Simulated Maximum Likelihood for Continuous ...

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)


dY (t) = AY (t)dt + GdW (t) Z t Y (t) = e A(t−t0 ) Y (t0 ) + e A(t−s) GdW (s) t0



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


Wiener process and stock index DAX

DAX-Zuwachs

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Nonlinear continuous/discrete state space model Linear stochastic differential equations (LSDE)

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Figure : German stock index (DAX) Simulierter Wiener-Prozeß

Parameter Estimation Langevin Sampling

Zuwächse: Simulierter Wiener-Prozeß 0.2

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Figure : Simulated Wiener process (random walk)

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Simulated Wiener processes Simulierte Wiener-Prozesse 4

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Nonlinear continuous/discrete state space model Linear stochastic differential equations (LSDE)

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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


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


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


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



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 )



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


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 +

√


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 pˆ(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 pˆ(η|z) = L−1


Langevin Sampling

X

k(η − ηl ; smooth)

l

Problem: – high dimensional state, – no structure imposed on p(η|z)


Estimation of importance density use Markov structure of state space model ηj+1 = f (ηj )δt + g (ηj )δWj zi


= 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 z¯i = (zi+1 , ...zT ) given η j : j

i

i

p(ηj +1 |η , z , z¯ ) i

i

p(ηj +1 |ηj , z , z¯ ) ji ≤

j

i

i

=

p(ηj +1 |η , z¯ ) = p(ηj +1 |ηj , z¯ )

=

p(ηj +1 |ηj , z¯ )

j

< ji +1

i

Euler transition kernel Euler density (discretization interval δt)


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


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


exact solution: set x = log y , use Itô’s lemma


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 )]

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Figure : Trajectory and log returns.


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State Space Models Parameter Estimation Langevin Sampling Simulated Likelihood Examples

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References


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

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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


Examples: Cameron-Martin formula

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References

Examples: Feynman-Kac-formula Schrödinger Equation (imaginary time t = iτ ) ut

=

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− 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


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Conclusion


smooth likelihood simulation for nonlinear continuous-discrete state space models.


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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