A novel fuzzy evidential reasoning paradigm for data fusion with applications in image processing

This paper presents a novel data fusion paradigm based on fuzzy evidential reasoning. A new fuzzy evidence structure model is first introduced to formulate probabilistic evidence and fuzzy evidence in a unified framework. A generalized Dempster’s rule is then utilized to combine fuzzy evidence structures associated with multiple information sources. Finally, an effective decision rule is developed to take into account uncertainty, quantified by Shannon entropy and fuzzy entropy, of probabilistic evidence and fuzzy evidence, to deal with conflict and to achieve robust decisions. To demonstrate the effectiveness of the proposed paradigm, we apply it to classifying synthetic images and segmenting multi-modality human brain MR images. It is concluded that the proposed paradigm outperforms both the traditional Dempster–Shafer evidence theory based approach and the fuzzy reasoning based approach     

Information fusion in data association applications

There is a limitation to process data fusion by means of traditional deterministic or probabilistic data association algorithm of multisensor data fusion (MSDF). Those methods for data association do not adequately account for quantitative and qualitative information in an automated fashion. Fuzzy logic offers an enabling technology for automated quantitative and qualitative information in the data fusion process. We propose the fusion system architecture, called fuzzy gating approach, coordinating both quantitative and qualitative information which is realized using a fuzzy-based reasoning approach. This approach is composed of two stages in cascade. The first stage implements available quantitative information, namely target range, azimuth, and elevation angle, to form a subset of statistically likely target solutions via fuzzy validation. The second stage of the fuzzy similarity utilizes available qualitative information, namely infrared image area and brightness, to form another subset of returns. Finally the fuzzy gating approach was tested in a dense clutter environment. Results of test show that performance of fuzzy gating approach is superior to JPDA based on a Bayesian approach. Moreover, adding qualitative information to the tracking algorithm can improve tracker performance effectively.     

Probabilistic models considering dependent relation in reasoning for decision-making

This paper proposes a fuzzy dependence-index for construction of the probabilistic models considering dependent relation for solving the reasoning problem. It is important for constructing the joint probability-distribution to consider the dependency of events. We consider that some vagueness is included in the dependency. Because causal relationship of among events is uncertain, it is difficult to express dependency as definite value. In this paper, we classify the dependent relations, and apply the fuzzy probability to calculation of the dependence-index. Then, the fuzzy dependence-index is defined to consider dependency with fuzziness. Using the fuzzy dependence-index, we calculate the joint probability of multi-events for constructing the probabilistic model. This work was presented in part at the 13th International Symposium on Artificial Life and Robotics, Oita, Japan, January 31–February 2, 2008     

Reducing false positives in anomaly detectors through fuzzy alert aggregation

In this paper we focus on the aggregation of IDS alerts, an important component of the alert fusion process. We exploit fuzzy measures and fuzzy sets to design simple and robust alert aggregation algorithms. Exploiting fuzzy sets, we are able to robustly state whether or not two alerts are “close in time”, dealing with noisy and delayed detections. A performance metric for the evaluation of fusion systems is also proposed. Finally, we evaluate the fusion method with alert streams from anomaly-based IDS.     

Numerical methods for interval and fuzzy number comparison based on the probabilistic approach and Dempster-Shafer theory

There are many different methods for interval and fuzzy number comparison proposed in the literature which provide the results of comparison in the form of a real or Boolean value. In this paper, we use the Dempster-Shafer theory of evidence with its probabilistic interpretation to justify and construct the method which provides the result of comparison in the form of an interval or fuzzy number. The complete and consistent set of expressions for inequality and equality relations between intervals and fuzzy numbers has been obtained in the framework of the probabilistic approach. These relations make it possible to compare intervals and fuzzy numbers with real values and may be considered as an asymptotic limit of the results we obtain using the Dempster-Shafer theory. A natural fuzzy extension of the proposed approach is considered and discussed using some illustrative examples.     

A probabilistic fuzzy approach to modeling nonlinear systems

Recently, the study of incorporating probability theory and fuzzy logic has received much interest. To endow the traditional fuzzy rule-based systems (FRBs) with probabilistic features to handle randomness, this paper presents a probabilistic fuzzy neural network (ProFNN) by introducing the probability of input linguistic terms and providing linguistic meaning into the connectionist architecture. ProFNN integrates the probabilistic information of fuzzy rules into the antecedent parts and quantifies the impacts of the rules on the consequent parts using mutual subsethood, which work in conjunction with volume defuzzification in a gradient descent learning frame work. Despite the increase in the number of parameters, ProFNN provides a promising solution to deal with randomness and fuzziness in a single frame. To evaluate the performance and applicability of the proposed approach, ProFNN is carried out on various benchmarking problems and compared with other existing models with a performance better than most of them.     

A fuzzy constraint-based approach to data reconciliation in material flow analysis

Data reconciliation consists in modifying noisy or unreliable data in order to make them consistent with a mathematical model (herein a material flow network). The conventional approach relies on least-squares minimization. Here, we use a fuzzy set-based approach, replacing Gaussian likelihood functions by fuzzy intervals, and a leximin criterion. We show that the setting of fuzzy sets provides a generalized approach to the choice of estimated values, that is more flexible and less dependent on oftentimes debatable probabilistic justifications. It potentially encompasses interval-based formulations and the least squares method, by choosing appropriate membership functions and aggregation operations. This paper also lays bare the fact that data reconciliation under the fuzzy set approach is viewed as an information fusion problem, as opposed to the statistical tradition which solves an estimation problem.     

概率犹豫模糊决策理论与方法综述

概率犹豫模糊集是在犹豫模糊集的基础上为每个隶属度添加与之相对应的概率值.与犹豫模糊集相比,它可以更加准确和全面地表达专家给出的初始决策信息,因此,基于概率犹豫模糊集的决策理论与方法更加可靠且符合实际.这里对概率犹豫模糊决策理论与方法进行综述.首先介绍其发展过程;然后分别对它的信息融合理论、偏好关系理论以及决策方法等进行阐述;最后展望了概率犹豫模糊决策理论与方法的未来研究方向.     

Multicriteria distribution network planning using simulated annealing

The classical long-range distribution network planning problem involves deciding network investments to meet future demand at a minimum cost while meeting technical restrictions (thermal limits and maximum voltage drop). The decision whether to construct facilities and branches leads to a mixed integer programming problem with a large number of decision variables. The great deal of uncertainty associated with data that cannot be modeled using probabilistic methods leads to the use of fuzzy models to capture the uncertainty. In addition, several criteria must be taken into account that is resulting in the problem being fuzzy multiobjective. The combinatorial nature of the problem limits the use of traditional mathematical tools to limited size problems. This contribution presents a methodology that generates a sample of efficient solutions for the fuzzy multiobjective problem, based on a meta-heuristic, simulated annealing (SA). The results obtained with this approach are shown to be satisfactory compared to other methods under similar conditions.     

A systematic approach to the assessment of fuzzy association rules

In order to allow for the analysis of data sets including numerical attributes, several generalizations of association rule mining based on fuzzy sets have been proposed in the literature. While the formal specification of fuzzy associations is more or less straightforward, the assessment of such rules by means of appropriate quality measures is less obvious. Particularly, it assumes an understanding of the semantic meaning of a fuzzy rule. This aspect has been ignored by most existing proposals, which must therefore be considered as ad-hoc to some extent. In this paper, we develop a systematic approach to the assessment of fuzzy association rules. To this end, we proceed from the idea of partitioning the data stored in a database into examples of a given rule, counterexamples, and irrelevant data. Evaluation measures are then derived from the cardinalities of the corresponding subsets. The problem of finding a proper partition has a rather obvious solution for standard association rules but becomes less trivial in the fuzzy case. Our results not only provide a sound justification for commonly used measures but also suggest a means for constructing meaningful alternatives.

SVM-FuzCoC: A novel SVM-based feature selection method using a fuzzy complementary criterion

An efficient filter feature selection (FS) method is proposed in this paper, the SVM-FuzCoC approach, achieving a satisfactory trade-off between classification accuracy and dimensionality reduction. Additionally, the method has reasonably low computational requirements, even in high-dimensional feature spaces. To assess the quality of features, we introduce a local fuzzy evaluation measure with respect to patterns that embraces fuzzy membership degrees of every pattern in their classes. Accordingly, the above measure reveals the adequacy of data coverage provided by each feature. The required membership grades are determined via a novel fuzzy output kernel-based support vector machine, applied on single features. Based on a fuzzy complementary criterion (FuzCoC), the FS procedure iteratively selects features with maximum additional contribution in regard to the information content provided by previously selected features. This search strategy leads to small subsets of powerful and complementary features, alleviating the feature redundancy problem. We also devise different SVM-FuzCoC variants by employing seven other methods to derive fuzzy degrees from SVM outputs, based on probabilistic or fuzzy criteria. Our method is compared with a set of existing FS methods, in terms of performance capability, dimensionality reduction, and computational speed, via a comprehensive experimental setup, including synthetic and real-world datasets.     

Safety assessment of structures in view of fuzzy randomness

Structural safety can be realistically assessed only if the uncertainty in the structural parameters is appropriately taken into consideration and realistic computational models are applied. Uncertainty must be accounted for in its natural form. Stochastic models are not always capable of fulfilling this task without restrictions, as uncertainty may also be characterized by fuzzy randomness or fuzziness. On the basis of the theory of the fuzzy random variables the fuzzy probabilistic safety concept is introduced and formulated as the fuzzy first order reliability method (FFORM). This concept permits fuzziness, randomness and fuzzy randomness to be accounted for simultaneously. FFORM is illustrated by way of an example; hereby, the influence of the computational model is also demonstrated.     

Induction of multiple fuzzy decision trees based on rough set technique

The integration of fuzzy sets and rough sets can lead to a hybrid soft-computing technique which has been applied successfully to many fields such as machine learning, pattern recognition and image processing. The key to this soft-computing technique is how to set up and make use of the fuzzy attribute reduct in fuzzy rough set theory. Given a fuzzy information system, we may find many fuzzy attribute reducts and each of them can have different contributions to decision-making. If only one of the fuzzy attribute reducts, which may be the most important one, is selected to induce decision rules, some useful information hidden in the other reducts for the decision-making will be losing unavoidably. To sufficiently make use of the information provided by every individual fuzzy attribute reduct in a fuzzy information system, this paper presents a novel induction of multiple fuzzy decision trees based on rough set technique. The induction consists of three stages. First several fuzzy attribute reducts are found by a similarity based approach, and then a fuzzy decision tree for each fuzzy attribute reduct is generated according to the fuzzy ID3 algorithm. The fuzzy integral is finally considered as a fusion tool to integrate the generated decision trees, which combines together all outputs of the multiple fuzzy decision trees and forms the final decision result. An illustration is given to show the proposed fusion scheme. A numerical experiment on real data indicates that the proposed multiple tree induction is superior to the single tree induction based on the individual reduct or on the entire feature set for learning problems with many attributes.     

模糊数学理论在图像处理中的应用

用计算机来处理图片已成为计算机研究的一个重要方向,基于模糊数学的图像处理技术是计算机图像处理中的重要计算。图像本质上具有模糊性,因此模糊信息处理技术在图像处理中的使用有其必然性。提出一种基于模糊数学的方法来融合多模图像。     

Fuzzy arithmetic-based interpolative reasoning for nonlinear dynamic fuzzy systems

FAIR (fuzzy arithmetic-based interpolative reasoning)—a fuzzy reasoning scheme based on fuzzy arithmetic, is presented here. Linguistic rules of the Mamdani type, with fuzzy numbers as consequents, are used in an inference mechanism similar to that of a Takagi–Sugeno model. The inference result is a weighted sum of fuzzy numbers, calculated by means of the extension principle. Both fuzzy and crisp inputs and outputs can be used, and the chaining of rule bases is supported without increasing the spread of the output fuzzy sets in each step. This provides a setting for modeling dynamic fuzzy systems using fuzzy recursion. The matching in the rule antecedents is done by means of a compatibility measure that can be selected to suit the application at hand. Different compatibility measures can be used for different antecedent variables, and reasoning with sparse rule bases is supported. The application of FAIR to the modeling of a nonlinear dynamic system based on a combination of knowledge-driven and data-driven approaches is presented as an example.     

一种概率聚类的新算法

提出了一种新的基于信息熵的概率聚类算法(Hierarchical Probabilistic Clustering Method,HPCM),HPCM算法和经典的模糊聚类算法FCM有着同样的聚类轨迹,因此,概率聚类和模糊聚类之间是紧密联系的.有关信息熵的大量研究成果可以帮助我们更深入地了解模糊聚类的本质.     

Conversion of probabilistic information into fuzzy sets for engineering decision analysis

To carry out seismic hazard analysis in the framework of fuzzy set theory, it may become necessary to convert probabilistic information regarding some of the variables into triangular or trapezoidal fuzzy sets. In this paper, three approaches for converting probabilistic information, represented by a probability distribution, into an equivalent triangular or trapezoidal fuzzy set are discussed. In all the three approaches, the probability distribution is first converted into a probabilistic fuzzy set, which is then converted into the equivalent triangular or trapezoidal fuzzy set. The first approach is based on the method of least-square curve fitting, the second approach is based on the conservation of uncertainty (represented by the entropy) associated with the probabilistic fuzzy set in a mean square sense, and the third approach is based on the minimisation of Hausdorff distance (HD) between the probabilistic and the equivalent fuzzy sets. The effectiveness of these approaches in preserving the entropy as well as in preserving the elements of the fuzzy set and their corresponding grades of membership are also discussed with the help of a numerical example of obtaining equivalent fuzzy set for peak ground acceleration. It is found that the approach based on minimisation of Hausdorff distance provides a simple and efficient way for converting the probabilistic information into an equivalent fuzzy set.     

Some remarks on the lattice of fuzzy intervals

In this paper we study the connections between three related concepts which have appeared in the fuzzy literature: fuzzy intervals, fuzzy numbers and fuzzy interval numbers (FIN’s). We show that these three concepts are very closely related. We propose a new definition which encompasses the three previous ones and proceeds to study the properties ensuing from this definition. Given a reference lattice (X, ?), we define fuzzy intervals to be the fuzzy sets such that their p-cuts are closed intervals of (X, ?). We show that, given a complete lattice (X, ?), the collection of its fuzzy intervals is a complete lattice. Furthermore we show that, if (X, ?) is completely distributive, then the lattice of its fuzzy intervals is distributive. Finally we introduce a new inclusion measure, which can be used to quantify the degree in which a fuzzy interval is contained in another, an approach which is particularly valuable in engineering applications.     

Note on the relationship between probabilistic and fuzzy clustering

In this short communication, based on Renyi entropy measure, a new Renyi information based clustering algorithm A is presented. Algorithm A and the well-known fuzzy clustering algorithm FCM have the same clustering track. This fact builds the very bridge between probabilistic clustering and fuzzy clustering, and fruitful research results on Renyi entropy measure may help us to further understand the essence of fuzzy clustering.This work was supported in part by the RGC CERG grant under project HongKong PolyU 5065/98E