# Operator Fitting for Parameter Estimation of Stochastic Differential Equations

@article{Riseth2017OperatorFF, title={Operator Fitting for Parameter Estimation of Stochastic Differential Equations}, author={Asbj{\o}rn Nilsen Riseth and Jake P. Taylor-King}, journal={arXiv: Statistics Theory}, year={2017} }

Estimation of parameters is a crucial part of model development. When models are deterministic, one can minimise the fitting error; for stochastic systems one must be more careful. Broadly parameterisation methods for stochastic dynamical systems fit into maximum likelihood estimation- and method of moment-inspired techniques. We propose a method where one matches a finite dimensional approximation of the Koopman operator with the implied Koopman operator as generated by an extended dynamic…

## 14 Citations

### Estimating Koopman operators for nonlinear dynamical systems: a nonparametric approach

- Computer Science, MathematicsIFAC-PapersOnLine
- 2021

### Data-driven approximation of the Koopman generator: Model reduction, system identification, and control

- Computer SciencePhysica D: Nonlinear Phenomena
- 2020

### Parameter Estimation and Identification of Nonlinear Systems with the Koopman Operator

- Mathematics
- 2020

This framework is based on the key idea that nonlinear system identification in the state space is equivalent to linear identification of the Koopman operator in the space of observables and is shown to be efficient with a general class of systems, including chaotic systems, and well suited to low sampling rate datasets.

### Extracting stochastic governing laws by non-local Kramers–Moyal formulae

- Computer SciencePhilosophical Transactions of the Royal Society A
- 2022

This work uses the normalizing flows technology to estimate the transition probability density function from data, and substitutes it into the recently proposed non-local Kramers–Moyal formulae to approximate Lévy jump measure, drift coefficient and diffusion coefficient, and demonstrates that this approach can learn the stochastic differential equation with LÉvy motion.

### Discovering transition phenomena from data of stochastic dynamical systems with Lévy noise.

- MathematicsChaos
- 2020

The purpose of this present work is to extract information about transition phenomena from data of stochastic differential equations with non-Gaussian Lévy noise by using the relation between the stochastics Koopman semigroup and the infinitesimal generator of a stochastically differential equation to learn the mean exit time and escape probability from data.

### Identification of Nonlinear Systems Using the Infinitesimal Generator of the Koopman Semigroup—A Numerical Implementation of the Mauroy–Goncalves Method

- MathematicsMathematics
- 2021

The subtle numerical details of the Koopman operator-based linearization method are addressed and a new implementation algorithm is proposed that alleviates these problems.

### Koopman-Based Lifting Techniques for Nonlinear Systems Identification

- MathematicsIEEE Transactions on Automatic Control
- 2020

A novel lifting technique for nonlinear system identification based on the framework of the Koopman operator is developed, an indirect method that does not require to compute time derivatives and is therefore well-suited to low-sampling rate data sets.

### Introduction to the Koopman Operator in Dynamical Systems and Control Theory

- Mathematics
- 2020

This introductory chapter provides an overview of the Koopman operator framework. We present basic notions and definitions, including those related to the spectral properties of the operator. We also…

### Learning the temporal evolution of multivariate densities via normalizing flows

- Computer Science, MathematicsChaos
- 2022

This work proposes a method to learn multivariate probability distributions using sample path data from stochastic differential equations and demonstrates that this approach can approximate probability density function evolutions in time from observed sampled data for systems driven by both Brownian and Lévy noise.

### Optim: A mathematical optimization package for Julia

- Computer ScienceJ. Open Source Softw.
- 2018

The aim of the Optim package is to enable researchers, users, and other Julia packages to solve optimization problems without writing such algorithms themselves.

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