Lecture

Parameter Estimation of SDEs with Linear Response Theory

Graph Chat

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.

Description

This lecture by the instructor covers the parameter estimation of Stochastic Differential Equations (SDEs) using Linear Response Theory. It discusses the problem statement, the use of quadratic variation and Girsanov formula for estimation, fitting equilibrium statistics, and the linear response operator. The lecture also explores the challenges, such as model error and admissible forcings, and presents examples like Langevin dynamics. It delves into least-square fitting of two-point statistics, orthogonal polynomial surrogate models, algorithms for parameter estimation, convergence issues, and the estimation of equilibrium density. The lecture concludes with discussions on nonparametric density estimation, reproducing kernel Hilbert space, and the consistency of linear response estimation.

Official source

Related concepts (32)

Maximum likelihood estimation

In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed statistical model, the observed data is most probable. The point in the parameter space that maximizes the likelihood function is called the maximum likelihood estimate. The logic of maximum likelihood is both intuitive and flexible, and as such the method has become a dominant means of statistical inference.

Orthogonal polynomials

In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product. The most widely used orthogonal polynomials are the classical orthogonal polynomials, consisting of the Hermite polynomials, the Laguerre polynomials and the Jacobi polynomials. The Gegenbauer polynomials form the most important class of Jacobi polynomials; they include the Chebyshev polynomials, and the Legendre polynomials as special cases.

Classical orthogonal polynomials

In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials: the Hermite polynomials, Laguerre polynomials, Jacobi polynomials (including as a special case the Gegenbauer polynomials, Chebyshev polynomials, and Legendre polynomials). They have many important applications in such areas as mathematical physics (in particular, the theory of random matrices), approximation theory, numerical analysis, and many others.

Least squares

The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the residuals (a residual being the difference between an observed value and the fitted value provided by a model) made in the results of each individual equation. The most important application is in data fitting.

Kernel (statistics)

The term kernel is used in statistical analysis to refer to a window function. The term "kernel" has several distinct meanings in different branches of statistics. In statistics, especially in Bayesian statistics, the kernel of a probability density function (pdf) or probability mass function (pmf) is the form of the pdf or pmf in which any factors that are not functions of any of the variables in the domain are omitted. Note that such factors may well be functions of the parameters of the pdf or pmf.

Related lectures (64)

Linear Estimation & Prediction: Models & Methods

Explores linear estimation and prediction in AR parametric models, focusing on Yule Walker equations and Wiener filter.

Nonparametric Statistics: Bayesian Approach

Explores non-parametric statistics, Bayesian methods, and linear regression with a focus on kernel density estimation and posterior distribution.

Clustering & Density Estimation

Covers dimensionality reduction, clustering, and density estimation techniques, including PCA, K-means, GMM, and Mean Shift.

Spectral & Parametric Estimation: Time Series

Covers spectral estimation techniques like tapering and parametric estimation, emphasizing the importance of AR models and Whittle likelihood in time series analysis.

Non-Parametric Estimation: Kernel Density Estimators