Optimal nonnegative definite approximations of estimated moving average covariance sequences

The problem of finding the closest nonnegative definite moving average covariance sequence to a given estimate which may not be nonnegative definite is considered. An algorithm is developed which is based on a set of constrained minimization problems, each parameterized by the zero frequencies of the spectral density function corresponding to the optimal solution. The algorithm entails first solving a simple minimization problem with linear constraints whose closed-form solution is given by a projection onto a subspace. These solutions lie either outside the set of nonnegative definite sequences, or on its boundary; if the solution lies on the boundary, it is the optimal solution. The problem is considered directly in the space of covariance sequence elements. As a result, the nonlinear maximization step is performed on sets of low dimension. By considering the minimization problem in this space, it is possible to characterize some of the geometrical properties of the optimal solution in terms of the locations of its zero frequencies