This paper is about the Compromise Ranking Problem (CRP), a well-known problem in the social choice theory. According to the famous Arrow’s theorem there is no voting method which is entirely satisfying and fairness if one accepts Arrow’s axioms. In this paper we formalize the problem as a minimisation problem in a discrete finite search space. We attempt to solve it based on the Least Squares (LS) approach thanks to some appealing metrics to get the optimal CRP solution. Surprisingly, we show that the optimal consensus (compromise) ranking solution disagrees with the commonsense solutions in four simple interesting examples. The search for an optimal solution in agreement with the commonsense appears to be an open very challenging question and our paper warns the users about the impossibility of the main current methods to provide acceptable solutions even for the rather simple examples considered in this work.

In self-navigation problems for autonomous vehicles, the variability of environmental conditions, complex scenes with vehicles and pedestrians, and the high-dimensional or real-time nature of tasks make segmentation challenging. Sensor fusion can representatively improve performances. Thus, this work highlights a late fusion concept used for semantic segmentation tasks in such perception systems. It is based on two approaches for merging information coming from two neural networks, one trained for camera data and one for LiDAR frames. The first approach involves fusing probabilities along with calculating partial conflicts and redistributing data. The second technique focuses on making individual decisions based on sources and fusing them later with weighted Shannon entropies. The two segmentation models are trained and evaluated on a particular KITTI semantic dataset. In the realm of multi-class segmentation tasks, the two fusion techniques are compared and evaluated with illustrative examples. Intersection over union metric and quality of decision are computed to assess the performance of each methodology.

In multimodal learning, multimodal coordinated representation is an important yet challenging issue, which establishes the interaction between different modalities to describe multimodal data more effectively. Existing coordinated representation methods are implemented in the deep feature space (or encoding space) of each modality. In this paper, based on the framework of evidence theory, we propose a novel coordinated representation method, where multimodal data is described as the basic belief assignment (BBA), and coordinated learning is implemented in the evidential space (i.e., the BBA-based space). That is, the information interaction between different modalities is implemented at the level of evidence modeling (or uncertainty modeling). To use the intra-class and inter-class difference information of multimodal data, we design an evidential coordinated constraint. Furthermore, to represent each modality clearly, we introduce an ambiguity constraint. Experimental results of multimodal classification show that our proposed method is rational and effective.

Dempster-Shafer evidence theory (DST) is a theoretical framework for uncertainty modeling and reasoning, with modeling the basic belief assignment (BBA) as one of its most crucial and challenging tasks. The prevailing BBA determination methods have their own pros and cons, and the joint use of them is expected to provide a better BBA. To realize an end-to-end BBA modeling without explicitly using various prevailing BBA modeling methods, a learning-based BBA modeling approach with multi-method fusion (LBMMF) is proposed in this paper. Deep learning is used to train a deep network which learns the mapping from the training samples to the comprehensive BBAs obtained by jointly using the prevailing BBA modeling methods as the generalized training labels. Given a test sample, the corresponding BBA can be obtained in an end-to-end manner, which is the output of the trained deep neural network. Experimental results show that to use the BBA obtained by our method can achieve better classification performance.

Traditional pattern recognition systems, tasked with categorizing inputs into known classes, often struggle when they encounter samples they haven’t been trained to recognize. This introduces the need for the open set recognition—enhancing models to reject unidentified samples effectively. The Openmax method represents a significant breakthrough in this field by leveraging deep learning to spot and handle these new, unseen classes, broadening the traditional Softmax layer to accommodate an “unknown” class and employing a single threshold to separate the known from the unknown. However, the reliance of the original Openmax method on a single threshold may result in incorrect classifications if the parameters are not selected appropriately. To address this, we introduce a dual-threshold fusion mechanism based on Dempster-Shafer evidence theory in this paper. This approach releases the difficulty of finding a precise threshold in the complex and dynamic real-world environments. By integrating deep networks with a novel evidence-based system, the refined approach can bolster the robustness of rejecting undefined classes.

Thisn paper introduces the concept of cross-entropy and relative entropy of two basic belief assignments. It is based on the new entropy measure presented recently. We prove that the cross-entropy satisfies a generalized Gibbs-alike inequality from which a generalized Kullback-Leibler divergence measure can be established in the framework of belief functions. We show on a simple illustrating example how these concepts can be used for decision-making under uncertainty.

Human Activity Recognition (HAR) based on wear-able device has become a hot topic of research due to its wide range of applications in health-care, fitness and smart homes. However, the classification of some activities with similar sensor readings, such as standing and sitting, is usually more challenging for the design of efficient activity recognition algorithms. Considering the inconsistent performance of different classifiers, which can provide information complementary for individual classifier, we propose a novel multi-classifier fusion method based on belief functions (BFs) theory for HAR. Specifically, at first, four classifiers are trained using time-domain and frequency-domain features to obtain basic belief assignments (BBA) of activity, respectively. Then, three assessment criteria are utilized to evaluate the reliability of the classifiers and a scoring matrix is constructed. Next, the algorithm of Belief Function based the Technique for Order Preference by Similarity to Ideal Solution (BF-TOPSIS) is employed to calculate the weighting coefficients for each classifier. Finally, the discounting and Dempster’s rules are adopted to combine the multiple classifiers and further decision making. Several experiments were conducted to illustrate the performance of the proposed method using the UCI smartphone dataset, and the results show that the proposed method is more accurate than the state-of-art methods.

In this paper, we propose a new fusion approach to combine basic belief assignments (BBAs) defined on a dichotomous frame of discernment based on their canonical decomposition. In a companion paper, we have already proved that the canonical decomposition of this type of BBA (called dichotomous BBA) is always possible and unique thanks to the proportional conflict redistribution rule No 5 (PCR5). More precisely, any dichotomous BBA is always the PCR5 combination of two simpler basic belief assignments named respectively the pro-evidence, and the contra-evidence. From this interesting canonical decomposition, we present a new way of combining many dichotomous BBAs and we show that the computational time for fusing these dichotomous BBAs based on their canonical decomposition is quasi-linear with the number of sources to combine, contrary to the direct fusion of the dichotomous BBAs altogether.

In this paper, we propose a new Multi-Criteria Decision-Making method (MCDM) which is rank reversal free. We call it the SPOTIS method standing for Stable Preference Ordering Towards Ideal Solution method. Our method is exempt of rank reversal because the preference ordering established from the score matrix of the MCDM problem under consideration does not require the relative comparisons between alternatives, but only comparisons with respect to the ideal solution chosen by the MCDM system designer after transforming the incomplete original MCDM problem into a well-defined one thanks to the specification of the min and max bounds of each criterion involved in the problem.

In this paper, we prove that any dichotomous basic belief assignment (BBA) m can be expressed as the combination of two simple belief assignments m p and m c called, respectively, the pros and cons BBAs thanks to the proportional conflict redistribution rule no 5 (PCR5). This decomposition always exists and is unique and we call it the canonical decomposition of the BBA m. We also show that canonical decompositions do not exist in general if we use the conjunctive rule, the disjunctive rule, Dempster's rule, Dubois and Prade's or Yager's rules, or even the averaging rule of combination. We give some numerical examples of canonical decompositions and discuss of the potential interest of this canonical decomposition for applications in information fusion.

This paper presents two new theoretical contributions for reasoning under uncertainty: 1) the Total Belief Theorem (TBT) which is a direct generalization of the Total Probability Theorem, and 2) the Generalized Bayes' Theorem drawn from TBT. A constructive justification of Fagin-Halpern belief conditioning formulas proposed in the nineties is also given. We also show how our new approach and formulas work through simple illustrative examples.

In this paper the notion of (probabilistic) independence of two events defined classically in the theory of probability is extended in the theory of belief functions as the credibilistic independence of two propositions. This new notion of independence which is compatible with the probabilistic independence as soon as the belief function is Bayesian, is defined from Fagin-Halpern belief conditioning formulas drawn from Total Belief Theorem (TBT) when working in the framework of belief functions to model epistemic uncertainties. We give some illustrative examples of this notion at the end of the paper.

The classifier based on rough sets is widely used in pattern recognition. However, in the implementation of rough set-based classifiers, there always exist the problems of uncertainty. Generally, information decision table in Rough Set Theory (RST) always contains many attributes, and the classification performance of each attribute is different. It is necessary to determine which attribute needs to be used according to the specific problem. In RST, such problem is regarded as attribute reduction problems which aims to select proper candidates. Therefore, the uncertainty problem occurs for the classification caused by the choice of attributes. In addition, the voting strategy is usually adopted to determine the category of target concept in the final decision making. However, some classes of targets cannot be determined when multiple categories cannot be easily distinguished (for example, the number of votes of different classes is the same). Thus, the uncertainty occurs for the classification caused by the choice of classes. In this paper, we use the theory of belief functions to solve two above mentioned uncertainties in rough set classification and rough set classifier based on Dezert-Smarandache Theory (DSmT) is proposed. It can be experimentally verified that our proposed approach can deal efficiently with the uncertainty in rough set classifiers.

Multi-source information fusion strategies in target recognition have been widely applied. Generally, each source is defined and modelled over a common frame composed of the hypotheses to discern. However, in practice, the independent sources of evidence can refer to distinct frames of discernment in terms of the hypotheses they consider. Under this condition, the classical combination process cannot be applied directly. Working with distinct frames of discernment for information fusion is a problem often encountered in the development of recognition systems which requires a particular attention. In order to combine such sources, this paper presents a new combination method which splits the process of fusion into two steps: construction of granular structure, calculation of belief mass, followed by the fusion process. Our simulations results show that the proposed method can effectively solve the problem of fusion of sources defined on distinct frames.

This paper presents a Credal algorithm to perform land cover classification from a pair of optical and radar remote sensing images. SAR (Synthetic Aperture Radar) /optical multispectral information fusion is investigated in this study for making the joint classification. The approach consists of two main steps: 1) relevant features extraction applied to each sensor in order to model the sources of information and 2) a Kohonen map-based estimation of Basic Belief Assignments (BBA) dedicated to heterogeneous data. This framework deals with co-registered images and is able to handle complete optical data as well as optical data affected by missing value due to the presence of clouds and shadows during observation. A pair of SPOT-5 and RADARSAT-2 real images is used in the evaluation, and the proposed experiment in a farming area shows very promising results in terms of classification accuracy and missing optical data reconstruction when some data are hidden by clouds.

One of the challenges of remote sensing image based building change detection is distinguishing building changes from other types of land cover alterations. Height information can be a great assistance for this task but its performance is limited to the quality of the height. Yet, the standard automatic methods for this task are still lacking. We propose a very high resolution stereo series data based building change detection approach that focuses on the use of time series information. In the first step, belief functions are explored to fuse the change features from the 2D and height maps to obtain an initial change detection result. In the second step, the building probability maps (BPMs) from the series data are adopted to refine the change detection results based on Dempster-Shafer theory. The final step is to fuse the series building change detection results in order to obtain a final change map. The advantages of the proposed approach are demonstrated by testing it on a set of time series data captured in North Korea.

In this paper, new theoretical results for reasoning with belief functions are obtained and discussed. After a judicious decomposition of the set of focal elements of a belief function, we establish the total belief theorem (TBT). which is the direct generalization of the total probability theorem when working in the framework of belief functions. The TBT is also generalized for dealing with different frames of discernments thanks to Cartesian product space. From TBT, we can derive and define formally the expressions of conditional belief functions, which are consistent with the bounds of imprecise conditional probability. This work provides a direct establishment and solid justification of Fagin–Halpern belief conditioning formulas. The well-known Bayes' theorem of probability theory is then generalized in the framework of belief functions, and we illustrate it with an example at the end of this paper.

In 2016 we developed a new approach for Multi-Criteria Decision-Making (MCDM) inspired by the technique for order preference by similarity to ideal solution (TOPSIS) and based on belief functions (BF). Our BF-TOPSIS (Belief Function based TOPSIS) approach assumes that the input score of each hypothesis for each criterion was a real precise number which is a quite restrictive assumption. In this paper we extend our BF-TOPSIS to deal with imprecise score values (intervals of real numbers) and we call it Imp-BF-TOPSIS. This new approach follows main ideas of BF-TOPSIS but extends its applicability for more realistic MCDM problems where the scores are given with a finite precision. Imp-BF-TOPSIS is based on Interval Arithmetic (IA), new probabilistic order relations between intervals and belief functions. We also present results of Imp-BF-TOPSIS for simple examples for illustrating its effectiveness.

This paper presents a comparative analysis of performances of two types of multi-target tracking algorithms: 1) the Joint Probabilistic Data Association Filter (JPDAF), and 2) classical Kalman Filter based algorithms for multi-target tracking improved with Quality Assessment of Data Association (QADA) method using optimal data association. The evaluation is based on Monte Carlo simulations for difficult maneuvering multiple-target tracking (MTT) problems in clutter.

Dempster-Shafer theory (DST) is an important theory for information fusion. However, in DST how to determinate the basic belief assignment (BBA) is still an open issue. The interval number based BBA determination method is simple and effective, where the features of different classes' samples are modeled using the interval numbers, i.e., an interval number model is constructed for each focal element. Then, the distances of interval numbers are used for measuring the similarity degrees between the testing sample and each focal element, and the similarity degrees are used for determinating the BBA. The definition of interval numbers' distance is crucial for the effectiveness of the interval number based BBA determination methods. In this paper, we use different interval numbers' distances for determinating BBAs. By using the artificial data set and the Iris date set of open UCI data base, respectively, we compare and analyze the determination of BBAs with different distances.

In many applications involving epistemic uncertainties usually modeled by belief functions, it is often necessary to approximate general (non-Bayesian) basic belief assignments (BBAs) to subjective probabilities (called Bayesian BBAs). This necessity occurs if one needs to embed the fusion result in a system based on the probabilistic framework and Bayesian inference (e.g. tracking systems), or if one wants to use classical decision theory to make a decision. There exists already several methods (probabilistic transforms) to approximate any general BBA to a Bayesian BBA. From a fusion standpoint, two approaches are usually adopted: 1) one can approximate at first each BBA in subjective probabilities and use Bayes fusion rule to get the final Bayesian BBA, or 2) one can fuse all the BBAs with a fusion rule, typically Dempster-Shafer's, or PCR6 rules (which is very costly in computations), and convert the combined BBA in a subjective probability measure. The former method is the simplest method but it generates a high loss of information included in original BBAs, whereas the latter is intractable for high dimension problems. This paper presents a new method to achieve this task based on hierarchical decomposition (coarsening) of the frame of discernment, which can be seen as an intermediary approach between the two aforementioned methods. After the presentation of this new method, we show through simulations how its performs with respect to other methods.

The study of alternative probabilistic transformation (PT) in DS theory has emerged recently as an interesting topic, especially in decision making applications. These recent studies have mainly focused on investigating various schemes for assigning both the mass of compound focal elements to each singleton in order to obtain Bayesian belief function for real-world decision making problems. In this paper, work by us also takes inspiration from both Bayesian transformation camps, with a novel evolutionary-based probabilistic transformation (EPT) to select the qualified Bayesian belief function with the maximum value of probabilistic information content (PIC) benefiting from the global optimizing capabilities of evolutionary algorithms. Verification of EPT is carried out by testing it on a set of numerical examples on 4D frames. On each problem instance, comparisons are made between the novel method and those existing approaches, which illustrate the superiority of the proposed method in this paper. Moreover, a simple constraint-handling strategy with EPT is proposed to tackle target type tracking (TTT) problem, simulation results of the constrained EPT on TTT problem prove the rationality of this modification.

Edge detection is one of the most important tasks in image processing and pattern recognition. Edge detector with multiple color channels can provide more edge information. However, the uncertainty occurring with the edge detection in each single channel and the discordance existing in the fusion of multiple channels edge detectors make the detection difficult. In this paper, we propose a new edge detection method in color images based on information fusion. We show that obtaining final edge through fusing the edge information in each channel is a challenging problem to make decision in the framework of Multi-Criteria decision making (MCDM). In this work, we propose to detect edges in color images using Cautious OWA with evidential reasoning (COWA-ER) and Fuzzy-Cautious OWA with evidential reasoning (FCOWA-ER) to handle the uncertainty and discordance. Experimental results show that the proposed approaches achieve better edge detection performance compared with the original edge detector.

A practical method for full-dimension attitude determination based on the combination of two-antenna global positioning system (GPS) and strapdown inertial navigation system (SINS) is presented. In view of the fact that two-antenna GPS can only provide two attitude angles using carrier phase difference measurements, not all of the SINS attitude errors can be directly corrected by the integrated navigation system. The proposed method makes use of the coordinate transformation about the attitude and the related measurements provided by two-antenna GPS and SINS to solve the full-dimension attitude determination problem through information fusion. The simulation results indicate that the proposed method can effectively determine the required attitude in real time.

Dempster-Shafer evidence theory (DST) is a theoretical framework for uncertainty modeling and reasoning. The determination of basic belief assignment (BBA) is crucial in DST, however, there is no general theoretical method for BBA determination. In this paper, a method of generating BBA using fuzzy numbers is proposed. First, the training data are modeled as fuzzy numbers. Then, the dissimilarities between each test sample and the training data are measured by the distance between fuzzy numbers. In the final, the BBAs are generated from the normalized dissimilarities. The effectiveness of this method is demonstrated by an application of classification problem. Experimental results show that the proposed method is robust to outliers.

The purpose of this paper is to focus on and discuss both: (i) the validity of the Dempster's rule of combination and foundations of Dempster–Shafer theory and (ii) the real compatibility (or not) of the Dempster's rule with the Bayes fusion rule. We analyze and explain, on the basis of a generic example, the inconsistent behavior of the Dempster's rule of combination, introduced by Shafer in his mathematical theory of evidence, as a valid method to combine sources of information. We identify the cause and the effect of the dictatorial power behavior of this rule and of its impossibility to manage the conflicts between the sources no matter of their level. For comparison purpose, we present the respective solutions obtained by the more efficient proportional conflict redistribution rule number 5 proposed originally in Dezert–Smarandache theory framework. The inherent contradiction of Dempster–Shafer theory foundations is identified and proved. Then, a deep analysis of the compatibility of the Dempster's fusion rule with the Bayes fusion rule (from a fusion standpoint) is made on the basis of proposed new interesting formulation of the Bayes rule. We prove that the Dempster's rule does not behave as the Bayes fusion rule in general, because both methods deal very differently with the prior information when it is really informative (not uniform). Only in the very particular case where the basic belief assignments to combine are Bayesian and when the prior information is uniform (or vacuous), the Dempster's rule remains consistent with the Bayes fusion rule. It is proved, that in more general cases, the Dempster's rule is incompatible with the Bayes rule and it is not a generalization of the Bayes fusion rule.

This paper defines and implements a non-Bayesian fusion rule for combining densities of probabilities estimated by local (non-linear) filters for tracking a moving target by passive sensors. This rule is the restriction to a strict probabilistic paradigm of the recent and efficient proportional conflict redistribution rule no 5 (PCR5) developed in the DSmT framework for fusing basic belief assignments. A sampling method for probabilistic PCR5 (p-PCR5) is defined. It is shown that p- PCR5 is more robust to an erroneous modeling and allows to keep the modes of local densities and preserve as much as possible the whole information inherent to each densities to combine. In particular, p-PCR5 is able of maintaining multiple hypotheses/modes after fusion, when the hypotheses are too distant in regards to their deviations. This new p-PCR5 rule has been tested on a simple example of distributed non-linear filtering application to show the interest of such approach for future developments. The non-linear distributed filter is implemented through a basic particles filtering technique. The results obtained in our simulations show the ability of this p-PCR5-based filter to track the target even when the models are not well consistent in regards to the initialization and real cinematic.

This paper deals with enriched qualitative belief functions for reasoning under uncertainty and for combining information expressed in natural language through linguistic labels. In this work, two possible enrichments (quantitative and/or qualitative) of linguistic labels are considered and operators (addition, multiplication, division, etc) for dealing with them are proposed and explained. We denote them qe-operators, qe standing for “qualitative-enriched” operators. These operators can be seen as a direct extension of the classical qualitative operators (q-operators) proposed recently in the Dezert-Smarandache Theory of plausible and paradoxist reasoning (DSmT). q-operators are also justified in details in this paper. The quantitative enrichment of linguistic label is a numerical supporting degree in [0,∞), while the qualitative enrichment takes its values in a finite ordered set of linguistic values. Quantitative enrichment is less precise than qualitative enrichment, but it is expected more close with what human experts can easily provide when expressing linguistic labels with supporting degrees. Two simple examples are given to show how the fusion of qualitative-enriched belief assignments can be done, and a simulation application is given to show its advantage in rough navigation map building of mobile robot.

In this paper we extend the new family of (quantitative) belief conditioning rules (BCR) recently developed in the Dezert-Smarandache theory (DSmT) to their qualitative counterpart for belief revision. Since the revision of quantitative as well as qualitative belief assignment given the occurrence of a new event (the conditioning constraint) can be done in many possible ways, we present here only what we consider as the most appealing qualitative belief conditioning rules (QBCR) which allow to revise the belief directly with words and linguistic labels and thus avoids the introduction of ad-hoc translations of quantitative beliefs into quantitative ones for solving the problem.

In this paper we consider and analyze the behavior of two combinational rules for temporal/sequential attribute data fusion for target type estimation. Our comparative analysis is based on Dempster's fusion rule proposed in Dempster-Shafer theory (DST) and on the proportional conflict redistribution rule no. 5 (PCR5) recently proposed in Dezert-Smarandache theory (DSmT). We show through very simple scenario and Monte-Carlo simulation, how PCR5 allows a very efficient target type Tracking and reduces drastically the latency delay for correct target type decision with respect to Dempster's rule. For cases presenting some short target type switches, Demspter's rule is proved to be unable to detect the switches and thus to track correctly the target type changes. The approach proposed here is totally new, efficient and promising to be incorporated in real-time generalized data association-multi target tracking systems (GDA-MTT) and provides an important result on the behavior of PCR5 with respect to Dempster's rule. The MatLab source code is also provided in the paper

We consider in this work evidential sources of information and propose a very general evidence supporting measure of similarity (ESMS) for selecting the most coherent subset of sources to combine among all sources available at each instant. The methodology proposed here coupled with a DSmT-based fusion machine is tested in robotics for the automatic estimation of an unknown simulated environment with obstacles where an autonomous mobile Pioneer II robot with sonar sensors evolves. Our simulation results are based on the fusion of similar and equireliable sensors but same approach can also be used with dissimilar sources as well by using a discounting method taking into account the reliability of each sensor. Our results show clearly the benefit of the selection of the sources as prerequisite for improvement of information fusion