General Bayes Filtering of Quantized Measurements

Transcription

1 4th International Conerence on Inormation Fusion Chicago, Illinois, UA, July 5-8, 20 General Bayes Filtering o Quantized Measurements Ronald Mahler Uniied Data Fusion ciences, Inc. Eagan, MN, 5522, UA. UDFRPM@gmail.com Abstract Quantized data is requently encountered when data must be compressed or eicient transmission over communication networs. ince quantized measurements are not precise but are, rather, subsets (cells, bins, quanta o measurement space, conventional iltering methods cannot be used to process them. In recent papers, Zhansheng Duan, X. Rong Li, and Vesselin Jilov have devised generalizations o the Kalman ilter that can process quantized measurements. In this paper I provide a theoretical oundation or processing such measurements, based on a Bayes iltering theory or generalized measurements mediated by generalized lielihood unctions. As a consequence, I also show that this theory ( results in a general Bayes-optimal approach or iltering quantized measurements; (2 generalizes the Duan-Li-Jilov iltering theory; and (3 can be extended to noncooperatively quantized measurements such as uzzy Dempster-haer (FD quantized measurements. I conclude by arguing that quantized measurements provide a concrete, applications-based conceptual bridge between probabilistic and nonprobabilistic orms o expertsystem reasoning. Keywords: Quantization, quantized measurements, Bayes ilter, random sets, generalized lielihood unction. Introduction In recent papers at this conerence [6,7], Zhansheng Duan, Vesselin Jilov, and X. Rong Li have addressed the problem o target tracing using inormation sources, such as communications networs, that supply quantized measurements. peciically, they have developed generalizations o the Kalman ilter that process quantized measurements [7], or both quantized and point measurements [6]. This paper, which was inspired by this wor, has the purpose o developing a general Bayes nonlinear iltering theory or. quantized measurements, and 2. uncooperatively quantized measurements i.e., measurements whose quantization scheme at the transmitter is not nown a priori at the receiver. My approach is based on the general theory o Bayes iltering o nontraditional measurements described in Chapters 3-5 o [5]. As applied in this paper, it results in Bayes-optimal ilters or quantized measurements in both single-target and multitarget situations (see ection 4. The primary result o this paper, however, is a demonstration that quantized measurements provide a concrete, applications-based conceptual bridge between probabilistic and nonprobabilistic orms o expertsystem reasoning. Generally speaing, the target-tracing community has tended to show little interest in expert-systems approaches such as uzzy logic or Dempster-haer theory. This has in part been because o suspicions about the theoretical trustworthiness o these approaches. But it has also been, in part, because o a lac o real-world, physics-rooted applications or which these approaches seem unavoidably necessary. On the other hand, the expert-systems community which tends to treat uzzy set theory and Dempster-haer theory as branches o abstract logic has tended to have limited interest in physics-bound applications. Quantized data, I claim, provides an example o a concrete application in which basic Dempster-haer concepts imprecision, nonstatistical uncertainty, etc. must unavoidably be addressed in the context o probabilistic physical modeling and physical intuition. But quantized data also provides an instance in which uzzy logic and Dempster-haer theory cannot be invoed without irst anchoring them in a concrete, physical-statistical context. This is because a quantized measurement is a speciic instance o what is nown as imprecise evidence in expert systems theory. My hope is that the wor presented in this paper will provoe urther investigation and discussion o the interrelationships between hard and sot orms o inormation usion.. Prologue: Digital voltmeters Begin with a simple example: the reading provided by a digital voltmeter when measuring the voltage o a battery. uch a reading is a real number or example, the number 99.98, rounded o to two digits. The voltage is not actually 99.98; rather, it is some value V v contained in the interval J (99.975, ] ( which has as its centroid. Though v is nown to be in J, aside rom this constraint it could equally well be any value in J Moreover, V is a random number o the orm V η(c U, U is a zero-mean random number and η(c is the nominal voltage supplied by the battery, which is o type c. Thus a dierent value J will be chosen depending on what value V taes IIF 346

2 Conventional practice is to treat [v] as the discretized value o v. This means that the discretization [V] o V will be a random number with a measurement model o the orm [V] [η(c] U. However, one can tae a dierent perspective: The actual measurement is the interval J rather than its centroid [v] 99.98; in which case J is an imprecise measurement. This means that the actual measurement model or the digital voltmeter has the orm J {V}. (2 That is, among all possible digitized readings, the speciic reading J will be returned only i it contains the value V. It ollows that the constraint (2 is equivalent to η(c Θ J (3 Θ J J U is the random interval (random subset o the real line deined by J U {v U v J}. The smaller the size o J, the more closely that model (3 will approximate the conventional model V η(c U..2 Quantized measurements Imprecise measurements are unavoidably encountered in communications theory because o quantization [2]. Quantization arises rom the act that communication channels have limited banidths, and consequently because o the act that data must be compressed. As stated by Gray and Neuho [2, p. 2326], the goal o quantization is to encode the data rom a source, characterized by its probability density unction, into as ew bits as possible (i.e., with a low rate in such a way that a reproduction may be recovered rom the bits with as high quality as possible (i.e., with small average distortion. Thus let z be a measurement-vector in some measurement space Z 0. Let Z 0 be partitioned ( quantized into cells i with centroids z i. Then any measurement Z z can be replaced by ( compressed into the unique quantized value z i such that z i. Typically, the cells are chosen so that the size i o i is smallest when z i is most probable, i.e., when the probability distribution Z (z o Z is largest. ee [2] or more details. In the approach taen in this paper, a quantized measurement is not z i but, rather, the subset i it is an imprecise measurement. In returning i as a measurement, we are stating that it is not possible to speciy the actual measurement z any more precisely than containment within i. That is: z is nown to be in i, but otherwise it could equally well be any element o i. It is also random, since dierent cells i will be selected depending on the value o Z. Thus as in the digital voltmeter example, the actual measurement model or the quantized measurement is {Z} (4 is restricted to the values i. Eq. (4 is to be contrasted with the inds o measurement models normally encountered in nonlinear iltering theory, such as the additive nonlinear model Z η( V. (5 The conventional practice in quantization is to replace this model with [Z] [η(] V, [η(] is the quantization o η(, [Z] is the quantization o Z, and V is a noise vector. Even i η is linear and V is Gaussian, [η(] will nonlinear and V will be non- Gaussian. However, given Eq. (5 it is clear that the model (4 is equivalent to the model η( Θ (6 or any, Θ V is the random subset o Z 0 deined as V {z V z }..3 ummary o main results The purpose o this paper is threeold..3. Equivalence o theoretical oundations. The iltering theory in [6,7] is based on a measure-theoretic approach due to Curry et al. [], which permits the computation o the posterior expected value E h z ] h( ( x z d (7 [ x o an arbitrary test unction h( with respect to an arbitrary quantized measurement (which is modeled not as an approximation o z, but as an imprecise event. Under the theory developed in [5], nontraditional measurements quantized measurements, or example are represented as random closed subsets Θ o Z 0, and they are mediated by generalized lielihood unctions (Θ Pr(η ( Θ (8 z η ( is a deterministic measurement model. Given this, one can compute the posterior expectation o a test unction h( in the usual manner: E h ] h( ( x Θ d. (9 [ Θ x Given this, I demonstrate that, when the underlying measurement model is additive, h z ] h Θ ]. (0 That is, in this case the two oundations are equivalent..3.2 General nonlinear iltering o quantized measurements. Because o Eq. (0, it will ollow that Bayes iltering o quantized measurements using the generalized lielihood approach is the ull nonlinear generalization o the minimum mean-squared error (MME approach o Duan et al..3.3 General nonlinear iltering o noncooperatively quantized measurements. Conventional quantization presumes that the quantization scheme imposed at the transmission node is nown at the receiving node. uppose, to the contrary, that the quantization scheme is not published and that, thereore, the receiver can only mae hypotheses about what it might be. The inal goal o this paper is to derive a Bayes nonlinear iltering theory or such measurements. peciically, the original treatment o generalized measurements and generalized lielihood unction in [5] 347

3 was oriented towards deterministic generalized measurements i.e., towards vague or uncertain or contingent measurements that were not instantiations o some random measurement process. Thus imprecise measurements were assumed to have the measurement models o the orm η( rather than, as in Eq. (6, measurement models o the orm η( V. Thus in this paper I derive ormulas or the generalized lielihood unctions or generalized measurement models o the orm η( Θ V ( Θ is the deterministic random set model o a uzzy Dempster-haer (FD quantized measurement..4 Organization o the paper In ection 2 I begin by summarizing the theoretical approach to quantization introduced by Curry et al. [] and then, in ection 3, the generalized lielihood approach to quantization. In ection 4 I briely introduce a general Bayes iltering theory or quantized measurements. ection 5 is devoted to a summary o uzzy measurements and uzzy Dempster-haer measurements, their generalized lielihood unctions, and Bayes iltering using such measurements. ection 6 describes the generalization o this theory to the representation and iltering o uncooperatively quantized measurements. Conclusions are in ection 7. It is necessary to point out that the primary purpose o this paper is not to propose a particular nonlinear iltering approach. Rather, its main purpose is to demonstrate that quantized measurements provide a concrete, application-based conceptual bridge between probabilistic and nonprobabilistic orms o reasoning. (In any case, the simulations reported in [6,7] already address this issue to some degree, since in this paper it is shown that the ilters implemented in those papers are approximations o the ilters proposed here. 2 Quantization: Measure-theoretic The approach o Curry et al. is based on the ollowing Bayesian, measure-theoretic identity (see Eq. (2 in []: h z ] h z] z ] (2 which is valid or any quantization cell and any test unction h(. The let-hand side is the posterior expected value o h(, conditioned on the quantized measurement. The right-hand side shows how to construct it by irst computing the conventional posterior expected value E h z] h( ( x z (3 [ o h( and then conditioning h z] on the quantized event z. Here, z x z (4 z is the posterior distribution conditional on the measurement z, ( is the prior distribution, (z is the lielihood unction or the unquantized source, and z z (5 is the Bayes normalization actor. Given this, it ollows that h( h z ] (6 ( x ( z (7 ( ( ( x. (8 To see this, let be a quantization cell with z. Then g h (z h z] (9 is a random number which is subject to the constraint z. (20 Its expected value is h z] z ] g h z ] (2 (see [], Eq. (3: g z ( z z h ( z z ( z (22 ( z. (23 w One then employs Eq. (2 to ind the posterior expectation o h( conditioned on the quantized measurement. ubstitution o Eqs. (3, (4, and (23 into Eq. (2 results in: h z] z ] h z] z z (24 Noting that h z] h( h( ( w ( z (25 ( z (26 ( w (. (27 ( w z we get, as claimed, h z] z ] ( z ( ( x h( ( ( (28 (29. (30 3 Quantization: Generalized lielihood In this section I will show that:. I Z η ( V is an additive measurement model or conventional measurements, then the model or a quantized measurement using the generalized lielihood unction approach is Θ V {z V z }. (3 2. The measure-theoretic and generalized unction oundations or computing posterior expectations produce the same result, h Θ ] h z ], (32 meaning that the two oundations or quantization are equivalent (or additive measurement models. 348

4 In the approach advocated in this paper, nontraditional measurements quantized measurements, in particular are represented as random closed subsets Θ o the measurement space Z 0, and they are mediated by generalized lielihood unctions o the orm (Θ Pr(η ( Θ. (33 Now, let Z η ( V be a conventional additive measurement model. Under the generalized lielihood approach, the undamental measurement model or quantization is {Z} or, equivalently, η( V Θ. Thus the generalized lielihood o Θ is (Θ Pr(η ( V. (34 Given this, one can compute the posterior expectation o a test unction h( in the usual manner: E h ] h( ( x Θ d (35 [ Θ x the posterior distribution conditioned on Θ is Θ x Θ (36 Θ Θ ( Θ ( x (37 is the Bayes normalization actor. I demonstrate that Eq. (35 produces the same result as Eq. (6: h( h Θ ]. (38 First note rom Eq. (34 that Θ x Pr( V η (39 V ( z (40 η x ( w w x d V ( η (4 ( w (42. (43 ubstituting this result into Eq. (37 yields Θ (44 (. (45 Thus Eq. (36 becomes x Θ (46 and so Eq. (35 becomes, as claimed, h( h Θ ]. ( (47 4 Bayes iltering o quantized measurements Let Z : z,, z n be a time-sequence o conventional measurements. Then the single-sensor, single-target Bayes ilter has the orm (x Z (x Z (x Z and is deined by the ollowing time-update and measurement-update equations: x Z ( x x' ( x' d (48 K ( x' ( x Z ( z z Z ( z Z z (49 (50 is the Bayes normalization actor, (z is the conventional lielihood unction, and (x x is the Marov transition density. uppose now that Z :,, n is a time-sequence o quantized measurements. From Eq. (42 it ollows that the generalized lielihood unction or a quantized measurement has the ormula ( z. (5 ( The generalized lielihood unction approach developed in [5] leads to iltering ormulas that are identical to Eqs. (48,49 except that conventional measurements z are replaced by quantized measurements, and (z is replaced by ( : K ( x Z ( x x' ( x' ' ( ( x x x Z ( Z ( Z (52 (53. (54 Eq. (0 guarantees that posterior distributions constructed in Eq. (53 using ( will produce the same posterior distributions constructed using conventional measuretheoretic methods. Consequently, it ollows that Eqs. (52-54 constitute the basis or a general Bayes iltering theory or quantized measurements (at least or nonlinearadditive measurement models. One result is that Eqs. (52-54 speciy a Bayesoptimal approach or iltering quantized measurements. For, we may apply any Bayes-optimal state estimator to the (x Z to get a Bayes-optimal reduction o the sequence Z o quantized measurements. It also ollows that Eqs. (52-54 generalize the iltering theory introduced in [6,7]. This is because, in those papers, the quantization cells i are taen to be rectangular in orm, and then various approximations are used to implement Eqs. ( The quantization approach advocated in this paper can be urther extended to the multisource-multitarget Bayes iltering theory described in [5]. In [5], all inormation sources are mediated by lielihood unctions, whether conventional or generalized (see ection o [5]. In particular, this means that PHD and CPHD ilters (see [3,4] and Chapter 6 o [5] can be extended to encompass quantized measurements. (O course, the conventional clutter spatial distribution c (z must be replaced by an analogous clutter unction c ( or quantized measurements, which must be unitless and thus not a density unction. 5 Generalized measurements: A review In Chapter 5 o [5], I introduced the concept o unambiguously generated ambiguous (UGA 349

5 measurements and their generalized lielihood unctions. Nontraditional measurements e.g., attributes, eatures, natural-language statements, and inerence rules are represented as random (closed subsets Θ o a measurement space Z 0 that is as generalized measurements. In turn, generalized measurements are mediated by generalized lielihood unctions, which are deined as (Θ Pr(η ( Θ (55 z η ( is a deterministic measurement model. Then I showed how various expert-system ormalisms or representing nontraditional measurements uzzy logic, Dempster-haer theory, rule-based inerence can be used to construct random set models. For example, consider an imprecise measurement. I it is understood as being deterministic, then Θ and the corresponding generalized lielihood unction is ( Pr(η ( (η ( (56 (z is the indicator unction o. As another example, consider a uzzy measurement i.e., a uzzy membership unction g(z on Z 0. uch a measurement can be understood as an imprecisely speciied imprecise measurement meaning that the measurement is imprecise, but that the speciic orm o the imprecision is unclear, taing many possible orms a {z a g(z} or 0 a. Using the random set representation Σ g {z A g(z} (57 o g(z, A is a uniormly distributed random number on the unit interval [0,], I showed that its corresponding generalized lielihood is ([5], Eq. 5.29: (g g(η (. (58 As a third example, consider a uzzy Dempster- haer (FD measurement. This is a basic mass assignment µ(g on the uzzy subsets g o Z 0, deined by the properties: µ(g 0 or all g; µ(g 0 i g 0; µ(g 0 or all but a inite number o g (the ocal uzzy sets o µ; and Σ g µ(g. (When the ocal uzzy sets o µ are all crisp, then µ is a conventional Dempster-haer basic mass assignment. A FD measurement is a urther generalization o the concept o an imprecise measurement, in which multiple hypotheses are required to represent the uncertainty in the choice o imprecise measurement. Employing a random set representation Σµ o µ, I showed that its corresponding generalized lielihood is ([5], Eq. (5.73: µ µ ( g g( η (. (59 ( g x I derived generalized lielihood unctions or other nontraditional measurements, such as uzzy inerence rules g gʹ ([5], Eq. (5.80: ( g g' x ( g g'( η ( g'( η (60 2 using a random set representation Σ g gʹ or g gʹ. 6 Noncooperatively quantized measurements Conventional quantization presumes that the quantization scheme imposed at the transmission node is nown at the receiving node. uppose, however, that the receiver node has only partial (or no a priori nowledge o this scheme. This could be because, or example, the transmission source is noncooperative or which reason I call such measurements noncooperatively quantized measurements (NQMs. In this case the best that the receiver node can do is pose hypotheses about what the actual scheme and thereore, the resulting quantized measurements might be. The simplest approach would be to pose a nested sequence n o guesses about the current quantized measurement, with associated degrees o belie w,,w n in the hypotheses, with w w n. Here, would represent the inest degree o hypothesized quantization and n the coarsest, with n Z 0 representing the possibility that all o the other hypotheses are mistaen. This ind o representation is actually a uzzy representation, with the corresponding uzzy membership unction being uniquely deined by g z w ( z... ( (6 ( w z n n and thus having n possible values. A general uzzy membership unction g(z on Z 0 is, thereore, a modeling o the quantized measurement with an ininite number o hypotheses. In this case g(z can be described as a uzzy quantized measurement. In lie ashion, a uzzy basic mass assignment µ on Z 0 can be interpreted as a model o the quantized measurement that has even more diverse hypotheses than a uzzy quantized measurement. In this case µ can be described as a FD quantized measurement. In what ollows, I ( deine the concept o generalized lielihood unctions or noncooperative quantized measurements, (2 derive a general ormula or such lielihoods, and then (3 derive speciic ormulas or uzzy quantized measurements and FD quantized measurements. 6. Generalized lielihoods or NQMs All o the lielihood unction ormulas in ection 5 were based on a common assumption: that the generalized measurement was deterministic. Even though random sets Σ g, Σµ, and Σ g gʹ, were used to represent the nontraditional measurements g, µ, and g gʹ, these measurements were themselves not random. That is, they did not arise as speciic instantiations o some random variable. (For a more complete discussion see, or example, that surrounding Eq. (4.23 o [5]. The concept o a quantized measurement orces us to extend this analysis. The random set representation o a quantized measurement is not Θ, but rather Θ 350

6 V as in Eq. (3. The same is the case or the obvious uzzy and FD generalizations o a quantized measurement. Consequently, let Θ denote the random set representation o a deterministic noncooperatively quantized measurement that is, a speciic instantiation o a quantized measurement that is actually random. Then as in Eq. (34, the undamental measurement model or quantization is Θ {Z} (62 or, equivalently, η( Θ V. (63 Thus the generalized lielihood o the quantized measurement, taing randomness into account, is (Θ Pr(η( Θ V. (64 as claimed. 6.2 The general case uppose that the generalized lielihood unction is as in Eq. (64. Then it is ( Θ µ Θ( z z (65 µθ(z Pr(z Θ (66 is Goodman s one-point covering unction o Θ (see [5], Eq. (4.20. Note that this equation is well deined, since µ z ( z ( z. (67 Θ( To prove Eq.(64, assume that Θ and V are independent. Then Θ Pr( η V Θ (68 Pr( η z Θ V ( z (69 Pr( w Θ V ( w η (70 µ w ( w (7 as claimed. Θ( 6.3 Fuzzy quantized measurements uppose that the quantized measurement is uzzy: g(z. Then its generalized lielihood unction is g g( z z. (72 This ollows directly rom the act that the one-point covering unction o the random set representation Σ g o g is g(z see Eq. (4.22 o [5]. As a simple closed-orm example, let z NR( z H (73 g( z det 2π C NC( z c. (74 Then Eq. (72 becomes g det 2π C NC( z c NR( z H (75 det 2π C N ( c H. (76 C R 6.4 FD quantized measurements uppose that the quantized measurement is FD: µ(g. Then its generalized lielihood unction is ( g g( z ( µ g µ x ( z. (77 This result immediately ollows rom the act that the onepoint covering unction o the random set representation Σµ o µ is (see Eq. (5.77 o [5] µ µ ( z µ ( g g( z. (78 7 Conclusion Σ g Recent papers at this conerence [6,7] have developed generalizations o the Kalman ilter that process quantized measurements [7], or both quantized and point measurements [6]. As a potential urther advancement o this wor, in this paper I have proposed a general Bayes iltering theory or quantized measurements. This theory generalizes to the multisource-multitarget case and, in particular, to the PHD and CPHD ilters. The quantization theory described in this paper is based on the generalized lielihood unction approach described in [5]. I demonstrated that, at the theoretically oundational level, it is equivalent to the measuretheoretic ormulation employed in [6,7] or the case o nonlinear-additive measurement models. (It is possible to extend this equivalence to nonadditive models. As an indication o its potential urther useulness, I also addressed the problem o representing and iltering uncooperatively quantized measurements. By this I mean quantized measurements whose quantization scheme is imperectly nown at the receiving node. I addressed two speciic special cases: uzzy-quantized measurements and uzzy Dempster-haer (FD- quantized measurements. ince, in general, quantization produces highly nonlinear measurement models, it is anticipated that sequential Monte Carlo (a..a. particle techniques will be required or implementation o the ilters described in this paper. This is a proper subject o uture research. However, the results o ection 4 already tell us that as a speciic example any implementation o the singletarget ilter o Eqs. (52-54 (such as those in [6,7] must be an approximation o a Bayes-optimal (and in this sense, best possible ilter. In any case, elucidation o these ilters was only a secondary purpose o this paper. Its primary purpose was to highlight quantized measurements as a previously unrecognized common ground that lins hard and sot inormation usion. Acnowledgement I am indebted to Brano Ristić or pointing out this problem i.e., iltering o quantized measurements as proposed in [6,7] as a potential area o investigation. Reerences [] R. E. Curry, W. E. vander Velde, and J. E. Potter, Nonlinear estimation with quantized measurements PCM, predictive quantization, and data compression," 35

7 IEEE Trans. Ino. Theory, Vol. IT-6, No. 2, pp. 52-6, March 970. [2] R. Gray and D. Neuho, Quantization, IEEE Trans. on Ino. Theory, Vol. 44, No. 6, 998. [3] R. Mahler, Multitarget iltering via irst-order multitarget moments, IEEE Trans. Aerospace and Electronics ys., Vol. 39, No. 4, pp , [4] R. Mahler, PHD ilters o higher order in target number, IEEE Trans. Aerospace and Electronic ys., Vol. 43, No. 4, pp , 2007 [5] R. Mahler, tatistical Multisource-Multitarget Inormation Fusion, Artech House, Boston, [6] Zhansheng Duan, X. Rong Li, and Vesselin P. Jilov, tate estimation with point and set measurements, Proc. 3th Int'l Con. on Inormation Fusion, Edinburgh, cotland, July 26-29, 200. [7] Zhansheng Duan, Vesselin P. Jilov, and X. Rong Li, tate estimation with quantized measurements: Approximate MME approach, Proc. th Int'l Con. on Inormation Fusion, pp , Cologne, Germany, June 30-July 3,