Vector estimates. In some applications, estimates are vectors. For example, the state of a mobile robot might be represented by a vector containing its position and velocity. Similarly, the vital signs of a person might be represented by a vector containing his temperature, pulse rate, and blood pressure. Here, we denote a vector by a boldfaced lowercase letter, and a matrix by an uppercase letter.

The covariance matrix ∑xx of a random variable x is the matrix E[(x − µx) (x − µx)T], where µx is the mean of x. Intuitively, entry (i,j) of this matrix is the covariance between the i and j components of vector x; in particular, entry (i,i) is the variance of the ith component of x. A random variable x with a pdf p whose mean is µx and covariance matrix is ∑xx is written as x∼p(µx, ∑xx). The inverse of the covariance matrix is called the precision or information matrix.

Uncorrelated random variables. The cross-covariance matrix ∑vw of two random variables v and w is the matrix E[(v−µv)(w−µw)T]. Intuitively, element (i, j) of this matrix is the covariance between elements v(i) and w( j). If the random variables are uncorrelated, all entries in this matrix are zero, which is equivalent to saying that every component of v is uncorrelated with every component of w. Lemma 2 generalizes Lemma 1.

 

Lemma 2. Let x1∼p1(µ1, ∑ 1), …, xn∼pn(µn, ∑n) be a set of pairwise uncorrelated random variables of length m. Let .

 

(i) The mean and covariance matrix of y are the following:

( 3)

( 4)

(ii) If random variable xn+ 1 is pairwise uncorrelated with x1, .., xn, it is uncorrelated with y.

 

The MSE of an unbiased estimator y is E[(y−µy)T(y−µy)], which is the sum of the variances of the components of y; if y has length 1, this reduces to variance as expected. The MSE is also the sum of the diagonal elements of ∑yy (this is called the trace of ∑yy).

Fusing Scalar Estimates

We now consider the problem of choosing the optimal values of the parameters α and β in the linear estimator β*x1 + α*x2 for fusing two estimates x1 and x2 from uncorrelated scalar-valued random variables.

The first reasonable requirement is
that if the two estimates x1 and x2 are
equal, fusing them should produce
the same value. This implies that α+β
= 1. Therefore, the linear estimators of
interest are of the form
( 5)

 

If x1 and x2 in Equation 5 are considered to be unbiased estimators of some quantity of interest, then yα is an unbiased estimator for any value of α. How should optimality of such an estimator be defined? One reasonable definition is that the optimal value of α minimizes the variance of yα as this will produce the highest-confidence fused estimates.

 

Theorem 1. Let and be uncorrelated random variables. Consider the linear estimator yα(x1,x2)=( 1−α)*x1+α*x2. The variance of the estimator is minimized for .

 

The proof is straightforward and is
given in the online appendix. The vari-
ance (MSE) of yα can be determined from
Lemma 1:
( 6)

 

Setting the derivative of with respect to α to zero and solving the resulting equation yield the required result.

In the literature, the optimal
value of α is called the Kalman gain K.
Substituting K into the linear fusion
model, we get the optimal linear esti-
mator y(x1, x2):
( 7)

As a step toward fusion of n> 2 estimates, it is useful to rewrite this as follows: ( 8)

 

Substituting the optimal value of α into Equation 6, we get

An unbiased
estimator is one
whose mean
is equal to the
unknown value
being estimated
and it is preferable
to a biased
estimator with
the same variance.