Multivariate Data Fusion Based on Fixed-Geometry Confidence Sets

The successful design and operation of autonomous or partially autonomous vehicles which are capable of traversing uncertain terrains requires the application of multiple sensors for tasks such as: local navigation, terrain evaluation, and feature recognition. In applications which include a teleoperation mode, there remains a serious need for local data reduction and decision-making to avoid the costly or impractical transmission of vast quantities of sensory data to a remote operator. There are several reasons to include multi-sensor fusion in a system design: (i) it allows the designer to combine intrinsically dissimilar data from several sensors to infer some property or properties of the environment, which no single sensor could otherwise obtain; and (ii) it allows the system designer to build a robust system by using partially redundant sources of noisy or otherwise uncertain information. At present, the epistemology of multi-sensor fusion is incomplete. Basic research topics include the following taskrelated issues: (i) the value of a sensor suite; (ii) the layout, positioning, and control of sensors (as agents); (iii) the marginal value of sensor information; the value of sensing-time versus some measure of error reduction, e.g., statistical efficiency; (iv) the role of sensor models, as well as a priori models of the environment; and (v) the calculus or calculi by which consistent sensor data are determined and combined. In our research on multi-sensor fusion, we have focused our attention on several of these issues. Specifically, we have studied the theory and application of robust fixed-size confidence intervals as a methodology for robust multi-sensor fusion. This work has been delineated and summarized in Kamberova and Mintz (1990) and McKendall and Mintz (1990a, 1990b). As we noted, this previous research focused on confidence intervals as opposed to the more general paradigm of confidence sets. The basic distinction here is between fusing data characterized by an uncertain scalar parameter versus fusing data characterized by an uncertain vector parameter, of known dimension. While the confidence set paradigm is more widely applicable, we initially chose to address the confidence interval paradigm, since we were simultaneously interested in addressing the issues of: (i) robustness to nonparametric uncertainty in the sampling distribution; and (ii) decision procedures for small sample sizes. Recently, we have begun to investigate the multivariate (confidence set) paradigm. The delineation of optimal confidence sets with fixed geometry is a very challenging problem when: (i) the a priori knowledge of the uncertain parameter vector is not modeled by a Cartesian product of intervals (a hyper-rectangle); and/or (ii) the noise components in the multivariate observations are not statistically independent. Although it may be difficult to obtain optimal fixed-geometry confidence sets, we have obtained some very promising approximation techniques. These approximation techniques provide: (i) statistically efficient fixed-size hyper-rectangular confidence sets for decision models with hyper-ellipsoidal parameter sets; and (ii) tight upper and lower bounds to the optimal confidence coefficients in the presence of both Gaussian and non-Gaussian sampling distributions. In both the univariate and multivariate paradigms, it is assumed that the a priori uncertainty in the parameter value can be delineated by a fixed set in an n-dimensional Euclidean space. It is further assumed, that while the sampling distribution is uncertain, the uncertainty class description for this distribution can be delineated by a given class of neighborhoods in the space of all n-dimensional probability distributions. The following sections of this paper: (i) present a paradigm for multi-sensor fusion based on position data; (ii) introduce statistical and set-valued models for sensor errors and a priori environmental uncertainty; (iii) explain the role of confidence sets in statistical decision theory and sensor fusion; (iv) relate fixed-size confidence intervals to fixedgeometry confidence sets; and (v) examine the performance of fixed-size hyper-cubic confidence sets for decision models with spherical parameter sets in the presence of both Gaussian and non-Gaussian sampling distributions