Semiparametric forecasting and filtering are introduced as a method ofaddressing model errors arising from unresolved physical phenomena. Whiletraditional parametric models are able to learn high-dimensional systems fromsmall data sets, their rigid parametric structure makes them vulnerable tomodel error. On the other hand, nonparametric models have a very flexiblestructure, but they suffer from the curse-of-dimensionality and are notpractical for high-dimensional systems. The semiparametric approach loosens thestructure of a parametric model by fitting a data-driven nonparametric modelfor the parameters. Given a parametric dynamical model and a noisy data set ofhistorical observations, an adaptive Kalman filter is used to extract atime-series of the parameter values. A nonparametric forecasting model for theparameters is built by projecting the discrete shift map onto a data-drivenbasis of smooth functions. Existing techniques for filtering and forecastingalgorithms extend naturally to the semiparametric model which can effectivelycompensate for model error, with forecasting skill approaching that of theperfect model. Semiparametric forecasting and filtering are a generalization ofstatistical semiparametric models to time-dependent distributions evolvingunder dynamical systems.
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