above. Vector-valued estimates The above generalizes easily to the case of taking the mean of vector-valued estimates. For example, estimates of position on a plane may have less certainty in one direction than another. As in the scalar case, the weighted mean of multiple estimates can provide a maximum likelihood estimate. We simply replace the variance by the covariance matrix and the arithmetic inverse by the matrix inverse (both denoted in the same way, via superscripts); the weight matrix then reads: The weighted mean in this case is: (where the order of the matrix–vector product is not commutative), in terms of the covariance of the weighted mean: For example, consider the weighted mean of the point [1 0] with high variance in the second component and [0 1] with high variance in the first component. Then then the weighted mean is: which makes sense: the [1 0] estimate is "compliant" in the second component and the [0 1] estimate is compliant in the first component, so the weighted mean is nearly [1 1]. Accounting for correlations In the general case, suppose that , is the covariance matrix relating the quantities , is the common mean to be estimated, and is a design matrix equal to a vector of ones (of length ). The Gauss–Markov theorem states that the estimate of the mean having minimum variance is given by: and where: Decreasing strength of interactions Consider the time series of an independent variable and a dependent variable , with observations sampled at discrete times . In many common situations, the value of at time depends not only on but also on its past values. Commonly, the strength of this dependence decreases as the separation of observations in time increases. To model this situation, one may replace the independent variable by its sliding mean for a window size . Exponentially decreasing weights In the scenario described in the previous section, most frequently the decrease in interaction strength obeys a negative exponential law. If the observations are sampled at equidistant times, then exponential decrease is equivalent to decrease by a constant fraction at each time step. Setting we can define normalized weights by where is the sum of the unnormalized weights. In this case is simply approaching for large values of . The damping constant must correspond to the actual decrease of interaction strength. If this cannot be determined from theoretical considerations, then