Proposed Combination Rule Research Articles

A novel evidence combination rule based on compromise conflict indicator and conflict focal element

The Dempster–Shafer evidence theory (DSET) has been commonly used in the fusion of multisource information in various fields. Owing to the normalization process based on the conflict measurement, the conclusion when fusing high conflict evidence is counter-intuitive when using Dempster’s Combination Rule (DCR). To solve the DCR issue, several combination rules have been developed. However, each of them has unreasonable behaviors in certain cases. In this study, the set of mathematical properties that must be satisfied by all the combinatorial rules is further refined. Under the framework of DSET, a novel combination rule is proposed by comprehensively considering the conflict focal elements and metrics. The conflict measurement is achieved through the newly proposed compromise conflict indicator. The advantages of the proposed combination rule over the existing methods are demonstrated by experiments on a few numerical examples and applications.

An evidence combination rule based on a new weight assignment scheme

In evidence theory, conflicting evidence and fuzzy evidence have a significant impact on the results of evidence combination. Nevertheless, the existing weight assignment methods can hardly reflect the significant influence of fuzzy evidence on the combination results. In this paper, we address this issue by proposing a new method for assigning evidence weights and the corresponding combination rule. The proposed weight assignment method strengthens the consideration of fuzzy evidence. We further incorporate the Wasserstein distance to compute the clarity degree of the evidence. This is an important reference index for the weight assignment in the proposed combination rule and effectively reduces the impact of ambiguous evidence. Using experiments, we illustrate the significant impact of fuzzy evidence on the results of combination. This justifies its integration in the weight assignment process. The proposed combination rule with the new weight assignment method is also examined on a set of numerical arithmetic and Iris datasets. Our results confirm that compared with the four existing methods, the proposed method improves decision accuracy, F1 score, computational convergence and achieves more reliable fusion results.

Multicomponent directional‐cum‐modal combination rule for seismic analysis: Theory and illustrations

AbstractSince spectral representation of ground motion can capture the epistemic uncertainty and aleatory variability better than time series representation, direction‐cum‐modal combination rule plays the pivotal rule when incorporating the multicomponent excitations in seismic design and in assessing the seismic resilience. CQC is generally believed to be the best modal combination rule for unidirectional seismic excitation, but the directional combination rule is still at its infancy when six‐component seismic excitations are considered. Though a few extensions have been reported to account for more than one component, six‐component directional combination rule is still an open call. Directional‐cum‐modal combination rule is developed in this paper from the first principle to combine the modal responses of a structure subjected to six‐component ground motion with spectral representation as the 5% damped acceleration response spectra. In addition, the framework requires the power spectral shape of each component of excitations, directional correlation function for each pair of excitations, and duration of the event. With linear time history analysis as the benchmark, the proposed combination rule is compared against CQC as the modal combination rule with SRSS for the directional combination. Two numerical examples are presented including a 30‐storied building to emphasize the role of rotational (rocking) components. The proposed combination rule works much better if the recorded components of ground motion are used for structural analysis. However, the difference reduces if the recorded components of ground motion are rotated along‐and‐normal to the principal plane prior to analysis. This is verified against several suites of recorded ground motion.

How to aggregate uncertain and incomplete cognitive evaluation information in lung cancer treatment plan selection? A method based on Dempster-Shafer theory

Multidisciplinary team is beneficial to select an appropriate treatment plan for a patient with lung cancer, where the collected cognitive evaluation information may be uncertain and incomplete. This study dedicates to dealing with a treatment plan selection problem of lung cancer through the multi-criteria analysis with generalised probabilistic linguistic term sets (GPLTSs) which are powerful in describing the uncertainty and incompleteness of subjective evaluations. The existing generalised probabilistic linguistic information aggregation method is based on the Dempster-Shafer combination rule, but the combined results may be counterintuitive. In addition, the GPLTS may not meet the conditions of applying the Dempster-Shafer combination rule. To make up for these gaps, a new combination rule based on Dempster-Shafer theory is introduced. Then, a multi-criteria decision-making (MCDM) process with generalised probabilistic linguistic information based on the proposed combination rule is formed and applied to select the treatment plans of lung cancer associated with a multidisciplinary team. Through the sensitivity analysis and comparative analysis, the advantages of the proposed method are highlighted.

Combining conflicting evidence based on Pearson correlation coefficient and weighted graph

Dempster–Shafer evidence theory (evidence theory) has been widely used as an efficient method for dealing with uncertainty. In evidence theory, Dempster's rule is the most well-known evidence combination method but it does not work well when the evidence is in high conflict. To improve the performance of combining conflicting evidence, an original and novel evidence combination method is presented based on the Pearson correlation coefficient and weighted graph. The proposed method can correctly recognize the alternative situation with a high accuracy. Besides, the convergence performance of this method is better when compared with other combination rules. In addition, the weighted graph generated by the proposed method can directly represent the relationship between different evidence, which can help researchers estimate the reliability of different body of evidence. Our experimental results indicate the advantages of our proposed evidence combination rule over existing methods, and the results are analyzed and discussed.

A recursive ensemble-based feature selection for multi-output models to discover patterns among the soil nutrients

Finding the reduced and the relevant subsets of the predictors is inevitable when it comes to predictive modelling. If the datasets involved are heterogeneous and heteroscedastic in nature such as the soil samples, the task turns out to be trickier and such scenarios demand ensemble-based feature selection approach. The proposed algorithm uses an ensemble of filter (RReliefF), wrapper (Adaptive Plus –l and Minus –r), and embedded (Neighbourhood Component Analysis) approaches and applies it to the datasets in a heterogeneous and homogeneous manner. The Adaptive Plus –l and Minus –r is an experiment done on the Plus –l and Minus –r wrapper method to enhance the performance of the algorithm. The proposed combination rule of the ensemble filters out the irrelevant predictors for each response variable. Further, this ensemble is recursively implemented using the floating set of predictors to estimate the optimal subsets for multiple response variables at one go. Each step in recursion ensures that only the best subset of features (in terms of length and weights) among the current and previous iterations is retained. The Akaikes Information Criteria and the predictors weights calculated, assessed the efficiency of the resultant predictor sets of the recursive ensemble. The adjusted R2, Median Absolute Deviation and Root Mean Square Error on the unseen datasets confirmed the suitability of the same for predictive modelling in ecological domains.

A Modified Approach to Conflict Management From the Perspective of Non-Conflicting Element Set

In Dempster-Shafer (D-S) evidence theory, how to deal with conflict is an important and open topic. Two key strategies for resolving conflicts of evidence are considered, namely information averaging and information focus. How to balance this relationship is still a question worth considering. Recently, Ma et al. studied evidence conflicts from the perspective of complete conflict set and proposed a flexible rule for conflict evidence combination. The proposed combination rule seems to take into account the above two strategies. However, through analysis, we find that Ma et al.'s method tends to use the average method to solve conflicting propositions, while Dempster's combined rule has a weak focusing function. In this paper, based on the concept of non-conflict element set, a new conflict handling method is proposed. First, the similarity between the evidences is characterized by the correlation coefficient; based on this, a new weighting scheme of evidence is developed. In addition, the propositional support is reasonably allocated through discounts.Through numerical examples, the applicability and superiority of the method are compared and analyzed. The results show that the proposed method takes two strategies of averaging and focusing into consideration, and the information variance is small.

Analysis of “mixing” combination rules and Smet’s combination rule

The process of solution of different practical and ecological problems, using hyperspectral satellite images usually includes a procedure of classification. Classification is one of the most difficult and important procedures. Some image classification methods were considered and analyzed in this work. These methods are based on the theory of evidence. Evidence theory can simulate uncertainty and process imprecise and incomplete information. It were considered such combination rules in this paper: “mixing” combination rule (or averaging), convolutive x-averaging (or c-averaging) and Smet’s combination rule. It was shown, that these methods can process the data from multiple sources or spectral bands, that provide different assessments for the same hypotheses. It was noted, that the purpose of aggregation of information is to simplify data, whether the data is coming from multiple sources or different spectral bands. It was shown, that Smet’s rule is unnormalized version of Dempster rule, that applied in Smet’s Transferable Belief Model. It also processes imprecise and incomplete data. Smet’s combination rule entails a slightly different formulation of Dempster-Shafer theory. Mixing (or averaging) rule was considered in this paper too. It is the averaging operation that is used for probability distributions. This rule uses basic probability assignments from different sources (spectral bands) and weighs assigned according to the reliability of the sources. Convolutive x-averaging (or c-averaging) rule was considered in this paper too. This combination rule is a generalization of the average for scalar numbers. This rule is commutative and not associative. It also was noted, that convolutive x-averaging (c-averaging) rule can include any number of basic probability assignments. It were also considered examples, where these proposed combination rules were used. Mixing, convolutive x-averaging (c-averaging) rule and Smet’s combination rule can be applied for analysis of hyperspectral satellite images, in remote searching for minerals and oil, solving different environmental and thematic problems.

Diffusion LMS Based on Message Passing Algorithm

Diffusion least-mean-square (LMS) is an adaptive algorithm that estimates an unknown global vector from its linear measurements obtained at all nodes in a distributed manner when each node in the network needs to track the unknown vector in real-time. The algorithm uses the conventional average consensus protocol in order to combine neighbors' estimates at each node, while another protocol, consensus propagation (CP), is known to achieve faster and exact average consensus when the network has a tree structure. This paper proposes a novel diffusion LMS algorithm using CP, which can be applied for any network by extracting a spanning tree from the original network and can achieve the same solution as the centralized LMS in a fully distributed manner. This paper also proposes an algorithm by using the idea of loopy CP, so that it can be directly applied even when the network is not a tree and shows that its special case results in the diffusion LMS using a novel combination rule. Moreover, we optimize the constants involved in the proposed combination rule in terms of the steady-state mean-square-deviation of the diffusion LMS and show an adaptive implementation of the proposed algorithm. The simulation results demonstrate that the proposed algorithm using CP is beneficial for large-scale networks, and the diffusion LMS with the proposed combination rule achieves better convergence performance than that with the conventional combination rules when the measurement noise power depends on nodes.

Linear Soft Combination for Cooperative Spectrum Sensing in Cognitive Radio Networks

In this letter, we study the cooperative energy-based combination for spectrum sensing in cognitive radio networks. Linear combination rule is investigated to minimize the probability of missed detection subject to an upper limit on the probability of false alarm. We propose a novel and practical linear combination rule, which only requires the mean and variance of the local test statistics. The performance of the proposed combination rule is verified in three typical fading scenarios: the slow fading, the block fading, and the fast fading. Simulations show that the proposed strategy has nearly the same performance as the optimal likelihood-ratio test approaches, and outperforms the conventional linear combination methods.

Evidence combination rule with contrary support in the evidential reasoning approach

An improved combination rule for evidence in the evidential reasoning approach is proposed.Combination algorithm for solving MCDM problems is established.Contrary support is used to construct the rule by bounding mutual influence.The rule separates contrary support from global ignorance to hold evidence specificity.Conjunctive reasoning process is constituted so that results can be easily explained. Three aspects of problems such as reasonable weight constraint, cumulative weight effect and relative weight equivalence cannot be well reflected in the evidential reasoning (ER) approach. In order to solve the above problems, a contrary support is defined to restrict the degree influenced by the evidence to be combined in combination process. Then pair-weighted and cumulative pair-weighted discounting methods are presented to generate basic probability assignment (BPA) for evidence. Pair-wised and recursive combination rules are established to make combination with the BPA of evidence by orthogonal sum operation and several theorems such as relative weight equivalence are proved. A combination algorithm is proposed to solve multiple criteria decision making (MCDM) problems by integrating pignistic probability and expected utility with the established combination rules. Finally, an illustrative example is provided to demonstrate the applicability of the proposed combination rules and algorithm.

An Improved Dempster-Shafer Algorithm Using a Partial Conflict Measurement

Multiple evidences based decision making is an important functionality for computers and robots. To combine multiple evidences, mathematical theory of evidence has been developed, and it involves the most vital part called Dempster’s rule of combination. The rule is used for combining multiple evidences. However, the combined result gives a counterintuitive conclusion when highly conflicting evidences exist. In particular, when we obtain two different sources of evidence for a single hypothesis, only one of the sources may contain evidence. In this paper, we introduce a modified combination rule based on the partial conflict measurement by using an absolute difference between two evidences’ basic probability numbers. The basic probability number is described in details in Section 2 “Mathematical Theory of Evidence”. As a result, the proposed combination rule outperforms Dempster’s rule of combination. More precisely, the modified combination rule provides a reasonable conclusion when combining highly conflicting evidences and shows similar results with Dempster’s rule of combination in the case of the both sources of evidence are not conflicting. In addition, when obtained evidences contain multiple hypotheses, our proposed combination rule shows more logically acceptable results in compared with the results of Dempster’s rule.

Evidence combination based on credibility and non-specificity

This paper addresses the combination of unreliable evidence sources which provide uncertain information in the form of basic probability assignment (BPA) functions. We proposed a novel evidence combination rule based on credibility and non-specificity of belief functions. Following a review of all existing non-specificity measures in evidence theory, a non-specificity measure for evidence theory is discussed. It is claimed that the non-specificity degree of a BPA is related to its ability of pointing to one and only one element. Based on the difference between the largest belief grades and other belief grades, a non-specificity measure is defined. Properties of the proposed non-specificity measure are put forward and proved mathematically. Illustrative examples are employed to show the properties of non-specificity measure. After providing a procedure for the evaluation of evidence credibility, we propose a novel evidence combination rule. Numerical example and application in target identification are applied to demonstrate the performance of our proposed evidence combination rule.

A Modified Combination Rule for D Numbers Theory

D numbers theory is an appropriate method to deal with the information of uncertainty and incompleteness when making a reasonable decision. Previous D numbers theory provides a rule to combine multiple D numbers. However, the commutative law is not satisfied in the rule of combining multiple D numbers. In this paper, a modified method for multiple D numbers combination is proposed. The proposed method defines a new function for multiple D numbers combination which is mainly determined by the original value of D numbers. Then the proposed combination rule is applied to environmental impact assessment (EIA); our results show that the proposed method is efficient for multiple D numbers combination and it is useful when dealing with uncertainty and incompleteness.

A new adaptive decentralized soft decision combining rule for distributed sensor systems with data fusion

A new adaptive decentralized soft decision combining rule for multiple-sensor distributed detection systems with data fusion is proposed. Unlike previously published rules, the proposed combining rule fuses soft decisions of sensors rather than hard decisions of sensors and does not require the knowledge of the false alarm and detection probabilities of the distributed sensors. Such a fusion rule is adaptive, insensitive to the instabilities of the sensor thresholds, and has the advantage of soft decision fusion. The proposed combination rule is derived: (1) for the case where the fusion center estimates the error probabilities of the sensors and (2) for the case where the fusion center does not estimate the error probabilities of the sensors. The performance of the proposed approach is evaluated, and illustrative examples are presented in the cases of Gaussian and Rayleigh distributed observations. Comparisons with the optimum centralized fusion, the optimum soft decision fusion, a soft decision fusion approach based on fusing confidence levels, and the optimum decentralized hard decision fusion are also presented. The results indicate that the proposed approach significantly outperforms the optimum decentralized hard decision fusion, is better than the approach based on fusing confidence levels, and has a performance similar to that of the optimum soft decision fusion.

A new DSmT combination rule in open frame of discernment and its application

A new combination rule based on Dezert-Smarandache theory (DSmT) is proposed to deal with the conflict evidence resulting from the non-exhaustivity of the discernment frame. A two-dimensional measure factor in Dempster-Shafer theory (DST) is extended to DSmT to judge the conflict degree between evidence. The original DSmT combination rule or new DSmT combination rule can be selected for fusion according to this degree. Finally, some examples in simultaneous fault diagnosis of motor rotor are given to illustrate the effectiveness of the proposed combination rule.

Classifier combination for sketch-based 3D part retrieval

In this paper, we present a search method with multi-class probability estimates for sketch-based 3D engineering part retrieval. The purpose of using probabilistic output from classification is to support high-quality part retrieval by motivating user relevance feedback from a ranked list of top categorical choices. Given a free-hand user sketch, we use an ensemble of classifiers to estimate the likelihood of the sketch belonging to each predefined category by exploring the strengths of various individual classifiers. Complementary shape descriptors are used to generate classifiers with probabilistic output using support vector machines (SVM). A weighted linear combination rule, called adapted minimum classification error (AMCE), is developed to concurrently minimize the classification errors and the log likelihood errors. Experiments are conducted using our Engineering Shape Benchmark database to evaluate the proposed combination rule. User studies show that users can easily identify the desired classes and then the parts under the proposed method and algorithms. Compared with the best individual classifier, the classification accuracy using AMCE increased by 7% for 3D models, and the average best rank improved by 11.6% for sketches.

A response spectrum method for seismic response analysis of structures under multi-support excitations

Based on the random vibration theory, a response spectrum method is developed for seismic response analysis of linear, multi-degree-of-freedom structures under multi-support excitations is developed. Various response quantities, including the mean and variance of the peak response, the response mean frequency, are obtained from proposed combination rules in terms of the mean response spectrum. This method makes it possible to apply the response spectrum to the seismic reliability analysis of structures subjected to multi-support excitations. Considering that the tedious numerical integration is required to compute the spectral parameters and correlation coefficients in above combination rules, this paper further offers simplified procedures for their computation, which enhance dramatically the computational efficiency of the suggested method. The proposed procedure is demonstrated for tow numerical examples: (1) two-span continuous beam; (2) two-tower cabled-stayed bridge by using Monte Carlo simulation (MC). For this purpose, this paper also presents an approach to simulation of ground motions, which can take into account both mean and variation properties of response spectrum. Computed results based on the response spectrum method are in good agreement with Monte Carlo simulation results. And compared with the MSRS method, a well-developed multi-support response spectrum method, the proposed method has an incomparable computational efficiency.

Alternate equation of state combining rules and interaction parameter generalizations for asymmetric mixtures

Following the work of Juris and Wenzel, an alternate combining rule is proposed for cubic equations of state (CEOS). A wide variety of interactions between unlike molecules can be represented effectively by this combining rule. The Soave–Redlich–Kwong (SRK) and Peng–Robinson (PR) equations of state (EOS) have been used to assess the usefulness of the alternate combining rule in describing the types of unlike interactions encountered in asymmetric mixtures. In addition, a study was undertaken to evaluate the predictive capability of both equations of state in representing vapor–liquid equilibrium (VLE) of asymmetric binary mixtures, involving methane, ethane, nitrogen, hydrogen, carbon monoxide and carbon dioxide in n-paraffins (C 4–C 44). EOS binary interaction parameters generated by the proposed combination rules are presented for the systems considered. The quality of the EOS representation is dependent on the level of complexity applied in the parameter regressions. Overall, average absolute deviations of 1–3% are realized from the various regression scenarios. In addition, generalized EOS parameter correlations for system-dependent parameters have been developed. These generalized interaction parameters represent the solubilities of the selected systems within 5%.

Study of interval belief combination

In this paper, a new mathematical procedure is presented for combining different pieces of evidence which are represented in the interval form to reflect our knowledge about the truth of a hypothesis. Evidence may be correlated to each other (dependent evidence) or conflicting in support (conflicting evidence). First, assuming independent evidence, we propose a methodology to construct combination rules which obey a set of essential properties. The method is based on a geometric model. We compare results obtained from the Dempster—Shafer rule, interval Bayes rule, and the proposed combination rules with both conflicting and nonconflicting data and show that the values generated by the proposed combining rules are in tune with our intuition in both cases. Secondly, in the case that evidence is known to be dependent, we consider extensions of the rules derived for handling conflicting evidence. The performance of proposed rules are shown by different examples. The results show that the proposed rules reasonably make decisions under dependent evidence.