International audience

Time-to-event analysis provides insights into clinical prognosis and treatment recommendations. However, this task is more challenging than standard regression problems due to the presence of censored observations. Additionally, the lack of confidence assessment, model robustness, and prediction calibration raises concerns about the reliability of predictions. To address these challenges, we propose an evidential regression model specifically designed for time-to-event prediction. Our approach computes a degree of belief for the event time occurring within a time interval, without any strict distribution assumption. Meanwhile, the proposed model quantifies both epistemic and aleatory uncertainties using Gaussian Random Fuzzy Numbers and belief functions, providing clinicians with uncertainty-aware survival time predictions. Experimental evaluations using simulated and real-world survival datasets highlight the potential of our approach for enhancing clinical decision-making in survival analysis. • Calculate survival time without strict distribution assumptions. • Quantifies epistemic and aleatory uncertainties via GRFN and belief functions. • Provides uncertainty-aware survival predictions for clinicians. • Validated on simulated & real-world survival datasets.

This paper proposes a tentative and original survey of meeting points between Knowledge Representation and Reasoning (KRR) and Machine Learning (ML), two areas which have been developed quite separately in the last four decades. First, some common concerns are identified and discussed such as the types of representation used, the roles of knowledge and data, the lack or the excess of information, or the need for explanations and causal understanding. Then, the survey is organised in seven sections covering most of the territory where KRR and ML meet. We start with a section dealing with prototypical approaches from the literature on learning and reasoning: Inductive Logic Programming, Statistical Relational Learning, and Neurosymbolic AI, where ideas from rule-based reasoning are combined with ML. Then we focus on the use of various forms of background knowledge in learning, ranging from additional regularisation terms in loss functions, to the problem of aligning symbolic and vector space representations, or the use of knowledge graphs for learning. Then, the next section describes how KRR notions may benefit to learning tasks. For instance, constraints can be used as in declarative data mining for influencing the learned patterns; or semantic features are exploited in low-shot learning to compensate for the lack of data; or yet we can take advantage of analogies for learning purposes. Conversely, another section investigates how ML methods may serve KRR goals. For instance, one may learn special kinds of rules such as default rules, fuzzy rules or threshold rules, or special types of information such as constraints, or preferences. The section also covers formal concept analysis and rough sets-based methods. Yet another section reviews various interactions between Automated Reasoning and ML, such as the use of ML methods in SAT solving to make reasoning faster. Then a section deals with works related to model accountability, including explainability and interpretability, fairness and robustness. Finally, a section covers works on handling imperfect or incomplete data, including the problem of learning from uncertain or coarse data, the use of belief functions for regression, a revision-based view of the EM algorithm, the use of possibility theory in statistics, or the learning of imprecise models. This paper thus aims at a better mutual understanding of research in KRR and ML, and how they can cooperate. The paper is completed by an abundant bibliography.

Gaussian random fuzzy numbers are random fuzzy sets generalizing Gaussian random variables and possibility distributions. They define belief functions on the real line that can be conveniently combined by the product-intersection rule under the independence assumption. In this paper, we introduce various extensions of this rule to account for dependence and partial reliability of the pieces of evidence. We first provide formulas for the combination of an arbitrary number of Gaussian random fuzzy numbers whose dependence is described by a correlation matrix, and we introduce a minimum-conflict combination operation. To account for partially reliable evidence, we then introduce two discounting operations called possibilistic and evidential discounting, as well as several combination operators based on different assumptions, each one parameterized by a correlation matrix and a vector of discounting coefficients. We demonstrate the application of these operators to the combination of predictions with different sets of inputs in machine learning, and show that performance can be enhanced by optimizing the parameters of the combination operators.

In this article, we introduce a neural network model for regression in which prediction uncertainty is quantified by Gaussian random fuzzy numbers (GRFNs), a newly introduced family of random fuzzy subsets of the real line that generalizes both Gaussian random variables and Gaussian possibility distributions. The output GRFN is constructed by combining GRFNs induced by prototypes using a combination operator that generalizes Dempster's rule of evidence theory. The three output units indicate the most plausible value of the response variable, variability around this value, and epistemic uncertainty. The network is trained by minimizing a loss function that generalizes the negative log-likelihood. Comparative experiments show that this method is competitive, both in terms of prediction accuracy and calibration error, with state-of-the-art techniques such as random forests or deep learning with Monte Carlo dropout. In addition, the model outputs a predictive belief function that can be shown to be calibrated, in the sense that it allows us to compute conservative prediction intervals with specified belief degree.

Multi-view deep learning is performed based on the deep fusion of data from multiple sources, i.e. data with multiple views. However, due to the property differences and inconsistency of data sources, the deep learning results based on the fusion of multi-view data may be uncertain and unreliable. It is required to reduce the uncertainty in data fusion and implement the trusted multi-view deep learning. Aiming at the problem, we revisit the multi-view learning from the perspective of opinion aggregation and thereby devise a trusted multi-view deep learning method. Within this method, we adopt evidence theory to formulate the uncertainty of opinions as learning results from different data sources and measure the uncertainty of opinion aggregation as multi-view learning results through evidence accumulation. We prove that accumulating the evidences from multiple data views will decrease the uncertainty in multi-view deep learning and facilitate to achieve the trusted learning results. Experiments on various kinds of multi-view datasets verify the reliability and robustness of the proposed multi-view deep learning method.

As an unsupervised ensemble learning strategy, clustering ensemble combines multiple base clusterings into a high-quality one and has achieved successful applications in image analysis and data mining. However, extant clustering ensemble methods are ineffective to handle the data uncertainty in clustering consensus process, which may mislead to poor clustering ensemble results. To tackle the problem, we propose a stable clustering ensemble (SCE) method based on evidence theory (Dempster–Shafer theory) in this paper. Specifically, we construct a belief function of cluster membership to measure the uncertainty and stability of data instances in clustering ensemble and thereby implement the stable clustering ensemble algorithm. We test the proposed stable clustering ensemble method in the tasks of structural data clustering and image segmentation. The experimental results validate the proposed method is effective to process the uncertain data and produce high-quality data clusterings.

Bladder cancer staging is crucial for operation planning and cancer assessment. Deep Convolutional Neural Networks (DCNNs) have been widely used to classify the bladder tumor images to identify cancer stages. However, the pure image-based deep learning methods over depend on the labeled data training and neglect the clinical priors. Human doctors judge the stage of a bladder tumor through checking whether the tumor infiltrating into bladder wall. The clinical priors of tumor infiltration are helpful to improve the DCNN-based bladder cancer staging and make the predictions coincide with the law of medicine. To involve clinical priors into deep learning for cancer staging, we propose a DCNN model with prior evidence to classify medical images of bladder tumors. Specifically, we measure the degree of tumor infiltrating into bladder wall to construct the prior evidence and integrate the prior evidence into the image-based prediction with evidential deep neural networks. We analyze the learning objective and prove that the prior evidences consistent with the ground truth will certainly reduce the prediction error and variance produced by image-based neural networks. The experiments on bladder cancer MR images datasets validate that involving prior evidences is effective to improve the DCNN-based cancer staging.

The Evidential K-Nearest Neighbor (EK-NN) classification rule provides a global treatment of imperfect knowledge in class labels, but still suffers from the curse of dimensionality as well as runtime and memory restrictions when performing nearest neighbors search, in particular for large and high-dimensional data. To avoid the curse of dimensionality, this article first proposes a rough evidential K-NN (REK-NN) classification rule in the framework of rough set theory. Based on a reformulated K-NN rough set model, REK-NN selects features and thus reduces complexity by minimizing a proposed neighborhood pignistic decision error rate, which considers both Bayes decision error and spatial information among samples in feature space. In contrast to existing rough set-based feature selection methods, REK-NN is a synchronized rule rather than a stepwise one, in the sense that feature selection and learning are performed simultaneously. In order to further handle data with large sample size, we derive a distributed REK-NN method and implement it in the Apache Spark. The theoretical analysis of the classifier generalization error bound is finally presented. It is shown that the distributed REK-NN achieves good performances while drastically reducing the number of features and consuming less runtime and memory. Numerical experiments conducted on real-world datasets validate our conclusions.

In applications of domain adaptation, there may exist multiple source domains, which can provide more or less complementary knowledge for pattern classification in the target domain. In order to improve the classification accuracy, a decision-level combination method is proposed for the multisource domain adaptation based on evidential reasoning. The classification results obtained from different source domains usually have different reliabilities/weights, which are calculated according to domain consistency. Therefore, the multiple classification results are discounted by the corresponding weights under belief functions framework, and then, Dempster's rule is employed to combine these discounted results. In order to reduce errors, a neighborhood-based cautious decision-making rule is developed to make the class decision depending on the combination result. The object is assigned to a singleton class if its neighborhoods can be (almost) correctly classified. Otherwise, it is cautiously committed to the disjunction of several possible classes. By doing this, we can well characterize the partial imprecision of classification and reduce the error risk as well. A unified utility value is defined here to reflect the benefit of such classification. This cautious decision-making rule can achieve the maximum unified utility value because partial imprecision is considered better than an error. Several real data sets are used to test the performance of the proposed method, and the experimental results show that our new method can efficiently improve the classification accuracy with respect to other related combination methods.

We consider the problem of quantifying prediction uncertainty in the Dempster-Shafer framework. Our approach assumes a parametric statistical model relating the variable of interest, the parameter and a pivotal random variable with known probability distribution. A predictive belief function is computed using this model and a belief function defined in the parameter space. In the case of multistep prediction, the quantity to be predicted is a vector, and the predictive belief function is defined in a multidimensional space, making its representation and manipulation difficult. To address this issue, we propose to approximate the focal sets of belief functions using point clouds, which allows us to approximate the belief and plausibility of arbitrary events with any accuracy. As an illustration, the approach is applied to the case of a first-order autoregressive process.

This paper introduces a new evidential clustering method based on the notion of “belief peaks” in the framework of belief functions. The basic idea is that all data objects in the neighborhood of each sample provide pieces of evidence that induce belief on the possibility of such sample to become a cluster center. A sample having higher belief than its neighbors and located far away from the other local maxima is then characterized as cluster center. Finally, a credal partition is created by minimizing an objective function with the fixed cluster centers. An adaptive distance metric is used to fit for unknown shapes of data structures. We show that the proposed evidential clustering procedure has very good performance with an ability to reveal the data structure in the form of a credal partition, from which hard, fuzzy, possibilistic, and rough partitions can be derived. Simulations on synthetic and real-world datasets validate our conclusions.

Precise delineation of target tumor is a key factor to ensure the effectiveness of radiation therapy. While hybrid positron emission tomography-computed tomography (PET-CT) has become a standard imaging tool in the practice of radiation oncology, many existing automatic/semi-automatic methods still perform tumor segmentation on mono-modal images. In this paper, a co-clustering algorithm is proposed to concurrently segment 3D tumors in PET-CT images, considering that the two complementary imaging modalities can combine functional and anatomical information to improve segmentation performance. The theory of belief functions is adopted in the proposed method to model, fuse, and reason with uncertain and imprecise knowledge from noisy and blurry PET-CT images. To ensure reliable segmentation for each modality, the distance metric for the quantification of clustering distortions and spatial smoothness is iteratively adapted during the clustering procedure. On the other hand, to encourage consistent segmentation between different modalities, a specific context term is proposed in the clustering objective function. Moreover, during the iterative optimization process, clustering results for the two distinct modalities are further adjusted via a belief-functions-based information fusion strategy. The proposed method has been evaluated on a data set consisting of 21 paired PET-CT images for non-small cell lung cancer patients. The quantitative and qualitative evaluations show that our proposed method performs well compared with the state-of-the-art methods.

In evidential clustering, cluster-membership uncertainty is represented by Dempster–Shafer mass functions. The notion of evidential partition generalizes other soft clustering structures such as fuzzy, possibilistic, or rough partitions. In this paper, we propose two extensions of the Rand index for evaluating and comparing evidential partitions, called similarity and consistency indices. The similarity index is suitable for measuring the closeness of two soft partitions, whereas the consistency index allows one to assess the agreement, or lack of conflict, between a soft partition and the true hard partition. Simulation experiments illustrate some applications of these indices.

In current interval-valued linear regression models, meaningless predictions may be generated because the lower bounds of the predicted intervals may be greater than their upper bounds. To avoid this problem, we propose a constrained interval-valued linear regression model based on random set theory. However, due to the introduction of constraints in this model, the expectation of the errors is no longer zero, and estimation provided by traditional least square may produce systematic bias. To address this issue, we introduce a two-step procedure: in the first step, a dummy variable is defined and plugged into the regression model to ensure that the expectation of errors is zero; least square estimation is then used in the second step. To show the validity of proposed method, experiments on simulated and real data are presented.

While hybrid PET/CT scanner is becoming a standard imaging technique in clinical oncology, many existing methods still segment tumor in mono-modality without consideration of complementary information from another modality. In this paper, we propose an unsupervised 3-D method to automatically segment tumor in PET images, where anatomical knowledge from CT images is included as critical guidance to improve PET segmentation accuracy. To this end, a specific context term is proposed to iteratively quantify the conflicts between PET and CT segmentation. In addition, to comprehensively characterize image voxels for reliable segmentation, informative image features are effectively selected via an unsupervised metric learning strategy. The proposed method is based on the theory of belief functions, a powerful tool for information fusion and uncertain reasoning. Its performance has been well evaluated by real-patient PET/CT images.

The evidential K-nearest-neighbor (EK-NN) method provided a global treatment of imperfect knowledge regarding the class membership of training patterns. It has outperformed traditional K-NN rules in many applications, but still shares some of their basic limitations, e.g., 1) classification accuracy depends heavily on how to quantify the dissimilarity between different patterns and 2) no guarantee for satisfactory performance when training patterns contain unreliable (imprecise and/or uncertain) input features. In this paper, we propose to address these issues by learning a suitable metric, using a low-dimensional transformation of the input space, so as to maximize both the accuracy and efficiency of the EK-NN classification. To this end, a novel loss function to learn the dissimilarity metric is constructed. It consists of two terms: the first one quantifies the imprecision regarding the class membership of each training pattern, while, by means of feature selection, the second one controls the influence of unreliable input features on the output linear transformation. The proposed method has been compared with some other metric learning methods on several synthetic and real datasets. It consistently led to comparable performance with regard to testing accuracy and class structure visualization.

In some real-world classification applications, such as target recognition, both training data collected by sensors and expert knowledge may be available. These two types of information are usually independent and complementary, and both are useful for classification. In this paper, a hybrid belief rule-based classification system (HBRBCS) is developed to make joint use of these two types of information. The belief rule structure, which is capable of capturing fuzzy, imprecise, and incomplete causal relationships, is used as the common representation model. With the belief rule structure, a data-driven belief rule base (DBRB) and a knowledge-driven belief rule base (KBRB) are learned from uncertain training data and expert knowledge, respectively. A fusion algorithm is proposed to combine the DBRB and KBRB to obtain an optimal hybrid belief rule base (HBRB). A belief reasoning and decision-making module is then developed to classify a query pattern based on the generated HBRB. An airborne target classification problem in the air surveillance system is studied to demonstrate the performance of the proposed HBRBCS for combining both uncertain sensor measurements and expert knowledge to make classification.

This paper presents a novel approach for predicting the energy consumption of a plug-in hybrid electric vehicle (PHEV). We propose to estimate energy consumption strategy from data via regression applied to trip recordings. Descriptors of the trip elements are obtained from both recordings and statistics provided by a GPS navigation system. Trips are then split into elementary units corresponding to an homogeneous driving context. For each trip element, the optimal energy consumption strategy is computed via (expensive) dynamic programming simulations. Here, data analysis is used so as to identify descriptors of this trip element that are relevant to predict the energy consumption. Then, a polynomial model is fit to the data so as to estimate, for each new trip element, the optimal energy consumption strategy from the expected driving condition, rather than using dynamic programming. Our approach distinguishes itself by the fact that road context, driver style, road slope and auxiliary electrical power are taken into account to estimate the energy consumption of a PHEV. The accuracy of the prediction process is evaluated over test data, and demonstrates the interest of our approach in predicting energy consumption.

In this paper, we present a distributed approach to build a dynamic map in the context of VANets (Vehicular Ad hoc Networks). It is based on the principle of cooperative perception where vehicles work as a team in order to extend their field of view. Each vehicle is equipped with sensors allowing it to detect its environment and to build its map, denoted by local map. It receives messages from other vehicles containing mobile objects detected in their surroundings. The algorithm of distributed dynamic map builds a map of the dynamic environment including objects in the sensor's field of view as well as those sent by other vehicles. This algorithm is developed under the belief functions framework. The implementation of such an application is complex and needs many treatments: temporal and spatial alignment, object association, fusion of messages and data dissemination. This approach has been validated by simulation on scenario involving several vehicles in traffic situation.

Object association is a crucial step in target tracking and data fusion applications. This task can be formalized as the search for a relation between two sets (e.g., a sets of tracks and a set of observations) in such a way that each object in one set is matched with at most one object in the other set. In this paper, this problem is tackled using the formalism of belief functions. Evidence about the possible association of each object pair, usually obtained by comparing the values of some attributes, is modeled by a Dempster-Shafer mass function defined in the frame of all possible relations. These mass functions are combined using Dempster's rule, and the relation with maximal plausibility is found by solving an integer linear programming problem. This problem is shown to be equivalent to a linear assignment problem, which can be solved in polynomial time using, for example, the Hungarian algorithm. This method is demonstrated using simulated and real data. The 3-D extension of this problem (with three object sets) is also formalized and is shown to be NP-Hard.

This paper addresses the problem of parameter estimation and state prediction in hidden Markov models (HMMs) based on observed outputs and partial knowledge of hidden states expressed in the belief function framework. The usual HMM model is recovered when the belief functions are vacuous. Parameters are learned using the evidential expectation-maximization algorithm, a recently introduced variant of the expectation-maximization algorithm for maximum likelihood estimation based on uncertain data. The inference problem, i.e., finding the most probable sequence of states based on observed outputs and partial knowledge of states, is also addressed. Experimental results demonstrate that partial information about hidden states, when available, may substantially improve the estimation and prediction performances.

In real world applications, data are often uncertain or imperfect. In most classification approaches, they are transformed into precise data. However, this uncertainty is an information in itself which should be part of the learning process. Data uncertainty can take several forms: probabilities, (fuzzy)sets of possible values, expert assessments, etc. We therefore need a flexible and generic enough model to represent and treat this uncertainty, such as belief functions. Decision trees are well known classifiers which are usually learned from precise datasets. In this paper we propose a methodology to learn decision trees from uncertain data in the belief function framework. In the proposed method, the tree parameters are estimated through the maximization of an evidential likelihood function computed from belief functions, using the recently proposed E2M algorithm that extends the classical EM. Some promising experiments compare the obtained trees with classical CART decision trees.

We consider the problem of parameter estimation in statistical models in the case where data are uncertain and represented as belief functions. The proposed method is based on the maximization of a generalized likelihood criterion, which can be interpreted as a degree of agreement between the statistical model and the uncertain observations. We propose a variant of the EM algorithm that iteratively maximizes this criterion. As an illustration, the method is applied to uncertain data clustering using finite mixture models, in the cases of categorical and continuous attributes.

The goal of map-matching algorithms is to identify the road taken by a vehicle and to compute an estimate of the vehicle position on that road using a digital map. In this paper, a map-matching algorithm based on interval analysis and the belief function theory is proposed. The method combines the outputs from existing bounded-error estimation techniques with piecewise rectangular roads that are selected using evidential reasoning. A set of candidate roads is first defined at each time step using the topology of the map and a similarity criterion, and a mass function on the set of candidate roads is computed. An overall estimate of the vehicle position is then derived after the most probable candidate road has been selected. This method allows multiple road junction hypotheses to efficiently be handled and can cope with missing data. In addition, the implementation of the method is quite simple, because it is based on geometrical properties of boxes and rectangular road segments. Experiments with simulated and real data demonstrate the ability of this method to handle junction situations and to compute an accurate estimate of the vehicle position.

Fuzzy or hard partitioning methods aim at grouping objects according to their similarity. Recently, a new concept of partition based on belief function theory, called credal partition, has been proposed and has been shown to generate meaningful description of the data. Hard, fuzzy or credal partitions are generally obtained using unsupervised learning methods, using only the numeric description between two objects to compute their similarity. However, in some applications, some kind of background knowledge about the objects or about the clusters is available. To integrate this auxiliary information, constraint-based (or semi-supervised) methods have been proposed. A popular type of constraints specifies whether two objects are in the same cluster (must-link) or in different clusters (cannot-link). We propose here a new algorithm, called CECM, which computes a credal partition using a constrained clustering method. We show how to translate the available information into constraints, and how to integrate them in the search of the credal partition. The paper ends with some experimental results. Results of CECM are compared to other constrained clustering algorithms. Then an application in image segmentation is described.

Multi-label learning is increasingly required by many applications where instances may belong to several classes at the same time. In this paper, we propose a fuzzy k-nearest neighbor method for multi-label classification using the veristic variable framework. Veristic variables are variables that can assume simultaneously multiple values with different degrees. In multi-label learning, class labels can be considered as veristic variables since each instance can belong simultaneously to more than one class. Several applications on benchmark datasets demonstrate the efficiency of our approach.

A new approach to nonlinear state estimation based on belief-function theory and interval analysis is presented. This method uses belief structures composed of a finite number of axis-aligned boxes with associated masses. Such belief structures can represent partial information on model and measurement uncertainties more accurately than can the bounded-error approach alone. Focal sets are propagated in system equations using interval arithmetics and constraint-satisfaction techniques, thus generalizing pure interval analysis. This model was used to locate a land vehicle using a dynamic fusion of Global Positioning System measurements with dead reckoning sensors. The method has been shown to provide more accurate estimates of vehicle position than does the bounded-error method while retaining what is essential: providing guaranteed computations. The performances of our method were also slightly better than those of a particle filter, with comparable running time. These results suggest that our method is a viable alternative to both bounded-error and probabilistic Monte Carlo approaches for vehicle-localization applications.

Knowledge of traffic demand at a junction is crucial for most transport systems. Generally, it is represented by an origin-destination (OD) matrix, where each element is a volume of vehicle flow between one of the OD pair of zones of a junction. This paper introduces a new method for a short-time estimation of OD matrices at a signalised junction with a complex structure and fitted out with video cameras. The estimation is made by taking into account the traffic lights and by fusion of different multisensor traffic data, which are subject to imprecision and uncertainty. The method proceeds in two steps. First, vehicle conservation law, expressed in terms of undertermined system of equations, is built dynamically for a short-time period. Second, four approaches, that overcome the system indetermination and model the data imperfection, are proposed to provide the best and unique estimates of the OD matrix. An experimental study has been made with data collected at the real signalised junction.

A method is proposed for converting a novelty measure such as produced by one-class SVMs or Kernel Principal Component Analysis (KPCA) into a belief function on a welldefined frame of discernment. This makes it possible to combine one-class classification or novelty detection methods with other information expressed in the same framework such as expert opinions or multi-class classifiers.

This paper addresses the problem of fault detection in a complex system made up of several spatially dependent subsystems. The diagnosis method consists of both detecting and localizing a defect on the system by combining the outputs scores of subclassifiers within the framework of belief function theory. This paper is focused on the coding and the combination of classifier outputs that can reflect the spatial relationship between the subsystems. In the particular case of upstream/downstream dependency, two strategies of output coding are detailed. The proposed methodology is illustrated on a railway device diagnosis application. It will be shown that the choice of an appropriate coding scheme improves the classification results

Dempster's rule plays a central role in the theory of belief functions. However, it assumes the items of evidence combined to be distinct, an assumption which is not always verified in practice. In this paper, a new operator, the cautious rule of combination, is introduced. This operator is commutative, associative and idempotent. This latter property makes it suitable to combine non distinct items of evidence. Extensions based on triangular norms (some of which allow to define operators whose behavior is intermediate between the Dempster's rule and the cautious rule) are also introduced

The discounting operation is a well known operation on belief functions, which has proved to be useful in many applications. However, the discounting operation only allows one to weaken a source, whereas it is sometimes useful to strengthen it when it is deemed to be too cautious. For that purpose, the de-discounting operation was introduced as the inverse operation of the discounting operation by Denoeux and Smets. From another point of view, Zhu and Basir introduced an extension of the classical discounting operation by allowing the discount rate to be out of the range [0,1]. This operation performs a discounting or a de-discounting of a belief function. A new interpretation of this scheme is presented in this paper. A more general form of reinforcement process, as well as a parameterized family of transformations encompassing all previous schemes, are also introduced

Whereas probability theory has been very successful as a conceptual framework for risk analysis in many areas where a lot of experimental data and expert knowledge are available, it presents certain limitations in applications where only weak information can be obtained. One such application investigated in this paper is water treatment, a domain in which key information such as input water characteristics and failure rates of various chemical processes is often lacking. An approach to handle such problems is proposed, based on the Dempster-Shafer theory of belief functions. Belief functions are used to describe expert knowledge of treatment process efficiency, failure rates, and latency times, as well as statistical data regarding input water quality. Evidential reasoning provides mechanisms to combine this information and assess the plausibility of various noncompliance scenarios. This methodology is shown to boil down to the probabilistic one where data of sufficient quality are available. This case study shows that belief function theory may be considered as a valuable framework for risk analysis studies in ill-structured or poorly informed application domains

The transferable belief model (TBM) is a model to represent quantified uncertainties based on belief functions, unrelated to any underlying probability model. In this framework, two main approaches to pattern classification have been developed: the TBM model-based classifier, relying on the general Bayesian theorem (GBT), and the TBM case-based classifier, built on the concept of similarity of a pattern to be classified with training patterns. Until now, these two methods seemed unrelated, and their connection with standard classification methods was unclear. This paper shows that both methods actually proceed from the same underlying principle, i.e., the GBT, and that they essentially differ by the nature of the assumed available information. This paper also shows that both methods collapse to a kernel rule in the case of precise and categorical learning data and for certain initial assumptions, and a simple relationship between basic belief assignments produced by the two methods is exhibited in a special case. These results shed new light on the issues of classification and supervised learning in the TBM. They also suggest new research directions and may help users in selecting the most appropriate method for each particular application, depending on the nature of the information at hand.

This paper describes an extension of principal component analysis (PCA) allowing the extraction of a limited number of relevant features from high-dimensional fuzzy data. Our approach exploits the ability of linear autoassociative neural networks to perform information compression in just the same way as PCA, without explicit matrix diagonalization. Fuzzy input values are propagated through the network using fuzzy arithmetics, and the weights are adjusted to minimize a suitable error criterion, the inputs being taken as target outputs. The concept of correlation coefficient is extended to fuzzy numbers, allowing the interpretation of the new features in terms of the original variables. Experiments with artificial and real sensory evaluation data demonstrate the ability of our method to provide concise representations of complex fuzzy data.

A new relational clustering method is introduced, based on the Dempster-Shafer theory of belief functions (or evidence theory). Given a matrix of dissimilarities between n objects, this method, referred to as evidential clustering (EVCLUS), assigns a basic belief assignment (or mass function) to each object in such a way that the degree of conflict between the masses given to any two objects reflects their dissimilarity. A notion of credal partition is introduced, which subsumes those of hard, fuzzy, and possibilistic partitions, allowing to gain deeper insight into the structure of the data. Experiments with several sets of real data demonstrate the good performances of the proposed method as compared with several state-of-the-art relational clustering techniques.

A methodology is proposed for assessing the risk to produce non-compliant potable water, taking into account the quality of the raw water, as well as characteristics of the treatment unit and different failure modes. Belief functions are used to describe expert knowledge of treatment process efficiency, failure rates, latency times and raw water quality. Evidential reasoning provides mechanisms to combine this information and assess the plausibility of non-compliant water production. This approach may be used by treatment plant designers to choose the optimal architecture, given a user-defined level of residual risk.

This paper presents and compares several evidential classifiers, i.e., classification rules based on the Dempster-Shafer theory of evidence. Three methods used in the majority of applications are compared, with emphasis on the techniques used to build belief functions from learning data. The methods are: the consonant method initially introduced by Shafer (1976) in the more general context of statistical inference, Appriou's separable method (1998), and the distance-based classifier introduced by Denoeux. These models can be derived with two decisions rules, based on the minimization of, respectively, lower and pignistic expected loss. Simulations on synthetic data demonstrate the performance of these techniques and allow to compare the behavior of the proposed models.

A hierarchical clustering approach is proposed for reducing the number of focal elements in a crisp or fuzzy belief function, yielding strong inner and outer approximations. At each step of the proposed algorithm, two focal elements are merged, and the mass is transfered to their intersection or their union. The resulting approximations allow the calculation of lower and upper bounds on the belief and plausibility degrees induced by the conjunctive or disjunctive sum of any number of belief structures. Numerical experiments demonstrate the effectiveness of this approach.

A new tree-structured classifier based on the Dempster-Shafer theory of evidence is presented. The entropy measure classically used to assess the impurity of nodes in decision trees is replaced by an evidence-theoretic uncertainty measure taking into account not only the class proportions, but also the number of objects in each node. The resulting algorithm allows the processing of training data whose class membership is only partially specified in the form of a belief function. Experimental results with EEG data are presented.

A new adaptive pattern classifier based on the Dempster-Shafer theory of evidence is presented. This method uses reference patterns as items of evidence regarding the class membership of each input pattern under consideration. This evidence is represented by basic belief assignments (BBA) and pooled using the Dempster's rule of combination. This procedure can be implemented in a multilayer neural network with specific architecture consisting of one input layer, two hidden layers and one output layer. The weight vector, the receptive field and the class membership of each prototype are determined by minimizing the mean squared differences between the classifier outputs and target values. After training, the classifier computes for each input vector a BBA that provides a description of the uncertainty pertaining to the class of the current pattern, given the available evidence. This information may be used to implement various decision rules allowing for ambiguous pattern rejection and novelty detection. The outputs of several classifiers may also be combined in a sensor fusion context, yielding decision procedures which are very robust to sensor failures or changes in the system environment. Experiments with simulated and real data demonstrate the excellent performance of this classification scheme as compared to existing statistical and neural network techniques.

Artificial neural network techniques are applied to the control of coagulant dosing in a drinking water treatment plant. Coagulant dosing rate is nonlinearly correlated to raw water parameters such as turbidity, conductivity, pH, temperature, etc. An important requirement of the application is robustness of the system against erroneous sensor measurements or unusual water characteristics. The hybrid system developed includes raw data validation and reconstruction based on the Kohonen self-organizing feature map, and prediction of coagulant dosage using multilayer perceptrons. A key feature of the system is its ability to take into account various sources of uncertainty, such as a typical input data, measurement errors and limited information content of the training set. Experimental results with real data are presented.

The paper presents a learning procedure for optimizing the parameters in the evidence-theoretic k-nearest neighbor rule, a pattern classification method based on the Dempster-Shafer theory of belief functions. In this approach, each neighbor of a pattern to be classified is considered as an item of evidence supporting certain hypotheses concerning the class membership of that pattern. Based on this evidence, basic belief masses are assigned to each subset of the set of classes. Such masses are obtained for each of the k-nearest neighbors of the pattern under consideration and aggregated using Dempster's rule of combination. In many situations, this method was found experimentally to yield lower error rates than other methods using the same information. However, the problem of tuning the parameters of the classification rule was so far unresolved. The authors determine optimal or near-optimal parameter values from the data by minimizing an error function. This refinement of the original method is shown experimentally to result in substantial improvement of classification accuracy.

In this paper, the problem of classifying an unseen pattern on the basis of its nearest neighbors in a recorded data set is addressed from the point of view of Dempster-Shafer theory. Each neighbor of a sample to be classified is considered as an item of evidence that supports certain hypotheses regarding the class membership of that pattern. The degree of support is defined as a function of the distance between the two vectors. The evidence of the k nearest neighbors is then pooled by means of Dempster's rule of combination. This approach provides a global treatment of such issues as ambiguity and distance rejection, and imperfect knowledge regarding the class membership of training patterns. The effectiveness of this classification scheme as compared to the voting and distance-weighted k-NN procedures is demonstrated using several sets of simulated and real-world data.>