Equation 13, showing that incremental fusion is optimal.

Summary. The results in this section can be summarized informally as follows. When using a linear estimator to fuse uncertain scalar estimates, the weight given to each estimate should be inversely proportional to the variance of the random variable from which that estimate is obtained. Furthermore, when fusing n> 2 estimates, estimates can be fused incrementally without any loss in the quality of the final result. These results are often expressed formally in terms of the Kalman gain K, as shown in Figure 3; the equations can be applied recursively to fuse multiple estimates. Note that if ν1ν2, K≈0 and y(x1,x2)≈x1; conversely if ν1ν2, K≈ 1 and y(x1,x2)≈x2.

Fusing Vector Estimates

The results for fusing scalar estimates can be extended to vectors by replacing variances with covariance matrices.

For vectors, the linear estimator is where . Here A stands for the matrix parameters (A1, …, An). All the vectors (xi) are assumed to be of the same length. To simplify notation, we omit the subscript n in yn,A in the discussion here as it is obvious from the context.

Optimality. The parameters A1, …, An in the linear data fusion model are chosen to minimize MSE(yA) which is E[(yA−µyA)T(yA−µyA)].

Theorem 3 generalizes Theorem 2 to the vector case. The proof of this theorem is given in the appendix. Comparing Theorems 2 and 3, we see that the expressions are similar, the main difference being that the role of variance in the scalar case is played by the covariance matrix in the vector case.

Theorem 3. Let xi∼pi(µi, ∑i) for ( 1≤i≤n)
be a set of pairwise uncorrelated
random variables. Consider the linear
estimator , where
. The value of MSE( yA) is mini-
mized for

( 23)

Therefore the optimal estimator is
( 24)

 

The covariance matrix of y can be computed by using Lemma 2.

 

( 25)

 

In the vector case, precision is the inverse of a covariance matrix, denoted by N. Equations 26–27 use precision to express the optimal estimator and its variance and generalize Equations 13–14 to the vector case.

 

( 26)

( 27)

 

As in the scalar case, fusion of n> 2 vector estimates can be done incrementally without loss of precision. The proof is similar to the scalar case and is omitted.

Figure 2. Dataflow graph for incremental fusion.

ν1
ν1+ν2

ν2 ν3 ν1+ν2

ν1+ν2 ν1+···+νn− 1

ν1+···+νn− 1 ν1+···+νn
νn− 1 νn
ν1+ν2+ν3

ν1+ν2+ν3
ν1+···+νn
(ν1 + ν2)
y2(x1, x2)
(ν1 + ν2 + ν3)
y2(y2(x1, x2), x3)

(ν1 + ··· + νn− 1) y2(y2(···), xn− 1)

(ν1 + ··· + νn) y2(y2(···), xn)

...

x2

(ν2)

(ν1) x1

x3

(ν3)

xn

(νn)

xn− 1

(νn− 1)

Figure 3. Optimal fusion of scalar estimates.

x1∼p1(µ1,σ1), x2∼p2(µ2,σ2) 2 2

K= σ12

σ1+ σ22 = ν2 ν1+ ν2

y(x1, x2)=x1 + K(x2 − x1)

2 2 σy =( 1−K )σ1 or νy = ν1+ ν2

( 17)

( 18)

( 19)

Figure 4. Optimal fusion of vector estimates.

Figure 5. BLUE line corresponding to Equation ( 31).

y

; y1

;

µx µy

; ; (y − µy)=Σyx Σxx(x − µx) − 1

x1 x