Efficient data association with multivariate Gaussian distributed states

We describe an efficient algorithm for evaluating the (weighted bipartite graph of) associations between two sets of data with gaussian error, e.g., between a set of measured state vectors and a set of estimated state vectors. First a general method is developed for determining, from the covariance matrix, minimal d-dimensional error ellipsoids for the state vectors which always overlap when a gating criterion is satisfied. Circumscribing boxes, or d-ranges, for the data ellipsoids are then found and whenever they overlap the association probability is computed. For efficiently determining the intersections of the d-ranges a multidimensional search tree method is used to reduce the overall scaling of the evaluation of associations. Very few associations that lie outside the predetermined error threshold or gate are evaluated. Empirical testing for variously distributed data in both three and eight dimensions indicate that the scaling is significantly reduced from N.sup.2, where N is the size of the data set. Computational loads for many large scale (N>10-100) data association tasks may therefore be significantly reduced by this or related methods.

FIELD OF THE INVENTION
 The invention pertains generally to tracking of objects, such as aircraft
 by radar or submarines by sonar, and particularly to computationally
 efficient apparatus and methods of tracking and data fusion.
 BACKGROUND OF THE INVENTION
 As demand for practical solutions to larger scale data fusion or data
 association tasks increases, issues of computational complexity of the
 algorithmic approaches used become topics of more serious interest.
 Theoretically well-defined and well-grounded analytical models for data
 association problems may exist without having easily computed solutions.
 When such situations arise it is desirable to determine efficient
 approximation procedures that, if possible, are well-conditioned such that
 certain bounds on errors can be determined.
 When data association tasks consist of merging, or fusing, two or more
 large sets of data that both represent a common underlying reality or
 ground truth, such issues of the computational complexity of the
 algorithms do arise. We assume that a general task of data association is
 to take two independent sets of state estimates, e.g., measurements and
 predicted state values, that each represent a common underlying set of
 objects and to determine the associations between the elements from the
 distinct sets of data. Fusion, or subsequent assignment of the elements
 from the independent representations to a single representation will
 depend on the determination of these associations.
 Typically in discussions of computational complexity, reference is made to
 the underlying graph theoretic structure of the problem at hand. The
 purpose of this exercise is to determine if the general problem has ever
 arisen in other applications. In fact, this is indeed the case with some
 data association problems. The graph structure associated with the fusion
 problem is a bipartite graph, since by assumption there are two sets of
 data, independently arrived at. From each element a.epsilon.A there may be
 an association with an element b.epsilon.B represented by an edge
 .omega..sub.ab.epsilon.E. On the graph, .GAMMA.{A, B, E} this corresponds
 to vertices of A connected to vertices of B with weighted edges, where the
 weight represents the weight of association. One may think of the
 association weights forming an association matrix, .OMEGA.. In the process
 of fusion there are commonly constraints that describe feasible joint
 events, i.e., each element of one set may be assigned to at most only one
 element of the other set and vice-versa. In such cases the joint event is
 properly termed a match or, defined on a bipartite graph, a bipartite
 match. If this match has maximum cardinality, generally corresponding to
 the constraint that a feasible joint event is a one-to-one mapping (always
 possible with false alarms), then it is called a perfect match.
 In general the representation of objects is a multidimensional state value
 with an associated error estimate. Commonly the state value is an
 estimated mean value and the error is represented as an associated
 covariance matrix defining a Gaussian probability distribution. When a
 probabilistic interpretation is given to the association weights, as we
 have assumed, the merging may be done in a variety of ways that have
 particular probabalistic interpretations. The nearest-neighbor approach,
 where the largest association weight at each node of one set alone
 determines the assignment, essentially assumes that there is virtually
 only one individual pairwise association of importance for every element,
 a condition of sparsity of the association matrix, .OMEGA.. The
 nearest-neighbor approach is the simplest of greedy heuristics. The
 optimum weighted bipartite match corresponds to the most probable, or
 modal, joint event, and is also used as a fusion method.
 In dense environments where individual elements appear to have many
 significant associations, a more appropriate approach to this data
 association problem is described by Bar-Shalom (Tracking and Data
 Association, ACADEMIC PRESS (1968)) and is referred to as JPDA (joint
 probabilistic data association). In this approach each association weight
 is taken as an individual pairwise association probability or marginal
 association probability. The relative probability of one match with
 respect to any other is taken as the product of the individual pairwise
 association probabilities (the edge weights) of the match. (Since
 objective functions are usually summed, this corresponds to the sum over
 the log of the pairwise association probabilities). The expected
 probability that a and b refer to the same object is determined by summing
 the joint probabilities of the perfect matches in which a and b are paired
 and normalizing by the sum of joint probabilities over all possible
 perfect matches. For an N.times.N association matrix, .OMEGA., this may be
 alternatively stated as
EQU &lt;p.sub.ab &gt;=.OMEGA..sub.ij *perM.sub.(i,j) /per.OMEGA. (1)
 where the permanent of .OMEGA.is defined as
 ##EQU1##
 which is similar to the definition of the determinant with the exception
 that the absolute value of the Levi-Civita symbol is used to represent the
 sum over all permutations, .sigma., of the indices 1 . . . N and the
 permanent--minor, or perM.sub.(i,j), is determined by taking the permanent
 of the matrix M.sub.(i,j), generated by crossing out the ith row and jth
 column of the matrix .OMEGA..
 Valiant (The Complexity of Computing the Permanent, THEORETICAL COMPUTER
 SCIENCE, 8, 189-201 (1979)) has shown that the evaluation of the permanent
 of a 0-1 matrix (similar to the validation matrix for JPDA), equivalent to
 counting the number of perfect matches, is NP-hard. Complete evaluation of
 the permanent of a general association matrix is equivalent to enumeration
 of all possible perfect matches, a significantly more difficult problem.
 This implies that the exact solution of the JPDA is indeed NP-hard.
 Approximation methods, however, do exist and generally reduce the
 computational scaling significantly. These methods themselves then become
 the important subjects of consideration in terms of computational scaling.
 Determining computational tractability in approaches to tracking for
 large-scale real-time problems is an exercise of putting practical upper
 bounds on algorithmic scaling. Clearly, any algorithm will scale at least
 linearly with the size of the input data. We would maintain, perhaps
 arbitrarily, that for large-scale problems quadratic algorithmic scaling
 forms a practical upper bound.
 Thresholding the elements of .OMEGA. is one approach that reduces the
 scaling of JPDA (or any other approach). The permanent may be evaluated
 recursively via a development through permanent-minors in the same manner
 as a determinant is evaluated by a Laplacian development through minors.
 For sparse matrices with O(kN) elements this evaluation method appears to
 scale at worst as N.sup.k. Clearly, by thresholding matrix elements to
 guarantee sparsity, one may both approximate the permanent and put a limit
 on the scaling behavior of the evaluation algorithm.
 Gating is the general problem in data fusion of determining significant
 association probabilities between two large sets of linear,
 multidimensional pattern elements (see S. S. Bleckman, Multi-Target
 Tracking with Radar Applications (Artech House 1986)), e.g., thresholding
 the association probability overlap integrals between a set of measured
 state vectors and a set of of estimated state vectors, both having
 associated error distributions. As a precursorial coarse assignment method
 for multitarget tracking, it reduces the computational complexity of the
 NP-hard exact JPDA, or any other data association method.
 Computing the full association matrix between two large data sets of size N
 scales as N.sup.2 in time. Also, computing the association probability
 overlap integral between two errored vectors is generally a
 computationally expensive procedure. For sparse systems, those where there
 are only order N significant associations, where significance is
 determined by a "gate ", this becomes a problem since a lot of work is
 performed relative to the significant pieces of information garnered. Two
 outstanding problems are determining a good or optimally scaling
 association algorithm and a method of evaluation that computes a minimum
 number of associations that are likely to satisfy the gating criterion.
 Solution of these problems contributes important computational labor
 savings to many data association tasks.
 An association probability is essentially the probability overlap integral
 between two (distributed) states. Significant associations exist, roughly
 speaking, for near-neighbors in state space. If a probability distribution
 for a state is not local, e.g., cannot be bounded to a small, connected
 and convex volume of the state space, then it will generally be necessary
 to evaluate a large number of overlap integrals and consideration of
 efficiently scaling algorithms may not be worthwhile. On the other hand,
 if the probability distributions do have these geometric properties, then
 preprocessing of the distributed states and the application of an
 efficiently scaling multidimensional search tree algorithm can
 significantly improve gating efficiency.
 In order to eliminate the need to individually consider every pairwise
 association it is necessary to determine if the thresholding imposed by
 the gating criterion on the pairwise associations of errored vectors
 induces or can be decomposed to a thresholding criterion on the individual
 distributed states themselves. If a thresholding criteria can be so
 determined so as to define finite volumes in state space for each state,
 then the existence of a significant association implies the geometric
 intersection of these volumes (or at least we can so define the induced
 thresholding criterion). The determination of the intersections of finite
 geometric objects can be done efficiently by the use of data structures
 known as multidimensional search trees (see Valiant and Blackman, supra).
 SUMMARY OF THE INVENTION
 Accordingly, an object of the invention is to track a plurality of objects,
 or fuse a plurality of data sets, in a computationally efficient manner.
 Another object is to track a plurality of N objects, or fuse a plurality of
 data sets of N members, in a manner which requires a number of
 computations significantly less than N.sup.2.
 In accordance with these and other objects made apparent hereinafter, the
 invention pertains to a method and apparatus of correlating a plurality of
 objects. At two different times, sets of d dimensional data vectors
 {.alpha.} and {.beta.} are produced, each of whose elements
 a.epsilon.{.alpha.} and b.epsilon.{.beta.} corresponds to one of the
 plurality of objects. The uncertainty of each element of each said data
 vector a or b is taken to be Gaussian, and thus has respective covariance
 matrices A and B. A numerical value .gamma..sub.AB is selected, which is
 used to determine which vector pairs a, b satisfy the gating criterion:
 .gamma..sub.A.gtoreq..gamma..sub.AB, and
 .gamma..sub.B.gtoreq..gamma..sub.AB, where
EQU .gamma..sub.A =(r-a).sup.T A.sup.-1 (r-a)
EQU .gamma..sub.B =(r-b).sup.T B.sup.-1 (r-b)
EQU .gamma..sub.AB =(a-b).sup.T (A+B).sup.-1 (a-b)
 and where r is some d dimensional vector. A relatedness criterion is
 selected for characterizing the degree of association between any such
 vector pair a,b, and the relatedness criterion applied to all vector pairs
 which satisfy the gating criterion. Finally, one uses these results to
 fuse the data vectors of sets {.alpha.} and {.beta.} into a fused set {c}
 of data vectors each of whose members corresponds to one of the objects.
 Because the gating criterion eliminated data pairs having small likelihood
 of correlation, the relatedness criterion need be applied only to those
 pairs not eliminated by the gating criterion, reducing the number of
 calculations necessary to correlate {.alpha.} and {.beta.}.

DETAILED DESCRIPTION
 An outline of the gating procedure required to implement an efficient
 search is as follows. A standard deviation threshold, or gate size, is
 chosen which each associated pair must satisfy. This induces a standard
 deviation thresholding for the independent elements, i.e., for each of the
 measured states and the estimated states. The resulting volume
 representing each errored vector is described by an ellipsoid in
 d-dimensional space. Detailed below is a justification showing that when
 the probability of association satisfies the gating criterion then the two
 ellipsoids of the associated data elements overlap. This is done by
 determining the thresholding criterion on each of the data element
 distributions that is induced by the gating criterion and that has this
 property. In fact, we show that the geometric property we seek turns out
 to be a consequence of a geometric inequality.
 We begin by defining a distributed state as a probability distribution
 which for multivariate normal distributions with a Euclidean distance
 measure can be represented as the pairing of a mean state vector and a
 covariance matrix. For example an estimated distributed state, A ={a, A},
 has an estimated mean state a and a covariance matrix A; a measured
 distributed state, B ={b, B}, has a measured state b and a covariance
 matrix B. Their probability distributions in d-dimensional state space, r,
 are
 ##EQU2##
 respectively, where A and B are positive definite and symmetric.
 The probability of association between the two object representations is
 proportional to the overlap integral
 ##EQU3##
 which, when integrated over state space reduces to
 ##EQU4##
 The probability of association, .rho..sub.AB, represents the distribution
 of measured states in the estimated-state state space (or vice-versa). For
 a given object, the corresponding probability that b will lie within the
 ellipsoidal surface
EQU E.sub.AB.vertline.(a-b').sup.T (A+B).sup.-1 (a-b')=.gamma..sub.AB (6)
 with probability, or rate, of correct associations
 ##EQU5##
 The gate is determined by choosing a threshold .gamma..sub.AB or a
 probability threshold p.sub.AB which are related by
EQU p.sub.AB =err(.gamma..sub.AB +L ,d). (8)
 where we define
 ##EQU6##
 Ellipsoidal surfaces, E.sub.A for the estimated position and E.sub.B for a
 measurement of the position, may also be defined by .gamma..sub.A and
 .gamma..sub.B like,
EQU E.sub.A.vertline.(r'-a).sup.T A.sup.-1 (r'-a)=.gamma..sub.A (10)
 and
EQU E.sub.B.vertline.(r"-b).sup.T B.sup.-1 (r"-b)=.gamma..sub.B (11)
 within which the actual object represented may be expected to lie with
 probabilities
 ##EQU7##
 respectively.
 To determine .gamma..sub.A and .gamma..sub.B such that when the gate is
 satisfied ellipsoids E.sub.A and E.sub.B overlap, we note that each must
 be at least greater than or equal to .gamma..sub.AB since as one
 distribution, say .rho..sub.A, tends to a Dirac .delta.-function then the
 gating criterion will only be satisfied if a falls within ellipse E.sub.B,
 or
EQU (a-b).sup.T B.sup.-1 (a-b).ltoreq..gamma..sub.AB (14)
 and vice-versa if .rho..sub.B tends to a Dirac .delta.-function. In other
 words .gamma..sub.A.gtoreq..gamma..sub.AB and
 .gamma..sub.B.gtoreq..gamma..sub.AB are necessary conditions to guarantee
 overlap.
 Now we show that .gamma..sub.A =.gamma..sub.B =.gamma..sub.AB is a
 sufficient condition such that when the gate is satisfied ellipsoids
 E.sub.A and E.sub.B overlap.
 The gating condition
EQU (a-b).sup.T (A+B).sup.-1 (a-b).ltoreq..gamma..sub.AB (15)
 can be interpreted to mean that a lies within ellipsoid E.sub.AB centered
 around b. A hyperplane tangent to the ellipsoid along any particular
 coordinate direction i (for any coordinate frame) is a distance
 .gamma..sub.AB +L (A+B).sub.ii +L from the center (a simple proof is
 detailed in Appendix I). Since a lies within the ellipsoid E.sub.AB, the
 projection of (a-b) along the same coordinate direction satisfies
EQU (a-b).sub.i.ltoreq..gamma..sub.AB +L (A+B).sub.ii +L (16)
 Because the diagonal elements of the matrices A and B are positive
 definite, we also know that
EQU .gamma..sub.AB +L (A+B).sub.ii +L .ltoreq..gamma..sub.AB +L A.sub.ii +L
 +.gamma..sub.AB +L B.sub.ii +L (17)
 which gives rise to the result
EQU (a-b).sub.i.ltoreq..gamma..sub.AB +L A.sub.ii +L +.gamma..sub.AB +L
 B.sub.ii +L . (16)
 This result implies that all projections of the ellipsoids E.sub.A and
 E.sub.B, centered at a and b respectively, overlap. This can only be true
 if the ellipsoids E.sub.A and E.sub.B themselves overlap.
 There is another way to understand this result. At a point r.sub.m the
 function
EQU .function.(r)=(r-a).sup.T A.sup.-1 (r-a)+(r-b).sup.T B.sup.-1 (r-b) (19)
 has a minimum value that is exactly .function.(r.sub.m)=(a-b).sup.T
 (A+B).sup.-1 (a-b).
 This allows the expression for all r
EQU (a-b).sup.T (A+B).sup.-1 (a-b).ltoreq.(r-a).sup.T A.sup.-1
 (r-a)+(r-b).sub.T B.sup.-1 (r-b) (20)
 which reflects an essential geometric property of the space of
 Gaussian-distributed, independent random variables. From this result
 follows the interpretation that at this point, r.sub.m, both
EQU (r-a).sup.T A.sup.-1 (r-a).ltoreq..function.(r.sub.m) and (r-b).sup.T
 B.sup.-1 (r-b).ltoreq..function.(r.sub.m) (21)
 are true, i.e., the ellipses E.sub.A and E.sub.B overlap.
 This proves the sufficiency condition that was sought; we therefore find
 that .gamma..sub.A =.gamma..sub.AB and .gamma..sub.B =.gamma..sub.AB are
 necessary and sufficient conditions to guarantee overlap of E.sub.A and
 E.sub.B if the gating criterion is satisfied. It should also be noted that
 since any .gamma..sub.A or .gamma..sub.B greater than .gamma..sub.AB is
 also sufficient, the condition .gamma..sub.A =.gamma..sub.B
 =.gamma..sub.AB is the minimum and therefore optimal criterion, resulting
 in the minimum number of possible tests for gate satisfaction.
 (Satisfaction of the gating criterion implies overlap but the reverse
 implication is not necessarily true and the probability overlap integrals
 corresponding to each overlapping pair of ellipsoids must be evaluated and
 compared to .gamma..sub.AB).
 As a final note, we point out that since
EQU (a-b).sub.i.ltoreq..gamma..sub.AB +L A.sub.ii +L (22)
 cannot be guaranteed in general, one cannot assume that any arbitrary
 finite gate chosen for one of the two sets will always determine
 ellipsoids that overlap the mean position of associated elements from the
 other set. This implies that without particular knowledge of the datasets
 a volumetric intersection approach is the only way to guarantee finding
 all associations that satisfy the gating criterion.
 For determining the intersection of two volumes efficient methods rely on
 comparisons of common reference coordinate values that bound a
 hyper-rectangular volume. These bounding coordinate values define
 circumscribing, coordinate aligned (isothetic) boxes or d-ranges in
 d-dimensions for each ellipsoid. A box oriented along the rectangular
 coordinate axes that circumscribes a d-ellipsoid described by the
 expression (a-r).sup.T A.sup.-1 (a-r)=.gamma. is determined by tangent
 hyperplanes defined
 by
EQU p.sub.k =.alpha..sub.k.+-..gamma.A.sub.kk +L (23)
 Overlapping box pairs are cheaply determined and for each of the
 overlapping box pairs a probability of association is calculated and
 thresholded.
 Circumscribing error ellipsoids with multidimensional ranges permits the
 application of efficient search algorithms (see: J. L. Bently et al., An
 Optimal Worst Case Algorithm for Reporting Intersections of Rectangles,
 IEEE TRANSACTIONS OF COMPUTERS, Vol. C-29, pp. 471-77; July 1980; B.
 Chazelle, A Functional Approach to Data Structures and its Use in
 Multidimensional Searching, Siam J. Comput., Vol 17, No. 3, pp. 427-62,
 June 1988) called multidimensional search trees. In our implementation of
 a multidimensional tree search-structure the set of d-dimensional boxes is
 divided (roughly) in half at each node of a ternary tree on one of the d
 coordinates. At each node the set is partitioned into a set of boxes which
 lie entirely to the left of the partitioning hyperplane, a set of boxes
 which lie entirely to the right of the partitioning hyperplane and a set
 of boxes which are intersected by the partitioning hyperplane, cycling
 through the coordinates for partitioning at progressively deeper nodes.
 The method for searching the tree then simply involves the determination
 of which set the search box would be assigned according to the
 partitioning hyperplane at each node. For example, if the search box lies
 entirely to the left of the partitioning hyperplane, then the sub-tree of
 boxes which lie entirely to the right of it can be ignored. Similarly, if
 the search box lies entirely to the right of the partitioning hyperplane,
 then the sub-tree of boxes which lie entirely to the left of it can be
 ignored. Only in the case where the partitioning hyperplane intersects the
 search box must all sub-trees be examined. (Note that the set of boxes
 intersected by the partitioning hyperplane represent a sub-problem of
 reduced dimensionality.) The depth of the tree is roughly log.sub.2 N,
 where N is the number of d-ranges in the tree. In the worst case a single
 search may require N.sup.1-1/d +k comparisons (where k is the average
 number of elements intersecting the search volume and d is the
 dimensionality of the data).
 Tests were performed on datasets ranging in size from 1K to 64K, and for
 data dimensions d=3 and d=8 (see figures). The mean positions of the boxes
 as well as their individual coordinate ranges were drawn independently
 from uniform distributions such that the average number of intersections
 per search query remained constant (five) independent of dataset size.
 Additional tests confirmed that the algorithm is relatively insensitive to
 the distribution of the mean positions; specifically, tests were performed
 using clustered and anisotropic distributions and similar results were
 obtained. These tests were carried out on a SUN4 architecture with an
 instruction rate of about sixteen MIPs. In our test cases for three and
 eight dimensions average scaling performance was consistent with a
 poly-log linear scaling, N(log.sub.2 N).sup.3 for three dimensions and
 N(log.sub.2 N).sup.4 for eight dimensions. We also note that the scaling
 performance was also consistent with a power-law scaling, N.sup.1.3 for
 three dimensions and N.sup.1.47 for eight dimensions. It appears that
 these two interpretations are numerically indistinguishable.
 This efficient gating may be done by only evaluating the association
 probabilities of those distributed state pairs whose coordinate aligned
 circumscribing boxes, or d-ranges, overlap. This is a result of the
 geometric properties of the space of Gaussian-distributed, independent
 random variables. The boxes are simply determined from the data vector, a,
 its covariance matrix, A, and the gating criterion, .gamma., to be defined
 by the set of hyperplanes
EQU p.sub.k =.alpha..sub.k.+-..gamma.A.sub.kk +L {k:k=1, . . . , d} (24)
 The efficient determination of overlaps between boxes can be done with a
 search tree algorithm. In our test cases average scaling performance was
 consistent with a poly-log linear scaling (indistinguishable from some
 power-law significantly less than 2) for the total association evaluation
 task, where N is the size of the data set. The variability of scaling
 appeared to vary only with the dimensionality of the data.
 The efficiency of this method is twofold: the overall scaling of the gating
 process is reduced by use of a multidimensional tree based search
 algorithm, and; the ellipsoidal volumes defined for each distributed state
 are the minimal volumes that can be so defined, minimizing the number of
 computationally expensive evaluations of the association probability.
 For few dimensions, using box intersections appears the most efficient
 method for determining ellipsoid intersections. As the dimensionality
 increases it may be worthwhile to investigate more sophisticated
 approaches to determining ellipsoid intersection other than testing
 ellipsoid pairs with intersecting d-ranges. (This difficulty arises from
 the fact that ellipsoids (L.sub.2 spheres) indicate distances calculated
 with a Euclidean norm and d-ranges (L.sub..infin. spheres) imply distances
 calculated with an L.sub..infin. vector space norm).
 While we have defined minimal volumes for gating of distributed states
 defined around a point, many tasks are defined for states distributed
 around such objects as a trajectory segment defined by a particular time
 period. One might define a suitable heuristic as follows: keeping in mind
 the minimal ellipsoidal volume of the distributed state at any particular
 point in time, the whole trajectory segment might either be circumscribed
 by a single box or represented by a chain of boxes. Once the objects are
 all decomposed into boxes gating as outlined above may proceed.
 FIG. 1 illustrates a situation in which one can use the invention. Platform
 10, such as a stationary ship, has radar 20 which scans a plurality 30 of
 hostile missiles approaching ship 10 radially from the same direction.
 Each radar return informs ship 10 of the radial distance of each object 30
 at the time of the return. Each radar pulse provides ship 10 with a return
 from each object (missile) 30, from which ship 10 can determine radial
 distance of each object 30 by conventional techniques. (Ship 10 could, of
 course, use conventional techniques to determine any of a number of other
 parameters about objects 30, such as doppler shift and velocity,
 acceleration, temperature, luminescence, or any other relevant and
 measurable parameter. Furthermore, object 30 need not approach ship 10
 radially for practice of the invention. These simplifications are made to
 simplify this example, most particularly to draw an example in which d=2,
 a condition more easily visualized than the hyperspace of d.gtoreq.3.)
 Thus, with each pulse ship 10 can record the radial distance for each
 object 30 at a specific time. This forms a data set comprised of two
 dimensional data vectors in which one element is radial position, the
 other time. For effective tracking, ship 10 wishes to know which of the
 data vectors resulting from different radar pulses correspond to the same
 object 30.
 Let the data vectors returned from one pulse be called {.alpha.}, having
 members a.epsilon.{.alpha.} and the set of data vectors returned from
 another pulse be {.beta.}, having members b.epsilon.{.beta.}. The radar
 cannot measure either time or distance perfectly, so the elements of each
 a and b have uncertainties which are considered to be Gaussian
 distributed. Thus associated with each vector a and b of {.alpha.} and
 {.beta.} is a corresponding covariance matrix A or B, and thus each {a,A},
 {b,B} forms a distributed state as defined above.
 FIG. 2 shows the probability density function 50 that one vector
 a.epsilon.{.alpha.} was measured accurately. The contour 50 is a function
 of each point r in the plane formed by orthogonal direction r.sub.1,
 r.sub.2 (which, in this example, corresponds to radial position and time,
 respectively). The point (0,0) in r.sub.1,r.sub.2 indicates an arbitrary
 origin (e.g. the time of the first pulse, and zero radial distance from
 ship 10), and contour 50 is centered about the vector a in
 r.sub.1,r.sub.2. (The axis notation r.sub.1 -a.sub.1, and r.sub.2 -a.sub.2
 merely indicate that these axes are linearly shifted in the
 r.sub.1,r.sub.2 plane to point a, i.e. a is the mean value of the
 distributed state {a,A}.
 Note also that contour 50 is Gaussian, but the spread of the distribution
 is different in the r.sub.1 and r.sub.2 direction. This is because the
 uncertainty in r.sub.1 and r.sub.2 (radial position and time in this
 example) are not identical.
 The most direct way for ship 20 to determine which members of {60 }
 correspond to which members of {.beta.} would be to perform the
 association integral for each vector pair between {.alpha.} and {.beta.}.
 However, it is more efficient to eliminate pairs which have little
 likelihood of correspondence to one another, i.e. which are far from one
 another in state space r.sub.1,r.sub.2. One can do this by selecting for
 each data vector an arbitrary magnitude based upon specific error rates
 required by the specific application at hand (gating criterion) of contour
 50, by cutting contour 50 at this magnitude by a plane 52 parallel to
 plane r.sub.1,r.sub.2, and projecting the intersection contour 50 and
 plane 52 onto the r.sub.1,r.sub.2 plane. This projection 54 is an ellipse
 (for d.gtoreq.3, an ellipsoid; for purposes of this invention, the term
 "ellipsoid " comprehends "ellipse "). Upon doing this for all vectors in
 {.alpha.} and {.beta.}, one can ignore all vector pairs a,b whose
 associated ellipses do not overlap. Thereafter, ship 10 can use any
 conventional method (e.g. evaluating the association integral) to further
 evaluate which vectors in {.alpha.} correspond to which in {.beta.}. by
 eliminating vector pairs which do not meet the gating criterion, one
 materially reduces the number of data pairs one must consider.
 By varying the height 56 of plane 52 above the r.sub.1,r.sub.2 plane, one
 makes the projected ellipse correspondingly larger or smaller. thus by a
 judicious placement of plane 56, one can guarantee overlap of ellipses
 corresponding to any two data vectors one wishes. Varying this height is
 the same as varying the values of .gamma..sub.A, .gamma..sub.B,
 .gamma..sub.AB, etc., as discussed above. The necessary and sufficient
 conditions for such .gamma.'s to ensure overlap are presented above. For
 example, if one knows a priori that two data vectors a.sub.1 and b.sub.1
 do in fact correspond to the same object, one can then calculate an
 optimal gating condition .gamma..sub.A1 =.gamma..sub.B1 =.gamma..sub.A1B1
 (the corresponding ellipses intersect at at least one point), and use this
 as the gating criterion for other data pairs a,b.
 For those data pairs which do meet the gating criterion, ship 10 applies
 conventional techniques for testing associations between data vector
 tracks. These techniques could include evaluation the association integral
 or any other association/correlation measure for each data pair.
 The invention has been described in what is considered to be the most
 practical and preferred embodiments. It is recognized, however, that
 obvious modifications may occur to those with skill in this art.
 Accordingly, the scope of the invention is to be discerned solely by
 reference to the appended claims, wherein: