Abstract. A theory of random Borel sets is presented, based on dyadic resolutions of compact metric spaces. The conditional expectation of the intersection of two independent random Borel sets is investigated. An example based on an embedding of Sierpiński's universal curve into the space of Borel sets is given.
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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, while 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.
classification with soft labels using the Evidential EM algorithm : linear discriminant analysis versus logistic regression.Abstract Partially supervised learning extends both supervised and unsupervised learning, by considering situations in which only partial information about the response variable is available. In this paper, we consider partially supervised classification and we assume the learning instances to be labeled by Dempster-Shafer mass functions, called soft labels. Linear discriminant analysis and logistic regression are considered as special cases of generative and discriminative parametric models. We show that the evidential EM algorithm can be particularized to fit the parameters in each of these models. We describe experimental results with simulated data sets as well as with two real applications: K-complex detection in sleep EEGs signals and facial expression recognition. These results confirm the interest of using soft labels for classification as compared to potentially erroneous crisp labels, when the true class membership is partially unknown or ill-defined.
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