  Extended Dynamic Mode Decomposition (EDMD) is a popular data-driven method to
approximate the action of the Koopman operator on a linear function space
spanned by a dictionary of functions. The accuracy of EDMD model critically
depends on the quality of the particular dictionary's span, specifically on how
close it is to being invariant under the Koopman operator. Motivated by the
observation that the residual error of EDMD, typically used for dictionary
learning, does not encode the quality of the function space and is sensitive to
the choice of basis, we introduce the novel concept of consistency index. We
show that this measure, based on using EDMD forward and backward in time,
enjoys a number of desirable qualities that make it suitable for data-driven
modeling of dynamical systems: it measures the quality of the function space,
it is invariant under the choice of basis, can be computed in closed form from
the data, and provides a tight upper-bound for the relative root mean square
error of all function predictions on the entire span of the dictionary.
