  The ensemble Gaussian mixture filter combines the simplicity and power of
Gaussian mixture models with the provable convergence and power of particle
filters. The quality of the ensemble Gaussian mixture filter heavily depends on
the choice of covariance matrix in each Gaussian mixture. This work extends the
ensemble Gaussian mixture filter to an adaptive choice of covariance based on
the parameterized estimates of the sample covariance matrix. Through the use of
the expectation maximization algorithm, optimal choices of the covariance
matrix parameters are computed in an online fashion. Numerical experiments on
the Lorenz '63 equations show that the proposed methodology converges to
classical results known in particle filtering. Further numerical results with
more advances choices of covariance parameterization and the medium-size Lorenz
'96 equations show that the proposed approach can perform significantly better
than the standard EnGMF, and other classical data assimilation algorithms.
