Abstract
This paper is concerned with the optimal approximation of a given multivariate Dirac mixture, i.e., a density comprising weighted Dirac distributions on a continuous domain, by a Dirac mixture with a reduced number of components. The parameters of the approximating density are calculated by numerically minimizing a smooth distance measure, a generalization of the well-known Cramér–von Mises-Distance to the multivariate case. This generalization is achieved by defining an alternative to the classical cumulative distribution, the Localized Cumulative Distribution (LCD), as a smooth characterization of discrete random quantities (on continuous domains). The resulting approximation method provides the basis for various efficient nonlinear estimation and control methods.
Zusammenfassung
Dieser Beitrag befasst sich mit der optimalen Approximation einer multivarianten Dirac-Mischdichte durch eine Dirac-Mischdichte mit einer geringeren Anzahl an Komponenten. Dirac-Mischdichten bestehen aus gewichteten Dirac-Distributionen auf einer kontinuierlichen Domäne. Die Parameter der approximierenden Dichte werden durch numerische Minimierung eines glatten Abstandsmaßes gewonnnen, welches eine Verallgemeinerung der bekannten Cramér–von Mises-Distanz darstellt. Diese Verallgemeinerung wird durch die Einführung einer Alternative zu klassischen kumulativen Verteilungen, den so genannten lokalisierten kumulativen Verteilungen, als eine glatte Charakterisierung von diskreten Zufallsgrößen (auf kontinuierlichen Domänen) erreicht. Die resultierende Approximationsmethode bildet die Grundlage für verschiedene effiziente nichtlineare Schätz- und Regelungsverfahren.
About the author
Uwe D. Hanebeck is a chaired professor of Computer Science at the Karlsruhe Institute of Technology (KIT) and director of the Intelligent Sensor-Actuator-Systems Laboratory (ISAS). Since 2005, he is the chairman of the Research Training Group RTG 1194 “Self-Organizing Sensor-Actuator-Networks” financed by the German Research Foundation. Prof. Hanebeck obtained his Ph.D. degree in 1997 and his habilitation degree in 2003, both in Electrical Engineering from the Technical University in Munich, Germany. His research interests are in the areas of information fusion, nonlinear state estimation, stochastic modeling, system identification, and control with a strong emphasis on theory-driven approaches based on stochastic system theory and uncertainty models. Research results are applied to various application topics like localization, human-robot-interaction, assistive systems, sensor-actuator-networks, medical engineering, distributed measuring system, and extended range telepresence. Uwe D. Hanebeck was the General Chair of the “2006 IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI 2006)”, Program Co-Chair of the “11th International Conference on Information Fusion (Fusion 2008)”, Program Co-Chair of the “2008 IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI 2008)”, Regional Program Co-Chair for Europe for the “2010 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2010)”, and will be General Chair of the “19th International Conference on Information Fusion (Fusion 2016)”. He is a Member of the Board of Directors of the International Society of Information Fusion (ISIF), Editor-in-chief of its Journal of Advances in Information Fusion (JAIF), and associate editor for the letter category of the IEEE Transactions on Aerospace and Electronic Systems (TAES). He is author and coauthor of more than 350 publications in various high-ranking journals and conferences.
Intelligent Sensor-Actuator-Systems Laboratory (ISAS), Institute for Anthropomatics and Robotics, Karlsruhe Institute of Technology (KIT), Adenauerring 2, 76131 Karlsruhe, Germany
Acknowledgement
The author would like to thank Dipl.-Inform. Henning Eberhardt for many fruitful discussions on this topic and the nice ideas for visualizing the performance of the proposed new reduction algorithm.
The author would also like to thank the anonymous reviewers for helpful comments that led to significant changes in this manuscript.
©2015 Walter de Gruyter Berlin/Boston
