Score-based parameter estimation for a class of continuous-time state
space models
space models
File(s)
Working paper
Author(s)
Beskos, Alexandros
Crisan, Dan
Jasra, Ajay
Kantas, Nikolas
Ruzayqat, Hamza
Type
Working Paper
Abstract
We consider the problem of parameter estimation for a class of
continuous-time state space models. 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 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.
continuous-time state space models. 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 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.
Date Issued
2021-07-15
Date Acceptance
2021-05-15
Citation
SIAM Journal on Scientific Computing, 2021
ISSN
1064-8275
Publisher
Society for Industrial and Applied Mathematics
Journal / Book Title
SIAM Journal on Scientific Computing
Copyright Statement
© 2021 The Author(s). This item is published under a CC BY license.
License URL
Identifier
http://arxiv.org/abs/2008.07803v2
Subjects
stat.CO
stat.CO
cs.NA
math.NA
math.PR
65C05, 65C35, 60G35, 60J60, 60J65, 60H10, 60H35, 65C30, 91G60,
93E11, 62F99
93E11, 62F99
Notes
32 pages, 32 figures
Publication Status
Published