with , we can also obtain its auto-covariance as Putting the expression for and , we get Lastly, the covariance of linear MMSE estimation error will then be given by The first term in the third line is zero due to the orthogonality principle. Since , we can re-write in terms of covariance matrices as This we can recognize to be the same as Thus the minimum mean square error achievable by such a linear estimator is . Univariate case For the special case when both and are scalars, the above relations simplify to where is the Pearson's correlation coefficient between and . The above two equations allows us to interpret the correlation coefficient either as normalized slope of linear regression or as square root of the ratio of two variances . When , we have and . In this case, no new information is gleaned from the measurement which can decrease the uncertainty in . On the other hand, when , we have and . Here is completely determined by , as given by the equation of straight line. Computation Standard method like Gauss elimination can be used to solve the matrix equation for . A more numerically stable method is provided by QR decomposition method. Since the matrix is a symmetric positive definite matrix, can be solved twice as fast with the Cholesky decomposition, while for large sparse systems conjugate gradient method is more effective. Levinson recursion is a fast method when is also a Toeplitz matrix. This can happen when is a wide sense stationary process. In such stationary cases, these estimators are also referred to as Wiener–Kolmogorov filters. Linear MMSE estimator for linear observation process Let us further model the underlying process of observation as a linear process: , where is a known matrix and is random noise vector with the mean and cross-covariance . Here the required mean and the covariance matrices will be Thus the expression for the linear MMSE estimator matrix further modifies to Putting everything into the expression for , we get Lastly, the error covariance is The significant difference between the estimation problem treated above and those of least squares and Gauss–Markov estimate is that the number of observations m, (i.e. the dimension of ) need not be at least as large as the number of unknowns, n, (i.e. the dimension of ). The estimate for the linear observation process exists