Simulated Maximum Likelihood for Continuous-Discrete State Space Models using Langevin Importance Sampling. Simulated Maximum Likelihood for Continuous-Discrete State Space Models using Langevin Importance Sampling. Hermann Singer State Space Models Parameter Estimation Langevin Sampling Simulated Likelihood Examples Hermann Singer FernUniversität in Hagen, [email protected] References presented at the 9th International Conference on Social Science Methodology (RC33) 11.-16. September 2016, Leicester, UK Simulated Maximum Likelihood for Continuous-Discrete State Space Models using Langevin Importance Sampling. Hermann Singer State Space Models Parameter Estimation Langevin Sampling Simulated Likelihood Examples Session: Estimation of Stochastic Differential Equations with Time Series, Panel and Spatial Data. References Nonlinear continuous/discrete state space model Simulated Maximum Likelihood for Continuous-Discrete State Space Models using Langevin Importance Sampling. Hermann Singer dY (t) = f (Y (t), t)dt + g (Y (t), t)dW (t) Zi = h(Yi , ti ) + i State Space Models Nonlinear continuous/discrete state space model Linear stochastic differential equations (LSDE) Parameter Estimation i = 0, ..., T Langevin Sampling Simulated Likelihood nonlinear drift and diffusion functions f , g f = f (Y (t), t, x(t), ψ) nonlinear diffusion g (Y ): Itô calculus Spatial models: Yn (t) = Y (xn , t), xn ∈ Rd : Random field Y (x, t, ω) Examples References Linear stochastic differential equations (LSDE) (exact ML, Singer; 1990, Kalman filter) Simulated Maximum Likelihood for Continuous-Discrete State Space Models using Langevin Importance Sampling. Hermann Singer dY (t) = AY (t)dt + GdW (t) Z t Y (t) = e A(t−t0 ) Y (t0 ) + e A(t−s) GdW (s) t0 State Space Models Nonlinear continuous/discrete state space model Linear stochastic differential equations (LSDE) Parameter Estimation W (t, ω): Wiener process: continuous time random walk (Brownian motion) Langevin Sampling Itô stochastic differential equations: dW /dt = ζ(t) = Gaussian white noise Examples A: drift matrix G : diffusion matrix Simulated Likelihood References Simulated Maximum Likelihood for Continuous-Discrete State Space Models using Langevin Importance Sampling. Wiener process and stock index DAX DAX-Zuwachs 8000 400 7000 6000 Out[467]= 200 5000 Out[454]= Hermann Singer 0 4000 -200 State Space Models 3000 Nonlinear continuous/discrete state space model Linear stochastic differential equations (LSDE) -400 2000 1995 2000 1995 2005 2000 2005 Figure : German stock index (DAX) Simulierter Wiener-Prozeß Parameter Estimation Langevin Sampling Zuwächse: Simulierter Wiener-Prozeß 0.2 3 Simulated Likelihood 2 Out[503]= Examples 0.1 1 Out[504]= References 0.0 0 -0.1 -1 0 1000 2000 3000 4000 5000 6000 7000 0 1000 2000 3000 4000 5000 Figure : Simulated Wiener process (random walk) 6000 7000 Simulated Maximum Likelihood for Continuous-Discrete State Space Models using Langevin Importance Sampling. Simulated Wiener processes Simulierte Wiener-Prozesse 4 3 0.5 0.0 2 3 1 2 Hermann Singer -0.5 -1.0 0 -1.5 -2.0 1 State Space Models -1 0 -2.5 0 1000 2000 3000 4000 5000 6000 7000 0 1000 2000 3000 4000 5000 6000 7000 0 5 7 4 6 1000 2000 3000 4000 5000 6000 7000 0 -1 5 3 Out[501]= 4 -2 Nonlinear continuous/discrete state space model Linear stochastic differential equations (LSDE) 2 3 -3 Parameter Estimation 1 2 -4 0 1000 2000 3000 4000 5000 6000 0 1 -1 0 7000 0 1000 2000 3000 4000 5000 6000 7000 0 1000 2000 3000 4000 5000 6000 7000 Langevin Sampling 3 2 Simulated Likelihood 1 2 1 Examples 0 1 0 -1 0 -1 References -2 -1 -2 -3 -2 0 1000 2000 3000 4000 5000 6000 7000 0 1000 2000 3000 4000 5000 6000 7000 0 1000 2000 Figure : Simulated Wiener processes 3000 4000 5000 6000 7000 Simulated Maximum Likelihood for Continuous-Discrete State Space Models using Langevin Importance Sampling. Exact discrete model (EDM) Bergstrom (1976, 1988) Hermann Singer Yi+1 = e A(ti +1 −ti ) Z Yi + ti +1 e A(ti +1 −s) GdW (s) ti restricted VAR(1) model State Space Models Nonlinear continuous/discrete state space model Linear stochastic differential equations (LSDE) Parameter Estimation Langevin Sampling Yi+1 = Φ(ti+1 , ti )Yi + ui Simulated Likelihood Examples References Φ: fundamental matrix of the system Yi := Y (ti ): sampled measurements dt = 0.5 Dt = 2 Simulated Maximum Likelihood for Continuous-Discrete State Space Models using Langevin Importance Sampling. Hermann Singer accumulated interaction Dt = 2 State Space Models Nonlinear continuous/discrete state space model Linear stochastic differential equations (LSDE) accumulated interaction Dt = 4 Parameter Estimation Langevin Sampling Simulated Likelihood Examples References Figure : 3 variable model: Product representation of matrix exponential within the measurement interval ∆t = 2. Latent states ηj , discretization interval δt = 2/4 = 0.5 (Singer; 2012) Maximum Likelihood Parameter Estimation likelihood function of all observations Simulated Maximum Likelihood for Continuous-Discrete State Space Models using Langevin Importance Sampling. Hermann Singer Z p(zT , ..., z0 ) = p(zT , ..., z0 |yT , ..., y0 )p(yT , ..., y0 )dy State Space Models Parameter Estimation Langevin Sampling High dimensional integration over latent variables smooth dependence on parameter vector ψ: L(ψ) = p(z; ψ) Problem: p(yT , ..., y0 ) not known Sampling interval ∆ti Transition density p(yi+1 , ∆ti |yi ) difficult to compute Use additional latent variables yT = ηJ , ..., η0 = y0 , y (ti ) = yi = ηji Simulated Likelihood Examples References Simulated Maximum Likelihood for Continuous-Discrete State Space Models using Langevin Importance Sampling. Integration latent states ηj Z p(zT , ..., z0 |ηJ , ..., η0 )p(ηJ , ..., η0 )d η h i = E p(zT , ..., z0 |ηJ , ..., η0 ) p(zT , ..., z0 ) = Hermann Singer State Space Models Parameter Estimation Langevin Sampling ηji = y (ti ) = yi , ti = measurement times even higher (infinite) dimensional integration over latent variables: Markov Chain Monte Carlo: simulate η Euler density (discretization interval δt) p(ηj+1 , δt|ηj+1 ) ≈ φ(ηj+1 ; ηj + fj δt, Ωj δt) fj = f (ηj , τj ), Ωj = (gg 0 )(ηj , τj ) Simulated Likelihood Examples References Simulated Maximum Likelihood for Continuous-Discrete State Space Models using Langevin Importance Sampling. Importance Sampling Z p(zT , ..., z0 ) = p(η) p(z|η) p2 (η)d η p2 (η) Hermann Singer State Space Models Parameter Estimation p2 : importance density Langevin Sampling Simulated Likelihood p2,optimal = p(z|η)p(η) p(z) = p(η|z) however, p(z) is unknown η → η(t, u): random field, u = simulation time Examples References Langevin Sampling Langevin (1908); Roberts and Stramer (2002) Langevin equation d η(u) = ∂η log p(η(u)|z)du + √ Simulated Maximum Likelihood for Continuous-Discrete State Space Models using Langevin Importance Sampling. Hermann Singer 2dW (u) State Space Models Parameter Estimation stationary distribution of conditional latent states pstat (η) = e −Φ(η) = p(η|z) drift = −gradient of potential Φ(η) −∂η Φ(η) = ∂η [log p(z|η) + log p(η) − log p(z)] sample from p(η|z), p(z) not needed! optimal nonlinear smoothing η(u) ∼ p(η|z) in equilibrium u → ∞ Langevin Sampling Simulated Likelihood Examples References Simulated Likelihood Variance reduced MC-integration p̂(zT , ..., z0 ) = L−1 X p(ηl ) p(z|ηl ) p2 (ηl ) ηl ∼ p(η|z) in equilibrium Simulated Maximum Likelihood for Continuous-Discrete State Space Models using Langevin Importance Sampling. Hermann Singer State Space Models Parameter Estimation Langevin Sampling Simulated Likelihood draw optimal ηl = η(ul ) from discretized Langevin equation (including Metropolis-Hastings mechanism) p2,optimal = p(z|η)p(η) p(z) = p(η|z) is unknown Idea: estimate p2 = p(η|z) from ηl ∼ p(η|z) Examples References Estimation of importance density use known (suboptimal) reference density p2 = p0 (η|z) = p0 (z|η)p0 (η)/p0 (z) Hermann Singer State Space Models Parameter Estimation kernel density estimate p̂(η|z) = L−1 Simulated Maximum Likelihood for Continuous-Discrete State Space Models using Langevin Importance Sampling. Langevin Sampling X k(η − ηl ; smooth) l Problem: – high dimensional state, – no structure imposed on p(η|z) Simulated Likelihood Examples References Estimation of importance density use Markov structure of state space model ηj+1 = f (ηj )δt + g (ηj )δWj zi Simulated Maximum Likelihood for Continuous-Discrete State Space Models using Langevin Importance Sampling. Hermann Singer = h(yi ) + i p(η|z) = p(ηJ |ηJ−1 , ..., η0 , z) ∗ p(ηJ−1 , ..., η0 |z) State Space Models Parameter Estimation Langevin Sampling Conditional Markov process Simulated Likelihood Examples p(ηj+1 |ηj , ..., η0 , z) = p(ηj+1 |ηj , z) References use conditional independence of past z i = (z0 , ..., zi ) and future z̄ i = (zi+1 , ...zT ) given η j : j i i p(ηj +1 |η , z , z̄ ) i i p(ηj +1 |ηj , z , z̄ ) ji ≤ j i i = p(ηj +1 |η , z̄ ) = p(ηj +1 |ηj , z̄ ) = p(ηj +1 |ηj , z̄ ) j < ji +1 i Euler transition kernel Euler density (discretization interval δt) Simulated Maximum Likelihood for Continuous-Discrete State Space Models using Langevin Importance Sampling. Hermann Singer p(ηj+1 , δt|ηj ) ≈ φ(ηj+1 ; ηj + fj δt, Ωj δt) modified drift and diffusion matrix conditional Euler density p(ηj+1 , δt|ηj , z) ≈ φ(ηj+1 ; ηj + (fj + δfj )δt, (Ωj + δΩj )δt) nonlinear regression for δfj and δΩj (parametric and nonparametric) draw data ηjl = η(τj , ul ) ∼ p(η|z) State Space Models Parameter Estimation Langevin Sampling Simulated Likelihood Examples References Kernel density Simulated Maximum Likelihood for Continuous-Discrete State Space Models using Langevin Importance Sampling. Hermann Singer conditional transition density p(ηj+1 , ηj |z) p(ηj+1 , δt|ηj , z) = p(ηj |z) State Space Models Parameter Estimation Langevin Sampling Simulated Likelihood estimate joint density p(ηj+1 , ηj |z) and p(ηj |z) with kernel density estimates variant: use φ(ηj+1 , ηj |z) and φ(ηj |z) Examples References Examples: Geometrical Brownian motion dy (t) = µy (t)dt + σy (t) dW (t) Simulated Maximum Likelihood for Continuous-Discrete State Space Models using Langevin Importance Sampling. Hermann Singer State Space Models nonlinear model (multiplicative noise) y ∗ dW Parameter Estimation exact solution: set x = log y , use Itô’s lemma Langevin Sampling Simulated Likelihood dx = dy /y + 1/2(−y −2 )dy 2 = (µ − σ 2 /2)dt + σdW Examples References exact discrete model y (t) = y (t0 )e (µ−σ 2 /2)(t−t 0 )+σ[W (t)−W (t0 )] ���� μ = �����σ = ���� ���(���) ���� ���� ���� ���� ���� ���� ���� ���� Simulated Maximum Likelihood for Continuous-Discrete State Space Models using Langevin Importance Sampling. ��� ������� ����� ����� ����� ����� -����� -����� ���� ���� � �� ��� ��� ��� ��� ��� ��� � �� ��� ��� ��� ��� ��� ��� -����� � �� ��� ��� ��� ��� ��� ��� Hermann Singer Figure : Trajectory and log returns. State Space Models Parameter Estimation �������� ���� Langevin Sampling ���� ��� ���� ���� ��� Simulated Likelihood ���� ���� ��� ���� ���� ��� ���� ���� ��� ���� � ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� � ��� �� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ���� ���� References ���� � ���� ���� Examples ���� ��� ���� ��� ���� ��� ���� ���� ���� ���� ���� ��� ���� ��� ���� � �� ��� ��� ��� ��� ��� ��� ��� � ��� ��� ��� ��� Figure : Langevin sampler, p̂2 = ��� Q ��� j ��� � ��� φ(ηj+1 , ηj |z)/φ(ηj |z) . ��� Simulated Maximum Likelihood for Continuous-Discrete State Space Models using Langevin Importance Sampling. �� ��� �� ��� ��� �� ��� �� ��� �� Hermann Singer ��� �� - ����� - ��� ����� ����� ����� ����� ����� -����� ����� ����� ����� ����� Figure : likelihood and score, p̂2 = conditional kernel density. State Space Models Parameter Estimation Langevin Sampling Simulated Likelihood Examples � References ��� -� - �� ��� - �� - �� ��� - �� - �� �� -����� ����� ����� ����� ����� ����� - ����� ����� ����� ����� Figure : likelihood and score, p̂2 = full kernel density. ����� Simulated Maximum Likelihood for Continuous-Discrete State Space Models using Langevin Importance Sampling. �� ��� �� ��� ��� �� ��� �� ��� �� Hermann Singer ��� �� -����� - ��� ����� ����� ����� ����� ����� -����� ����� ����� ����� ����� Figure : likelihood and score, p̂2 = conditionally Gaussian. State Space Models Parameter Estimation Langevin Sampling Simulated Likelihood Examples �� ��� �� References ��� ��� �� ��� �� ��� �� �� - ����� ��� - ��� ����� ����� ����� ����� ����� -����� ����� ����� ����� Figure : likelihood and score, p̂2 = linear GLS. ����� Simulated Maximum Likelihood for Continuous-Discrete State Space Models using Langevin Importance Sampling. Hermann Singer �� � �� � State Space Models � Parameter Estimation �� � Langevin Sampling � Simulated Likelihood �� - ����� ����� ����� ����� ����� ����� -����� ����� ����� ����� ����� Examples Figure : likelihood and score, p̂2 = linear GLS, constant diffusion matrix. References Simulated Maximum Likelihood for Continuous-Discrete State Space Models using Langevin Importance Sampling. Examples: Cameron-Martin formula p λ2 R T 2 E e − 2 0 W (t) dt = 1/ cosh(T λ) Cameron and Martin (1945); Gelfand and Yaglom (1960) Hermann Singer State Space Models �������� ��� � � ��� � Parameter Estimation � ��� � � ��� -� -� ��� � ��� ��� ��� ��� ��� Langevin Sampling -� ��� -� � ��� ��� ��� Simulated Likelihood ��� � � � � �� Examples ��� � ��� ��� � References ��� ��� -� ��� ��� ��� -�� ��� ��� � ��� ��� ��� ��� � ��� ��� ��� ��� � ��� ��� ��� ��� Figure : Cameron-Martin formula. Simulation using a conditionally gaussian importance density. Tp = 1, λ = 1, dt = 0.1 and L = 500 replications. Exact value 1/ cosh(1) = 0.805018 Simulated Maximum Likelihood for Continuous-Discrete State Space Models using Langevin Importance Sampling. Examples: Cameron-Martin formula Hermann Singer �������-������ �������� �������� ��������� �� = ���� �������-������ �������� �������� ��������� �� = ���� ��� ��� ��� ��� ��� ��� ��� State Space Models Parameter Estimation ��� �������� Langevin Sampling �������� ��� ��� ��� ��� ��� ��� ��� Simulated Likelihood Examples ��� � � � � � � � � � � � � Figure : Expectation value as a function of T . Right: Ω = fix. References Examples: Feynman-Kac-formula Schrödinger Equation (imaginary time t = iτ ) ut = 1 2 uxx − φ(x)u Simulated Maximum Likelihood for Continuous-Discrete State Space Models using Langevin Importance Sampling. Hermann Singer State Space Models u(x, t = 0) = δ(x − z) φ(x) = 1 2 2 2γ x Parameter Estimation Langevin Sampling Simulated Likelihood Examples Feynman-Kac-formula h γ2 R t i 2 u(x, t) = Ex e − 2 0 W (u) du δ(W (t) − z) = r γ × 2π sinh(γt) γ 2 2 exp 2xz − (x + z ) cosh(γt) 2 sinh(γt) References Simulated Maximum Likelihood for Continuous-Discrete State Space Models using Langevin Importance Sampling. Feynman-Kac-formula ������� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� - ��� ��� ��� ��� ��� � ��� ��� ��� ���� � ��� ��� ���� �� �� �� �� ��� � State Space Models ��� � ���� ��� ���� Parameter Estimation � ���� �� ���� � �� ���� ���� Hermann Singer -��� � Langevin Sampling � ��� ��� ��� ��� � ��� ��� ��� ��� � ��� ��� ��� ��� Simulated Likelihood �������-������ �������� �������� ��������� �� = ���� Examples ���� References ���� ���� ���� ���� -��� -��� ��� ��� ��� ��� ��� � Figure : Expectation value as a function of z, x = 1, t = 1, γ = 1. Conclusion Simulated Maximum Likelihood for Continuous-Discrete State Space Models using Langevin Importance Sampling. Hermann Singer smooth likelihood simulation for nonlinear continuous-discrete state space models. State Space Models Parameter Estimation Nonlinear smoothing of latent variables between measurements. Langevin Sampling Simulated Likelihood Examples References Variance reduced MC estimation of functional integrals in finance, statistics and quantum theory (Feynman-Kac formula). Bergstrom, A. (1988). The history of continuous-time econometric models, Econometric Theory 4: 365–383. Bergstrom, A. (ed.) (1976). Statistical Inference in Continuous Time Models, North Holland, Amsterdam. Borodin, A. and Salminen, P. (2002). Handbook of Brownian Motion – Facts and Formulae, second edn, Birkhäuser-Verlag, Basel. Cameron, R. H. and Martin, W. T. (1945). Transformations of Wiener Integrals Under a General Class of Linear Transformations, Transactions of the American Mathematical Society 58(2): 184–219. Feynman, R. and Hibbs, A. (1965). Quantum Mechanics and Path Integrals, McGraw-Hill, New York. Gelfand, I. and Yaglom, A. (1960). Integration in Functional Spaces and its Application in Quantum Physics, Journal of Mathematical Physics 1(1): 48–69. Langevin, P. (1908). Sur la théorie du mouvement brownien [on the theory of brownian motion], Comptes Rendus de l’Academie des Sciences (Paris) 146: 530–533. Oud, J. and Singer, H. (2008). Special issue: Continuous time modeling of panel data. Editorial introduction, Statistica Neerlandica 62, 1: 1–3. Roberts, G. O. and Stramer, O. (2002). Langevin Diffusions and Metropolis-Hastings Algorithms, Methodology And Computing In Applied Probability 4(4): 337–357. Simulated Maximum Likelihood for Continuous-Discrete State Space Models using Langevin Importance Sampling. Hermann Singer State Space Models Parameter Estimation Langevin Sampling Simulated Likelihood Examples References Singer, H. (1990). Parameterschätzung in zeitkontinuierlichen dynamischen Systemen [Parameter estimation in continuous time dynamical systems; Ph.D. thesis, University of Konstanz, in German], Hartung-Gorre-Verlag, Konstanz. Singer, H. (2012). SEM modeling with singular moment matrices. Part II: ML-Estimation of Sampled Stochastic Differential Equations., Journal of Mathematical Sociology 36(1): 22–43. Simulated Maximum Likelihood for Continuous-Discrete State Space Models using Langevin Importance Sampling. Hermann Singer State Space Models Parameter Estimation Langevin Sampling Simulated Likelihood Examples References
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