Filtering the maximum likelihood for multiscale problems

Andrew Papanicolaou, Konstantinos Spiliopoulos

Research output: Contribution to journalArticle

Abstract

Filtering and parameter estimation under partial information for multiscale diffusion problems are studied in this paper. The nonlinear filter converges in the mean-square sense to a filter of reduced dimension. Based on this result, we establish that the conditional (on the observations) log-likelihood process has a correction term given by a type of central limit theorem. We prove that an appropriate normalization of the log-likelihood minus a log-likelihood of reduced dimension converges weakly to a normal distribution. In order to achieve this we assume that the operator of the (hidden) fast process has a discrete spectrum and an orthonormal basis of eigenfunctions. We then propose to estimate the unknown model parameters using the reduced log-likelihood, which is beneficial because reduced dimension means that there is significantly less runtime for this optimization program. We also establish consistency and asymptotic normality of the maximum likelihood estimator. Simulation results illustrate our theoretical findings.

Original languageEnglish (US)
Pages (from-to)1193-1229
Number of pages37
JournalMultiscale Modeling and Simulation
Volume12
Issue number3
DOIs
StatePublished - 2014

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Multiscale Problems
Normal distribution
Eigenvalues and eigenfunctions
Parameter estimation
Maximum likelihood
Maximum Likelihood
Likelihood
Filtering
filter
normality
nonlinear filters
Converge
normal density functions
estimators
Nonlinear Filters
Orthonormal basis
Partial Information
Diffusion Problem
Discrete Spectrum
eigenvectors

Keywords

  • Central limit theory
  • Ergodic filtering
  • Fast mean reversion
  • Homogenization
  • Maximum likelihood estimation
  • Zakai equation

ASJC Scopus subject areas

  • Modeling and Simulation
  • Chemistry(all)
  • Computer Science Applications
  • Ecological Modeling
  • Physics and Astronomy(all)

Cite this

Filtering the maximum likelihood for multiscale problems. / Papanicolaou, Andrew; Spiliopoulos, Konstantinos.

In: Multiscale Modeling and Simulation, Vol. 12, No. 3, 2014, p. 1193-1229.

Research output: Contribution to journalArticle

Papanicolaou, Andrew ; Spiliopoulos, Konstantinos. / Filtering the maximum likelihood for multiscale problems. In: Multiscale Modeling and Simulation. 2014 ; Vol. 12, No. 3. pp. 1193-1229.
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