Zhun-ga Liu

Combination of sources of evidence with different discounting factors based on a new dissimilarity measure

The sources of evidence may have different reliability and importance in real applications for decision making. The estimation of the discounting (weighting) factors when the prior knowledge is unknown have been regularly studied until recently. In the past, the determination of the weighting factors focused only on reliability discounting rule and it was mainly dependent on the dissimilarity measure between basic belief assignments (bbas) represented by an evidential distance. Nevertheless, it is very difficult to characterize efficiently the dissimilarity only through an evidential distance. Thus, both a distance and a conflict coefficient based on probabilistic transformations BetP are proposed to characterize the dissimilarity. The distance represents the difference between bbas, whereas the conflict coefficient reveals the divergence degree of the hypotheses that two belief functions strongly support. These two aspects of dissimilarity are complementary in a certain sense, and their fusion is used as the dissimilarity measure. Then, a new estimation method of weighting factors is presented by using the proposed dissimilarity measure. In the evaluation of weight of a source, both its dissimilarity with other sources and their weighting factors are considered. The weighting factors can be applied in the both importance and reliability discounting rules, but the selection of the adapted discounting rule should depend on the actual application. Simple numerical examples are given to illustrate the interest of the proposed approach.

A new belief-based K-nearest neighbor classification method

The K-nearest neighbor (K-NN) classification method originally developed in the probabilistic framework has serious difficulties to classify correctly the close data points (objects) originating from different classes. To cope with such difficult problem and make the classification result more robust to misclassification errors, we propose a new belief-based K-nearest neighbor (BK-NN) method that allows each object to belong both to the specific classes and to the sets of classes with different masses of belief. BK-NN is able to provide a hyper-credal classification on the specific classes, the rejection classes and the meta-classes as well. Thus, the objects hard to classify correctly are automatically committed to a meta-class or to a rejection class, which can reduce the misclassification errors. The basic belief assignment (bba) of each object is defined from the distance between the object and its neighbors and from the acceptance and rejection thresholds. The bbas are combined using a new combination method specially developed for the BK-NN. Several experiments based on simulated and real data sets have been carried out to evaluate the performances of the BK-NN method with respect to several classical K-NN approaches.

Adaptive imputation of missing values for incomplete pattern classification

In classification of incomplete pattern, the missing values can either play a crucial role in the class determination, or have only little influence (or eventually none) on the classification results according to the context. We propose a credal classification method for incomplete pattern with adaptive imputation of missing values based on belief function theory. At first, we try to classify the object (incomplete pattern) based only on the available attribute values. As underlying principle, we assume that the missing information is not crucial for the classification if a specific class for the object can be found using only the available information. In this case, the object is committed to this particular class. However, if the object cannot be classified without ambiguity, it means that the missing values play a main role for achieving an accurate classification. In this case, the missing values will be imputed based on the K-nearest neighbor (K-NN) and Self-Organizing Map (SOM) techniques, and the edited pattern with the imputation is then classified. The (original or edited) pattern is classified according to each training class, and the classification results represented by basic belief assignments are fused with proper combination rules for making the credal classification. The object is allowed to belong with different masses of belief to the specific classes and meta-classes (which are particular disjunctions of several single classes). The credal classification captures well the uncertainty and imprecision of classification, and reduces effectively the rate of misclassifications thanks to the introduction of meta-classes. The effectiveness of the proposed method with respect to other classical methods is demonstrated based on several experiments using artificial and real data sets. HighlightsMissing values are adaptively imputed in classification according to context.SOM and K-NN are used for the imputation with admissible computation burden.Ensemble classifier is introduced for credal classification.The imprecision of classification can be well captured using belief functions.The proposed method has been tested by artificial and real data sets.

Credal c-means clustering method based on belief functions

The recent credal partition approach allows the objects to belong to not only the singleton clusters but also the sets of clusters (i.e. meta-clusters) with different masses of belief. A new credal c-means (CCM) clustering method working with credal partition has been proposed in this work to effectively deal with the uncertain and imprecise data. In the clustering problem, one object simultaneously close to several clusters can be difficult to correctly classify, since these close clusters appear not very distinguishable for this object. In such case, the object will be cautiously committed by CCM to a meta-cluster (i.e. the disjunction of these close clusters), which can be considered as a transition cluster among these different close clusters. It can well characterize the imprecision of the class of the object and can also reduce the misclassification errors thanks to the use of meta-cluster. CCM is robust to the noisy data because of the outlier cluster. The clustering centers and the mass of belief on each cluster for any object are obtained by the optimization of a proper objective function in CCM. The effectiveness of CCM has been demonstrated by three experiments using synthetic and real data sets with respect to fuzzy c-means (FCM) and evidential c-means (ECM) clustering methods.

Credal classification rule for uncertain data based on belief functions

Abstract In this paper we present a new credal classification rule (CCR) based on belief functions to deal with the uncertain data. CCR allows the objects to belong (with different masses of belief) not only to the specific classes, but also to the sets of classes called meta-classes which correspond to the disjunction of several specific classes. Each specific class is characterized by a class center (i.e. prototype), and consists of all the objects that are sufficiently close to the center. The belief of the assignment of a given object to classify with a specific class is determined from the Mahalanobis distance between the object and the center of the corresponding class. The meta-classes are used to capture the imprecision in the classification of the objects when they are difficult to correctly classify because of the poor quality of available attributes. The selection of meta-classes depends on the application and the context, and a measure of the degree of indistinguishability between classes is introduced. In this new CCR approach, the objects assigned to a meta-class should be close to the center of this meta-class having similar distances to all the involved specific classes׳ centers, and the objects too far from the others will be considered as outliers (noise). CCR provides robust credal classification results with a relatively low computational burden. Several experiments using both artificial and real data sets are presented at the end of this paper to evaluate and compare the performances of this CCR method with respect to other classification methods.

Median evidential c-means algorithm and its application to community detection

Median clustering is of great value for partitioning relational data. In this paper, a new prototype-based clustering method, called Median Evidential C-Means (MECM), which is an extension of median c-means and median fuzzy c-means on the theoretical framework of belief functions is proposed. The median variant relaxes the restriction of a metric space embedding for the objects but constrains the prototypes to be in the original data set. Due to these properties, MECM could be applied to graph clustering problems. A community detection scheme for social networks based on MECM is investigated and the obtained credal partitions of graphs, which are more refined than crisp and fuzzy ones, enable us to have a better understanding of the graph structures. An initial prototype-selection scheme based on evidential semi-centrality is presented to avoid local premature convergence and an evidential modularity function is defined to choose the optimal number of communities. Finally, experiments in synthetic and real data sets illustrate the performance of MECM and show its difference to other methods.

Evidential classifier for imprecise data based on belief functions

A new evidential classifier (EC) based on belief functions is developed in this paper for the classification of imprecise data using K-nearest neighbors. EC works with credal classification which allows to classify the objects either in the specific classes, in the meta-classes defined by the union of several specific classes, or in the ignorant class for the outlier detection. The main idea of EC is to not classify an object in a particular class whenever the object is simultaneously close to several classes that turn to be indistinguishable for it. In such case, EC will associate the object with a proper meta-class in order to reduce the misclassification errors. The full ignorant class is interpreted as the class of outliers representing all the objects that are too far from the other data. The K basic belief assignments (bbas) associated with the object are determined by the distances of the object to its K-nearest neighbors and some chosen imprecision thresholds. The classification of the object depends on the global combination results of these K bbas. The interest and potential of this new evidential classifier with respect to other classical methods are illustrated through several examples based on artificial and real data sets.

Belief C-Means: An extension of Fuzzy C-Means algorithm in belief functions framework

The well-known Fuzzy C-Means (FCM) algorithm for data clustering has been extended to Evidential C-Means (ECM) algorithm in order to work in the belief functions framework with credal partitions of the data. Depending on data clustering problems, some barycenters of clusters given by ECM can become very close to each other in some cases, and this can cause serious troubles in the performance of ECM for the data clustering. To circumvent this problem, we introduce the notion of imprecise cluster in this paper. The principle of our approach is to consider that objects lying in the middle of specific classes (clusters) barycenters must be committed with equal belief to each specific cluster instead of belonging to an imprecise meta-cluster as done classically in ECM algorithm. Outliers object far away of the centers of two (or more) specific clusters that are hard to be distinguished, will be committed to the imprecise cluster (a disjunctive meta-cluster) composed by these specific clusters. The new Belief C-Means (BCM) algorithm proposed in this paper follows this very simple principle. In BCM, the mass of belief of specific cluster for each object is computed according to distance between object and the center of the cluster it may belong to. The distances between object and centers of the specific clusters and the distances among these centers will be both taken into account in the determination of the mass of belief of the meta-cluster. We do not use the barycenter of the meta-cluster in BCM algorithm contrariwise to what is done with ECM. In this paper we also present several examples to illustrate the interest of BCM, and to show its main differences with respect to clustering techniques based on FCM and ECM.

A New Incomplete Pattern Classification Method Based on Evidential Reasoning

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.

Dynamic evidential reasoning for change detection in remote sensing images

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.