Linear MMSE Estimator
In many cases, it is not possible to determine a closed form expression for the conditional expectation required to obtain the MMSE estimator. Direct numerical evaluation of the conditional expectation is computationally expensive, since they often require multidimensional integration. In such cases, one possibility is to abandon the full optimality requirements and seek a technique minimizing the MSE within a particular class of estimators, such as the class of linear estimators. Thus we postulate that the conditional expectation of given is a simple linear function of, where the measurement is a random vector, is a matrix and is a vector. The linear MMSE estimator is the estimator achieving minimum MSE among all estimators of such form. Such linear estimator only depends on the first two moments of the probability density function. The expression for linear MMSE estimator, its mean, and its auto-covariance is given by
where, the is cross-covariance matrix between and, the is auto-covariance matrix of, and the is cross-covariance matrix between and . The minimum mean square error achievable by such estimator is
These estimators are sometimes referred to as Wiener filters.
Read more about this topic: Minimum Mean Square Error