Semiparametric forecasting and filtering: correcting low-dimensional model error in parametric models

TL;DR

This paper introduces a semiparametric approach combining parametric models with nonparametric techniques to correct model errors in high-dimensional dynamical systems, improving forecasting accuracy.

Contribution

It proposes a novel semiparametric method that integrates data-driven nonparametric models with parametric dynamical models for improved forecasting.

Findings

Effective correction of model error demonstrated

Forecasting skill approaches that of perfect models

Method applicable to high-dimensional systems

Abstract

Semiparametric forecasting and filtering are introduced as a method of addressing model errors arising from unresolved physical phenomena. While traditional parametric models are able to learn high-dimensional systems from small data sets, their rigid parametric structure makes them vulnerable to model error. On the other hand, nonparametric models have a very flexible structure, but they suffer from the curse-of-dimensionality and are not practical for high-dimensional systems. The semiparametric approach loosens the structure of a parametric model by fitting a data-driven nonparametric model for the parameters. Given a parametric dynamical model and a noisy data set of historical observations, an adaptive Kalman filter is used to extract a time-series of the parameter values. A nonparametric forecasting model for the parameters is built by projecting the discrete shift map onto a data-driven basis of smooth functions. Existing techniques for filtering and forecasting algorithms extend naturally to the semiparametric model which can effectively compensate for model error, with forecasting skill approaching that of the perfect model. Semiparametric forecasting and filtering are a generalization of statistical semiparametric models to time-dependent distributions evolving under dynamical systems.