Random Set

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Olaf Wolkenhauer - One of the best experts on this subject based on the ideXlab platform.

  • Random Set system identification
    IEEE Transactions on Fuzzy Systems, 2002
    Co-Authors: J. Nunez-garcia, Olaf Wolkenhauer
    Abstract:

    The paper gives a brief review of the basic mathematical aspects of Random Set theory. Concepts such as a Random Set mapping and its coverage function are introduced in a comprehensive way, avoiding too much detail. We adapt this theory to system identification and forecasting of time series. This is achieved by using the one-point coverage function of a Random Set as a possibility measure of the process which generates such a time series. The coverage function of a Random Set defines a fuzzy Set, and we thereby establish the relationship between statistical objects and fuzzy systems. The possibility measure obtained in this way can be used for either prediction or to evaluate the quality of a model with respect to the training data. The technique is adapted to nonlinear time series analysis. A practical application of a nonlinear dynamic plant is presented.

  • Random-Sets: theory and applications
    Granular Computing, 2001
    Co-Authors: Javier Nuñez-garcia, Olaf Wolkenhauer
    Abstract:

    The relevance, applicability and importance of fuzzy Sets is generally linked to successful applications in the domain of engineering, especially when subjective notions are modelled and matched with data. For problems in which uncertainty has been modelled using probability theory in the past, discussions on what approach is right, frequently conclude that both should complement each other. In the present text, we consider such synergy of fuzzy Sets, probability and possibility distributions provided by the concept of a Random-Set. Following a brief review of basic mathematical and semantic aspects of Random-Set theory, we introduce an application of the theory to time series analysis.

  • FUZZ-IEEE - Random Sets and histograms
    10th IEEE International Conference on Fuzzy Systems. (Cat. No.01CH37297), 1
    Co-Authors: J. Nunez-garcia, Olaf Wolkenhauer
    Abstract:

    One of the main reasons why histograms are the most used density estimators is that they are easier to implement and interpret than other density estimators. Some people have already exploited the connection between probability theory and possibility theory or fuzzy Sets to Set up membership functions and to create fuzzy Sets models. Two different ways have been used: 1) transform the density function of a Random variable into a possibility measure, which is an almost automatic operation; and 2) calculate the coverage function of a Random Set, which is a possibility measure. In this paper, we show that a histogram is the coverage function of a determined Random Set. This suggests other methods to create more accurate or different featured histograms by using the Random Set theory. One example of a histogram with overlapping classes is provided.

Pedro Gil - One of the best experts on this subject based on the ideXlab platform.

  • A Random Set characterization of possibility measures
    Information Sciences, 2004
    Co-Authors: Enrique Miranda, Inés Couso, Pedro Gil
    Abstract:

    Several authors have pointed out the relationship between consonant Random Sets and possibility measures. However, this relationship has only been proven for the finite case, where the inverse Mobius of the upper probability induced by the Random Set simplifies the computations to a great extent. In this paper, we study the connection between both concepts for arbitrary referential spaces. We complete existing results about the lack of an implication in general with necessary and sufficient conditions for the most interesting cases.

  • Imprecise distribution function associated to a Random Set
    Information Sciences, 2003
    Co-Authors: Inés Couso, Luciano Sánchez, Pedro Gil
    Abstract:

    Some different extensions to Random Sets of the most common parameters of a Random variable share a common rationale: a Random Set represents the imprecise observation of a Random variable, hence the generalized parameter contains the available information about the respective parameter of the imprecisely observed variable. Following the same principles, in this paper it is proposed a new definition of the distribution function of a Random Set. This definition is simpler in its formulation and it can be used in more general cases than previous proposals. The properties of the distribution function defined here are discussed: some issues about continuity, convergence of the images of the distribution function, monotonocity and measurability are studied. It is also stated that not all the information conveyed by the Random Set about the original probability measure (the probability measure induced by the imprecisely observed Random variable) is kept by its distribution function.

  • Relationships between possibility measures and nested Random Sets
    International Journal of Uncertainty Fuzziness and Knowledge-Based Systems, 2002
    Co-Authors: Enrique Miranda, Inés Couso, Pedro Gil
    Abstract:

    Different authors have observed some relationships between consonant Random Sets and possibility measures, specially for finite universes. In this paper, we go deeply into this matter and propose several possible definitions for the concept of consonant Random Set. Three of these conditions are equivalent for finite universes. In that case, the Random Set considered is associated to a possibility measure if and only if any of them is satisfied. However, in a general context, none of the six definitions here proposed is sufficient for a Random Set to induce a possibility measure. Moreover, only one of them seems to be necessary.

  • Upper Probabilities and Selectors of Random Sets
    Advances in Intelligent and Soft Computing, 2002
    Co-Authors: Enrique Miranda, Inés Couso, Pedro Gil
    Abstract:

    We investigate the probabilistic information given by a Random Set when it represents the imprecise observation of a Random variable. We compare the information given by the distributions of the selectors with that provided by the upper and lower probabilities induced by the Random Set. In particular, we model the knowledge on both the probability of an event and the probability distribution of the original Random variable. Some characterizations and examples are given for the case of a finite final space, and the main difficulties for the infinite case are commented.

Fredrik Gustafsson - One of the best experts on this subject based on the ideXlab platform.

  • http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-62849 Random Set BASED ROAD MAPPING USING RADAR MEASUREMENTS
    2013
    Co-Authors: Christian Lundquist, Fredrik Gustafsson, Lars Danielsson, Om Set Based Road
    Abstract:

    This work is concerned with the problem of multi-sensor multitarget tracking of stationary road side objects, i.e. guard rails and parked vehicles, in the context of automotive active safety systems. Advanced active safety applications, such as collision avoidance by steering, rely on obtaining a detailed map of the surrounding infrastructure to accurately assess the situation. Here, this map consists of the position of objects, represented by a Random finite Set (RFS) of multi-target states and we propose to describe the map as the spatial stationary object intensity. This intensity is the first order moment of a multi-target RFS representing the position of stationary objects and it is calculated using a Gaussian mixture probability hypothesis density (GM-PHD) filter. 1

Peter A Torrione - One of the best experts on this subject based on the ideXlab platform.

  • Random Set framework for multiple instance learning
    Information Sciences, 2011
    Co-Authors: Jeremy Bolton, Paul D Gader, Hichem Frigui, Peter A Torrione
    Abstract:

    Multiple instance learning (MIL) is a technique used for learning a target concept in the presence of noise or in a condition of uncertainty. While standard learning techniques present the learner with individual samples, MIL alternatively presents the learner with Sets of samples. Although Sets are the primary elements used for analysis in MIL, research in this area has focused on using standard analysis techniques. In the following, a Random Set framework for multiple instance learning (RSF-MIL) is proposed that can directly perform analysis on Sets. The proposed method uses Random Sets and fuzzy measures to model the MIL problem, thus providing a more natural mathematical framework, a more general MIL solution, and a more versatile learning tool. Comparative experimental results using RSF-MIL are presented for benchmark data Sets. RSF-MIL is further compared to the state-of-the-art in landmine detection using ground penetrating radar data.

Wijerupage Sardha Wijesoma - One of the best experts on this subject based on the ideXlab platform.

  • A Random Set formulation for Bayesian SLAM
    2008 IEEE/RSJ International Conference on Intelligent Robots and Systems, 2008
    Co-Authors: John Mullane, Martin D. Adams, Ba-ngu Vo, Wijerupage Sardha Wijesoma
    Abstract:

    This paper presents an alternative formulation for the Bayesian feature-based simultaneous localisation and mapping (SLAM) problem, using a Random finite Set approach. For a feature based map, SLAM requires the joint estimation of the vehicle location and the map. The map itself involves the joint estimation of both the number of features and their states (typically in a 2D Euclidean space), as an a priori unknown map is completely unknown in both landmark location and number. In most feature based SLAM algorithms, so-called dasiafeature managementpsila algorithms as well as data association hypotheses along with extended Kalman filters are used to generate the joint posterior estimate. This paper, however, presents a recursive filtering algorithm which jointly propagates both the estimate of the number of landmarks, their corresponding states, and the vehicle pose state, without the need for explicit feature management and data association algorithms. Using a finite Set-valued joint vehicle-map state and Set-valued measurements, the first order statistic of the Set, called the intensity, is propagated via the probability hypothesis density (PHD) filter, from which estimates of the map and vehicle can be jointly extracted. Assuming a mildly non-linear Gaussian system, an extended-Kalman Gaussian Mixture implementation of the recursion is then tested for both feature-based robotic mapping (known location) and SLAM. Results from the experiments show promising performance for the proposed SLAM framework, especially in environments of high spurious measurements.