Parameterized Joint Densities with Gaussian and Gaussian Mixture Marginals

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1 Parameterized Joint Densities with Gaussian and Gaussian Miture Marginals Feli Sawo, Dietrich Brunn, and Uwe D. Hanebeck Intelligent Sensor-Actuator-Sstems Laborator Institute of Computer Science and Engineering Universität Karlsruhe (TH) Karlsruhe, German Abstract - In this paper we attempt to la the foundation for a novel filtering technique for the fusion of two random vectors with imprecisel known stochastic dependenc. This problem mainl occurs in decentralized estimation, e.g., of a distributed phenomenon, where the stochastic dependencies between the individual states are not stored. Thus, we derive parameterized joint densities with both Gaussian marginals and Gaussian miture marginals. These parameterized joint densities contain all information about the stochastic dependencies between their marginal densities in terms of a parameter vector ξ, which can be regarded as a generalized correlation parameter. Unlike the classical correlation coefficient, this parameter is a sufficient measure for the stochastic dependenc even characterized b more comple densit functions such as Gaussian mitures. Once this structure and the bounds of these parameters are known, bounding densities containing all possible densit functions could be found. Kewords: Stochastic sstems, data fusion, decentralized estimation, parameterized joint densit, generalized correlation parameter 1 Introduction In a wide variet of applications estimating the state of a dnamic sstem b fusing uncertain information is a topic of etraordinar importance, e.g., robot localization [1], multiple target tracking [2], and decentralized observation of a distributed phenomenon, to name just a few. In most cases, an appropriate sstem model together with a stochastic noise model is given and then the state is estimated b means of a Kalman filter or one of its variations [3]. Problems mainl occur when the stochastic dependenc between the states and/or the sensor noises are not precisel known. In that case, classical concepts like the Kalman filter convenientl assume stochastic independence or precisel known correlation, which automaticall leads to unjustified improvement of estimation results. For understanding the source of unknown stochastic dependenc a tpical application problem is considered: observation of a distributed phenomenon Local Estimate (Marginal Densit) Sensor Node Parameterized Joint Densit Sensor Node Local Estimate (Marginal Densit) Sensor Node Figure 1: Simple eample scenario for a decentralized sensor-actuator-network used for the observation of a distributed phenomenon. b means of a sensor-actuator-network, see Fig. 1. This scenario suffers from two tpes of imprecisel known stochastic dependencies. The first tpe is immanent to the sstem and caused b partiall stochastic dependent noise sources for different sensor nodes. In other words, there are usuall additional eternal disturbances affecting more than one sensor, e.g. sunshine, wind, or the same origin of a pollutant cloud. Even for ideal sensor properties, this alread would lead to a partiall stochastic dependenc between the measured states. In general, using a centralized approach, the distributed phenomenon can be easil observed b appling a standard estimator to the augmented state vector containing all the phsical quantities estimated b individual sensor nodes; in the linear case this can be achieved b a Kalman filter, see [4]. In that case, the estimator stores the associated stochastic dependencies and use them for the net estimation step. However, for practical applications it is often desirable to reduce the heav computational burden and to reduce communication activities between the individual nodes to a minimum. This leads to a decentralized estimation approach, which implies that just parts of the state vector are manipulated at each update step. Hence, the decentralized estimation process itself causes a second source of imprecisel known stochastic dependenc. Let us assume that a phsical quantit is measured and processed onl locall. In order to estimate the phsical quantit at non-measurement points, i.e., b means of a distributed parameterized sstem (PDE), the individual sensor nodes echange

2 and fuse their local estimates. In this case the resulting estimates become automaticall stochastic dependent after the first fusion of the individual estimates. Unfortunatel, appling the Kalman filter for decentralized problems while ignoring the eisting dependencies between the individual states leads to overoptimistic estimation results. Coping with such problems is one of the main justifications for robust filter design [1]. Based on the correlation coefficient r, which is a measure for the stochastic dependenc of random variables, two robust estimators have been introduced to solve the previous mentioned problems, namel Covariance Intersection [5, 6] and Covariance Bounds [7]. Basicall, these filters do not neglect the unknown stochastic dependenc, but consider them b producing conservative estimates for prediction and measurement step, which are compatible with all correlations within an assumed correlation structure. Besides, the generalization of the Covariance Intersection which is based on a minimization of the Chernoff information is worthwhile mentioning [8]. Although robust filters like Covariance Intersection and Covariance Bounds are efficient for linear statespace models and linear measurement models, the cannot be directl applied to nonlinear models, e.g. of a distributed phenomenon. In addition, these filters are not able to work with more comple densit functions such as Gaussian mitures, which are known as universal approimators and thus, well-suited for nonlinear estimation [9]. In this paper, we attempt to la the foundation for a novel filtering technique, which we believe could possibl cope with all previousl mentioned drawbacks. The main problem of a robust decentralized estimation based on more comple densit functions, such as Gaussian mitures, is that for these functions the classical correlation coefficient r is not a sufficient measure for the stochastic dependenc. This means that for such densit functions no obvious correlation coefficient eists that can be bounded and thus, no bounding densit can be found in the classical sense. The main contribution of this paper is to derive parameterized joint densities with both Gaussian marginals and Gaussian miture marginals. These parameterized joint densities contain all information about the stochastic dependenc between their marginal densit functions in terms of a parameter vector ξ. This parameter vector can be regarded as a kind of generalized correlation parameter for the assumed structure of stochastic dependenc. Once this structure and the bound of the correlation parameter vector ξ are known, bounding densities which are compatible with all stochastic dependenc structures can be found. In [10], a definition for lower and upper bounding densities is given, and their potential use in robust estimation is shown. The remainder of this paper is structured as follows. In Section 2, a rigorous formulation of the problem of state estimation with unknown correlations is given. Section 3 then derives several tpes of parameterized joint densities with Gaussian marginals. These joint densities are useful for robust linear estimation. Furthermore, the are helpful to gain some insight about the generalized correlation parameter vector ξ. Whereas, in section 4 we generalize these ideas to parameterize joint densities with Gaussian miture marginals. Bounding the parameter vector of these joint densities, it is possible to find a conservative estimation even for nonlinear problems and for more comple densit functions. Throughout this paper, we use the notation N (z, C(r)) to represent a joint Gaussian densit, which is defined as N (z, C(r)) = where [ ] ˆ z = ŷ 1 2π C(r) ep [ C, C(r) = r C C { 1 } 2 zt C(r) z, r ] C C are state vector and covariance matri, respectivel. The epected value of the states are denoted b ˆ and ŷ, and r [, 1] denotes the classical correlation coefficient. 2 Problem Formulation As mentioned in the introduction, there are man sources of stochastic dependencies. In this section, we take up the previousl mentioned eample, the observation of a distributed phenomenon with a sensoractuator-network. B this means we are able to clarif the main problem common to all sources of imprecisel known stochastic dependenc: imprecisel known joint densities with given marginal densities. For the sake of simplicit, consider a simple discretetime dnamic model with the sstem state k IR, and the sstem input u k IR according to C k+1 = a k ( k ) + b k (u k ). (1) The corresponding densit functions of the estimates are given b f e ( k ) and f e u(u k ). In the case of a precisel known joint densit f e k ( k, u k ), the predicted densit f p k+1 ( k+1) is given b f p k+1 ( k+1) = δ( k+1 a k ( k ) b k (u k ))fk( e k, u k )d k du k, IR 2, (2) where δ denotes the Dirac delta distribution. With the justification of the considered eample scenario we assume that the state estimate k and the sstem input u k are stochasticall dependent with an imprecisel known structure. Hence, although the marginal densit functions f e ( k ) and f e u(u k ) are known, the joint densit f e k ( k, u k ), which contains all information about the stochastic dependenc, is unknown. However, as it can be seen in (2), the knowledge of the joint densit or at least its parameterization in terms of a correlation parameter is essential for the estimation process.

3 If the joint densit, i.e., the correlation structure, would be known precisel, this correlation could easil be tracked and be considered in the net processing step. However, when the correlation structure is not precisel known, the joint densities need somehow to be reconstructed. In the net section, the reconstruction of joint densities f(, ) based on known marginal densities f () and f () is discussed in more detail. For all introduced tpes this leads to a joint densit parameterized b a generalized correlation parameter vector ξ containing the information about the stochastic dependenc between the considered random variables. 3 Gaussian Marginals for Linear Estimation In this section, we describe possible parameterizations of joint densities for given Gaussian marginal densities. We have identified three tpes of parameterized joint densities. The first two tpes mainl serve for didactic purposes for understanding the problem of imprecisel known stochastic dependenc. The third tpe, however, is useful for robust decentralized linear estimation and thus, is of practical relevance. 3.1 Piecewise Gaussian Densities with Different Weighting Factors This section consists of a description for the simplest non-gaussian joint densit with Gaussian marginals; this tpe frequentl appears in informal discussions on the internet. Basicall, a jointl Gaussian densit is sliced into different pieces, which are raised or lowered, respectivel. The visualization of two eamples for this simple tpe of parameterized joint densit is shown in Fig. 2. THEOREM 3.1 Given two Gaussian marginal densities f () = N (ˆ, C ) and f () = N (ŷ, C ), a famil of possible joint densities f(, ) can be parameterized b f(, ) = { ξ 1 f(, ) 0 ξ 2 f(, ) < 0 (3) where ξ 1 = 2 (1 ξ) and ξ 2 = 2 ξ. The free parameter ξ [0, 1] can be regarded as a generalized correlation parameter, which specifies the member of the famil. To assure that the parameterized joint densit f(, ) is a valid joint densit for the given marginals f () and f (), the densit f(, ) must be defined according to f(, ) = N ( ˆ, C ) N ( ŷ, C ). Proof. To prove this, it must be shown that the marginal densities of f(, ) in (3) are represented b f () = N ( ˆ, C ) and f () = N ( ŷ, C ), respectivel. The marginal densit f () can be derived (a) f () f () f () f () ξ 2 f(, ) ξ 1 f(, ) ξ 2 f(, ) ξ 1 f(, ) Figure 2: Gaussian densit with piecewise different weighting factors; sliced in (a) four pieces and nine pieces with different weighting factors. b direct integration over. Assuming > 0, it follows f () = = = 2(1 ξ) ξ 1 f(, ) d f(, )d 2ξ 0 f(, ) d + 2ξ f(, )d + 2ξ ξ 2 f(, ) d 0 0 f(, ) d f(, )d Due to the smmetr propert of the Gaussian densit this integral can be simplified, which ields f () = f(, ) d = N ( ˆ, C ). Similar calculations for 0 justif the choices for ξ 1 and ξ 2. This proves Theorem 3.1. This simple parameterized joint densit is depicted in Fig. 2 (a). It is possible to etend this tpe of parameterized joint densit to more comple ones, i.e., the joint densit could be sliced into more pieces, as shown in Fig Sum of Positive and Negative Gaussians A second tpe of non-gaussian densit with Gaussian marginals could be constructed b the sum of positive and negative jointl Gaussian densities. The mean values, variances, and weighting factors of the individual joint densities need to be chosen appropriatel, i.e., in such wa that the joint densit is both a valid densit function and the marginals are Gaussian densities. For the sake of simplicit we consider onl three components in the following theorem. THEOREM 3.2 Given two Gaussian densities f () = N ( ˆ, C ) and f () = N ( ŷ, C ), the unknown joint densit f(, ) can be defined b the sum of positive and negative Gaussians according to f(, ) = f 1 (, ) + f 2 (, ) f 3 (, ), (4)

4 (a) f () f () ˆ,ŷ ˆ,ŷ ˆ,ŷ THEOREM 3.3 Given two Gaussian marginal densities f () = N (ˆ, C ) and f () = N (ŷ, C ), a famil of possible joint densities, depending on the generalized correlation function ξ(r), can be parameterized b f(, ) = ξ(r) N ([ ˆ ŷ ], C(r) ) dr, (5) where ξ(r) is defined on r [, 1]. The parameterized continuous Gaussian miture f(, ) is a valid normalized densit function for f () negative Gaussian positive Gaussian f () Figure 3: Joint densit consisting of the sum of positive and negative Gaussian densities, (a) three components, eight components. where the individual densities f 1 (, ), f 2 (, ), and f 3 (, ) are given b f 1 (, ) = N ( ˆ, C ) N ( ŷ, C ), f 2 (, ) = N ( ˆ, C ) N ( ŷ, C ), f 3 (, ) = N ( ˆ, C ) N ( ŷ, C ). To assure that the parameterized joint densit f(, ) is a valid joint densit, the mean values and variances of the individual joint densities need to be chosen so that f(, ) 0 holds. Proof. The marginal densit f () can be derived b direct integration over according to f () = f(, )d IR =N ( ˆ, C ) + N ( ˆ, C ) N ( ˆ, C ) =N ( ˆ, C ). Similar calculations for the marginal densit f () justif the choices for individual mean values and variances of the parameterized joint densit, and finall conclude the proof. Fig. 3 (a) visualizes the simplest case for this tpe of parameterized joint densit consisting of three components. It is possible to etend this tpe to joint densities with more components, for eample with eight components as depicted in Fig Mitures of Correlated Jointl Gaussian Densities In this section, we derive a parameterized joint densit with practical relevance, which is based on the integral of jointl Gaussian densities with different classical correlation coefficients. The weighting factors of the individual joint densities need to be chosen in such a wa that the marginals are represented b the given Gaussian marginal densities. ξ(r) 0, ξ(r)dr = 1. Proof. The results directl follow from the integration of the joint densit f(, ) over and, respectivel. Hence, the marginal densit f () can be derived b direct integration of the joint densit f(, ) over according to ([ ] ) ˆ f () = ξ(r) N, C(r) dr d = IR ξ(r) IR N ŷ ([ ˆ ŷ ], C(r) ) d dr. With reference to [3] it can be shown that the solution of the integral does not depend on the correlation coefficient r at all. Thus, it can easil be obtained f () = ξ(r) N ( ˆ, C ) dr = N ( ˆ, C ), which justifies the condition ξ(r)dr = 1. Similar calculations for the marginal densit f () leads to the same condition and finall concludes the proof. Two eamples for this tpe of parameterized joint densit are depicted in Fig. 4. In the following, a rough idea is shown on how this tpe of parameterized joint densit can be used for the development of a novel estimator for an imprecisel known correlation coefficient. Consider two marginal densities f () and f (). Let us assume that just the mean value ˆr and variance C r of their classical correlation coefficient r is known. In this case, a densit function ξ(r) for the correlation coefficient can be defined according to ξ(r) = c n N (r ˆr, C r ) r [, 1], with normalization constant c n (c n ) = N (r ˆr, C r ) dr. The densit function ξ(r), which is depicted in the right column of Fig. 4, can be regarded as a generalized correlation function. That means, this function contains all information about the correlation structure. Using this information, the joint densit f(, ) can be parameterized b f(, ) = c n N (r ˆr, C r ) N (z, C(r)) dr. (6)

5 (a) f () f () -1 ξ(r) ξ(r) 1 r ŷ 2 C,2 w,2 ŷ 1 C,1 w,1 w 12 w 22 w 11 w 21 f () f () Figure 4: Joint densit consisting of sum of correlated Gaussian densit for different generalized correlation function ξ(r), (a) ξ(r) = 7 2 r6, and ξ(r) = N (r ˆr, C r ). It is possible to use this parameterized joint densit as the estimated joint densit f e k ( k, u k ) in (2). The predicted densit f p k+1 ( k+1) depending on the parameters of the generalized correlation function ξ(r) can then be derived. This idea could la the foundation for a novel filtering technique taking into account the imprecisel known classical correlation coefficient r and gives a bounding densit for the prediction result. The actual calculation of a bounding densit is left for future research work. 4 Gaussian Miture Marginals for Nonlinear Estimation So far, we have onl considered parameterized joint densities with Gaussian marginals. In this section, we generalize these ideas to the parameterization of joint densities with Gaussian miture marginals. Gaussian mitures consist of the conve sum of weighted Gaussian densities. Due to the fact that Gaussian mitures are universal approimators, the are well-suited for nonlinear estimation problems [11]. Thus, finding a parameterization for the imprecisel known joint densit with Gaussian miture marginals, it is possible to develop a novel filtering technique, which could possibl cope with both nonlinear sstem models and nonlinear measurement models in a robust manner. As it was mentioned in the introduction, the challenge of a robust decentralized estimation based on Gaussian mitures is that the classical correlation coefficient r is not a sufficient measure for the stochastic dependenc of Gaussian mitures. Therefore, we define in this section a generalized correlation parameter vector ξ for Gaussian mitures. Finding bounding densities, which are compatible with all stochastic dependenc structures in terms of ξ, it is possible to derive a robust filtering technique for distributed nonlinear problems. For the sake of simplicit consider two scalar -1 1 r ˆ 1 C,1 w,1 ˆ 2 C,2 w,2 Figure 5: Notation for marginal densities and joint densit for Gaussian mitures. Gaussian mitures marginals according to f () = w,i N ( ˆ i, C,i ), (7) f () = i=1 n w,i N ( ŷ i, C,i ), (8) i=1 where ˆ i and ŷ i are the individual means, C,i and C,i are the individual variances, and w,i and w,i are the individual weighting factors, which must be positive and sum to one. The notation for the marginals and their joint densit are visualized in Fig. 5. THEOREM 4.1 Given two Gaussian miture marginal densities f () and f (), a famil of possible joint densities f(, ) depending on the weighting factors w ij is defined b n ([ ] ) ˆi f(, ) = w ij N, C ŷ ij (r ij ). (9) j To assure that the parameterized Gaussian miture f(, ) is a valid normalized densit function, the weighting factors w ij must be positive and sum to one n w ij 0, w ij = 1. (10) In addition, the weighting factors must satisf n w ij = w,i, w ij = w,j, (11) j=1 i=1 to ensure that the parameterized joint densit f(, ) is a valid joint densit for the marginal densit functions f () and f (). Proof. These properties and conditions can be proven b showing that the marginal densities of f(, ) are represented b f () and f (), given b (7) and (8), respectivel. The marginal densit f () can be derived b direct integration over according to n ([ ] ) ˆi f () = w ij N, C ŷ ij (r ij ) d j = IR n w ij IR N ([ ] ) ˆi, C ŷ ij (r ij ) d j

6 (a) f () (c) (d) f () f () f () f () f () f () f () Figure 6: (a) Gaussian miture marginal densities with two individual components (w,i = w,i = 0.5), f () with ˆ 1 = 3, ˆ 2 = 1, C,1 = 2, C,2 = 1.6, and f () with ŷ 1 = 4, ŷ 2 =, C = 2.1, C = 1.4. Parameterized joint densities for various generalized correlation parameter ξ, ξ = 0, (c) ξ =, (d) ξ = 1. Refering to [3] it can be shown that the integral solution does not depend on the correlation coefficient r ij at all. Thus, f () = n w ij N ( ˆ i, C,i ) can easil be obtained. Now, it is obvious that for the condition n j=1 w ij = w,i this leads to the desired Gaussian miture marginals f () = w,i N ( ˆ i, C,i ). i=1 Similar calculations for marginal densit function f () leads to the desired condition m i=1 w ij = w,j. This concludes the proof. For the following calculations it is more convenient to rearrange the weighting factors of the joint Gaussian miture densit w ij from matri form to vector form according to w = [ w 11 w 1n, w 21 w 2n, w mn ] T The weighting factors of marginals are given b w = [ w,1 w,m ] T, w = [ w,1 w,n ] T. The conditions (11) for a valid joint densit f(, ) with marginals f () and f () can be epressed b a linear equation, which transforms weighting factors of joint densit to weighting factors of marginals. This linear transformation is given b [ ] w T w =. w Due to the fact that the matri T IR (n+m) n m in general is not a square matri and does not have full rank, there eists a null space (kernel) ker T, which is given b (ker T) w = ξ. The null space ker T is spanned b the free parameter vector ξ. Thus, for the calculation of valid weighting factors w we propose following Lemma. LEMMA 4.2 The weighting factors w of the joint densit f(, ) can be derived according to w [ ] w = T e w T, T e =, ker T ξ where T e describes a unique transformation of weighting factors for valid joint densities to weighting factors of given marginal densities. The pseudo-inverse is denoted b T e. Similar to the other tpes of parameterized joint densities the free parameter vector ξ can be regarded as a kind of generalized correlation parameter for Gaussian mitures. This parameter vector needs to be specified in order to define the joint densit f(, ) uniquel. Here, the individual components of the Gaussian miture are assumed to be uncorrelated, i.e., r ij = 0. Thats wh the resulting generalized correlation vector ξ is derived onl b the weighting factors w; and not the correlation coefficients r ij. Eample 4.1 Now, we will consider possible joint densities f(, ) for two given Gaussian miture marginals f () and f (), depicted in Fig. 6 (a). The weighting factors for components of the joint densit are obtained b w 11 w 12 w 21 w 22 = w,1 w,2 w,1 w,2 ξ, where ξ is the free parameter. Possible joint densities for various parameters ξ are depicted in Fig. 6 (d). Furthermore, it is possible to combine all the tpes of parameterized joint densities introduced in this paper. The most relevant combination is appling the mitures of correlated jointl Gaussian (Sec. 3.3) to the individual components of a Gaussian miture. For simplicit we consider the previous eample and assume the generalized correlation function ξ ij (r) for the ij-th component to be given b ξ ij (r) = δ(r α), where δ is the Dirac delta distribution and α denotes the classical correlation coefficient. In Fig. 7 the parameterized joint densit is depicted for different α and ξ.

7 (a) f () f () f () f () Figure 7: Combination of two tpes of parameterized joint densities. (a) ξ =, α =, ξ = 1, α = 1. It turns out that in the case of α = and ξ = the two considered random variables, which are described b Gaussian mitures, are full correlated, i.e., similar to the Gaussian case r. Similar to the previous eample and with reference to (2), let us consider possible joint densities f e k ( k, u k ) for two given Gaussian miture marginals, f e ( k ) and f e u(u k ). The Gaussian mitures consisting of three components is visualized in Fig. 8 (a). It turns out that the free parameter vector is given b ξ = [ξ (1),..., ξ (4) ] T, which completel describes the stochastic dependenc between the random variables k and u k. In Fig. 9 (a) (d) the parameterized joint densit is depicted for variation of the individual free parameters ξ (i) from their minimum value to their maimum value, respectivel. In order to visualize that all possible joint densities can be described just b the generalized correlation parameter vector ξ, various joint densities are depicted in Fig. 10 (a) (d). Furthermore, the predicted densit f p k+1 ( k+1) depending on the generalized correlation parameter vector ξ can be derived, which is shown in the lower row in Fig. 9 (a) (d) and Fig. 10 (a) (d). For a net step toward a nonlinear decentralized filtering technique robust against unknown correlation it would be necessar to find a bounding densit which somehow contains the densit functions of all possible, regarding ξ and/or ξ(r). 5 Conclusion and Future Work This paper focuses on the parameterization of different tpes of joint densities both with Gaussian marginals and Gaussian miture marginals. It is shown that assuming a specific stochastic dependenc structure these joint densities contain all information about this dependenc in terms of a generalized correlation parameter vector ξ and/or a generalized correlation function ξ(r). Unlike the classical correlation coefficient r the generalized correlation parameter vector ξ is a sufficient measure for the stochastic dependenc between two random variables represented b Gaussian mitures. More detailed prediction results and measurement results depending on the generalized correlation parameter will be covered in a forthcoming paper. Finding a bounding densit fulfilling the stochastic dependenc constraints, and thus containing all possible values of the generalized correlation parameter ξ, is left for future research work; a possible direction for solving this problem can be found in [10]. It is obvious that once such a bounding densit is found, a filtering technique can be derived, which can cope with nonlinear models and is robust against imprecisel known stochastic dependencies. 6 Acknowledgement This work was partiall supported b the German Research Foundation (DFG) within the Research Training Group GRK 1194 Self-organizing Sensor- Actuator-Networks. References [1] J. A. Castellanos, J. D. Tardos, G. Schmidt, Building a Global Map of the Environment of a Mobile Robot: The Importance of Correlations, IEEE International Conference on Robotics and Automation (ICRA 97), Albuquerque, New Meico, USA, [2] S. Oh, S. Sastr, L. Schenato, A Hierarchical Multiple- Target Tracking Algorithm for Sensor Networks, Proc. of the 2005 IEEE International Conference on Robotics and Automation (ICRA 05), Barcelona, Spain, [3] A. Papoulis, Probabilit, Random Variables, and Stochastic Processes, McGraw Hill Book Compan, [4] F. Sawo, K. Roberts, U. D. Hanebeck, Baesian Estimation of Distributed Phenomena Using Discretized Representations of Partial Differential Equations, Proc. of the 3rd International Conference on Informatics in Control, Automation and Robotics (ICINCO 06), Setúbal, Portugal, [5] J. Uhlmann, S. Julier, M. Csorba, Nondivergent Simultaneous Map Building and Localization Using Covariance Intersection, Proc. of the 1997 SPIE Aerosense Conference, Orlando, Florida, [6] C. Y. Chong, S. Mori, Conve Combination and Covariance Intersection Algorithms in Distributed Fusion, Proc. of the 4th International Conference on Information Fusion 2001, Montreal, Canada, [7] U. D. Hanebeck, K. Briechle, J. Horn, A Tight Bound for the Joint Covariance of Two Random Vectors with Unknown but Constrained Cross-Correlation, IEEE Conference on Multisensor Fusion and Integration for Intelligent Sstems (MFI 01), Baden-Baden, German, [8] M. B. Hurle, An Information Theoretic Justification for Covariance Intersection and Its Generalization, Proc. of the 5th International Conference on Information Fusion 2002, Annapolis, Marland, USA, [9] D. L. Alspach, H. W. Sorenson, Nonlinear Baesian Estimation Using Gaussian Sum Approimation, IEEE Transactions on Automatic Control, Vol. 17, No. 4, pp , [10] V. Klumpp, D. Brunn, U. D. Hanebeck, Approimate Nonlinear Baesian Estimation Based on Lower and Upper Densities, Proc. of the 9th International Conference on Information Fusion 2006, Florence, Ital, [11] U. D. Hanebeck, K. Briechle, A. Rauh, Progressive Baes: A New Framework for Nonlinear State Estimation, Proceedings of SPIE, AeroSense Smposium, 2003.

8 (a) f e k( k ) (c) f p k+1 ( k+1) f e k(u k ) 0 u k 10 f e k( k ) f e k(u k ) 20 k+1 20 Figure 8: (a) Gaussian miture marginal densities f e k ( k) and f e k (u k), and their joint densit f( k, u k ) for the uncorrelated case, i.e. ξ (i) = 0, (c) predicted densit f p k+1 ( k+1). (a) (c) (d) f p k+1 ( k+1) f p k+1 ( k+1) f p k+1 ( k+1) f p k+1 ( k+1) 20 k k k k+1 20 Figure 9: Parameterized joint densit f( k, u k ) and predicted densit f p k+1 ( k+1) for variation of the individual free parameter ξ (i) from their minimum values to their maimum values, ξ (i) = 0 ecept (a) ξ (1) = /3... 2/3, ξ (2) = /3... 2/3, (c) ξ (3) = /3... 2/3, and (d) ξ (4) = /3... 2/3. (a) (c) (d) f p k+1 ( k+1) f p k+1 ( k+1) f p k+1 ( k+1) f p k+1 ( k+1) 20 k k k k+1 20 Figure 10: Parameterized joint densit f( k, u k ) and predicted densit f p k+1 ( k+1) for various free parameter vectors, (a) ξ = [ 2 3, 1 3, 1 3, 2 3 ]T, ξ = [ 2 3, 1 3, 1 3, 1 3 ]T, (c) ξ = [ 1 3, 2 3, 2 3, 1 3 ]T, and (d) ξ = [ 1 3, 1 3, 1 3, 2 3 ]T.