control//state estimation//correlated measurements//covariance intersection
Covariance intersection is a rule for fusing two estimates of the same quantity whose errors are correlated by an unknown amount, and it produces a covariance that never claims more certainty than the data justify, whatever that correlation is. It is the tool for estimates that may already contain each other: two drones exchanging their tracks of a target, each of which may have been fed earlier by the other, two navigation solutions that share a correction service, nodes of a sensor network whose messages have circulated. The ordinary fusion formula assumes independence and, applied to such estimates, counts shared information twice (correlated measurements).
Covariance intersection is a rule for fusing two estimates of the same quantity whose errors are correlated by an unknown amount, and it produces a covariance that never claims more certainty than the data justify, whatever that correlation is. It is the tool for estimates that may already contain each other: two drones exchanging their tracks of a target, each of which may have been fed earlier by the other, two navigation solutions that share a correction service, nodes of a sensor network whose messages have circulated. The ordinary fusion formula assumes independence and, applied to such estimates, counts shared information twice (correlated measurements).
Julier and Uhlmann proposed it in 1997. It adds the two inverse covariances with weights that sum to one:
P−1=ω Pa−1+(1−ω) Pb−1,P−1x^=ω Pa−1x^a+(1−ω) Pb−1x^b,P^{-1}=\omega\,P_a^{-1}+(1-\omega)\,P_b^{-1},\qquad P^{-1}\hat x=\omega\,P_a^{-1}\hat x_a+(1-\omega)\,P_b^{-1}\hat x_b ,P−1=ωPa−1+(1−ω)Pb−1,P−1x^=ωPa−1x^a+(1−ω)Pb−1x^b,
where x^a,Pa\hat x_a, P_ax^a,Pa and x^b,Pb\hat x_b, P_bx^b,Pb are the two estimates and ω∈[0,1]\omega\in[0,1]ω∈[0,1] is chosen to make the result as small as possible, usually by minimizing the trace or the determinant of PPP. Drawn as ellipses, the fused one passes through the points where the two ellipses cross and encloses their intersection, which is where the name comes from; whatever the true correlation, the fused covariance is never smaller than the real spread of the fused error.
It loses some precision to stop lying. When the two estimates were in fact independent, the ordinary fusion would have been tighter, and covariance intersection leaves that gain on the table.
Its gain comes from geometry. For a single scalar the best ω\omegaω simply keeps the better of the two estimates; the benefit appears when the two are precise in different directions, as two drones seeing a target from different angles are, each sharp across its line of sight and vague along it, and their intersection is small in both.
It is the safe fallback for exchanges where bookkeeping of who knows what is impossible (distributed Kalman filter); when the correlation is known, the full covariance of the stacked readings does better.