Skip to main content

Advertisement

Log in

An improved belief \(\chi ^2\) divergence for Dempster–Shafer theory and its applications in pattern recognition

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

Dempster–Shafer (D–S) evidence theory is widely used in the field of uncertain information processing, but because of its defective fusion rules, it often results in visual violations when dealing with highly conflicting evidence. To solve the conflict management problem of D–S theory, an improved belief \(\chi 2\) divergence (called \({\mathcal {I}}{\mathcal {B}}\chi ^2\)) is proposed in this paper. The \({\mathcal {I}}{\mathcal {B}}\chi ^2\) divergence takes into account the amount of all possible hypotheses, which allow it be a more credible and efficient solution to measure the dissimilarity between evidences. Moreover, it has good mathematical properties including symmetry, boundedness and non-degeneracy. Next, we designed a novel multi-resource information fusion algorithm based on \({\mathcal {I}}{\mathcal {B}}\chi ^2\) divergence. Finally, application in automobile system fault recognition and iris feature recognition prove the effectiveness and accuracy of the proposed multi-resource information fusion algorithm.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

Notes

  1. A Generalized \(\chi \)2 Divergence for multisource Information Fusion.

References

  • Babajanyan S, Allahverdyan A, Cheong KH (2020) Energy and entropy: path from game theory to statistical mechanics. Phys Rev Res 2(4):043055

    Article  Google Scholar 

  • Cao Z, Chuang C-H, King J-K, Lin C-T (2019) Multi-channel EEG recordings during a sustained-attention driving task. Sci Data. https://doi.org/10.1038/s41597-019-0027-4

    Article  Google Scholar 

  • Chang L, Zhang L, Fu C, Chen Y-W (2021) Transparent digital twin for output control using belief rule base. IEEE Trans Cybern. https://doi.org/10.1109/TCYB.2021.3063285

    Article  Google Scholar 

  • Chen L, Deng Y, Cheong KH (2021) Probability transformation of mass function: a weighted network method based on the ordered visibility graph. Eng Appl Artif Intell 105:104438

    Article  Google Scholar 

  • Cheng C, Xiao F (2021) A distance for belief functions of orderable set. Pattern Recogn Lett 145:165–170

    Article  Google Scholar 

  • Cui H, Zhou L, Li Y, Kang B (2022) Belief entropy-of-entropy and its application in the cardiac interbeat interval time series analysis. Chaos Solitons Fractals 20:20

    MathSciNet  Google Scholar 

  • Dempster AP (1967) Upper and lower probabilities induced by a multivalued mapping. Ann Math Stat 38(2):325–339

    Article  MathSciNet  MATH  Google Scholar 

  • Deng Y (2020) Uncertainty measure in evidence theory. Sci China Inf Sci 63(11):210201

    Article  MathSciNet  Google Scholar 

  • Deng Y (2020) Information volume of mass function. Int J Comput Commun Control 15(6):3983

    Article  Google Scholar 

  • Deng X, Deng Y (2019) D-AHP method with different credibility of information. Soft Comput 23(2):683–691

    Article  Google Scholar 

  • Deng J, Deng Y (2021) Information volume of fuzzy membership function. Int J Comput Commun Control 16(1):4106

    Article  Google Scholar 

  • Deng Y, Shi W, Zhu Z, Liu Q (2004) Combining belief functions based on distance of evidence. Decis Support Syst 38(3):489–493

    Article  Google Scholar 

  • Ejegwa PA, Agbetayo JM (2022) Similarity-distance decision-making technique and its applications via intuitionistic fuzzy pairs. J Comput Cogn Eng 20:20

    Google Scholar 

  • Eymard R, Guichard C (2018) Discontinuous galerkin gradient discretisations for the approximation of second-order differential operators in divergence form. Comput Appl Math 37(4):4023–4054

    Article  MathSciNet  MATH  Google Scholar 

  • Fei L, Feng Y (2021) Intuitionistic fuzzy decision-making in the framework of Dempster–Shafer structures. Int J Intell Syst 36(10):5419–5448

    Article  Google Scholar 

  • Fu C, Chang W, Yang S (2020) Multiple criteria group decision making based on group satisfaction. Inf Sci 518:309–329

    Article  MathSciNet  MATH  Google Scholar 

  • Fujita H, Ko Y-C (2020) A heuristic representation learning based on evidential memberships: case study of UCI-SPECTF. Int J Approx Reason 20:120

    MathSciNet  MATH  Google Scholar 

  • Fujita H, Gaeta A, Loia V, Orciuoli F (2020) Hypotheses analysis and assessment in counter-terrorism activities: a method based on OWA and fuzzy probabilistic rough sets. IEEE Trans Fuzzy Syst 28:831–845

    Article  Google Scholar 

  • Gao X, Deng Y (2020) The pseudo-pascal triangle of maximum Deng entropy. Int J Comput Commun Control 15(1):1006

    Article  MathSciNet  Google Scholar 

  • Gao X, Pan L, Deng Y (2021) A generalized divergence of information volume and its applications. Eng Appl Artif Intell 108:104584. https://doi.org/10.1016/j.engappai.2021.104584

    Article  Google Scholar 

  • Gao X, Su X, Qian H, Pan X (2021) Dependence assessment in human reliability analysis under uncertain and dynamic situations. Nucl Eng Technol. https://doi.org/10.1016/j.net.2021.09.045

    Article  Google Scholar 

  • Gao Q, Wen T, Deng Y (2021) Information volume fractal dimension. Fractals. https://doi.org/10.1142/S0218348X21502637

    Article  MATH  Google Scholar 

  • Garg H (2021) Cn-q-rofs: connection number-based q-rung orthopair fuzzy set and their application to decision-making process. Int J Intell Syst 36(7):3106–3143

    Article  Google Scholar 

  • Garg H (2022) Svnmpr: a new single-valued neutrosophic multiplicative preference relation and their application to decision-making process. Int J Intell Syst 37(3):2089–2130

    Article  Google Scholar 

  • Garg H, Kaur G (2022) Algorithm for solving the decision-making problems based on correlation coefficients under cubic intuitionistic fuzzy information: a case study in watershed hydrological system. Complex Intell Syst 8(1):179–198

    Article  Google Scholar 

  • Garg H, Rani D (2021) Novel exponential divergence measure of complex intuitionistic fuzzy sets with an application to the decision-making process. Sci Iran 28(4):2439–2456

    Google Scholar 

  • Garg H, Rani D (2022) An efficient intuitionistic fuzzy multimoora approach based on novel aggregation operators for the assessment of solid waste management techniques. Appl Intell 52(4):4330–4363

    Article  Google Scholar 

  • Garon EM, Lambers JV (2018) Modeling the diffusion of heat energy within composites of homogeneous materials using the uncertainty principle. Comput Appl Math 37(3):2566–2587

    Article  MathSciNet  MATH  Google Scholar 

  • Han D, Dezert J, Yang Y (2016) Belief interval-based distance measures in the theory of belief functions. IEEE Trans Syst Man Cybern Syst 48(6):833–850

    Article  Google Scholar 

  • Jiang W, Wei B, Xie C, Zhou D (2016) An evidential sensor fusion method in fault diagnosis. Adv Mech Eng 8(3):1687814016641820

    Article  Google Scholar 

  • Jiang W, Cao Y, Deng X (2020) A novel Z-network model based on Bayesian network and Z-number. IEEE Trans Fuzzy Syst 28(8):1585–1599. https://doi.org/10.1109/TFUZZ.2019.2918999

    Article  Google Scholar 

  • Joshi R, Kumar S (2018) An (R, S)-norm fuzzy information measure with its applications in multiple-attribute decision-making. Comput Appl Math 37(3):2943–2964

    Article  MathSciNet  MATH  Google Scholar 

  • Khan F, Shakeel M, Abdullah S (2019) Ranking methodology of irrigation problems based on pythagorean trapezoidal fuzzy aggregations operators. Comput Appl Math 38(3):1–20

    Article  MathSciNet  MATH  Google Scholar 

  • Lai JW, Chang J, Ang L, Cheong KH (2020) Multi-level information fusion to alleviate network congestion. Inf Fusion 63:248–255

    Article  Google Scholar 

  • Liu Z, Liu Y, Dezert J, Cuzzolin F (2020) Evidence combination based on credal belief redistribution for pattern classification. IEEE Trans Fuzzy Syst 28(4):618–631

    Article  Google Scholar 

  • Liu P, Shen M, Teng F, Zhu B, Rong L, Geng Y (2021) Double hierarchy hesitant fuzzy linguistic entropy-based TODIM approach using evidential theory. Inf Sci 547:223–243

    Article  Google Scholar 

  • Meng D, Lv Z, Yang S, Wang H, Xie T, Wang Z (2021) A time-varying mechanical structure reliability analysis method based on performance degradation. Structures, vol 34. Elsevier, New York, pp 3247–3256

    Google Scholar 

  • Meng D, Wang H, Yang S, Lv Z, Hu Z, Wang Z (2022) Fault analysis of wind power rolling bearing based on EMD feature extraction. Comput Model Eng Sci 130(1):543–558

    Google Scholar 

  • Murphy CK (2000) Combining belief functions when evidence conflicts. Decis Support Syst 29(1):1–9

    Article  Google Scholar 

  • Ni L, Chen Y-W, de Brujin O (2021) Towards understanding socially influenced vaccination decision making: an integrated model of multiple criteria belief modelling and social network analysis. Eur J Oper Res 293(1):276–289

    Article  MathSciNet  MATH  Google Scholar 

  • Qian J, Guo X, Deng Y (2017) A novel method for combining conflicting evidences based on information entropy. Appl Intell 46(4):876–888

    Article  Google Scholar 

  • Riaz M, Tehrim ST (2019) Cubic bipolar fuzzy ordered weighted geometric aggregation operators and their application using internal and external cubic bipolar fuzzy data. Comput Appl Math 38(2):1–25

    Article  MathSciNet  MATH  Google Scholar 

  • Saeed M, Ahmad MR, Rahman AU (2022) Refined pythagorean fuzzy sets: properties, set-theoretic operations and axiomatic results. J Comput Cogn Eng 20:20

    Google Scholar 

  • Singh P (2017) Distance and similarity measures for multiple-attribute decision making with dual hesitant fuzzy sets. Comput Appl Math 36(1):111–126

    Article  MathSciNet  MATH  Google Scholar 

  • Song Y, Deng Y (2021) Entropic explanation of power set. Int J Comput Commun Control 16(4):4413

    Article  Google Scholar 

  • Song X, Xiao F (2022) Combining time-series evidence: a complex network model based on a visibility graph and belief entropy. Appl Intell. https://doi.org/10.1007/s10489-021-02956-5

    Article  Google Scholar 

  • Song Y, Fu Q, Wang Y-F, Wang X (2019) Divergence-based cross entropy and uncertainty measures of Atanassov’s intuitionistic fuzzy sets with their application in decision making. Appl Soft Comput 84:105703

    Article  Google Scholar 

  • Tang S-W, Zhou Z-J, Hu C-H, Yang J-B, Cao Y (2021) Perturbation analysis of evidential reasoning rule. IEEE Trans Syst Man Cybern Syst 51(8):4895–4910

    Article  Google Scholar 

  • Tao R, Liu Z, Cai R, Cheong KH (2021) A dynamic group MCDM model with intuitionistic fuzzy set: perspective of alternative queuing method. Inf Sci 555:85–103

    Article  MathSciNet  MATH  Google Scholar 

  • Wang Z, Xiao F (2019) An improved multi-source data fusion method based on the belief entropy and divergence measure. Entropy 21(6):611

    Article  MathSciNet  Google Scholar 

  • Wang Z, Wang C, Li X, Gao C, Li X, Zhu J (2020) Evolutionary Markov dynamics for network community detection. IEEE Trans Knowl Data Eng. https://doi.org/10.1109/TKDE.2020.2997043

    Article  Google Scholar 

  • Wang Z, Wang C, Gao C, Li X, Li X (2020) An evolutionary autoencoder for dynamic community detection. Sci China Inf Sci 63(11):1–16

    Article  MathSciNet  Google Scholar 

  • Wang Z, Li Z, Wang R, Nie F, Li X (2021) Large graph clustering with simultaneous spectral embedding and discretization. IEEE Trans Pattern Anal Mach Intell 43(12):4426–4440. https://doi.org/10.1109/TPAMI.2020.3002587

    Article  Google Scholar 

  • Wang Z, Dai X, Zhu P, Wang R, Li X, Nie F (2021) Fast optimization of spectral embedding and improved spectral rotation. IEEE Trans Knowl Data Eng. https://doi.org/10.1109/TKDE.2021.3098806

    Article  Google Scholar 

  • Wang Z, Xiao F, Ding W (2022) Interval-valued intuitionistic fuzzy Jenson–Shannon divergence and its application in multi-attribute decision making. Appl Intell. https://doi.org/10.1007/s10489-022-03347-0

    Article  Google Scholar 

  • Wen T, Cheong KH (2021) The fractal dimension of complex networks: a review. Inf Fusion 73:87–102

    Article  Google Scholar 

  • Wu Z, Liao H (2021) A consensus reaching process for large-scale group decision making with heterogeneous preference information. Int J Intell Syst. https://doi.org/10.1002/int.22469

    Article  Google Scholar 

  • Xiao F (2019) Multi-sensor data fusion based on the belief divergence measure of evidences and the belief entropy. Inf Fusion 46:23–32

    Article  Google Scholar 

  • Xiao F, Pedrycz W (2022) Negation of the quantum mass function for multisource quantum information fusion with its application to pattern classification. IEEE Trans Pattern Anal Mach Intell. https://doi.org/10.1109/TPAMI.2022.3167045

    Article  Google Scholar 

  • Xiao F, Wen J, Pedrycz W (2022) Generalized divergence-based decision making method with an application to pattern classification. IEEE Trans Knowl Data Eng. https://doi.org/10.1109/TKDE.2022.3177896

    Article  Google Scholar 

  • Xie D, Xiao F, Pedrycz W (2021) Information quality for intuitionistic fuzzy values with its application in decision making. Eng Appl Artif Intell. https://doi.org/10.1016/j.engappai.2021.104568

    Article  Google Scholar 

  • Xiong L, Su X, Qian H (2021) Conflicting evidence combination from the perspective of networks. Inf Sci 580:408–418

    Article  MathSciNet  Google Scholar 

  • Yager RR (2019) Generalized Dempster–Shafer structures. IEEE Trans Fuzzy Syst 27(3):428–435

    Article  MathSciNet  Google Scholar 

  • Yager RR (2020) Using fuzzy measures for modeling human perception of uncertainty in artificial intelligence. Eng Appl Artif Intell 87:103228

    Article  Google Scholar 

  • Ye J, Zhan J, Ding W, Fujita H (2021) A novel fuzzy rough set model with fuzzy neighborhood operators. Inf Sci 544:266–297

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang L, Xiao F (2022) A novel belief \(\chi \)2 divergence for multisource information fusion and its application in pattern classification. Int J Intell Syst. https://doi.org/10.1002/int.22912

    Article  Google Scholar 

  • Zhang Z, Xu X, Chen P, Wu X, Xu X, Wang G (2021) A novel nonlinear causal inference approach using vector-based belief rule base. Int J Intell Syst. https://doi.org/10.1002/int.22500

    Article  Google Scholar 

  • Zhou Q, Mo H, Deng Y (2020) A new divergence measure of pythagorean fuzzy sets based on belief function and its application in medical diagnosis. Mathematics 8:1. https://doi.org/10.3390/math8010142

    Article  Google Scholar 

  • Zhou M, Chen Y-W, Liu X-B, Cheng B-Y, Yang J-B (2020) Weight assignment method for multiple attribute decision making with dissimilarity and conflict of belief distributions. Comput Ind Eng 147:106648

    Article  Google Scholar 

  • Zhu C, Xiao F, Cao Z (2022) A generalized Rényi divergence for multi-source information fusion with its application in EEG data analysis. Inf Sci. https://doi.org/10.1016/j.ins.2022.05.012

    Article  Google Scholar 

Download references

Acknowledgements

The authors greatly appreciate the reviewers’ suggestions and the editor’s encouragement. This research is supported by the National Natural Science Foundation of China (No. 62003280), Chongqing Talents: Exceptional Young Talents Project (cstc2022ycjh-bgzxm0070), and Chongqing Overseas Scholars Innovation Program (No. cx2022024).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Fuyuan Xiao.

Additional information

Communicated by Graçaliz Pereira Dimuro.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This paper is an extension of conference paper A Generalized \(\chi ^2\) Divergence for Multisource Information Fusion

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gao, X., Xiao, F. An improved belief \(\chi ^2\) divergence for Dempster–Shafer theory and its applications in pattern recognition. Comp. Appl. Math. 41, 277 (2022). https://doi.org/10.1007/s40314-022-01975-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-022-01975-3

Keywords

Mathematics Subject Classification

Navigation