Weighted Mean - Vector-valued Estimates

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 by the covariance matrix:

$W_i = \Sigma_i^{-1}.$

The weighted mean in this case is:

$\bar{\mathbf{x}} = \left(\sum_{i=1}^n \Sigma_i^{-1}\right)^{-1}\left(\sum_{i=1}^n \Sigma_i^{-1} \mathbf{x}_i\right),$

and the covariance of the weighted mean is:

$\Sigma_{\bar{\mathbf{x}}} = \left(\sum_{i=1}^n \Sigma_i^{-1}\right)^{-1},$

For example, consider the weighted mean of the point with high variance in the second component and with high variance in the first component. Then

then the weighted mean is:

which makes sense: the estimate is "compliant" in the second component and the estimate is compliant in the first component, so the weighted mean is nearly .

Read more about this topic: Weighted Mean

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