## Kalman filter and joint tracking and classification based on belief functions

Venue: | in the TBM framework. Information Fusion, 2005. In |

Citations: | 4 - 0 self |

### BibTeX

@INPROCEEDINGS{Ristic_kalmanfilter,

author = {Branko Ristic},

title = {Kalman filter and joint tracking and classification based on belief functions},

booktitle = {in the TBM framework. Information Fusion, 2005. In},

year = {},

publisher = {Press}

}

### OpenURL

### Abstract

The paper presents an approach to joint tracking and classification based on belief func-tions as understood in the transferable belief model (TBM). The TBM model is identical to the classical model except all probability functions are replaced by belief functions, which are more flexible for representing uncertainty. It is felt that the tracking phase is well han-dled by the classical Kalman filter but that the classification phase deserves amelioration. For the tracking phase, we derive a minimal set of assumptions needed in the TBM ap-proach in order to recover the classical relations. For the classification phase, we distinguish between the observed target behaviors and the underlying target classes which are usually not in one-to-one correspondence. We feel the results obtained with the TBM approach are more reasonable than those obtained with the corresponding Bayesian classifiers.

### Citations

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Citation Context ...oncepts related to the TBM. (19) (20) p(zi|Zi−1) (21) The transferable belief model (TBM) [9, 17, 6] is a model to represent quantified beliefs based on the belief function theory developed by Shafer =-=[18]-=-, but completely unrelated to any underlying probabilistic constraints as it is the case with the model of Dempster [19] and with the hint model [14]. These differences are not important here. The ess... |

422 |
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Citation Context ...serves amelioration. For the tracking phase, we determine under which ‘minimal’ assumptions the TBM approach produces the classical KF relations (as a typical building block of class-matched filters) =-=[10]-=-. For this derivation we assume: • for the dynamic equations, that the uncertainty in the additive noise is represented by belief functions whose pignistic transformations are Gaussian probability dis... |

367 | The transferable belief model
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Citation Context ...e to inadequate mapping between target behavior and target class. This was a motivation to consider the classification phase of the JTC problem in the framework of the transferable belief model (TBM) =-=[6, 7, 8, 9]-=-. It was hoped - and in fact observed - that the use of belief functions, which are more flexible than the probability functions, could produce better classification results. In order to use the TBM f... |

119 | Belief functions: the disjunctive rule of combination and the generalized Bayesian theorem. International Journal of Approximate reasoning 9:1–35
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(Show Context)
Citation Context ...n the observed target behaviors and the underlying target classes which are usually not in one-to-one correspondence. Within the TBM, the classification phase is based on the General Bayesian Theorem =-=[11, 12]-=-. The classification results differ significantly from those derived within the classical framework. It is due to the fact that the TBM provides more flexible ways to represent adequately the prior be... |

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Citation Context ... on the belief function theory developed by Shafer [18], but completely unrelated to any underlying probabilistic constraints as it is the case with the model of Dempster [19] and with the hint model =-=[14]-=-. These differences are not important here. The essential tool is the basic belief assignment (bba) m Ω which maps subsets of its domain Ω to [0, 1]. Its value m Ω (A) for A ⊆ Ω is called the basic be... |

88 |
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Citation Context ...e to inadequate mapping between target behavior and target class. This was a motivation to consider the classification phase of the JTC problem in the framework of the transferable belief model (TBM) =-=[6, 7, 8, 9]-=-. It was hoped - and in fact observed - that the use of belief functions, which are more flexible than the probability functions, could produce better classification results. In order to use the TBM f... |

58 |
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Citation Context ...as : BetP (Y ) = � X⊆Ω |Y ∩ X| |X| 15 (25) mΩ (X) 1 − mΩ , ∀Y ⊆ Ω (26) (∅)sThe nature of the pignistic transformation given by (26) is presented in [20, 9]. Its detailed justification is presented in =-=[21]-=-. When the credal variable X is defined on R, we end up with a pignistic probability density function Betf which is defined as: Details can be found in [15]. � x=a � y=∞ Betf(a) = lim ε→0 x=−∞ y=a+ε 3... |

42 |
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Citation Context ...ef funtions are said to be isopignistic. In order to apply the belief function theory, one needs to formulate a method of building a bbd from the pignistic density. The least commitment principle [6],=-=[24]-=- suggests to choose among all the isopignistic bbds, the one which maximizes the commonality function q. The q-least committed belief density is a consonant bbd. On the real axis R this means that all... |

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Citation Context ...e to inadequate mapping between target behavior and target class. This was a motivation to consider the classification phase of the JTC problem in the framework of the transferable belief model (TBM) =-=[6, 7, 8, 9]-=-. It was hoped - and in fact observed - that the use of belief functions, which are more flexible than the probability functions, could produce better classification results. In order to use the TBM f... |

29 |
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Citation Context ...i−1, Si). Thus: 3 TBM Background i=1,...,t p(Zt) = p(Zt−1)N (zt; ˆzt|t−1, St). We introduce some preliminary concepts related to the TBM. (19) (20) p(zi|Zi−1) (21) The transferable belief model (TBM) =-=[9, 17, 6]-=- is a model to represent quantified beliefs based on the belief function theory developed by Shafer [18], but completely unrelated to any underlying probabilistic constraints as it is the case with th... |

27 |
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Citation Context |

26 | Decision making in a context where uncertainty is represented by belief functions
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(Show Context)
Citation Context ...tic probability function (denoted BetP ) is defined as : BetP (Y ) = � X⊆Ω |Y ∩ X| |X| 15 (25) mΩ (X) 1 − mΩ , ∀Y ⊆ Ω (26) (∅)sThe nature of the pignistic transformation given by (26) is presented in =-=[20, 9]-=-. Its detailed justification is presented in [21]. When the credal variable X is defined on R, we end up with a pignistic probability density function Betf which is defined as: Details can be found in... |

21 | Target identification based on the Transferable Belief Model interpretation of Dempster-Shafer model
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(Show Context)
Citation Context ...n the observed target behaviors and the underlying target classes which are usually not in one-to-one correspondence. Within the TBM, the classification phase is based on the General Bayesian Theorem =-=[11, 12]-=-. The classification results differ significantly from those derived within the classical framework. It is due to the fact that the TBM provides more flexible ways to represent adequately the prior be... |

17 |
Upper and lower probabilities induced by a multiple valued mapping
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(Show Context)
Citation Context ...sent quantified beliefs based on the belief function theory developed by Shafer [18], but completely unrelated to any underlying probabilistic constraints as it is the case with the model of Dempster =-=[19]-=- and with the hint model [14]. These differences are not important here. The essential tool is the basic belief assignment (bba) m Ω which maps subsets of its domain Ω to [0, 1]. Its value m Ω (A) for... |

14 | Efficient Particle Filters for Joint Tracking and Classification
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(Show Context)
Citation Context ...: branko.ristic@dsto.defence.gov.au.s1 Introduction The paper is devoted to joint tracking and classification (JTC) of targets based on kinematic data. The optimal Bayesian estimator for this problem =-=[2, 3, 4, 5]-=- consists of a bank of filters that match the expected dynamic behavior of each class (class-matched filters). In some numerical examples, however, this type of classifier may produce unsatisfactory p... |

14 |
Construction and local computation aspects of network belief functions. Pp. 121-141 of Influence diagrams, belief nets and decision analysis
- Dempster
- 1990
(Show Context)
Citation Context ...ained with the TBM approach are more reasonable than those obtained with the corresponding Bayesian classifiers. Historically, the KF has already been described as an example of evidential network by =-=[13]-=-. In fact the special nature of the belief functions we assume for the KF make it possible to solve easily the filtering without using the whole machinery of the TBM. Nevertheless for the more general... |

9 |
Joint Target Tracking and Classification Using Radar and
- Challa, Pulford
- 2001
(Show Context)
Citation Context ...: branko.ristic@dsto.defence.gov.au.s1 Introduction The paper is devoted to joint tracking and classification (JTC) of targets based on kinematic data. The optimal Bayesian estimator for this problem =-=[2, 3, 4, 5]-=- consists of a bank of filters that match the expected dynamic behavior of each class (class-matched filters). In some numerical examples, however, this type of classifier may produce unsatisfactory p... |

9 | Belief function independence: I. the marginal case
- Yaghlane, Smets, et al.
(Show Context)
Citation Context ...a) = lim ε→0 x=−∞ y=a+ε 3.5 Doxastic independence mR (X ∈ [x, y]) dydx. (27) y − x The concept of doxastic independence is the extension of the concept of stochastic independence in the TBM framework =-=[22, 23]-=-. Its syntactical definition is given by: Definition 3.5 Let X and Y be two credal variables defined on R n1 and R n2 , respectively. Let m{X} and m{Y } be their bbds. The credal variables X and Y are... |

8 | Belief function theory on the continuous space with an application to model based classification - Ristic, Smets |

5 |
Belief function independence: II. The conditional case
- Yaghlane, Smets, et al.
(Show Context)
Citation Context ...a) = lim ε→0 x=−∞ y=a+ε 3.5 Doxastic independence mR (X ∈ [x, y]) dydx. (27) y − x The concept of doxastic independence is the extension of the concept of stochastic independence in the TBM framework =-=[22, 23]-=-. Its syntactical definition is given by: Definition 3.5 Let X and Y be two credal variables defined on R n1 and R n2 , respectively. Let m{X} and m{Y } be their bbds. The credal variables X and Y are... |

4 |
Joint tracking and identification algorithms for multisensor data
- Farina, Lombardo, et al.
- 2002
(Show Context)
Citation Context ...: branko.ristic@dsto.defence.gov.au.s1 Introduction The paper is devoted to joint tracking and classification (JTC) of targets based on kinematic data. The optimal Bayesian estimator for this problem =-=[2, 3, 4, 5]-=- consists of a bank of filters that match the expected dynamic behavior of each class (class-matched filters). In some numerical examples, however, this type of classifier may produce unsatisfactory p... |

2 |
On target classification using kinematic data. Information Fusion, 2003
- Ristic, Gordon, et al.
(Show Context)
Citation Context |

1 | 2(x − µ)N (x; µ, Σ) + = 2(µ − y)N (y; µ, Σ) + � t=∞ t=x � t=y t=−∞ 2N (t; µ - pl |

1 |
Kalman filters and joint tracking and classification
- Smets, Ristic
- 2004
(Show Context)
Citation Context ...get classification, Kalman filter, belief function theory. A short version of this study was presented at the Seventh International Conference on Information Fusion (FUSION 2004) in Stockholm, Sweden =-=[1]-=-. February 25, 2005 † IRIDIA, Université libre de Bruxelles, Av. F. Roosevelt 50, CP 194/6, 1050 Bruxelles, Belgium; tel: +32 (2) 344 82 96; email: psmets@ulb.ac.be ‡ DSTO, ISRD - 200 Labs, PO Box 150... |

1 |
Belief functions on real numbers,” Submitted: see http://iridia.ulb.ac.be/∼psmets
- Smets
- 2004
(Show Context)
Citation Context ...based on probability theory, whereas the TBM does not assume the existence of any underlying probabilities. Belief functions used in this papers are defined on continuous spaces, a topic presented in =-=[15]-=-, used in [16] and summarized in appendix A.1. The paper is organized as follows. In section 2, we present the classical JTC and an example that motivated this work as its classification results seem ... |