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I'd like to combine several vector-valued estimates of a physical quantity in order to obtain a better estimate with less uncertainty.
As in the scalar case, the weighted mean of multiple estimates can provide a maximum likelihood estimate. For independent estimates 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 (see https://en.wikipedia.org/wiki/Weight...lued_estimates)
,
where stands for the covariance matrix of the vector-valued quantity .
The weighted mean in this case is:
(where the order of the matrix-vector product is not commutative).
The covariance of the weighted mean is:
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:
On the other hand, for scalar quantities it is well known that correlations between estimates can be easily accounted. In the general case (see https://en.wikipedia.org/wiki/Weight...r_correlations), suppose that , is the covariance matrix relating the quantities , is the common mean to be estimated, and is the design matrix (of length ). The Gauss–Markov theorem states that the estimate of the mean having minimum variance is given by:
with
The question is, how can correlated vector-valued estimates be combined?
In our case, how to proceed if and are not independent and all the terms in the covariance matrix are known?
In other words, are there analogous expressions to the last two for vector-valued estimates?
Any suggestion or reference, please?
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