istribution Statement : roved for ublic release; distribution is unlimited. istributed Rule-ased Inference in the Presence of Redundant Information June 8, 004 William J. Farrell III Lockheed Martin dvanced Technology Laboratories herry Hill, New Jersey 0800 wfarrell@atl.lmco.com STRT The roblem of rocessing redundant information (also known as double counting ) has been addressed in distributed Level Fusion systems over the ast decade. Some aroaches for avoiding the ill-effects of rocessing redundant data include: tagging all data items with a unique identifier, controlling the flow of data throughout the network or tracking edigree information for each data item. Recently, ovariance Intersection rovides an alternative solution to Level Fusion in the resence of redundant information. This technique allows fusion rocesses to combine redundant information without the results becoming inconsistent and overconfident. s the fusion community attacks the higher levels of information fusion, we are still faced with rocessing data items in the resence of redundant information. Lockheed Martin dvanced Technology Laboratories has develoed a technique analogous to ovariance Intersection for combining information using roositional logic. s a result, if a roositional inference engine consumes data from distributed sources that have redundant inuts, the outut of that inference engine will still be consistent and will not overestimate the confidence. This aroach simlifies roositional inference in the resence of redundant information, allowing for inference over a distributed network of fusion nodes.. Motivation One of the biggest difficulties in the develoment of distribution fusion systems is the otential for redundant information to corrut fusion comutations. voiding redundant information rocessing has led to designs involving: elaborate data tagging schemes, control over network toology, and edigree tracking. With the develoment of ovariance Intersection [], distributed Level Fusion could be erformed within an arbitrary network toology without edigree tracking. Realizing that the risk of rocessing redundant information will drive the develoment of higher-level fusion architectures, Lockheed Martin dvanced Technology Laboratories (TL) has develoed an algorithm analogous to ovariance Intersection for roositional logic. Since roositional logic is being
alied to the Level and 3 Fusion roblem sace, it seems fitting to begin investigating the imlications of rocessing redundant information in this context.. Motivational Examle onsider a network consisting of three latforms as shown in Figure. Platform # consumes information x and y roducing inferred results,, and : () Platform # consumes information x and z in order to deduce,, and E : E () Subsequently, the information deduced by Platform # and # are transmitted to Platform #3, where the following logic is alied: E F (3) Figure. istributed Inference Examle omlications arise in Equation 3. The reciient at Platform #3 does not know whether the antecedents in Equation 3 are statistically indeendent. Even if all variables (,, and ) are statistically indeendent, it is clear that and E are correlated because they are both derived from which relies on x. Further analysis illustrates the effects of incororating the redundant information into Equation 3. ssuming that Platform # and Platform # have also rovided Platform #3 with robabilities for their inferred results, Platform #3 would naively comute the robability of F as follows []: F + + + E E (4) However, if Platform #3 had full knowledge of all of the contributing sources, the actual robability could be comuted as: F ( + ) + (5)
using the fact that Equations -3 yield: ( ) ( ) ( ) F F (6) Equation 5 yields a higher robability than Equation 6, illustrating the roblem of consuming redundant information. That is, without full knowledge of how Platform # and Platform # arrived at their deductions, Platform #3 would naively derive a robability for its deduction F that is overotimistic. The algorithm resented in this aer mitigates the effect of redundant information as illustrated by the examle in Figure. 3. Fuzzy Logic One aroach for mitigating the effects of redundant information is Fuzzy Logic. The key concet in Fuzzy Logic is fuzzy set membershi [3], which is defined by a membershi function. Figure illustrates a membershi function for the set lose. lose (TTG) 0 Time-To-Go Range Time To Go Range Rate Figure. Examle Fuzzy Set Membershi Function In addition to the concet of fuzzy set membershi, Fuzzy Logic also defines logical connectives for erforming logical oerations on fuzzy sets. These connectives are [3]: min max [ ( x), ( x) ] [ ( x), ( x) ] ( x) (7) Since the min and max oerators are idemotent, redundant information does not affect Fuzzy logical oerations: min [ ( x), ( x), ( x) ] [ ( x), ( x) ] min (8) 3
Thus, Fuzzy Logic is an aroach to avoiding the ill effects of redundant information within the context of roositional inference. There is a difficulty in taking the Fuzzy Logic aroach for handling redundant information. It leads to essimistic results for conjunction and otimistic results for disjunction. lying Fuzzy Logic would be analogous to a Level Fusion solution where a single source track is selected as the fused outut. Of course, in the context of Level Fusion, there is at least some covariance matrix to determine which single source is the best to choose. This analogy begs the question, how can uncertainty in set membershi be exressed within Fuzzy Logic. The set membershi concet (Figure ) does not account for uncertainty in the observable quantity. For examle, the uncertainty in the quantity Time-To-Go does not affect the resulting set membershi lose. There is no indication of the set membershi value's uncertainty. nd this shortcoming eliminates the ossibility of fusing inferred results in any non-trivial way. Our aroach catures the affects of uncertain observables by emloying an Unscented Transform [4]: lose (TTG) 0 Transformed Sigma Points Set Membershi Mean and Variance (, σ ) lose lose Time-To-Go Sigma Points Figure 3. Unscented Transform alied to Fuzzy Set Membershi Function Now that the uncertainty in set membershi is attainable, fusion of inferred results can be erformed. It is imortant to note that the Unscented Transform can be alied to more than just the set membershi function. It can also be alied to an entire inference network. s a result, the outut nodes of an inference network (fuzzy or otherwise) can have uncertainties assigned to them. 4. ovariance Intersection ovariance Intersection was recently develoed for otimal state estimate in the resence of otentially redundant information []. It is successful in obtaining consistent state estimates where a standard Kalman filter roduces inconsistent results. s a result, most Level Fusion system develoers consider ovariance Intersection an essential algorithm comonent in distributed environments. From an imlementation standoint, ovariance Intersection is a simle modification to the standard Kalman filter equations. The ovariance Intersection equations are a modification of the information form of the Kalman Filter [5]: P P Fused Fused ω P xˆ Fused + ω P ( ω) P ˆ x + ( ω) P ˆ x (9) 4
where P is the state estimate s error covariance and xˆ is the state estimate. Thus, Equation 9 reresents the fusion of two state estimates having error covariance matrices. In addition, ovariance Intersection guarantees that the error covariance for the fused result is consistent as long as the inut error covariances are consistent []. The arameter ω is found by convex one-dimensional otimization of an objective function on the interval [0,]. Tyically, this objective function is chosen to be either the determinant or the trace of the fused error covariance. If the trace of the fused error covariance is chosen as the objective function, a closed form solution exists [6]. 5. Stochastic Fuzzy Logic With insiration from the Kalman filter, Equations 4 and 5 are examined to determine where the assumtion of indeendence enters. From classical robability theory, it is clear that the indeendence assumtion has been alied when erforming the conjunction oeration. That is, it has been assumed that: (0) Taking the natural logarithm of Equation 0, an alternate form is: ( ) ln( ) ln( ) ln + () which looks very similar to Equation 9. Furthermore, recalling that the Kalman filter is simly a ayesian filter restricted by the assumtion that the robability density functions are Gaussian [7] and statistically indeendent, the information form of the Kalman filter is roortional to the natural logarithm of the ayesian filter equations: ln ( osterior ) ln( likelihood ariori) ln( ) + ln( ) likelihood ariori () which is strikingly similar to Equation. Taking this analogy one ste further, and considering the modification made by ovariance Intersection, a modified Equation 0 yields: ω ω (3) TL has alied a generalization of Equation 3 to the roblem of ayesian Target Identification in a distributed fusion environment where redundant target identification declarations are ossible. Here, the exonential arameter ω was alied to the target identification likelihood function rovided by each fusion source. The results show that target identification confidences remained consistent even in the resence of redundant inuts. With the establishment of Equation 3, the Stochastic Fuzzy Logic oerators are: ω ω + ω ω (4) 5
where the arameter ω is the solution of a convex one-dimensional otimization roblem restricted to the interval [0,], just as with ovariance Intersection. In articular, ω requires minimizing the uncertainty in the resulting logical oeration and uses the uncertainties for fuzzy membershi comuted from the Unscented Transform (Figure 3). Equation 4 rovides interesting results when comared to conventional Fuzzy Logic. First, the logical connective functions are similar when lotted (Figures 4 and 5). The logical connectives have the same concavity, however the logical connectives given in Equation 4 do not exhibit the otimism/essimism as the Fuzzy Logic connectives in Equation 7. In addition, since the logical connectives in Stochastic Fuzzy Logic minimize a meaningful objective function (the uncertainty in set membershi), the results of Equation 4 may be more desirable. Figure 4. Fuzzy Logic onnectives Figure 5. Stochastic Fuzzy Logic onnectives 6
In Figure 5, Equation 4 is shown with the arameter ω. This is because ω reresents the situation where the uncertainties in the set membershi are equal. Thus, it is a fair comarison to Fuzzy Logic connectives. In addition to the similarity between the functions, Equation 4 also has desirable bounding roerties: ω ω [, ] min (5) and + max + [, ] ω ω (6) 6. onclusions This aer has resented an alternative to classical Fuzzy Logic for distributed inference in the resence of otentially redundant information. y combining the Unscented Transform and ovariance Intersection, Stochastic Fuzzy Logic yields several desirable roerties. First, the uncertainty in fuzzy set membershi is quantified through the use of the Unscented Transform. Secondly, set membershis are combined in a way to minimize the resulting fuzzy set membershi uncertainty. In addition, the Stochastic Fuzzy Logic connectives are bounded above and below by the classical Fuzzy Logic connectives and the classical robabilistic connectives. Overall, Stochastic Fuzzy Logic yields results that are not as extreme as Fuzzy Logic, yet not as naïve as classical robability theory. References [] Julier, S. and Uhlmann J., Non-divergent Estimation lgorithm in the Presence of Unknown orrelations, 998. [] Saeed Ghahramani. Fundamentals of Probability, Prentice Hall, 996. [3] Yaakov ar-shalom. Multitarget-Multisensor Tracking: lications and dvances, Volume II, rtech House, 99. [4] Julier, J.S. The Scaled Unscented Transformation, Proc. Of the merican ontrol onference, 00. [5] Mutambara,. ecentralized Estimation and ontrol for Multisensor Systems, R Press, 998. [6] Neihsen, W. and osch, R. Information Fusion based on Fast ovariance Intersection Filtering, Proceedings of the International Society of Information Fusion, 00. [7] Stone, L.., arlow,.., and orwin, T.L. ayesian Multile Target Tracking, rtech House, 999. 7