ��� - �� - ��� - ���� ���� ���� ���� - ���� ���� ���� ���� Likelihood surface Zakai filter { sample size,beta,alpha,method } = { 100, 1.5, 1.5, { UT, 2, _}} { ���� ���� ��� } ����� - �� ��� ��� - �� ��� - �� ��� - �� ��� - �� - ��� - �� - ��� - ���� ���� ���� ���� - ���� ���� ���� ���� Figure: Likelihood for ZKF (unscented transform UT), GHF, TKF. N = 20 , 100. Increment d β = 0 . 0025. 16/21
Conclusions Continuous-discrete filtering with continuous time measurement equation Feynman-Kac representation of backward Zakai equation Variance reduced simulation of unnormalized filter density at supporting points No resampling required Smooth likelihood approximation using quadrature formulas at supporting points 17/21
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