  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.
