( 13)

( 14)

 

Equations 13 and 14 generalize Equations 10 and 11.

Incremental fusing is optimal. In many applications, the estimates x1, x2, …, xn become available successively over a period of time. Although it is possible to store all the estimates and use Equations 13 and 14 to fuse all the estimates from scratch whenever a new estimate becomes available, it is possible to save both time and storage if one can do this fusion incrementally. We show that just as a sequence of numbers can be added by keeping a running sum and adding the numbers to this running sum one at a time, a sequence of n> 2 estimates can be fused by keeping a “running estimate” and fusing estimates from the sequence one at a time into this running estimate without any loss in the quality of the final estimate. In short, we want to show that yn(x1,..,xn)=y2(y2(..y2(x1,x2)…),xn). A little bit of algebra shows that if n> 2, Equations 13 and 14 for the optimal linear estimator and its precision can be expressed as shown in Equations 15 and 16.

 

( 15)

( 16)

 

This shows that yn(x1,.., xn) = y2(yn− 1
(x1,..,xn− 1), xn). Using this argument
recursively gives the required result.d
To make the connection to Kalman
filtering, it is useful to derive the
same result using a pictorial argu-
ment. Figure 2 shows the process of
incrementally fusing the n estimates.
In this picture, time progresses from
left to right, the precision of each esti-
mate is shown in parentheses next to
it, and the weights on the edges are
the weights from Equation 10. The
contribution of each xi to the final
value y2(y2(…), xn) is given by the prod-
uct of the weights on the path from xi
to the final value, and this product is
obviously equal to the weight of xi in
d We thank Mani Chandy for showing us this
approach to proving the result.

( 9)

 

The expressions for y and are
complicated because they contain the
reciprocals of variances. If we let ν1 and
ν2 denote the precisions of the two dis-
tributions, the expressions for y and νy
can be written more simply as follows:
( 10)

 

( 11)

 

These results say the weight we should give to an estimate is proportional to the confidence we have in that estimate, and that we have more confidence in the fused estimate than in the individual estimates, which is intuitively reasonable. To use these results, we need only the variances of the distributions. In particular, the pdfs pi, which are usually not available in applications, are not needed, and the proof of Theorem 1 does not require these pdfs to have the same mean.

Fusing multiple scalar estimates. These results can be generalized to optimally fuse multiple pairwise uncorrelated estimates x1, x2, …, xn. Let yn,α(x1, .., xn) denote the linear estimator for fusing the n estimates given parameters α1, .., αn, which we denote by α (the notation yα(x1, x2) introduced previously can be considered to be an abbreviation of y2,α(x1, x2) ).

Theorem 2. Let for ( 1≤i≤n) be a set of pairwise uncorrelated random variables. Consider the linear estimator where

. The variance of the estimator is minimized for

 

The minimal variance is given by the
following expression:
( 12)

 

As before, these expressions are more intuitive if the variance is replaced with precision: the contribution of xi to the value of yn(x1, .., xn) is proportional to xi’s confidence.

Kalman filtering
can be seen as a
particular approach
to combining
approximations of
an unknown value
to produce a better
approximation.