  We develop a method to estimate space-time flow statistics from a limited set
of known data. While previous work has focused on modeling spatial or temporal
statistics independently, space-time statistics carry fundamental information
about the physics and coherent motions of the flow and provide a starting point
for low-order modeling and flow control efforts. The method is derived using a
statistical interpretation of resolvent analysis. The central idea of our
approach is to use known data to infer the statistics of the nonlinear terms
that constitute a forcing on the linearized Navier-Stokes equations, which in
turn imply values for the remaining unknown flow statistics through application
of the resolvent operator. Rather than making an a priori rank-1 assumption,
our method allows the known input data to select the most relevant portions of
the resolvent operator for describing the data, making it well-suited for
high-rank turbulent flows. We demonstrate the predictive capabilities of the
method using two examples: the Ginzburg-Landau equation, which serves as a
convenient model for a convectively unstable flow, and a turbulent channel flow
at low Reynolds number.
