Covariance Projection as a General Framework of Data Fusion and Outlier Removal

Abstract

A fundamental issue in sensor fusion is to detect and remove outliers as sensors often produce inconsistent measurements that are difficult to predict and model. The detection and removal of spurious data is paramount to the quality of sensor fusion by avoiding their inclusion in the fusion pool. In this paper, a general framework of data fusion is presented for distributed sensor networks of arbitrary redundancies, where inconsistent data are identified simultaneously within the framework. By the general framework, we mean that it is able to fuse multiple correlated data sources and incorporate linear constraints directly, while detecting and removing outliers without any prior information. The proposed method, referred to here as Covariance Projection (CP) Method, aggregates all the state vectors into a single vector in an extended space. The method then projects the mean and covariance of the aggregated state vectors onto the constraint manifold representing the constraints among state vectors that must be satisfied, including the equality constraint. Based on the distance from the manifold, the proposed method identifies the relative disparity among data sources and assigns confidence measures. The method provides an unbiased and optimal solution in the sense of Minimum Mean Square Error (MMSE) for distributed fusion architectures and is able to deal with correlations and uncertainties among local estimates and/or sensor observations across time. Simulation results are provided to show the effectiveness of the proposed method in identification and removal of inconsistency in distributed sensors system.

Appendices

Appendix 1

The fused mean and covariance of Covariance Projection (CP) Method are given as,

$$ \tilde{x} = W^{ - 1} P_{r} W\hat{x} $$

(A1)

$$ \tilde{P} = W^{ - 1} P_{r} WPW^{T} P_{r}^{T} W^{ - T} $$

(A2)

Putting $W = D^{ - 1/2} E^{T} ,$ $P_{r} = M^{W} \left( {M^{{W^{T} }} M^{W} } \right)^{ - 1} M^{{W^{T} }}$ and $M^{W} = WM$ in (A2), we get,

$$ \tilde{P} = W^{ - 1} \left( {WM\left( {M^{T} W^{T} WM} \right)^{ - 1} M^{T} W^{T} } \right) \times \left( {WM\left( {M^{T} W^{T} WM} \right)^{ - 1} M^{T} W^{T} } \right)^{T} W^{ - T} $$

Let $\alpha = M^{T} W^{T} WM$, then,

$$ \begin{aligned} \tilde{P} & = W^{ - 1} WM\alpha^{ - 1} M^{T} W^{T} WM\alpha^{ - T} M^{T} W^{T} W^{ - T} \\ \tilde{P} & = M\alpha^{ - T} M^{T} \\ \end{aligned} $$

(A3)

Putting the value of $\alpha$ in (A3) and simplifying, we get,

$$ \tilde{P} = M\left( {M^{T} ED^{ - 1} E^{T} M} \right)^{ - 1} M^{T} $$

(A4)

$$ \tilde{P} = M\left( {M^{T} P^{ - 1} M} \right)^{ - 1} M^{T} $$

(A5)

The $\tilde{P}$ in (A5) is the projection of the ellipsoid on the equality constraint. Projecting it on the subspace of individual data source will result in fused covariance as,

$$ \tilde{P} = \left( {M^{T} P^{ - 1} M} \right)^{ - 1} $$

(A6)

Similarly, using definitions of various components in fused mean (A1), we have,

$$ \tilde{x} = M\left( {M^{T} W^{T} WM} \right)^{ - 1} M^{T} W^{T} W\hat{x} $$

(A7)

$$ \tilde{x} = M\left( {M^{T} P^{ - 1} M} \right)^{ - 1} M^{T} P^{ - 1} \hat{x} $$

(A8)

The fused mean on the subspace of individual data source can then be obtained as,

$$ \tilde{x} = \left( {M^{T} P^{ - 1} M} \right)^{ - 1} M^{T} P^{ - 1} \hat{x} $$

(A9)

Appendix 2

The weighted distance from the joint mean of two data sources to the point on manifold can be calculated as,

$$ \begin{aligned} d & = \left( {\hat{x}_{1} - \tilde{x}} \right)^{T} P_{1}^{ - 1} \left( {\hat{x}_{1} - \tilde{x}} \right) + \left( {\hat{x}_{2} - \tilde{x}} \right)^{T} P_{2}^{ - 1} \left( {\hat{x}_{2} - \tilde{x}} \right) \\ \hat{x}_{1} - \tilde{x} & = \hat{x}_{1} - P_{2} \left( {P_{1} + P_{2} } \right)^{ - 1} \hat{x}_{1} + P_{1} \left( {P_{1} + P_{2} } \right)^{ - 1} \hat{x}_{2} \\ \hat{x}_{1} - \tilde{x} & = \left[ {I - P_{2} \left( {P_{1} + P_{2} } \right)^{ - 1} } \right]\hat{x}_{1} - \left[ {P_{1} \left( {P_{1} + P_{2} } \right)^{ - 1} } \right]\hat{x}_{2} \\ \end{aligned} $$

(B1)

Since $P_{1} \left( {P_{1} + P_{2} } \right)^{ - 1} + P_{2} \left( {P_{1} + P_{2} } \right)^{ - 1} = I$

$$ \hat{x}_{1} - \tilde{x} = P_{1} \left( {P_{1} + P_{2} } \right)^{ - 1} \left[ {\hat{x}_{1} - \hat{x}_{2} } \right] $$

(B2)

Similarly

$$ \begin{aligned} \hat{x}_{2} - \tilde{x} & = \left[ {I - P_{1} \left( {P_{1} + P_{2} } \right)^{ - 1} } \right]\hat{x}_{2} - \left[ {P_{2} \left( {P_{1} + P_{2} } \right)^{ - 1} } \right]\hat{x}_{1} \\ \hat{x}_{2} - \tilde{x} & = - P_{2} \left( {P_{1} + P_{2} } \right)^{ - 1} \left[ {\hat{x}_{1} - \hat{x}_{2} } \right] \\ \end{aligned} $$

(B3)

Putting (B2) and (B3) in (B1) and simplifying, we get,

$$ \begin{aligned} d & = \left( {\left[ {\hat{x}_{1} - \hat{x}_{2} } \right]^{T} \left( {P_{1} + P_{2} } \right)^{ - 1} P_{1} } \right)P_{1}^{ - 1} \left( {P_{1} \left( {P_{1} + P_{2} } \right)^{ - 1} \left[ {\hat{x}_{1} - \hat{x}_{2} } \right]} \right) \\ \quad & + \left( {\left[ {\hat{x}_{1} - \hat{x}_{2} } \right]^{T} \left( {P_{1} + P_{2} } \right)^{ - 1} P_{2} } \right)P_{2}^{ - 1} \left( {P_{2} \left( {P_{1} + P_{2} } \right)^{ - 1} \left[ {\hat{x}_{1} - \hat{x}_{2} } \right]} \right) \\ d & = \left[ {\hat{x}_{1} - \hat{x}_{2} } \right]^{T} \left[ {\left( {P_{1} + P_{2} } \right)^{ - 1} \left( {P_{1} + P_{2} } \right)\left( {P_{1} + P_{2} } \right)^{ - 1} } \right]\left[ {\hat{x}_{1} - \hat{x}_{2} } \right] \\ d & = \left[ {\hat{x}_{1} - \hat{x}_{2} } \right]^{T} \left( {P_{1} + P_{2} } \right)^{ - 1} \left[ {\hat{x}_{1} - \hat{x}_{2} } \right] \\ \end{aligned} $$

(B4)