TY - JOUR

T1 - Score-Based Parameter Estimation for a Class of Continuous-Time State Space Models

AU - Beskos, Alexandros

AU - Crisan, Dan

AU - Jasra, Ajay

AU - Kantas, Nikolas

AU - Ruzayqat, Hamza Mahmoud

N1 - KAUST Repository Item: Exported on 2021-11-21
Acknowledgements: The work of the third and fifth authors was supported by KAUST baseline funding. The work of the first author was supported by a Leverhulme Trust Prize. The work of the second author was partially supported by EU Synergy project STUOD - DLV-856408. The work of the fourth author was supported by a JP Morgan A.I. Faculty award.

PY - 2021/7/15

Y1 - 2021/7/15

N2 - We consider the problem of parameter estimation for a class of continuous-time state space models (SSMs). In particular, we explore the case of a partially observed diffusion, with data also arriving according to a diffusion process. Based upon a standard identity of the score function, we consider two particle filter based methodologies to estimate the score function. Both methods rely on an online estimation algorithm for the score function, as described, e.g., in [P. Del Moral, A. Doucet, and S. S. Singh, M$2$AN Math. Model. Numer. Anal., 44 (2010), pp. 947--975], of $\mathcal{O}(N^2)$ cost, with $N\in\mathbb{N}$ the number of particles. The first approach employs a simple Euler discretization and standard particle smoothers and is of cost $\mathcal{O}(N^2 + N\Delta_l^{-1})$ per unit time, where $\Delta_l=2^{-l}$, $l\in\mathbb{N}_0$, is the time-discretization step. The second approach is new and based upon a novel diffusion bridge construction. It yields a new backward-type Feynman--Kac formula in continuous time for the score function and is presented along with a particle method for its approximation. Considering a time-discretization, the cost is $\mathcal{O}(N^2\Delta_l^{-1})$ per unit time. To improve computational costs, we then consider multilevel methodologies for the score function. We illustrate our parameter estimation method via stochastic gradient approaches in several numerical examples.

AB - We consider the problem of parameter estimation for a class of continuous-time state space models (SSMs). In particular, we explore the case of a partially observed diffusion, with data also arriving according to a diffusion process. Based upon a standard identity of the score function, we consider two particle filter based methodologies to estimate the score function. Both methods rely on an online estimation algorithm for the score function, as described, e.g., in [P. Del Moral, A. Doucet, and S. S. Singh, M$2$AN Math. Model. Numer. Anal., 44 (2010), pp. 947--975], of $\mathcal{O}(N^2)$ cost, with $N\in\mathbb{N}$ the number of particles. The first approach employs a simple Euler discretization and standard particle smoothers and is of cost $\mathcal{O}(N^2 + N\Delta_l^{-1})$ per unit time, where $\Delta_l=2^{-l}$, $l\in\mathbb{N}_0$, is the time-discretization step. The second approach is new and based upon a novel diffusion bridge construction. It yields a new backward-type Feynman--Kac formula in continuous time for the score function and is presented along with a particle method for its approximation. Considering a time-discretization, the cost is $\mathcal{O}(N^2\Delta_l^{-1})$ per unit time. To improve computational costs, we then consider multilevel methodologies for the score function. We illustrate our parameter estimation method via stochastic gradient approaches in several numerical examples.

UR - http://hdl.handle.net/10754/664791

UR - https://epubs.siam.org/doi/10.1137/20M1362942

U2 - 10.1137/20m1362942

DO - 10.1137/20m1362942

M3 - Article

VL - 43

SP - A2555-A2580

JO - SIAM Journal on Scientific Computing

JF - SIAM Journal on Scientific Computing

SN - 1064-8275

IS - 4

ER -