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 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.

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.

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