Unified fusion rules for multisensor multihypothesis network decision ...
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Unified Fusion Rules for Multisensor Multihypothesis Network Decision Systems Yunmin Zhu and X. Rong Li, Senior Member, IEEE
Abstract—In this paper, we present a fusion rule for distributed multihypothesis decision systems where communication patterns among sensors are given and the fusion center may also observe data. It is a specific form of the most general fusion rule, independent of statistical characteristics of observations and decision criteria, and thus, is called a unified fusion rule of the decision system. To achieve globally optimum performance, only sensor rules need to be optimized under the proposed fusion rule for the given conditional distributions of observations and decision criterion. Following this idea, we present a systematic and efficient scheme for generating optimum sensor rules and hence, optimum fusion rules, which reduce computation tremendously as compared with the commonly used exhaustive search. Numerical examples are given, which support the above results and provide a guideline on how to assign sensors to nodes in a signal detection networks with a given communication pattern. In addition, performance of parallel and tandem networks is compared. Index Terms—Distributed decision, global optimization, optimum sensor rule, unified fusion rule.
I. INTRODUCTION
T
HE DISTRIBUTED decision problem continues to attract much research interest, as evidenced by recent publications such as [1]–[21]. This is because a system with multiple sensors has many advantages over one with a single sensor, such as, the increase in the reliability, robustness, and survivability of the system [4], [9]. Consider the following distributed system. Each local sensor observes data and possibly a number of compressed binary messages (information bits) from other sensors simultaneously; it locally fuses/compresses all its data and received messages, which are then transmitted to other sensors; finally, the sensor at the top level node (i.e., the fusion center) in the network makes a final decision by combining all the received information using some fusion (final decision) rule. Communications between each sensor and the fusion center, as well as among sensors are permitted. These networks are of a parallel, tandem, or, their hybrid–tree–topology and thus, are more general than those considered in [1]–[14], formulated in [15] and [16], and reviewed in [17] and [18]. To optimize the performance of the
Manuscript received January 25, 2002; revised September 18, 2002. This work was supported in part by the National Key Project and NNSF of China (#60074017) and by ONR under Grant N00014-00-1-0677, NSF under Grant ECS-9734285, and NASA/LEQSF Grant (2001-4)-01. This paper was recommended by Associate Editor S. E. Shimony. Y. Zhu is with the Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China (e-mail: [email protected]). X. R. Li is with the Department of Electrical Engineering, University of New Orleans, New Orleans, LA 70148 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TSMCA.2003.809211
system globally, it is customary to find an optimum fusion rule from all possible fusion rules (see the formulation in [16] and [18]) and then determine the corresponding set of optimal local sensor compression rules under a given communication pattern. Usually, to evaluate the merits of two different fusion rules, we determine the two corresponding sets of optimal sensor rules and then compare the two final costs. However, the number of possible fusion rules increases exponentially with the number of sensors or the number of bits received by the fusion center. An exhaustive method obviously is computationally intractable. To the authors’ knowledge, however, there has been hardly any theoretical result on how to find a globally optimum fusion rule more efficiently. In [1], we discussed the optimal sensor rules for a fixed fusion rule and unified fusion rules in a parallel binary Bayesian decision system, where no communications among local sensors are permitted, without the commonly used data independence assumption. In this paper, we extend the results to the aforementioned general distributed multihypothesis decision system. When the network architecture and its communication pattern are given, we propose a unified fusion rule, which is actually a specific form of the most general fusion rule, as it is independent of the statistical characteristics of observations and decision criteria. To achieve the globally optimum performance, we now only need to optimize sensor rules under this single fusion rule for the given conditional distribution of observations and decision criterion. In other words, this rule is optimum if the sensor rules are optimized. This reflects a huge saving in computation. For example, for a three-sensor 4-ary parallel decision system with a total of ten sensor bits, using the proposed unified fusion rule, we only need to optimize the ten sensor bits to achieve the globally optimum performance. Using the exhaustive search, however, we need to compare more than trillions different fusion of sensor bits because there are rules! While the above network decision systems are also discussed in the literature [15]–[18], no alternative to the intractable exhaustive search for a globally optimum fusion rule from a large number of possible fusion rules has been proposed prior to this work. Performance comparisons between the parallel and tandem networks were presented in [15] and [17], where the main conclusions are: In the two-sensor case the tandem network is dominant; in the case with more than two sensors, one does not dominate the other in general, but the parallel network outperforms the tandem network asymptotically as the number of sensors goes to infinity, although the number of sensors at which the parallel network becomes superior is not known. In this
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ZHU AND LI: UNIFIED FUSION RULES FOR MULTISENSOR MULTI-HYPOTHESIS NETWORK
paper, we give a more reasonable and precise comparison. For instance, we conclude that the aforementioned asymptotic superiority of the parallel network to the tandem network does not hold generally. We also provide a number of numerical examples. They support the above results and illustrate how to assign compression rates to local sensors by comparing their signal-to-noise ratios — a lower compression rate should be assigned to a sensor with a larger SNR. The examples also demonstrate that a parallel network outperforms a tandem network while detecting a Gaussian signal in Gaussian noise. The rest of the paper is organized as follows. In Section II, we formulate the problems associated with the three types of network decision systems. In Section III, we discuss unified fusion rules under different information structures for distributed decision systems. Section IV is dedicated to an iterative algorithm for searching for optimal sensor compression rules given a fusion rule for a general distributed -ary decision problem. In Section V, a performance comparison between the parallel and tandem networks with the same amount of communication is presented. In Section VI, several numerical examples are given, which substantiate the analytic results in the previous sections. Finally, we provide conclusions in Section VII.
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Fig. 1. Parallel network.
II. REPRESENTATIONS OF NETWORK STRUCTURES AND DECISION RULES In this section, we consider two types of elementary distributed systems: parallel and tandem. A. Parallel Network The parallel network is a basic information structure of a distributed decision system. We consider a distributed decihypotheses, , and sion problem with sensors, , with multiple observation data in space . A set of local compression rules, , where
compresses data to information bits at each sensor . Obviously, binary digits can correspond to different integers, that is, the th sensor quantizes its observation to different integers. Then the local sensors transmit possibly their compressed binary messages .. .
Fig. 2. Modified parallel network.
sensor compresses its observation to information bits. ” implies Moreover, “ in “{}” transmits information bits to the that each sensor fusion center , where an -ary decision is made. In addition, assume that known conditional probability density funcare of arbitions trary forms. for a parallel network is given by an A fusion rule of -valued function (2) In practice, we could build one of the sensors, say, , and the fusion center in the same station to save communication between this sensor and the fusion center. We call such a network a modified parallel network (see Fig. 2). The corresponding expression is (3)
to a fusion center . Let be the total information tuple), bits. Upon the receipt of the local message (an , the fusion center makes a final decision under some fusion rule (see Fig. 1). We denote the above information structure by (1) denotes that all sensors inside “{}” are in where ” means that parallel without mutual communications and “
In this structure, we do not care how large is because no trans, mission is required for sensor . Furthermore, when where is an integer, we can also regard sensor , which compresses its received information to bits, as an intermediate node in the network, and accordingly view the fusion rule as a local compression rule. Also, note that the results on the unified fusion rule in [1] depend not on any cost functional but only on the set of final decision partitions of the joint observation space generated by this fusion rule together with all possible local compression rules. This point of view is useful for dealing
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Fig. 3.
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Fig. 4.
Tandem network.
with (local or final) compression rules at either an intermediate node or the fusion center. B. Tandem Network The sensor network is a team of sensors in tandem. That is, the first sensor compresses its observation data to inforand transmits them mation bits to the second sensor . Then due to communication bandwidth compresses its observation and the received meslimit, to bits and transsage mits them to the next sensor , and so on. This is repeated until th sensor . The last sensor uses its observathe tion together with the received message to make a final -ary decision under a fu. In this model, sensor and the fusion center are sion rule co-located (see Fig. 3). is also a binary Notice that function, which will be referred to as a local fusion rule of sensor and can be expressed equivalently (see [1]) by binary , as functions of , follows:
(4) are binary functions and thus, because all different -tuples . Let . there are It turns out that we can view these as local compression rules for at the th sensor. A final for the tandem network now is dependent on fusion rule . Since the local fusion rules further depend on and local all previous local fusion rules , finally can be written as compression rules . an -valued function of Similarly as for the parallel network, we denote the tandem network by (5)
Tree network.
where “ data and the to the next sensor alently as
” means that sensor compresses its own received bits to bits and transmits them . Similar to (4), we can rewrite (5) equiv-
(6) and “ ” means that sensor comwhere bits, fuses them and the presses its own data to received bits to bits, and transmits these bits to the . Besides, as derived in Section III, we can use next sensor without information loss, where is an integer sat. However, it must be kept isfying inequality in general are comin mind that the local fusion results ), that is, from pressed from ( bits to bits; hence, there are at most (rather ) subsets in possibly observed by than the fusion center in the tandem network. Clearly, the tandem network is another basic information structure in distributed decision systems. Combining the above two basic structures, an arbitrary tree network can be constructed (see Fig. 4). C. Cost Functionals Since the focus of this paper is on presenting unified fusion rules (so as to obtain optimum fusion rules) and comparing relative merits of fusion rules, suppose that we can always find optimum local compression rules for any given cost functional and any fusion rule (see [1] and Section IV below for details that support this statement). Thus, minimization of a cost functional by a fusion rule depends only on the set of possible final decision partitions produced by the rule. When this set of a fusion rule contains those of all fusion rules, this fusion rule is a unified one. It is optimum if the corresponding sensor rules are optimum given this fusion rule. It follows that the so-obtained optimum fusion rules of this paper do not depend on what test (e.g., Bayes or Neyman-Pearson test) is considered. However, for simplicity of presentation, we only consider Bayesian cost functionals in the sequel.
ZHU AND LI: UNIFIED FUSION RULES FOR MULTISENSOR MULTI-HYPOTHESIS NETWORK
The Bayesian cost is (7) is some suitable cost coefficients; is a priori where each ; and each is the probability for hypothesis conditional probability of the event that the final decision is is true, , where equal to while denotes , , and for the parallel, tandem, and tree networks, respectively. Substituting the conditional probabiliinto (7) and simplifying yield ties given
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globally optimum final decision rule consisting of disjoint de, respectively, that is, an cision regions for . It optimum partition of the set of possible -tuples follows from the above analysis that every final decision region uniquely corresponds to a summafor tion polynomial of some local-message polynomials, denoted . Obviously, they are still indicator functions by since the regions determined by of a region in different local-message polynomials are disjoint—for a given one and only one of the values of observation data the local-message polynomials equals 1 and the others are all equal to 0. In other words, we have
(8) and a correOur goal is to select an optimum fusion rule sponding set of optimum local compression rules that jointly minimize the cost functional. D. Polynomial Representations of Decision Rules In this subsection, we extend the formulation presented in [1] for a distributed binary decision system to the corresponding -ary decision system. , , , are actually All local compression rules in . Let and indicator functions of the set represent the regions over which is compressed to digit 1 and , 0, respectively, by the sensor using the compression rule that is (9) (10) As such, there is a bijective relationship between a pair and a pair of simple polyof local regions . Furthermore, a local message nomials , where and or , and , corresponds uniquely to of the above simple a product polynomial polynomials, where
(11) as the decision polynomial. As such, we We refer to have established a correspondence between a final decision rule and a set of final decision polynomials of local compression rules. Now, let us give two examples to show how to write fusion rules as their polynomial versions. Example 2.1: Consider a 3-sensor ternary parallel network, where there is only one compression rule at each sensor
A fusion rule
is given by making decision (i.e., ) if any of the four local messages (111), (110), (101), (011) is received by the fusion , center, where decision region is defined by {(000)}. and so on; and the , , and decision polynomials are Thus, the
Example 2.2: Consider a tandem network of the following structure
, We call a local-message polynomial, which is also an . Obviously indicator function of a region in
A fusion rule partitions the set of (recall ) into disjoint subsets generdifferent -tuples different such partitions for a parallel netally. There are work and normally may be fewer for a tandem network (see the analysis at the last part of Subsection II-B). In addition, a fixed fusion rule combining all possible local compression rules can produce infinitely many different final decision regions in a continuous model. The goal of our distributed decision is to find a
where a local fusion rule at sensor is defined as if and only if the local message is either or at sensor , and otherwise. An decision region is given by
which is actually, using the definition of
above
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Thus, the corresponding
Likewise, an
decision polynomial
is
Now suppose an -sensor parallel network is given by (3). , that is, the fusion center Consider a special case with compresses all received and observed information to a binary or of length . This case code is similar to that of the intermediate node in the analysis around (4). Since is also the fusion center, the general fusion rule is
decision polynomial can be written as where sub-rules of fusion:
(12) . Clearly, this fusion rule consists of
.. .
Obviously, two regions and are disjoint. Finally, decision polynomial must be . the Thus, it can be seen that using decision polynomials to represent a fusion rule is much more convenient and simpler than using local messages. Example 2.3: Now we use the parallel and tandem structures to construct a tree structure as follows:
A fusion rule for this structure is given in Example 6.3, which is actually a unified fusion rule by Theorems 3.1 and 3.2. III. UNIFIED FUSION RULES In this section, extending the results in [1] we present a unified form of the most general fusion rule, referred to as a unified fusion rule, for the aforementioned distributed decision systems. This form includes any fusion rule under the information structure considered as a special case and thus the globally optimum fusion rule also has this form. A fusion rule in this form that optimizes the cost functional is therefore an optimal fusion rule. In addition, this unified fusion rule does not depend on the statistical characteristics of sensor observations or decision criteria. To achieve the global optimal performance, traditionally we find an optimum fusion rule and determine the corresponding optimum sensor rules. As seen from the simple 3-sensor ternary decision examples in Section II, however, the number of possible fusion rules is very large and thus finding optimum fusion rules is computationally intensive and often intractable. Using the result presented in this section, we only need to optimize sensor rules under the unified fusion rule for the given decision criterion and conditional distributions of observations. Therefore, using this fusion rule can save huge computation. We consider the unified fusion rule for the parallel network first, and then treat each level in a tandem or tree network as a parallel network and obtain the corresponding unified local fusion rule. Finally, the global unified fusion rule for the whole tandem or tree network is composed by all unified local fusion rules.
(13) is a general fusion rule for an -sensor biNotice that every nary decision system. Recall the basic idea and technique of [1], in fact defines sensor compression rules at where every the th sensor as follows:
(14)
.. .
different -tuples for a fixed set of . In other words, the general information structure considered with a comamounts to sensor compressing into pletely known bits. The corresponding general fusion rule is given in the following unified form
since there are
.. . (15) is the th binary decision region for for where the general fusion rule given in (12). This fusion rule has the same optimum performance as that of the fusion center using and all compressed local messages from sensors the uncompressed data to make a binary decision. It follows from Subsection II-B that the corresponding decision polynoand can be easily obtained from mials . Now we extend the above idea and technique to an -ary . The sub-rules define binary sensor system with compression rules
ZHU AND LI: UNIFIED FUSION RULES FOR MULTISENSOR MULTI-HYPOTHESIS NETWORK
at the th sensor in terms of their polynomials
. We can write them
.. .
(16) decision polynomials It follows that the corresponding are given by , . Using polynomials, we define the -ary final decision rule these as follows. First, define a 1-1 correspondence between integers and an -tuple binary code . is defined by a given for any Suppose , . Then we can obtain a polynomial so that
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some of observation data arbitrarily. Then a unified to fuse and tolocal/global rule for in this case) gether to different digits (or bits with is given above. An immediate consequence of Theorem 3.1(b) is the following. Corollary 3.1: For a multisensor -ary decision system, makes observation and suppose that a sensor, say, simultaneously, receives information bits may depend on some of observation where arbitrarily. Then needs to send at most data bits to the fusion center, where satisfies inequality . These two theorems present a unified fusion rule for a parthe fusion allel network. As mentioned before, when center integrated with a local sensor in a parallel network can be viewed as an intermediate node in a multisensor network. In addition, the unified fusion rule here and in [1] is nothing but the most general fusion rule that includes all possible fusion rules as special cases in the sense mentioned in Theorems 4.1 and 4.2 of [1]. Consequently, we propose a unified fusion rule for a tandem network as follows. Theorem 3.2: For the following information structure
(17) where
and
is given by (18)
, i.e., , we In the case of local compression rules at still need to determine the th sensor. The only modification is to define some just for times so that the now is rather than . It follows total number of in (15), (16), and (17) that the from the definition of decision regions determined by these polynomials and can form any possible partition with members. Therefore, this fusion rule is a unified -ary fusion rule. Note that we do not increase communication by doing so, although we need to determine more optimal local compression rules, which is still tractable in practice. From the above argument, we have the following theorem. Theorem 3.1(a): For the following parallel structure
or
the fusion rule constructed as above is a unified one: Any fusion rule is a special case of it with some of the local compression or ). rules identically zero or unity (i.e., For convenience of treating tandem and tree networks, we rewrite Theorem 3.1(a) in the following more general version. Theorem 3.1(b): Suppose that a sensor, say, makes obserand receives information bits vation simultaneously, where may depend on
where “ ” means that sensor compresses its to bits ( , ), own data a general fusion rule of a unified form can be constructed sequentially as follows: Starting with the last sensor , at each ” (if ,“ ” is replaced sub-structure “ ”), we determine a set of unified local by “ (or final fusion rule ) by the fusion rules above method for a parallel network, where is replaced by from in the tandem netif . work and is replaced by Proof: Using Theorem 3.1(b), we know that the above fu” gives the unision rule at each sub-structure “ ) fusion rule in the sense that any fied local (or final if specific local fusion rule in the sub-structure is a special case of it. Therefore, at the top level sensor , the final fusion rule given in this theorem must be also a unified fusion rule. By Theorem 3.2, it can be verified that the fusion rule given in Example 2.2 is actually a unified fusion rule for the given information structure. More examples are given in the next section. Using Theorems 3.1 and 3.2 together, the corresponding unified fusion rule for a tree network can be given. We illustrate in Example 6.3 how to construct such a unified fusion rule for Example 2.3. IV. OPTIMUM SENSOR RULES Given the unified fusion rule of Section III, to have an optimal decision system we need to find the corresponding optimal
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sensor rules. In this section we extend the results in Sections II and III of [1] to the network decision models here. We first present a necessary condition for local compression rules to be optimum for any fixed fusion rule in the above distributed decision systems, and then propose a Gauss-Seidel iterative algorithm and its discrete version. Finally we give some convergence results to show that the discrete algorithm converges in finite steps to a minimum of the discrete cost functional and that under a mild assumption on the integrand of the cost functional, the global minimum of the discrete cost functional converges to the infimum of the original continuous cost functional as the discretization step size tends to zero. Since the extension here is straightforward except for the formulation, we present the relevant results without argument. A. A Necessary Condition for Optimum Local Compression Rules
.. .
.. . (20) where
is a indicator function defined by .
(21)
This necessary condition is of a fixed-point type. It is an extension of Theorem 2.1 in [1]. In particular, if we define the mapping
and
Let
. It follows from Eq. (11) and the same argument as in [1] ) that we can write the (i.e., integrand in (8) in various forms as follows:
.. . .. . (22) .. .
.. . then Theorems 4.1 shows that a set of optimal local compression rules must be a fixed point of the map in (22), which is a solution of the integral in (20).
.. .
B. Iterative Algorithms and Convergence .. . (19) and are independent of , where functions , . Now we propose the following necessary condition for optimum local compression rules. Theorem 4.1: Suppose we have a distributed decision system employing fusion rule (11). A set of optimal local decision that minimizes the rules cost functional (8) must satisfy the following integral equations
Let the local sensor rules at the th stage of iteration be denoted by with a given initial rules . Suppose the given fusion rule of (11) is employed. We now consider the following Gauss-Seidel iterative algorithm for the mapping .
.. .
.. . .. . .. .
ZHU AND LI: UNIFIED FUSION RULES FOR MULTISENSOR MULTI-HYPOTHESIS NETWORK
(23) To facilitate computer implementation of this iteration, we need to discretize the variables. Let the discretization of be given, respectively, by
For each iteration and for , let the -vector denote the values of at the discretization points such that , . Thus, the iteration of (23) can be approximated as, for
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Theorem 4.2: For any positive discretization step size of the elements of and any initial ( . ), the algorithm of (24) terminates at satisfying (25) after a finite number of some iterations. Outline of proof: The proof consists of two steps. First, we prove that the Bayesian cost decreases as iteration goes. Hence, the cost reaches a minimum after a finite number of iterations. However, this does not imply either the finite convergence of local sensor rules or the attainment of the minimal cost by the algorithm without the convergence of local sensor rules. Second, we can prove the latter convergence by contradiction. A detailed proof follows the same argument as that of Theorem 2.1 in [1]. Let
denote the discretization step size of each element of and be the minimum of the discrete verof . Similarly, we have sion a convergence result corresponding to Theorem 2.2 of [1], which states that under mild assumption on the integrand in the above exists and is cost functional, as tends to zero the limit of . We omit the details equal to the infimum of here.
.. .
V. PERFORMANCE COMPARISON OF PARALLEL AND TANDEM NETWORKS .. .
.. .
(24) are the step sizes for discretizing the vecwhere , respectively. Iteration (24) is the corresponding tors discretized version of the continuous iteration (23) and is readily implementable. A simple termination criterion for this iteration , and is, for all
In [15] and [17], some performance comparison results between the parallel and tandem networks were presented. Their main conclusions are: The tandem network is dominant in two sensor case; for cases with more than two sensors, one does not dominate the other in general, but the parallel network outperforms the tandem network asymptotically as the number of sensors goes to infinity, although the value of sensor number at which the parallel network becomes superior is not known. In this section, we give a more reasonable and rigorous comparison. We only consider cases with more than two sensors because two-sensor parallel and tandem networks are the same if the second sensor in the parallel network and the fusion center are colocated. To fairly compare the performance of various networks in the sequel, we assume that the fusion center in the parallel network is also a sensor and that all other corresponding sensors in both networks transmit the same number of information bits to the fusion center (in the parallel network case) or to the next sensor (in the tandem network case). For notational simplicity, we further assume that every sensor is one-bit sensor. In other words, we compare the performance of the following two decision networks:
(25) An alternative is
(27) and
(26)
(28)
is some prespecified tolerance. where We now examine the convergence of the iteration (23). We have the following theorem on the finite convergence of its discrete Gauss-Seidel iteration process.
Our analysis shows that in general, a dominance between the parallel and tandem networks does not exist no matter how large the number is: There is no dominance between two sets of possible partitions of the joint observation space generated by the
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parallel and tandem networks, respectively—there always exist partitions generated by one network that cannot be generated by the other—and we can easily construct examples in which the optimal decision region for the centralized is the region that can be generated by one network but not the other. It follows from (15), (16), and (17) that the above tandem network is equivalent to the following tandem network:
A. Three-Sensor System The hypotheses are
where the signals and and the noise mutually independent, and
,
and
are all
(29) Hence, from the second sensor on, each sensor actually can compress its own observation to two bits first and then fuses them and the received single bit from the previous sensor to a single bit and transmit it out. However, in the above parallel network, every local sensor only compresses its own observation into a single bit and thus there always exist some decision regions generated by the tandem network that in general cannot be generated by the parallel network, such as
On the other hand, while the fusion center in an -sensor parinformation bits from the allel network can receive local sensors, the fusion center in the tandem network of the same number of sensors can only receive a total of a single bit from all other local sensors. It turns out that some final decision regions that can be generated by the parallel network cannot be decision generated by the tandem network. For instance, the region for the parallel network
is an example, which cannot be generated by the tandem network since the fusion center here uses more than a single bit information from the first two sensor. Obviously, this situation cannot change even as the number of sensors goes to infinity. In practice, it is rare that a better decision region can only be generated by the tandem network but not by the parallel network because information is compressed much more in the tandem network than in the parallel network. Therefore, in most practical situations, the parallel network often outperforms the corresponding tandem network (see the numerical examples in the next section). However, the tandem network has better survivability than the parallel network, particularly in a war situation: When a fusion center is destroyed, the damaged parallel network above is a set of single-sensor decision makers, but a tandem network would become a smaller tandem network, which would perform in general better than a single-sensor decision maker.
Hence, the three conditional Probability Density Functions (PDFs) are
In all examples below, we take for , , , . As such, the Bayessian cost functional actually becomes decision error probability . Example 6.1: The information structures are
and
which can be obtained by adding seven local compression rules in Example 2.1. By Theorem 3.1 the unified (and to sensor thus optimum) fusion rule is defined by the following three decision polynomials:
It can be seen from (19) and this fusion rule that the following , , , ) are sufficient to six initial values ( , , run the iterative algorithm to achieve the optimal performance. For performance evaluation, we compare its final decision cost, denoted by , with the centralized decision cost for initial values
VI. NUMERICAL EXAMPLES We consider 3- and 4-sensor detection systems for Gaussian signals in additive Gaussian noise.
for initial values
ZHU AND LI: UNIFIED FUSION RULES FOR MULTISENSOR MULTI-HYPOTHESIS NETWORK
where is a indicator function defined by (21). From this example, we see that the optimum distributed decision (ODD) is slightly more costlier than the centralized decision (CD) but not sensitive to the initial values. Now we interchange the positions and in the information structure, that is, of sensors with the smallest signal-to-noise ratio (SNR) now is with the largest SNR is compressed to 8 bits and compressed to 1 bit. This yields for initial values
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Example 6.3: We consider the same information structure as that of Example 2.3.
A set of optimum decision polynomials can be constructed sequentially in three steps. , , and we construct an Step I. Using optimum fusion rule at the last sensor :
for initial values This case has significantly inferior performance to the previous case, as expected. Example 6.2: The information structure is
The unified (and thus optimum) fusion rule by Theorem 3.2 is just the fusion rule shown in Example 2.2. It can be seen from (4.1) and the above fusion rule that the following five initial , , , ) are sufficient to run the iterative values ( , algorithm to achieve the optimal performance. The results for similar initial values as in Example 6.1 are Step II. Using , , and we construct the four , at polynomials of the two optimum local fusion rules the sensor
for initial values for initial values The ODD costs here are still not far away from the CD cost but larger than the ODD cost in Example 6.1. Similar to Example 6.1, we interchange the positions of senand . This yields, for the same initial values sors with initial values with initial values As before, the performance of the first case is significantly better than that of the second case.
Step III. Substituting the above two polynomials into , and , we get the final decision polynomials with respect to all local compression rules . It can be seen from (15) and the prior fusion rule that the following 18 initial values for are sufficient to run the iterative algorithm. The decision error , for initial values probabilities are
B. Four-Sensor System Now we add a sensor with additive noise the above three-sensor systems. The three PDFs become
to and
for initial values
The ODD costs here are still not far away from CD cost and not sensitive to the initial values. Example 6.4: To compare the performance of the above information structure with that of a parallel system, we consider the following structure:
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The optimum fusion rule can be constructed similarly as in Example 6.1 and thus is omitted. To run the iterative algorithm, 11 initial local compression rules are needed. for initial values The results are
of data in the latter. All these results have been substantiated by numerical examples. Further, extension to other distributed decision systems, such as Neyman-Pearson and sequential decision systems, can be made (see [20] and [21]).
ACKNOWLEDGMENT
and
for initial values
The authors would like to acknowledge many valuable suggestions of the referees and editor to this work.
REFERENCES Here it is required to communicate 3 bits and compute 19 local compression rules. The performance appears to be better than that of Example 6.3, where communication of 4 bits and computation of 18 local compression rules are required. In summary, although the distributed decision systems employing our proposed rules have slightly worse performance than the optimum centralized decision systems, the performance differences appear acceptable since they are minor and the distributed systems require much less communication than the centralized; the sensor with the largest SNR should have the least data compression and be built at the same place with the fusion center; and the modified parallel network has a superior performance to the tandem network in most practical situations.
VII. CONCLUSIONS In this paper, we have extended the results on unified fusion rules and optimum local compression rules of [1] to the general distributed network (Bayesian) decision problem. We have formulated two types of elementary multisensor network structures, and shown via examples how to construct an arbitrary tree network by these structures. For networks in which (global or local) fusion center can observe data, we have presented a general fusion rule in a unified form, called unified fusion rule, which includes all possible fusion rules as special cases. In essence, its generality stems from a guarantee of a certain number of (fusion or compression) rules at the (global or local) fusion center, which is achieved by the specific form proposed. We have shown that the proposed fusion rule is unified and general in that it depends on neither the statistical properties of data nor decision criteria. To achieve globally optimum performance, we only need to optimize sensor rules under this unified fusion rule for a given decision criterion and conditional distributions of observations. In this sense, this unified fusion rule is also optimum. We have also made a comparison analysis on the performance of a modified parallel network and the tandem network. In contrast to some statements in the literature, there exists no general performance dominance between them no matter how large the number of sensors is. In practice, however, the modified parallel network often has better performance than the tandem network due to the higher compression
[1] Y. M. Zhu, R. S. Blum, Z. Q. Luo, and K. M. Wong, “Unexpected properties and optimum distributed sensor detectors for dependent observation cases,” IEEE Trans. Automat. Contr., vol. 45, pp. 62–72, Jan. 2000. [2] Y. M. Zhu, Multisensor Decision and Estimation Fusion. Norwell, MA: Kluwer, 2003. [3] R. R. Tenney and N. R. Sandell Jr., “Detection with distributed sensors,” IEEE Trans. Aerosp. Electron. Syst., vol. AES-17, pp. 501–510, 1981. [4] B. V. Dasarathy, Decision Fusion. Los Alamitos, CA: IEEE Comput. Soc. Press, 1994. [5] L. A. Klein, Sensor and Data Fusion Concepts and Applications. Bellingham, WA: SPIE Press, 1993. [6] S. Chaudhuri, A. Crandall, and D. Reidy, “Multisensor data fusion for mine detection,” in Proc. SPIE—Sensor Fusion III: 3-D Perception and Recognition, vol. 1306. Bellingham, WA, 1990, pp. 187–204. [7] J. Han, P. K. Varshney, and V. C. Vannicola, “Distributed detection of moving optical objects,” in Proc. SPIE—Sensor Fusion III: 3-D Perception and Recognition, vol. 1306. Bellingham, WA, 1990, pp. 115–124. [8] R. C. Luo and M. G. Kay, “Multisensor integration and fusion in intelligent systems,” IEEE Trans. Syst., Man, Cybern., vol. 19, pp. 901–931, Sept./Oct. 1989. [9] D. D. Freedman and P. A. Smyton, “Overview of data fusion activities,” in Proc. Amer.Control Conf., San Francisco, CA, June 1993. [10] H. Delic, H. Papantoni-Kazakos, and D. Kazakos, “Fundamental structures and asymptotic performance criteria in decentralized binary hypothesis testing,” IEEE Trans. Commun., vol. 43, pp. 32–43, Jan, 1995. [11] Z. B. Tang, K. R. Pattipati, and D. K. Kleinman, “An algorithm for determining the decision threshold in a distributed detection problem,” IEEE Trans. Syst., Man, Cybern., vol. 21, pp. 231–237, Jan./Feb. 1991. [12] J. N. Tsitsiklis and M. Athans, “On the complexity of decentralized decision making and detection problems,” IEEE Trans. Automat. Contr., vol. AC-30, pp. 440–446, Apr. 1985. [13] V. Venugopal, V. Veeravalli, T. Basar, and H. V. Poor, “Decentralized sequential detection with a fusion center performing the sequential test,” IEEE Trans. Inform. Theory, vol. 37, pp. 433–442, Mar. 1993. [14] V. Veeravalli, T. Basar, and H. V. Poor, “Minimax Robust decentralized detection,” IEEE Trans. Inform. Theory, vol. 38, pp. 35–40, Jan. 1994. [15] J. D. Papastavrou and M. Athans, “Distributed detection by a large team of sensors in tandem,” IEEE Trans. Aerosp. Electron. Syst., vol. 28, pp. 639–652, July 1992. [16] J. N. Tsitsiklis, “Decentralized detection,” in Advances in Statistical Signal Processing, H. V. Poor and J. B. Thomas, Eds. Greenwich, CT: JAI, 1993, 2. [17] R. Vismanathan and P. K. Varshney, “Distributed detection with multiple sensors: Part I—Fundamentals,” Proc. IEEE, vol. 85, pp. 54–63, Jan. 1997. [18] P. K. Varshney, Distributed Detection and Data Fusion. New York: Springer-Verlag, 1997. [19] R. S. Blum, S. A. Kassam, and H. V. Poor, “Distributed detection with multiple sensors: Part II—Advanced topics,” Proc. IEEE, vol. 85, pp. 64–79, Jan. 1997. [20] Y. M. Zhu, C. Y. Liu, and Y. Gan, “Multisensor distributed Neyman-Pearson decision with correlated sensor observations,” in Proc. Int. Conf. Multisource-Multisensor Information Fusion, vol. 1, 1998, pp. 35–39. [21] Y. M. Zhu, Y. Gan, K. S. Zhang, and C. Y. Liu, “Multisensor NeymanPearson type sequential decision with correlated sensor observations,” in Proc. Int. Conf. Multisource-Multisensor Information Fusion, vol. 2, 1998, pp. 787–793.
ZHU AND LI: UNIFIED FUSION RULES FOR MULTISENSOR MULTI-HYPOTHESIS NETWORK
Yunmin Zhu received the B.S. degree in mathematics and mechanics, Beijing University, Beijing, China, in 1968. From 1968 to 1978, he was with Luoyang Tractor Factory, Luoyang, Henan, China, as a Steel Worker and a Machine Engineer. From 1981 to 1994 he was with the Institute of Mathematical Sciences, Chengdu Institute of Computer Applications, Chengdu Branch, Academia Sinica. Since 1995, he has been with the Department of Mathematics, Sichuan University as Professor. During 1986 to 1987, 1989 to 1990, 1993 to 1996, 1998 to 1999, and 2001 to 2002, he was a Visiting Associate, or Visiting Professor at Lefschetz Centre for Dynamical Systems and Division of Applied Mathematics, Brown University, Providence, RI; Department of Electrical Engineering, McGill University, Montreal, QC, Canada; Communications Research Laboratory, McMaster University, Hamilton, ON, Canada; Department of Electrical Engineering, University of New Orleans, New Orleans, LA. His research interests include stochastic approximations, adaptive filtering, other stochastic recursive algorithms and their applications in estimations, optimizations, and decisions for dynamic system as well as for signal processing, information compression. In particular, his present major interest is multisensor distributed estimation and decision fusion. He is the author or coauthor of over 50 papers in international and Chinese journals. He is the author of Multisensor Decision and Estimation Fusion (Norwell, MA: Kluwer, 2002) and Multisensor Distributed Statistical Decision (Beijing: Science Press, Chinese Academy of Science, 2000), and coauthor (with Prof. H. F. Chen) of Stochastic Approximations (Shanghai Scientific & Technical Publishers, 1996). He is on the editorial board of the Journal of Control Theory and Applications, South China University of Technology, Guangzhou.
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X. Rong Li (S’90-M’92-SM’95) received the B.S. degree and M.S. degree from Zhejiang University, Hangzhou, Zhejiang, China., in 1982 and 1984, respectively, and the M.S. degree and Ph.D. degree from University of Connecticut, Storrs, in 1990 and 1992, respectively. He joined the Department of Electrical Engineering, University of New Orleans, New Orleans, LA, in 1994, where he is now University Research Professor and Department Chair. From 1986 to 1987, he did research on electric power at University of Calgary, Calgary, AB, Canada. He was an Assistant Professor at University of Hartford, West Hartford, CT, from 1992 to 1994. He has authored or coauthored four books: Estimation and Tracking (with Yaakov Bar-Shalom), (Norwood, MA: Artech House, 1993), Multitarget-Multisensor Tracking (with Yaakov Bar-Shalom), (Storrs, CT: YBS, 1995), Probability, Random Signals, and Statistics (Boca Baton, FL: CRC Press, 1999), and Estimation with Applications to Tracking and Navigation (with Yaakov Bar-Shalom and T. Kirubarajan), (New York, Wiley, 2001); six book chapters, and more than 160 journal articles and conference papers. His current research interests include signal and data processing, target tracking and information fusion, stochastic systems, statistical inference, and electric power. Dr. Li has served as Associate Editor for IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS from 1995 to 1996, and as Editor since 1996. He received a Career award and an RIA award from the U.S. National Science Foundation. He received a 1996 Early Career Award for excellence in Research from University of New Orleans and has given numerous seminars and short courses in the U.S., Europe and Asia. He won several outstanding paper awards, is listed in Marquis’ Who’s Who in America and Who’s Who in Science and Engineering, and consulted for several companies. He has served the International Society of Information Fusion as the President, from 2003 to the present, Vice President, from 1998 to 2002, and a member of Board of Directors since 1998; served as General Chair for 2002 International Conference on Information Fusion, and Steering Chair or General Vice-Chair for 1998, 1999, and 2000 International Conferences on Information Fusion. He has served as Editor for Communications in Information and Systems since 2001.
IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART A: SYSTEMS AND HUMANS, VOL. 33, NO. 4, JULY 2003
Unified Fusion Rules for Multisensor Multihypothesis Network Decision Systems Yunmin Zhu and X. Rong Li, Senior Member, IEEE
Abstract—In this paper, we present a fusion rule for distributed multihypothesis decision systems where communication patterns among sensors are given and the fusion center may also observe data. It is a specific form of the most general fusion rule, independent of statistical characteristics of observations and decision criteria, and thus, is called a unified fusion rule of the decision system. To achieve globally optimum performance, only sensor rules need to be optimized under the proposed fusion rule for the given conditional distributions of observations and decision criterion. Following this idea, we present a systematic and efficient scheme for generating optimum sensor rules and hence, optimum fusion rules, which reduce computation tremendously as compared with the commonly used exhaustive search. Numerical examples are given, which support the above results and provide a guideline on how to assign sensors to nodes in a signal detection networks with a given communication pattern. In addition, performance of parallel and tandem networks is compared. Index Terms—Distributed decision, global optimization, optimum sensor rule, unified fusion rule.
I. INTRODUCTION
T
HE DISTRIBUTED decision problem continues to attract much research interest, as evidenced by recent publications such as [1]–[21]. This is because a system with multiple sensors has many advantages over one with a single sensor, such as, the increase in the reliability, robustness, and survivability of the system [4], [9]. Consider the following distributed system. Each local sensor observes data and possibly a number of compressed binary messages (information bits) from other sensors simultaneously; it locally fuses/compresses all its data and received messages, which are then transmitted to other sensors; finally, the sensor at the top level node (i.e., the fusion center) in the network makes a final decision by combining all the received information using some fusion (final decision) rule. Communications between each sensor and the fusion center, as well as among sensors are permitted. These networks are of a parallel, tandem, or, their hybrid–tree–topology and thus, are more general than those considered in [1]–[14], formulated in [15] and [16], and reviewed in [17] and [18]. To optimize the performance of the
Manuscript received January 25, 2002; revised September 18, 2002. This work was supported in part by the National Key Project and NNSF of China (#60074017) and by ONR under Grant N00014-00-1-0677, NSF under Grant ECS-9734285, and NASA/LEQSF Grant (2001-4)-01. This paper was recommended by Associate Editor S. E. Shimony. Y. Zhu is with the Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China (e-mail: [email protected]). X. R. Li is with the Department of Electrical Engineering, University of New Orleans, New Orleans, LA 70148 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TSMCA.2003.809211
system globally, it is customary to find an optimum fusion rule from all possible fusion rules (see the formulation in [16] and [18]) and then determine the corresponding set of optimal local sensor compression rules under a given communication pattern. Usually, to evaluate the merits of two different fusion rules, we determine the two corresponding sets of optimal sensor rules and then compare the two final costs. However, the number of possible fusion rules increases exponentially with the number of sensors or the number of bits received by the fusion center. An exhaustive method obviously is computationally intractable. To the authors’ knowledge, however, there has been hardly any theoretical result on how to find a globally optimum fusion rule more efficiently. In [1], we discussed the optimal sensor rules for a fixed fusion rule and unified fusion rules in a parallel binary Bayesian decision system, where no communications among local sensors are permitted, without the commonly used data independence assumption. In this paper, we extend the results to the aforementioned general distributed multihypothesis decision system. When the network architecture and its communication pattern are given, we propose a unified fusion rule, which is actually a specific form of the most general fusion rule, as it is independent of the statistical characteristics of observations and decision criteria. To achieve the globally optimum performance, we now only need to optimize sensor rules under this single fusion rule for the given conditional distribution of observations and decision criterion. In other words, this rule is optimum if the sensor rules are optimized. This reflects a huge saving in computation. For example, for a three-sensor 4-ary parallel decision system with a total of ten sensor bits, using the proposed unified fusion rule, we only need to optimize the ten sensor bits to achieve the globally optimum performance. Using the exhaustive search, however, we need to compare more than trillions different fusion of sensor bits because there are rules! While the above network decision systems are also discussed in the literature [15]–[18], no alternative to the intractable exhaustive search for a globally optimum fusion rule from a large number of possible fusion rules has been proposed prior to this work. Performance comparisons between the parallel and tandem networks were presented in [15] and [17], where the main conclusions are: In the two-sensor case the tandem network is dominant; in the case with more than two sensors, one does not dominate the other in general, but the parallel network outperforms the tandem network asymptotically as the number of sensors goes to infinity, although the number of sensors at which the parallel network becomes superior is not known. In this
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paper, we give a more reasonable and precise comparison. For instance, we conclude that the aforementioned asymptotic superiority of the parallel network to the tandem network does not hold generally. We also provide a number of numerical examples. They support the above results and illustrate how to assign compression rates to local sensors by comparing their signal-to-noise ratios — a lower compression rate should be assigned to a sensor with a larger SNR. The examples also demonstrate that a parallel network outperforms a tandem network while detecting a Gaussian signal in Gaussian noise. The rest of the paper is organized as follows. In Section II, we formulate the problems associated with the three types of network decision systems. In Section III, we discuss unified fusion rules under different information structures for distributed decision systems. Section IV is dedicated to an iterative algorithm for searching for optimal sensor compression rules given a fusion rule for a general distributed -ary decision problem. In Section V, a performance comparison between the parallel and tandem networks with the same amount of communication is presented. In Section VI, several numerical examples are given, which substantiate the analytic results in the previous sections. Finally, we provide conclusions in Section VII.
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Fig. 1. Parallel network.
II. REPRESENTATIONS OF NETWORK STRUCTURES AND DECISION RULES In this section, we consider two types of elementary distributed systems: parallel and tandem. A. Parallel Network The parallel network is a basic information structure of a distributed decision system. We consider a distributed decihypotheses, , and sion problem with sensors, , with multiple observation data in space . A set of local compression rules, , where
compresses data to information bits at each sensor . Obviously, binary digits can correspond to different integers, that is, the th sensor quantizes its observation to different integers. Then the local sensors transmit possibly their compressed binary messages .. .
Fig. 2. Modified parallel network.
sensor compresses its observation to information bits. ” implies Moreover, “ in “{}” transmits information bits to the that each sensor fusion center , where an -ary decision is made. In addition, assume that known conditional probability density funcare of arbitions trary forms. for a parallel network is given by an A fusion rule of -valued function (2) In practice, we could build one of the sensors, say, , and the fusion center in the same station to save communication between this sensor and the fusion center. We call such a network a modified parallel network (see Fig. 2). The corresponding expression is (3)
to a fusion center . Let be the total information tuple), bits. Upon the receipt of the local message (an , the fusion center makes a final decision under some fusion rule (see Fig. 1). We denote the above information structure by (1) denotes that all sensors inside “{}” are in where ” means that parallel without mutual communications and “
In this structure, we do not care how large is because no trans, mission is required for sensor . Furthermore, when where is an integer, we can also regard sensor , which compresses its received information to bits, as an intermediate node in the network, and accordingly view the fusion rule as a local compression rule. Also, note that the results on the unified fusion rule in [1] depend not on any cost functional but only on the set of final decision partitions of the joint observation space generated by this fusion rule together with all possible local compression rules. This point of view is useful for dealing
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Fig. 3.
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Fig. 4.
Tandem network.
with (local or final) compression rules at either an intermediate node or the fusion center. B. Tandem Network The sensor network is a team of sensors in tandem. That is, the first sensor compresses its observation data to inforand transmits them mation bits to the second sensor . Then due to communication bandwidth compresses its observation and the received meslimit, to bits and transsage mits them to the next sensor , and so on. This is repeated until th sensor . The last sensor uses its observathe tion together with the received message to make a final -ary decision under a fu. In this model, sensor and the fusion center are sion rule co-located (see Fig. 3). is also a binary Notice that function, which will be referred to as a local fusion rule of sensor and can be expressed equivalently (see [1]) by binary , as functions of , follows:
(4) are binary functions and thus, because all different -tuples . Let . there are It turns out that we can view these as local compression rules for at the th sensor. A final for the tandem network now is dependent on fusion rule . Since the local fusion rules further depend on and local all previous local fusion rules , finally can be written as compression rules . an -valued function of Similarly as for the parallel network, we denote the tandem network by (5)
Tree network.
where “ data and the to the next sensor alently as
” means that sensor compresses its own received bits to bits and transmits them . Similar to (4), we can rewrite (5) equiv-
(6) and “ ” means that sensor comwhere bits, fuses them and the presses its own data to received bits to bits, and transmits these bits to the . Besides, as derived in Section III, we can use next sensor without information loss, where is an integer sat. However, it must be kept isfying inequality in general are comin mind that the local fusion results ), that is, from pressed from ( bits to bits; hence, there are at most (rather ) subsets in possibly observed by than the fusion center in the tandem network. Clearly, the tandem network is another basic information structure in distributed decision systems. Combining the above two basic structures, an arbitrary tree network can be constructed (see Fig. 4). C. Cost Functionals Since the focus of this paper is on presenting unified fusion rules (so as to obtain optimum fusion rules) and comparing relative merits of fusion rules, suppose that we can always find optimum local compression rules for any given cost functional and any fusion rule (see [1] and Section IV below for details that support this statement). Thus, minimization of a cost functional by a fusion rule depends only on the set of possible final decision partitions produced by the rule. When this set of a fusion rule contains those of all fusion rules, this fusion rule is a unified one. It is optimum if the corresponding sensor rules are optimum given this fusion rule. It follows that the so-obtained optimum fusion rules of this paper do not depend on what test (e.g., Bayes or Neyman-Pearson test) is considered. However, for simplicity of presentation, we only consider Bayesian cost functionals in the sequel.
ZHU AND LI: UNIFIED FUSION RULES FOR MULTISENSOR MULTI-HYPOTHESIS NETWORK
The Bayesian cost is (7) is some suitable cost coefficients; is a priori where each ; and each is the probability for hypothesis conditional probability of the event that the final decision is is true, , where equal to while denotes , , and for the parallel, tandem, and tree networks, respectively. Substituting the conditional probabiliinto (7) and simplifying yield ties given
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globally optimum final decision rule consisting of disjoint de, respectively, that is, an cision regions for . It optimum partition of the set of possible -tuples follows from the above analysis that every final decision region uniquely corresponds to a summafor tion polynomial of some local-message polynomials, denoted . Obviously, they are still indicator functions by since the regions determined by of a region in different local-message polynomials are disjoint—for a given one and only one of the values of observation data the local-message polynomials equals 1 and the others are all equal to 0. In other words, we have
(8) and a correOur goal is to select an optimum fusion rule sponding set of optimum local compression rules that jointly minimize the cost functional. D. Polynomial Representations of Decision Rules In this subsection, we extend the formulation presented in [1] for a distributed binary decision system to the corresponding -ary decision system. , , , are actually All local compression rules in . Let and indicator functions of the set represent the regions over which is compressed to digit 1 and , 0, respectively, by the sensor using the compression rule that is (9) (10) As such, there is a bijective relationship between a pair and a pair of simple polyof local regions . Furthermore, a local message nomials , where and or , and , corresponds uniquely to of the above simple a product polynomial polynomials, where
(11) as the decision polynomial. As such, we We refer to have established a correspondence between a final decision rule and a set of final decision polynomials of local compression rules. Now, let us give two examples to show how to write fusion rules as their polynomial versions. Example 2.1: Consider a 3-sensor ternary parallel network, where there is only one compression rule at each sensor
A fusion rule
is given by making decision (i.e., ) if any of the four local messages (111), (110), (101), (011) is received by the fusion , center, where decision region is defined by {(000)}. and so on; and the , , and decision polynomials are Thus, the
Example 2.2: Consider a tandem network of the following structure
, We call a local-message polynomial, which is also an . Obviously indicator function of a region in
A fusion rule partitions the set of (recall ) into disjoint subsets generdifferent -tuples different such partitions for a parallel netally. There are work and normally may be fewer for a tandem network (see the analysis at the last part of Subsection II-B). In addition, a fixed fusion rule combining all possible local compression rules can produce infinitely many different final decision regions in a continuous model. The goal of our distributed decision is to find a
where a local fusion rule at sensor is defined as if and only if the local message is either or at sensor , and otherwise. An decision region is given by
which is actually, using the definition of
above
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Thus, the corresponding
Likewise, an
decision polynomial
is
Now suppose an -sensor parallel network is given by (3). , that is, the fusion center Consider a special case with compresses all received and observed information to a binary or of length . This case code is similar to that of the intermediate node in the analysis around (4). Since is also the fusion center, the general fusion rule is
decision polynomial can be written as where sub-rules of fusion:
(12) . Clearly, this fusion rule consists of
.. .
Obviously, two regions and are disjoint. Finally, decision polynomial must be . the Thus, it can be seen that using decision polynomials to represent a fusion rule is much more convenient and simpler than using local messages. Example 2.3: Now we use the parallel and tandem structures to construct a tree structure as follows:
A fusion rule for this structure is given in Example 6.3, which is actually a unified fusion rule by Theorems 3.1 and 3.2. III. UNIFIED FUSION RULES In this section, extending the results in [1] we present a unified form of the most general fusion rule, referred to as a unified fusion rule, for the aforementioned distributed decision systems. This form includes any fusion rule under the information structure considered as a special case and thus the globally optimum fusion rule also has this form. A fusion rule in this form that optimizes the cost functional is therefore an optimal fusion rule. In addition, this unified fusion rule does not depend on the statistical characteristics of sensor observations or decision criteria. To achieve the global optimal performance, traditionally we find an optimum fusion rule and determine the corresponding optimum sensor rules. As seen from the simple 3-sensor ternary decision examples in Section II, however, the number of possible fusion rules is very large and thus finding optimum fusion rules is computationally intensive and often intractable. Using the result presented in this section, we only need to optimize sensor rules under the unified fusion rule for the given decision criterion and conditional distributions of observations. Therefore, using this fusion rule can save huge computation. We consider the unified fusion rule for the parallel network first, and then treat each level in a tandem or tree network as a parallel network and obtain the corresponding unified local fusion rule. Finally, the global unified fusion rule for the whole tandem or tree network is composed by all unified local fusion rules.
(13) is a general fusion rule for an -sensor biNotice that every nary decision system. Recall the basic idea and technique of [1], in fact defines sensor compression rules at where every the th sensor as follows:
(14)
.. .
different -tuples for a fixed set of . In other words, the general information structure considered with a comamounts to sensor compressing into pletely known bits. The corresponding general fusion rule is given in the following unified form
since there are
.. . (15) is the th binary decision region for for where the general fusion rule given in (12). This fusion rule has the same optimum performance as that of the fusion center using and all compressed local messages from sensors the uncompressed data to make a binary decision. It follows from Subsection II-B that the corresponding decision polynoand can be easily obtained from mials . Now we extend the above idea and technique to an -ary . The sub-rules define binary sensor system with compression rules
ZHU AND LI: UNIFIED FUSION RULES FOR MULTISENSOR MULTI-HYPOTHESIS NETWORK
at the th sensor in terms of their polynomials
. We can write them
.. .
(16) decision polynomials It follows that the corresponding are given by , . Using polynomials, we define the -ary final decision rule these as follows. First, define a 1-1 correspondence between integers and an -tuple binary code . is defined by a given for any Suppose , . Then we can obtain a polynomial so that
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some of observation data arbitrarily. Then a unified to fuse and tolocal/global rule for in this case) gether to different digits (or bits with is given above. An immediate consequence of Theorem 3.1(b) is the following. Corollary 3.1: For a multisensor -ary decision system, makes observation and suppose that a sensor, say, simultaneously, receives information bits may depend on some of observation where arbitrarily. Then needs to send at most data bits to the fusion center, where satisfies inequality . These two theorems present a unified fusion rule for a parthe fusion allel network. As mentioned before, when center integrated with a local sensor in a parallel network can be viewed as an intermediate node in a multisensor network. In addition, the unified fusion rule here and in [1] is nothing but the most general fusion rule that includes all possible fusion rules as special cases in the sense mentioned in Theorems 4.1 and 4.2 of [1]. Consequently, we propose a unified fusion rule for a tandem network as follows. Theorem 3.2: For the following information structure
(17) where
and
is given by (18)
, i.e., , we In the case of local compression rules at still need to determine the th sensor. The only modification is to define some just for times so that the now is rather than . It follows total number of in (15), (16), and (17) that the from the definition of decision regions determined by these polynomials and can form any possible partition with members. Therefore, this fusion rule is a unified -ary fusion rule. Note that we do not increase communication by doing so, although we need to determine more optimal local compression rules, which is still tractable in practice. From the above argument, we have the following theorem. Theorem 3.1(a): For the following parallel structure
or
the fusion rule constructed as above is a unified one: Any fusion rule is a special case of it with some of the local compression or ). rules identically zero or unity (i.e., For convenience of treating tandem and tree networks, we rewrite Theorem 3.1(a) in the following more general version. Theorem 3.1(b): Suppose that a sensor, say, makes obserand receives information bits vation simultaneously, where may depend on
where “ ” means that sensor compresses its to bits ( , ), own data a general fusion rule of a unified form can be constructed sequentially as follows: Starting with the last sensor , at each ” (if ,“ ” is replaced sub-structure “ ”), we determine a set of unified local by “ (or final fusion rule ) by the fusion rules above method for a parallel network, where is replaced by from in the tandem netif . work and is replaced by Proof: Using Theorem 3.1(b), we know that the above fu” gives the unision rule at each sub-structure “ ) fusion rule in the sense that any fied local (or final if specific local fusion rule in the sub-structure is a special case of it. Therefore, at the top level sensor , the final fusion rule given in this theorem must be also a unified fusion rule. By Theorem 3.2, it can be verified that the fusion rule given in Example 2.2 is actually a unified fusion rule for the given information structure. More examples are given in the next section. Using Theorems 3.1 and 3.2 together, the corresponding unified fusion rule for a tree network can be given. We illustrate in Example 6.3 how to construct such a unified fusion rule for Example 2.3. IV. OPTIMUM SENSOR RULES Given the unified fusion rule of Section III, to have an optimal decision system we need to find the corresponding optimal
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sensor rules. In this section we extend the results in Sections II and III of [1] to the network decision models here. We first present a necessary condition for local compression rules to be optimum for any fixed fusion rule in the above distributed decision systems, and then propose a Gauss-Seidel iterative algorithm and its discrete version. Finally we give some convergence results to show that the discrete algorithm converges in finite steps to a minimum of the discrete cost functional and that under a mild assumption on the integrand of the cost functional, the global minimum of the discrete cost functional converges to the infimum of the original continuous cost functional as the discretization step size tends to zero. Since the extension here is straightforward except for the formulation, we present the relevant results without argument. A. A Necessary Condition for Optimum Local Compression Rules
.. .
.. . (20) where
is a indicator function defined by .
(21)
This necessary condition is of a fixed-point type. It is an extension of Theorem 2.1 in [1]. In particular, if we define the mapping
and
Let
. It follows from Eq. (11) and the same argument as in [1] ) that we can write the (i.e., integrand in (8) in various forms as follows:
.. . .. . (22) .. .
.. . then Theorems 4.1 shows that a set of optimal local compression rules must be a fixed point of the map in (22), which is a solution of the integral in (20).
.. .
B. Iterative Algorithms and Convergence .. . (19) and are independent of , where functions , . Now we propose the following necessary condition for optimum local compression rules. Theorem 4.1: Suppose we have a distributed decision system employing fusion rule (11). A set of optimal local decision that minimizes the rules cost functional (8) must satisfy the following integral equations
Let the local sensor rules at the th stage of iteration be denoted by with a given initial rules . Suppose the given fusion rule of (11) is employed. We now consider the following Gauss-Seidel iterative algorithm for the mapping .
.. .
.. . .. . .. .
ZHU AND LI: UNIFIED FUSION RULES FOR MULTISENSOR MULTI-HYPOTHESIS NETWORK
(23) To facilitate computer implementation of this iteration, we need to discretize the variables. Let the discretization of be given, respectively, by
For each iteration and for , let the -vector denote the values of at the discretization points such that , . Thus, the iteration of (23) can be approximated as, for
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Theorem 4.2: For any positive discretization step size of the elements of and any initial ( . ), the algorithm of (24) terminates at satisfying (25) after a finite number of some iterations. Outline of proof: The proof consists of two steps. First, we prove that the Bayesian cost decreases as iteration goes. Hence, the cost reaches a minimum after a finite number of iterations. However, this does not imply either the finite convergence of local sensor rules or the attainment of the minimal cost by the algorithm without the convergence of local sensor rules. Second, we can prove the latter convergence by contradiction. A detailed proof follows the same argument as that of Theorem 2.1 in [1]. Let
denote the discretization step size of each element of and be the minimum of the discrete verof . Similarly, we have sion a convergence result corresponding to Theorem 2.2 of [1], which states that under mild assumption on the integrand in the above exists and is cost functional, as tends to zero the limit of . We omit the details equal to the infimum of here.
.. .
V. PERFORMANCE COMPARISON OF PARALLEL AND TANDEM NETWORKS .. .
.. .
(24) are the step sizes for discretizing the vecwhere , respectively. Iteration (24) is the corresponding tors discretized version of the continuous iteration (23) and is readily implementable. A simple termination criterion for this iteration , and is, for all
In [15] and [17], some performance comparison results between the parallel and tandem networks were presented. Their main conclusions are: The tandem network is dominant in two sensor case; for cases with more than two sensors, one does not dominate the other in general, but the parallel network outperforms the tandem network asymptotically as the number of sensors goes to infinity, although the value of sensor number at which the parallel network becomes superior is not known. In this section, we give a more reasonable and rigorous comparison. We only consider cases with more than two sensors because two-sensor parallel and tandem networks are the same if the second sensor in the parallel network and the fusion center are colocated. To fairly compare the performance of various networks in the sequel, we assume that the fusion center in the parallel network is also a sensor and that all other corresponding sensors in both networks transmit the same number of information bits to the fusion center (in the parallel network case) or to the next sensor (in the tandem network case). For notational simplicity, we further assume that every sensor is one-bit sensor. In other words, we compare the performance of the following two decision networks:
(25) An alternative is
(27) and
(26)
(28)
is some prespecified tolerance. where We now examine the convergence of the iteration (23). We have the following theorem on the finite convergence of its discrete Gauss-Seidel iteration process.
Our analysis shows that in general, a dominance between the parallel and tandem networks does not exist no matter how large the number is: There is no dominance between two sets of possible partitions of the joint observation space generated by the
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parallel and tandem networks, respectively—there always exist partitions generated by one network that cannot be generated by the other—and we can easily construct examples in which the optimal decision region for the centralized is the region that can be generated by one network but not the other. It follows from (15), (16), and (17) that the above tandem network is equivalent to the following tandem network:
A. Three-Sensor System The hypotheses are
where the signals and and the noise mutually independent, and
,
and
are all
(29) Hence, from the second sensor on, each sensor actually can compress its own observation to two bits first and then fuses them and the received single bit from the previous sensor to a single bit and transmit it out. However, in the above parallel network, every local sensor only compresses its own observation into a single bit and thus there always exist some decision regions generated by the tandem network that in general cannot be generated by the parallel network, such as
On the other hand, while the fusion center in an -sensor parinformation bits from the allel network can receive local sensors, the fusion center in the tandem network of the same number of sensors can only receive a total of a single bit from all other local sensors. It turns out that some final decision regions that can be generated by the parallel network cannot be decision generated by the tandem network. For instance, the region for the parallel network
is an example, which cannot be generated by the tandem network since the fusion center here uses more than a single bit information from the first two sensor. Obviously, this situation cannot change even as the number of sensors goes to infinity. In practice, it is rare that a better decision region can only be generated by the tandem network but not by the parallel network because information is compressed much more in the tandem network than in the parallel network. Therefore, in most practical situations, the parallel network often outperforms the corresponding tandem network (see the numerical examples in the next section). However, the tandem network has better survivability than the parallel network, particularly in a war situation: When a fusion center is destroyed, the damaged parallel network above is a set of single-sensor decision makers, but a tandem network would become a smaller tandem network, which would perform in general better than a single-sensor decision maker.
Hence, the three conditional Probability Density Functions (PDFs) are
In all examples below, we take for , , , . As such, the Bayessian cost functional actually becomes decision error probability . Example 6.1: The information structures are
and
which can be obtained by adding seven local compression rules in Example 2.1. By Theorem 3.1 the unified (and to sensor thus optimum) fusion rule is defined by the following three decision polynomials:
It can be seen from (19) and this fusion rule that the following , , , ) are sufficient to six initial values ( , , run the iterative algorithm to achieve the optimal performance. For performance evaluation, we compare its final decision cost, denoted by , with the centralized decision cost for initial values
VI. NUMERICAL EXAMPLES We consider 3- and 4-sensor detection systems for Gaussian signals in additive Gaussian noise.
for initial values
ZHU AND LI: UNIFIED FUSION RULES FOR MULTISENSOR MULTI-HYPOTHESIS NETWORK
where is a indicator function defined by (21). From this example, we see that the optimum distributed decision (ODD) is slightly more costlier than the centralized decision (CD) but not sensitive to the initial values. Now we interchange the positions and in the information structure, that is, of sensors with the smallest signal-to-noise ratio (SNR) now is with the largest SNR is compressed to 8 bits and compressed to 1 bit. This yields for initial values
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Example 6.3: We consider the same information structure as that of Example 2.3.
A set of optimum decision polynomials can be constructed sequentially in three steps. , , and we construct an Step I. Using optimum fusion rule at the last sensor :
for initial values This case has significantly inferior performance to the previous case, as expected. Example 6.2: The information structure is
The unified (and thus optimum) fusion rule by Theorem 3.2 is just the fusion rule shown in Example 2.2. It can be seen from (4.1) and the above fusion rule that the following five initial , , , ) are sufficient to run the iterative values ( , algorithm to achieve the optimal performance. The results for similar initial values as in Example 6.1 are Step II. Using , , and we construct the four , at polynomials of the two optimum local fusion rules the sensor
for initial values for initial values The ODD costs here are still not far away from the CD cost but larger than the ODD cost in Example 6.1. Similar to Example 6.1, we interchange the positions of senand . This yields, for the same initial values sors with initial values with initial values As before, the performance of the first case is significantly better than that of the second case.
Step III. Substituting the above two polynomials into , and , we get the final decision polynomials with respect to all local compression rules . It can be seen from (15) and the prior fusion rule that the following 18 initial values for are sufficient to run the iterative algorithm. The decision error , for initial values probabilities are
B. Four-Sensor System Now we add a sensor with additive noise the above three-sensor systems. The three PDFs become
to and
for initial values
The ODD costs here are still not far away from CD cost and not sensitive to the initial values. Example 6.4: To compare the performance of the above information structure with that of a parallel system, we consider the following structure:
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The optimum fusion rule can be constructed similarly as in Example 6.1 and thus is omitted. To run the iterative algorithm, 11 initial local compression rules are needed. for initial values The results are
of data in the latter. All these results have been substantiated by numerical examples. Further, extension to other distributed decision systems, such as Neyman-Pearson and sequential decision systems, can be made (see [20] and [21]).
ACKNOWLEDGMENT
and
for initial values
The authors would like to acknowledge many valuable suggestions of the referees and editor to this work.
REFERENCES Here it is required to communicate 3 bits and compute 19 local compression rules. The performance appears to be better than that of Example 6.3, where communication of 4 bits and computation of 18 local compression rules are required. In summary, although the distributed decision systems employing our proposed rules have slightly worse performance than the optimum centralized decision systems, the performance differences appear acceptable since they are minor and the distributed systems require much less communication than the centralized; the sensor with the largest SNR should have the least data compression and be built at the same place with the fusion center; and the modified parallel network has a superior performance to the tandem network in most practical situations.
VII. CONCLUSIONS In this paper, we have extended the results on unified fusion rules and optimum local compression rules of [1] to the general distributed network (Bayesian) decision problem. We have formulated two types of elementary multisensor network structures, and shown via examples how to construct an arbitrary tree network by these structures. For networks in which (global or local) fusion center can observe data, we have presented a general fusion rule in a unified form, called unified fusion rule, which includes all possible fusion rules as special cases. In essence, its generality stems from a guarantee of a certain number of (fusion or compression) rules at the (global or local) fusion center, which is achieved by the specific form proposed. We have shown that the proposed fusion rule is unified and general in that it depends on neither the statistical properties of data nor decision criteria. To achieve globally optimum performance, we only need to optimize sensor rules under this unified fusion rule for a given decision criterion and conditional distributions of observations. In this sense, this unified fusion rule is also optimum. We have also made a comparison analysis on the performance of a modified parallel network and the tandem network. In contrast to some statements in the literature, there exists no general performance dominance between them no matter how large the number of sensors is. In practice, however, the modified parallel network often has better performance than the tandem network due to the higher compression
[1] Y. M. Zhu, R. S. Blum, Z. Q. Luo, and K. M. Wong, “Unexpected properties and optimum distributed sensor detectors for dependent observation cases,” IEEE Trans. Automat. Contr., vol. 45, pp. 62–72, Jan. 2000. [2] Y. M. Zhu, Multisensor Decision and Estimation Fusion. Norwell, MA: Kluwer, 2003. [3] R. R. Tenney and N. R. Sandell Jr., “Detection with distributed sensors,” IEEE Trans. Aerosp. Electron. Syst., vol. AES-17, pp. 501–510, 1981. [4] B. V. Dasarathy, Decision Fusion. Los Alamitos, CA: IEEE Comput. Soc. Press, 1994. [5] L. A. Klein, Sensor and Data Fusion Concepts and Applications. Bellingham, WA: SPIE Press, 1993. [6] S. Chaudhuri, A. Crandall, and D. Reidy, “Multisensor data fusion for mine detection,” in Proc. SPIE—Sensor Fusion III: 3-D Perception and Recognition, vol. 1306. Bellingham, WA, 1990, pp. 187–204. [7] J. Han, P. K. Varshney, and V. C. Vannicola, “Distributed detection of moving optical objects,” in Proc. SPIE—Sensor Fusion III: 3-D Perception and Recognition, vol. 1306. Bellingham, WA, 1990, pp. 115–124. [8] R. C. Luo and M. G. Kay, “Multisensor integration and fusion in intelligent systems,” IEEE Trans. Syst., Man, Cybern., vol. 19, pp. 901–931, Sept./Oct. 1989. [9] D. D. Freedman and P. A. Smyton, “Overview of data fusion activities,” in Proc. Amer.Control Conf., San Francisco, CA, June 1993. [10] H. Delic, H. Papantoni-Kazakos, and D. Kazakos, “Fundamental structures and asymptotic performance criteria in decentralized binary hypothesis testing,” IEEE Trans. Commun., vol. 43, pp. 32–43, Jan, 1995. [11] Z. B. Tang, K. R. Pattipati, and D. K. Kleinman, “An algorithm for determining the decision threshold in a distributed detection problem,” IEEE Trans. Syst., Man, Cybern., vol. 21, pp. 231–237, Jan./Feb. 1991. [12] J. N. Tsitsiklis and M. Athans, “On the complexity of decentralized decision making and detection problems,” IEEE Trans. Automat. Contr., vol. AC-30, pp. 440–446, Apr. 1985. [13] V. Venugopal, V. Veeravalli, T. Basar, and H. V. Poor, “Decentralized sequential detection with a fusion center performing the sequential test,” IEEE Trans. Inform. Theory, vol. 37, pp. 433–442, Mar. 1993. [14] V. Veeravalli, T. Basar, and H. V. Poor, “Minimax Robust decentralized detection,” IEEE Trans. Inform. Theory, vol. 38, pp. 35–40, Jan. 1994. [15] J. D. Papastavrou and M. Athans, “Distributed detection by a large team of sensors in tandem,” IEEE Trans. Aerosp. Electron. Syst., vol. 28, pp. 639–652, July 1992. [16] J. N. Tsitsiklis, “Decentralized detection,” in Advances in Statistical Signal Processing, H. V. Poor and J. B. Thomas, Eds. Greenwich, CT: JAI, 1993, 2. [17] R. Vismanathan and P. K. Varshney, “Distributed detection with multiple sensors: Part I—Fundamentals,” Proc. IEEE, vol. 85, pp. 54–63, Jan. 1997. [18] P. K. Varshney, Distributed Detection and Data Fusion. New York: Springer-Verlag, 1997. [19] R. S. Blum, S. A. Kassam, and H. V. Poor, “Distributed detection with multiple sensors: Part II—Advanced topics,” Proc. IEEE, vol. 85, pp. 64–79, Jan. 1997. [20] Y. M. Zhu, C. Y. Liu, and Y. Gan, “Multisensor distributed Neyman-Pearson decision with correlated sensor observations,” in Proc. Int. Conf. Multisource-Multisensor Information Fusion, vol. 1, 1998, pp. 35–39. [21] Y. M. Zhu, Y. Gan, K. S. Zhang, and C. Y. Liu, “Multisensor NeymanPearson type sequential decision with correlated sensor observations,” in Proc. Int. Conf. Multisource-Multisensor Information Fusion, vol. 2, 1998, pp. 787–793.
ZHU AND LI: UNIFIED FUSION RULES FOR MULTISENSOR MULTI-HYPOTHESIS NETWORK
Yunmin Zhu received the B.S. degree in mathematics and mechanics, Beijing University, Beijing, China, in 1968. From 1968 to 1978, he was with Luoyang Tractor Factory, Luoyang, Henan, China, as a Steel Worker and a Machine Engineer. From 1981 to 1994 he was with the Institute of Mathematical Sciences, Chengdu Institute of Computer Applications, Chengdu Branch, Academia Sinica. Since 1995, he has been with the Department of Mathematics, Sichuan University as Professor. During 1986 to 1987, 1989 to 1990, 1993 to 1996, 1998 to 1999, and 2001 to 2002, he was a Visiting Associate, or Visiting Professor at Lefschetz Centre for Dynamical Systems and Division of Applied Mathematics, Brown University, Providence, RI; Department of Electrical Engineering, McGill University, Montreal, QC, Canada; Communications Research Laboratory, McMaster University, Hamilton, ON, Canada; Department of Electrical Engineering, University of New Orleans, New Orleans, LA. His research interests include stochastic approximations, adaptive filtering, other stochastic recursive algorithms and their applications in estimations, optimizations, and decisions for dynamic system as well as for signal processing, information compression. In particular, his present major interest is multisensor distributed estimation and decision fusion. He is the author or coauthor of over 50 papers in international and Chinese journals. He is the author of Multisensor Decision and Estimation Fusion (Norwell, MA: Kluwer, 2002) and Multisensor Distributed Statistical Decision (Beijing: Science Press, Chinese Academy of Science, 2000), and coauthor (with Prof. H. F. Chen) of Stochastic Approximations (Shanghai Scientific & Technical Publishers, 1996). He is on the editorial board of the Journal of Control Theory and Applications, South China University of Technology, Guangzhou.
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X. Rong Li (S’90-M’92-SM’95) received the B.S. degree and M.S. degree from Zhejiang University, Hangzhou, Zhejiang, China., in 1982 and 1984, respectively, and the M.S. degree and Ph.D. degree from University of Connecticut, Storrs, in 1990 and 1992, respectively. He joined the Department of Electrical Engineering, University of New Orleans, New Orleans, LA, in 1994, where he is now University Research Professor and Department Chair. From 1986 to 1987, he did research on electric power at University of Calgary, Calgary, AB, Canada. He was an Assistant Professor at University of Hartford, West Hartford, CT, from 1992 to 1994. He has authored or coauthored four books: Estimation and Tracking (with Yaakov Bar-Shalom), (Norwood, MA: Artech House, 1993), Multitarget-Multisensor Tracking (with Yaakov Bar-Shalom), (Storrs, CT: YBS, 1995), Probability, Random Signals, and Statistics (Boca Baton, FL: CRC Press, 1999), and Estimation with Applications to Tracking and Navigation (with Yaakov Bar-Shalom and T. Kirubarajan), (New York, Wiley, 2001); six book chapters, and more than 160 journal articles and conference papers. His current research interests include signal and data processing, target tracking and information fusion, stochastic systems, statistical inference, and electric power. Dr. Li has served as Associate Editor for IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS from 1995 to 1996, and as Editor since 1996. He received a Career award and an RIA award from the U.S. National Science Foundation. He received a 1996 Early Career Award for excellence in Research from University of New Orleans and has given numerous seminars and short courses in the U.S., Europe and Asia. He won several outstanding paper awards, is listed in Marquis’ Who’s Who in America and Who’s Who in Science and Engineering, and consulted for several companies. He has served the International Society of Information Fusion as the President, from 2003 to the present, Vice President, from 1998 to 2002, and a member of Board of Directors since 1998; served as General Chair for 2002 International Conference on Information Fusion, and Steering Chair or General Vice-Chair for 1998, 1999, and 2000 International Conferences on Information Fusion. He has served as Editor for Communications in Information and Systems since 2001.
