Estimation under unknown correlation: covariance intersection revisited

Abstract

This note addresses the problem of obtaining a consistent estimate (or upper bound) of the covariance matrix when combining two quantities with unknown correlation. The combination is defined linearly with two gains. When the gains are chosen a priori, a family of consistent estimates is presented in the note. The member in this family having minimal trace is said to be "family-optimal." When the gains are to be optimized in order to achieve minimal trace of the family-optimal estimate of the covariance matrix, it is proved that the global optimal solution is actually given by the Covariance Intersection algorithm, which conducts the search only along a one-dimensional curve in the n-squared-dimensional space of combination gains.

The covariance intersection fusion estimation algorithm weighted by diagonal matrix based on genetic simulated annealing algorithm and machine learning

“…In this section, the optimization process of the DCI is implemented by the GSA algorithm. The 𝑗 × n vector w k = [ 𝜔 (11) k , … , 𝜔 (1n) k , 𝜔 (21)…”

“…From Figure 5, when p c ∈ (0.6, 0.9) and p m ∈ (0.001, 0.2), a more satisfactory result can be obtained. Figure 6 shows that the larger the annealing FIGURE 10 The diagonal elements 𝜔 (11) k , 𝜔 (12) k , 𝜔 (13) k , 𝜔 (14) k of the diagonal matrix weights w (1) k of the subsystem-1…”

Conservative Quantization of Covariance Matrices with Applications to Decentralized Information Fusion

“…Both approaches require additional data to be transmitted. If the error covariance matrices C aa and C bb are not known, but upper boundsĈ aa C aa andĈ bb C bb are available, applying ( 25) and (26) using the estimates yields a conservative error covariance matrix estimate C f f C(x f − x) [42]. In the common case where the cross-covariance is unknown but not negligible, setting C ab = 0 in ( 25) and ( 26) generally does not produce a conservative error covariance matrix estimate, i.e., C f f C(x f − x).…”

Multisensor Parallel Largest Ellipsoid Distributed Data Fusion with Unknown Cross-Covariances

“…The second one is that the LE algorithm can’t guarantee its consistency, which implies that ${\overline{P}}^{LE}\le P^{LE}$ will be unsatisfied in some cases. In the two-sensor case, according to [23], for any point $x\in {\Re }_{false({\overline{\hat{x}}}_1,P_{1}false)}\cap {\Re }_{false({\overline{\hat{x}}}_2,P_{2}false)}$, there is a feasible cross-covariance $P_{12}$ that lets $x\in P_{false(2false)}^O$. As mentioned above, the covariance ellipsoid of $P^{LE}$ is the largest one contained within the intersection region ${\Re }_{false({\overline{\hat{x}}}_1,P_{1}false)}\cap {\Re }_{false({\overline{\hat{x}}}_2,P_{2}false)}$, but it generally doesn’t cover the whole intersection region.…”