Quantum Dempster-Shafer theory (QDST) derives a quantum mass function (QMF), a fuzzy metric obtained from multiple information sources based on quantum interference. In general, QMF effectively represents and processes uncertain information, but managing conflicts among multiple QMFs remains challenging. To address this issue, we propose a novel quantum conflict indicator (QCI) within the QDST framework. It is the first metric satisfying ideal conflict measurement properties, including non-negativity, symmetry, boundedness, extreme consistency, and insensitivity to refinement. Based on QCI, a novel quantum conflict fusion method (QCI-Fusion) is introduced to fuse highly conflicting QMFs. Moreover, traditional methods, including QCI-Fusion, typically constructs the Quantum Frame of Discernment (QFoD) based on predicted labels, which makes it difficult to cover unseen classes. Therefore, a new decision architecture, QCI-Decision, is proposed for unsupervised detection that rejects out-of-distribution (OOD) samples while maintaining in-distribution (ID) classification. Experimental results show that the classification accuracy of QCI-Decision deviates from the original model predictions by at most 1.49%. Meanwhile, compared with the latest OOD detection methods, QCI-Decision improves the Area Under the Receiver Operating Characteristic Curve (AUC) by up to 0.6% and reduces the False Positive Rate at 95% True Negative Rate (FPR) by up to 1.63%. Moreover, compared with QCI-Fusion, QCI-Decision achieves approximately threefold faster fusion speed with negligible performance degradation, offering a promising solution for open-world quantum information decision.

Classifiers fusion can be seen as a kind of multisource information fusion (MSIF), and classifiers fusion based on Dempster–Shafer (DS) evidence theory is an effective approach to improve the accuracy of classification tasks. However, different classifiers usually exhibit varying performances, making it challenging to achieve enhanced classification accuracy through direct fusion. Simultaneously, when the frame of discernment (FoD) of the target class expands, the number of focal elements involved in the fusion increases, resulting in a rapid growth in computational complexity. To enhance the classification performance while reducing the time cost of fusion, a novel weighted fusion of classifiers method based on approximate reasoning and reliability evaluation (WFC-AR-RE) is proposed in this article. Specifically, at first, the key focal elements are determined based on the outputs of classifiers, and an approximate basic belief assignment (BBA) is generated. Subsequently, the validation set is utilized to evaluate the performance of each classifier, thus obtaining the self-reliability of each BBA. Afterward, a novel divergence measure is introduced to quantify the discrepancy between BBAs, determining the relative reliability of each BBA. Finally, the fusion weight of each BBA is derived from its self-reliability and relative reliability, and Dempster’s rule is applied to combine the weighted BBA. The proposed WFC-AR-RE algorithm is applied to the MSIF system, and its effectiveness is demonstrated on 12 public datasets.

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Despite advances in integrating reasoning based on belief functions to generalise probabilistic representations, distance-to-prototype-based evidential deep neural networks are still emerging and require further consolidation. Existing studies in segmentation or classification tasks typically perform prior initialisation and do not address or mitigate the potential conflicts that may arise during fusion. This work investigates high-conflict scenarios within an evidential neural network for segmentation in autonomous driving, focusing on the distance-to-prototypes component, where prototypes, derived from feature maps, serve as sources of evidence and may yield contradictory information. Conflict is mitigated through parameter adjustments within the evidential reasoning, enhancing consistency before fusion. This enables more reliable data integration and a valid application of fusion rules and decision-making processes. The proposed rectification is validated on two prototype configurations of a deep evidential lidar-camera cross-fusion architecture, using two distance-based decision strategies and adapted metrics. The impact on the network's predictions is demonstrated through qualitative and quantitative results on road detection tasks with the KITTI dataset.

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.

The belief functions (BFs) introduced by Shafer in the mid of 1970s are widely applied in information fusion to model epistemic uncertainty and to reason about uncertainty. Their success in applications is however limited because of their high-computational complexity in the fusion process, especially when the number of focal elements is large. To reduce the complexity of reasoning with BFs, we can envisage as a first method to reduce the number of focal elements involved in the fusion process to convert the original basic belief assignments (BBAs) into simpler ones, or as a second method to use a simple rule of combination with potentially a loss of the specificity and pertinence of the fusion result, or to apply both methods jointly. In this article, we focus on the first method and propose a new BBA granulation method inspired by the community clustering of nodes in graph networks. This article studies a novel efficient multigranular belief fusion (MGBF) method. Specifically, focal elements are regarded as nodes in the graph structure, and the distance between nodes will be used to discover the local community relationship of focal elements. Afterward, the nodes belonging to the decision-making community are specially selected, and then the derived multigranular sources of evidence can be efficiently combined. To evaluate the effectiveness of the proposed graph-based MGBF, we further apply this new approach to combine the outputs of convolutional neural networks + attention (CNN + Attention) in the human activity recognition (HAR) problem. The experimental results obtained with real datasets prove the potential interest and feasibility of our proposed strategy with respect to classical BF fusion methods.

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 pattern classification, there may not exist labeled patterns in the target domain to train a classifier. Domain adaptation (DA) techniques can transfer the knowledge from the source domain with massive labeled patterns to the target domain for learning a classification model. In practice, some objects in the target domain are easily classified by this classification model, and these objects usually can provide more or less useful information for classifying the other objects in the target domain. So a new method called distribution adaptation based on evidence theory (DAET) is proposed to improve the classification accuracy by combining the complementary information derived from both the source and target domains. In DAET, the objects that are easy to classify are first selected as easy-target objects, and the other objects are regarded as hard-target objects. For each hard-target object, we can obtain one classification result with the assistance of massive labeled patterns in the source domain, and another classification result can be acquired based on the easy-target objects with confidently predicted (pseudo) labels. However, the weights of these classification results may vary because the reliabilities of the used information sources are different. The weights are estimated by mean difference reflecting the information source quality. Then, we discount the classification results with the corresponding weights under the framework of the evidence theory, which is expert at dealing with uncertain information. These discounted classification results are combined by an evidential combination rule for making the final class decision. The effectiveness of DAET for cross-domain pattern classification is evaluated with respect to some advanced DA methods, and the experiment results show DAET can significantly improve the classification accuracy.

In recent years, wearable sensor-based human activity recognition (HAR) is becoming more and more attractive, especially in health monitoring and sports management. However, in order to obtain high-quality HAR, it is often necessary to get sufficient labeled activity data, which is very difficult, time-consuming, and costly in a natural environment. To tackle this problem, multisource domain adaptation (DA) is a promising method that aims to learn enough multisource prior knowledge from labeled activity data, and then transfer this learned knowledge to the target unlabeled dataset. Thus, this article presents a novel multisource weighted DA with evidential reasoning (w-MSDAER) for HAR, which can effectively utilize complementary knowledge between multiple sources. Specifically, we first use the strategy of distribution alignment to learn local domain-invariant classifiers based on multisource domains. And then the reliabilities of these derived classifiers are comprehensively evaluated according to the belief function based technique for order preference by similarity to ideal solution (BF-TOPSIS). Finally, the discounting fusion method is used to fuse the local classification results. Comprehensive experiments are conducted on two open-source datasets, and the results show that the proposed w-MSDAER significantly outperforms other state-of-art methods.

In the original belief function (BF) theory, a precise-valued belief structure has been widely used to represent uncertain information. However, this mentioned belief structure is difficult to effectively measure the specific hesitant situation, especially when decision makers have a set of possible values for the belief assignments of focal elements. In order to model the hesitant nature of the behavior of people to make a decision under uncertainty, we propose a hesitant fuzzy belief structure (HFBS) that is based on the BF theory and the recent hesitant fuzzy set theory. We also present the novel rule of combination of HFBS that is used and evaluated in a wearable human activity recognition (HAR) system coupled with an extreme learning machine. The evaluation of this new HFBS approach is done from two benchmark datasets. We clearly show its effectiveness and its superiority compared to various methods used classically for the wearable HAR.

Classifier fusion remains an effective method to improve classification performance. In applications, the classifiers learnt using different attributes may work with various frames of discernment (FoD) of classification. There generally exist more or less complementary knowledge among these classifiers. However, how to efficiently combine such classifiers under different FoD is a challenging problem. In this article, we propose a new method for classifier fusion with different FoD based on the belief functions, which allow to well represent and deal with uncertain information. The credal transformation rules are developed to map the various FoD into a common one. It allows to transfer the probability (or mass of belief) of one class in the given FoD not only to several singleton classes but also to the metaclasses (i.e., disjunction of several classes) and the ignorance in other chosen FoD according to a transformation matrix, which is estimated based on the training (pairwise) data by minimizing a certain error criteria. Thus, we can well characterize the uncertainty and imprecision during the transformation of FoD. After that, the outputs of different classifiers represented by basic belief assignments (BBAs) can be transformed to a common FoD. Then, the well-known Dempster's rule is employed to combine these transformed BBA to obtain final classification result under the chosen FoD. Several real data sets are used in the experiment to evaluate the performance of the proposed method. Our experimental results show that this new method can efficiently improve the classification accuracy with respect to other related methods.

In this paper, we present a fast Belief Function based Inter-Criteria Analysis (BF-ICrA) method based on the canonical decomposition of basic belief assignments defined on a dichotomous frame of discernment.This new method is then applied for evaluating the Multiple-Objective Ant Colony Optimization (MO-ACO) algorithm for Wireless Sensor Networks (WSN) deployment.

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.

Evidence theory, also called belief function theory, provides an efficient tool to represent and combine uncertain information for pattern classification. Evidence combination can be interpreted, in some applications, as classifier fusion. The sources of evidence corresponding to multiple classifiers usually exhibit different classification qualities, and they are often discounted using different weights before combination. In order to achieve the best possible fusion performance, a new credal belief redistribution (CBR) method is proposed to revise such evidence. The rationale of CBR consists of transferring belief from one class not just to other classes, but also to the associated disjunctions of classes (i.e., meta-classes). As classification accuracy for different objects in a given classifier can also vary, the evidence is revised according to prior knowledge mined from its training neighbors. If the selected neighbors are relatively close to the evidence, a large amount of belief will be discounted for redistribution. Otherwise, only a small fraction of belief will enter the redistribution procedure. An imprecision matrix estimated based on these neighbors is employed to specifically redistribute the discounted beliefs. This matrix expresses the likelihood of misclassification (i.e., the probability of a test pattern belonging to a class different from the one assigned to it by the classifier). In CBR, the discounted beliefs are divided into two parts. One part is transferred between singleton classes, whereas the other is cautiously committed to the associated meta-classes. By doing this, one can efficiently reduce the chance of misclassification by modeling partial imprecision. The multiple revised pieces of evidence are finally combined by the Dempster-Shafer rule to reduce uncertainty and further improve classification accuracy. The effectiveness of CBR is extensively validated on several real datasets from the UCI repository and critically compared with that of other related fusion methods.

Multisensor fusion strategies have been widely applied in human activity recognition (HAR) in body sensor networks (BSNs). However, the sensory data collected by BSNs systems are often uncertain or even incomplete. Thus, designing a robust and intelligent sensor fusion strategy is necessary for high-quality activity recognition. In this article, Dezert-Smarandache theory (DSmT) is used to develop a novel sensor fusion strategy for HAR in BSNs, which can effectively improve the accuracy of recognition. Specifically, in the training stage, the kernel density estimation (KDE)-based models are first built and then precisely selected for each specific activity according to the proposed discriminative functions. After that, a structure of basic belief assignment (BBA) can be constructed, using the relationship between the test data of unknown class and the selected KDE models of all considered types of activities. In order to deal with the conflict between the obtained BBAs, proportional conflict redistribution-6 (PCR6) is applied to fuse the acquired BBAs. Moreover, the missing data of the involved sensors are addressed as ignorance in the framework of the DSmT without manual interpolation or intervention. Experimental studies on two real-world activity recognition datasets (The OPPORTUNITY dataset; Daily and Sports Activity Dataset (DSAD)) are conducted, and the results shows the superiority of our proposed method over some state-of-the-art approaches proposed in the literature.

In this paper we present an application of a new Belief Function-based Inter-Criteria Analysis (BF-ICrA) approach for Global Positioning System (GPS) Surveying Problems (GSP). GPS surveying is an NP-hard problem. For designing Global Positioning System surveying network, a given set of earth points must be observed consecutively. The survey cost is the sum of the distances to go from one point to another one. This kind of problems is hard to be solved with traditional numerical methods. In this paper we use BF-ICrA to analyze an Ant Colony Optimization (ACO) algorithm developed to provide near-optimal solutions for Global Positioning System surveying problem.

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.

This paper presents a study on human perception of the heading on the base of motion and form visual cues integration. The authors examine how human age influences this process. Because the visual stimuli are in general uncertain, or in some cases even conflicting, the process of combination is estimated on the base on the well known Normalized Conjunctive Consensus fusion rule, as well as on the base of the more efficient Dezert-Smarandache Theory (DSmT) of plausible and paradoxical reasoning, and more precisely on the probabilistic Proportional Conflict Redistribution rule no.5 defined within it. The main goal is focused on how these fusion rules succeed to model consistent and adequate predictions about both individuals' behavior, and age-contingent groups of individuals1.

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.

Classifier fusion is an efficient strategy to improve the classification performance for the complex pattern recognition problem. In practice, the multiple classifiers to combine can have different reliabilities and the proper reliability evaluation plays an important role in the fusion process for getting the best classification performance. We propose a new method for classifier fusion with contextual reliability evaluation (CF-CRE) based on inner reliability and relative reliability concepts. The inner reliability, represented by a matrix, characterizes the probability of the object belonging to one class when it is classified to another class. The elements of this matrix are estimated from the -nearest neighbors of the object. A cautious discounting rule is developed under belief functions framework to revise the classification result according to the inner reliability. The relative reliability is evaluated based on a new incompatibility measure which allows to reduce the level of conflict between the classifiers by applying the classical evidence discounting rule to each classifier before their combination. The inner reliability and relative reliability capture different aspects of the classification reliability. The discounted classification results are combined with Dempster-Shafer's rule for the final class decision making support. The performance of CF-CRE have been evaluated and compared with those of main classical fusion methods using real data sets. The experimental results show that CF-CRE can produce substantially higher accuracy than other fusion methods in general. Moreover, CF-CRE is robust to the changes of the number of nearest neighbors chosen for estimating the reliability matrix, which is appealing for the applications.

In pattern classification problem, different classifiers learnt using different training data can provide more or less complementary knowledge, and the combination of classifiers is expected to improve the classification accuracy. Evidential reasoning (ER) provides an efficient framework to represent and combine the imprecise and uncertain informations. In this paper, we want to focus on the weighted combination of classifiers based on ER. Because each classifier may have different performance on the given dataset, the classifiers to combine are considered with different weights. A new weighted classifier combination method is proposed based on ER to enhance the classification accuracy. The optimal weighting factors of classifiers are obtained by minimizing the distances between fusion results obtained by Dempster's rule and the target output in training data space to fully take advantage of the complementarity of the classifiers. A confusion matrix is additionally introduced to characterize the probability of the object belonging to one class but classified to another class by the fusion result. This matrix is also optimized using training data jointly with classifier weight, and it is used to modify the fusion result to make it as close as possible to truth. Moreover, the training patterns are considered with different weights for the parameter optimization in classifier fusion, and the patterns hard to classify are committed with bigger weight than the ones easy to deal with. The pattern weight and the other parameters (i.e., classifier weight and confusion matrix) are iteratively optimized for obtaining the highest classification accuracy. A cautious decision making strategy is introduced to reduce the errors, and the pattern hard to classify will be cautiously committed to a set of classes, because the partial imprecision of decision is considered better than error in certain case. The effectiveness of the proposed method is demonstrated with various real datasets from UCI repository, and its performances are compared with those of other classical methods.

In belief functions related fields, the distance measure is an important concept, which represents the degree of dissimilarity between bodies of evidence. Various distance measures of evidence have been proposed and widely used in diverse belief function related applications, especially in performance evaluation. Existing definitions of strict and nonstrict distance measures of evidence have their own pros and cons. In this paper, we propose two new strict distance measures of evidence (Euclidean and Chebyshev forms) between two basic belief assignments based on the Wasserstein distance between belief intervals of focal elements. Illustrative examples, simulations, applications, and related analyses are provided to show the rationality and efficiency of our proposed measures for distance of evidence.

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.

In classification problem, several different classes may be partially overlapped in their borders. The objects in the border are usually quite difficult to classify. A hybrid classification system (HCS) is proposed to adaptively utilize the proper classification method for each object according to the K-nearest neighbors (K-NNs), which are found in the weighting vector space obtained by self-organizing map (SOM) in each class. If the K-close weighting vectors (nodes) are all from the same class, it indicates that this object can be correctly classified with high confidence, and the simple hard classification will be adopted to directly classify this object into the corresponding class. If the object likely lies in the border of classes, it implies that this object could be difficult to classify, and the credal classification working with belief functions is recommended. The credal classification allows the object to belong to both singleton classes and sets of classes (meta-class) with different masses of belief, and it is able to well capture the potential imprecision of classification thanks to the meta-class and also reduce the errors. Fuzzy classification is selected for the object close to the border and hard to clearly classify, and it associates the object with different classes by different membership (probability) values. HCS generally takes full advantage of the three classification ways and produces good performance. Moreover, it requires quite low computational burden compared with other K-NNs-based methods due to the use of SOM. The effectiveness of HCS is demonstrated by several experiments with synthetic and real datasets.

Dempster-Shafer evidence theory is a primary methodology for multisource information fusion because it is good at dealing with uncertain information. This theory provides a Dempster's rule of combination to synthesize multiple evidences from various information sources. However, in some cases, counter-intuitive results may be obtained based on that combination rule. Numerous new or improved methods have been proposed to suppress these counter-intuitive results based on perspectives, such as minimizing the information loss or deviation. Inspired by evolutionary game theory, this paper considers a biological and evolutionary perspective to study the combination of evidences. An evolutionary combination rule (ECR) is proposed to help find the most biologically supported proposition in a multievidence system. Within the proposed ECR, we develop a Jaccard matrix game to formalize the interaction between propositions in evidences, and utilize the replicator dynamics to mimick the evolution of propositions. Experimental results show that the proposed ECR can effectively suppress the counter-intuitive behaviors appeared in typical paradoxes of evidence theory, compared with many existing methods. Properties of the ECR, such as solution's stability and convergence, have been mathematically proved as well.

In the framework of the evidence theory, several approaches for estimating belief functions are proposed. However, they generally suffer from the problem of masses attribution in the case of compound hypotheses that lose much conceptual contribution of the theory. In this paper, an original method for estimating mass functions using Kohonen's map derived from the initial feature space and an initial classifier is proposed. Our approach allows a smart mass belief assignment, not only for simple hypotheses but also for disjunctions and conjunctions of hypotheses. Thus, it can model at the same time ignorance, imprecision, and paradox. The proposed method for a basic belief assignment (BBA) is of interest for solving estimation mass functions problems where a large quantity of multivariate data is available. Indeed, the use of Kohonen's map simplifies the process of assigning mass functions. The proposed method has been compared with the state-of-the-art BBA technique on benchmark database and applied on remote sensing data for image classification purpose. Experimentation shows that our approach gives similar or better results than other methods presented in the literature so far, with an ability to handle a large amount of data.

The transformation of belief function into probability is one of the most important and common ways for decision making under the framework of evidence theory. In this paper, we focus on the evaluation of such probability transformations (PTs), which are crucial for their proper applications and the design of new ones. Shannon entropy or probabilistic information content (PIC) measure is traditionally used in evaluating PTs. The transformation having the lowest entropy or highest PIC is considered as the best one. This standpoint is questioned in this paper by comparing a PT based on uncertainty minimization with other available PTs. It shows experimentally that entropy or PIC is not comprehensive to evaluate a PT. To make a comprehensive evaluation, some new approaches are proposed by the joint use of PIC and the distance of evidence according to the value- and rank-based fusion. A pattern classification application oriented evaluation approach for PTs is also proposed. Some desired properties for PTs are also discussed. Experimental results and related analysis are provided to show the rationality of the new evaluation approaches.

The classification of incomplete patterns is a very challenging task because the object (incomplete pattern) with different possible estimations of missing values may yield distinct classification results. The uncertainty (ambiguity) of classification is mainly caused by the lack of information of the missing data. A new prototype-based credal classification (PCC) method is proposed to deal with incomplete patterns thanks to the belief function framework used classically in evidential reasoning approach. The class prototypes obtained by training samples are respectively used to estimate the missing values. Typically, in a c -class problem, one has to deal with c prototypes, which yield c estimations of the missing values. The different edited patterns based on each possible estimation are then classified by a standard classifier and we can get at most c distinct classification results for an incomplete pattern. Because all these distinct classification results are potentially admissible, we propose to combine them all together to obtain the final classification of the incomplete pattern. A new credal combination method is introduced for solving the classification problem, and it is able to characterize the inherent uncertainty due to the possible conflicting results delivered by different estimations of the missing values. The incomplete patterns that are very difficult to classify in a specific class will be reasonably and automatically committed to some proper meta-classes by PCC method in order to reduce errors. The effectiveness of PCC method has been tested through four experiments with artificial and real data sets.

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.

We present a multidimensional evidential reasoning (MDER) approach to estimate change detection from the fusion of heterogeneous remote sensing images. MDER is based on a multidimensional (M-D) frame of discernment composed by the Cartesian product of the separate frames of discernment used for the classification of each image. Every element of the M-D frame is a basic joint state that allows to describe precisely the possible change occurrences between the heterogeneous images. Two kinds of rules of combination are proposed for working either with the free model, or with a constrained model depending on the integrity constraints one wants to take into account in the scenario under study. We show the potential interest of the MDER approach for detecting changes due to a flood in the Gloucester area in the U.K. from two real ERS and SPOT images.

A credal classification rule (CCR) is proposed to deal with the uncertain data under the belief functions framework. CCR allows the objects to belong to not only the specific classes, but also any set of classes (i.e. meta-class) with different masses of belief. In CCR, each specific class is characterized by a class center. Specific class consists of the objects that are very close to the center of this class. A meta-class is used to capture imprecision of the class of the object that is simultaneously close to several centers of specific classes and hard to be correctly committed to a particular class. The belief assignment of the object to a meta-class depends both on the distances to the centers of the specific class included in the meta-class, and on the distance to the meta-class center. Some objects too far from the others will be considered as outliers (noise). CCR provides the robust classification results since it reduces the risk of misclassification errors by increasing the non-specificity. The effectiveness of CCR is illustrated by several experiments using artificial and real data sets.

Theories of evidence have already been applied more or less successfully in the fusion of remote sensing images. These attempts were based on the classical evidential reasoning which works under the condition that all sources of evidence and their fusion results are related to the same invariable (static) frame of discernment. When working with multitemporal remote sensing images, some change occurrences are possible between two images obtained at a different period of time, and these changes need to be detected efficiently in particular applications. The classical evidential reasoning is adapted for working with an invariable frame of discernment over time, but it cannot efficiently detect nor represent the occurrence of change from heterogeneous remote sensing images when the frame is possibly changing over time. To overcome this limitation, dynamic evidential reasoning (DER) is proposed for the sequential fusion of multitemporal images. A new state-transition frame is defined in DER, and the change occurrences can be precisely represented by introducing a statetransition operator. Two kinds of dynamical combination rules working in the free model and in the constrained model are proposed in this new framework for dealing with the different cases. Moreover, the prior probability of state transitions is taken into account, and the link between DER and Dezert–Smarandache theory is presented. The belief functions used in DER are defined similarly to those defined in the Dempster–Shafer theory. As shown in the last part of this paper, DER is able to estimate efficiently the correct change detections as a postprocessing technique. Two applications are given to illustrate the interest of DER: The first example is based on a set of two SPOT images acquired before and after a flood, and the second example uses three QuickBird images acquired during an earthquake event.

This paper introduces two new fusion rules for combining quantitative basic belief assignments. These rules although very simple have not been proposed in literature so far and could serve as useful alternatives because of their low computation cost with respect to the recent advanced Proportional Conflict Redistribution rules developed in the DSmT framework.

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