^{1}

^{2}

^{1}

^{1}

^{2}

This paper addresses the problem of distributed fusion when the conditional independence assumptions on sensor measurements or local estimates are not met. A new data fusion algorithm called Copula fusion is presented. The proposed method is grounded on Copula statistical modeling and Bayesian analysis. The primary advantage of the Copula-based methodology is that it could reveal the unknown correlation that allows one to build joint probability distributions with potentially arbitrary underlying marginals and a desired intermodal dependence. The proposed fusion algorithm requires no a priori knowledge of communications patterns or network connectivity. The simulation results show that the Copula fusion brings a consistent estimate for a wide range of process noises.

The rapid growth of sensing and computational capabilities has advanced the development of distributed estimation [

In a distributed estimation scenario where the local nodes process their local information and exchange their results (local tracks) to each other for fusion, the common process noise from the target of interest causes the dependency between the local estimation errors [

Most traditional fusion approaches rely on assumptions of conditionally independent measurements. When such condition is not met, the Bayesian fusion equation is not optimal. In that case, a complete knowledge of the joint probability distribution of the observations is required in order to produce the optimal fusion results. However, the derivation of the joint distribution is intractable in general. We propose using Copula theory to model the correlations between local estimates in a consistent manner.

The primary advantage of the Copula-based methodology is that it can characterize modal dependencies regardless of the respective marginal distributions [

This paper proposes a new fusion method to deal with unknown correlation in a distributed sensing network. The method will function in a mathematically consistent manner while limiting data exchange and processing requirements. The paper is organized as follows. Section

The Wiener process is widely used to model unknown inputs (maneuvers) in state estimation/tracking problem. Consider a 1D Wiener random process (or a discrete time Brownian motion) as follows:

The sensor configuration involves two range measuring sensors, where each sensor

Typically, each sensor computes an estimate of the state of interest

When the conditional independence assumption is not met, the fusion equation (

However, the derivation of the joint distribution

Let

Theorem

Sklar’s Theorem states that a multivariate joint distribution can be written in terms of univariate marginal distribution functions and a Copula function which describes the dependence structure between the variables. Besides, when the marginal distribution functions are continuous, then the Copula function is unique.

While there exists a unique Copula, its exact form is not available to us when we want to construct a joint distribution function with only marginal distribution functions. In this regard, we are supposed to choose a suitable Copula function to “approximate” the unknown true one. There are families of Copula functions, like Gaussian Copula, Archimedean Copula, and Student-t Copula. In this work, we choose Gaussian Copula since the local estimates are also Gaussian. The bivariate Gaussian Copula function [

This result implies that the Copula fusion equation (

The exact implementation of fusion rule represented by (

The next issue to consider is how to derive the “consistent” correlation coefficient

In this paper, we conduct an off-line process to approximate the coefficient

Note that (

Although it is hard to prove the equivalence analytically, we show that the Copula fusion and MAP fusion obtain very similar results in a numerical manner. Here we conduct simulations at different fusion rate, where the coefficient

As shown in Tables

Fusion rate = 1/2,

Estimation variance | Averaged MSE | |
---|---|---|

Copula | 0.3754 | 0.3749 |

MAP | 0.3704 | 0.3723 |

Fusion rate = 1/4,

Estimation variance | Averaged MSE | |
---|---|---|

Copula | 0.3834 | 0.3822 |

MAP | 0.3818 | 0.3806 |

Fusion rate = 1/8,

Estimation variance | Averaged MSE | |
---|---|---|

Copula | 0.3849 | 0.3841 |

MAP | 0.3826 | 0.3829 |

In this section, we summarize the Copula fusion algorithm. As shown in Figure

Copula fusion algorithm description.

Suppose that, at time step

To validate the proposed Copula fusion algorithm, a simulation scenario is developed with target and sensors models described in (

A hierarchical fusion scenario with a fusion period of

To verify the performance, we compare the results with the information matrix (IM) fusion algorithm. It is well known that, except in the full-rate fusion case (fusion after each measurement update), the IM fusion is only suboptimal and could be inconsistent [

Figures

Positional MSE for Copula fusion and IMF at full rate (

Positional MSE for Copula fusion and IMF at 1/2 rate (

Positional MSE for Copula fusion and IMF at 1/4 rate (

Positional MSE for Copula fusion and IMF at 1/8 rate (

Figures

Probability density of fused result at steady state for Copula fusion and IMF at 1/2 rate (

Probability density of fused result at steady state for Copula fusion and IMF at 1/4 rate (

Probability density of fused result at steady state for Copula fusion and IMF at 1/8 rate (

Figures

The ratio of perceived variance over the true variance (1/2 rate for Copula fusion).

The ratio of perceived variance over the true variance (1/4 rate for Copula fusion).

The ratio of perceived variance over the true variance (1/8 rate for Copula fusion).

With the Copula fusion algorithm, the results shown in Figures

This paper presents a novel sensor fusion methodology based on Copula and Bayesian probabilistic theory for track-to-track fusion with unknown dependency. In the method, a mathematical characterization of the dependence structure of the local estimates is constructed using Copula statistical modeling. The recursive version of distributed Copula fusion is implemented by approximating the full pedigree of the local measurements with a pseudo measurement. The simulation demonstrates that, unlike other traditional approaches, the resulting fused estimates are consistent in the sense that the perceived uncertainty characterized by the estimation error variance is close to the true uncertainty. This is particularly important because overly optimistic uncertainty assessment could mislead a critical decision. Compared with the existing work, the proposed Copula fusion requires no detailed knowledge of communications patterns and is potentially applicable to fusion process with networked disparate sensors. A natural future research direction is to generalize the methodology coupled with the proven scalable “channel filter” algorithm to the fusion problems of heterogeneous and correlated measurements in ad hoc sensing networks.

Given the target and measurement models in (

The authors declare that they have no competing interests.

Research is partially supported by ARO under Grant no. W911NF-15-1-0409 (K. C. Chang).