  In many applications it is desirable to infer coarse-grained models from
observational data. The observed process often corresponds only to a few
selected degrees of freedom of a high-dimensional dynamical system with
multiple time scales. In this work we consider the inference problem of
identifying an appropriate coarse-grained model from a single time series of a
multiscale system. It is known that estimators such as the maximum likelihood
estimator or the quadratic variation of the path estimator can be strongly
biased in this setting. Here we present a novel parametric inference
methodology for problems with linear parameter dependency that does not suffer
from this drawback. Furthermore, we demonstrate through a wide spectrum of
examples that our methodology can be used to derive appropriate coarse-grained
models from time series of partial observations of a multiscale system in an
effective and systematic fashion.
