Skip to main content
Springer Nature Link
Log in

On Assessments of Agreement Between Fuzzy Partitions

  • Published:
Journal of Classification Aims and scope Submit manuscript

Abstract

We extend the literature regarding assessments of agreement between soft/fuzzy/probabilistic cluster allocations by providing closed-form approaches for two measures which behave as fuzzy generalizations of the popular adjusted Rand index (ARI): one novel and one previously requiring a Monte Carlo estimation process. Both of these measures retain the reflexive property of the ARI—an arguably essential property for the interpretability of a cluster agreement measure—and both are feasible in their closed-form for sample sizes ranging into five digits or more using standard consumer computers. We describe the approximate computational complexity in each case, and apply both measures in simulated and real data contexts.

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

Access this article

Log in via an institution

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.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  • Albatineh, A.N., Niewiadomska-Bugaj, M., & Mihalko, D. (2006). On similarity indices and correction for chance agreement. Journal of Classification, 23 (2), 301–313.

    Article  MathSciNet  Google Scholar 

  • Anderson, D.T., Bezdek, J.C., Popescu, M., & Keller, J.M. (2010). Comparing fuzzy, probabilistic, and possibilistic partitions. IEEE Transactions on Fuzzy Systems, 18(5), 906–918.

    Article  Google Scholar 

  • Banfield, J.D., & Raftery, A.E. (1993). Model-based Gaussian and non-Gaussian clustering. Biometrics, 49(3), 803–821.

    Article  MathSciNet  Google Scholar 

  • Bezdek, J.C., Ehrlich, R., & Full, W. (1984). Fcm: The fuzzy c-means clustering algorithm. Computers & Geosciences, 10(2-3), 191–203.

    Article  Google Scholar 

  • Brouwer, R.K. (2009). Extending the rand, adjusted rand and Jaccard indices to fuzzy partitions. Journal of Intelligent Information Systems, 32(3), 213–235.

    Article  Google Scholar 

  • Campello, R.J. (2007). A fuzzy extension of the rand index and other related indexes for clustering and classification assessment. Pattern Recognition Letters, 28(7), 833–841.

    Article  Google Scholar 

  • Core Team, R. (2019). R: A language and environment for statistical computing Vienna Austria: R Foundation for Statistical Computing.

  • D’Ambrosio, A., Amodio, S., Iorio, C., Pandolfo, G., & Siciliano, R. (2020). Adjusted concordance index: an extensionl of the adjusted rand index to fuzzy partitions. Journal of Classification 1–17.

  • Flynt, A., Dean, N., & Nugent, R. (2019). Sari: a soft agreement measure for class partitions incorporating assignment probabilities. Advances in Data Analysis and Classification, 13(1), 303–323.

    Article  MathSciNet  Google Scholar 

  • Fowlkes, E.B., & Mallows, C.L. (1983). A method for comparing two hierarchical clusterings. Journal of the American statistical association, 78(383), 553–569.

    Article  Google Scholar 

  • Fraley, C., & Raftery, A.E. (2002). Model-based clustering, discriminant analysis, and density estimation. Journal of the American Statistical Association, 97(458), 611–631.

    Article  MathSciNet  Google Scholar 

  • Frigui, H., Hwang, C., & Rhee, F.C.-H. (2007). Clustering and aggregation of relational data with applications to image database categorization. Pattern Recognition, 40(11), 3053–3068.

    Article  Google Scholar 

  • Hubert, L., & Arabie, P. (1985). Comparing partitions. Journal of Classification, 2, 193–218.

    Article  Google Scholar 

  • Hullermeier, E., Rifqi, M., Henzgen, S., & Senge, R. (2012). Comparing fuzzy partitions: a generalization of the rand index and related measures. IEEE Transactions on Fuzzy Systems, 20(3), 546–556.

    Article  Google Scholar 

  • Johnson, S.C. (1967). Hierarchical clustering schemes. Psychometrika, 32(3), 241–254.

    Article  Google Scholar 

  • MacQueen, J., et al. (1967). Some methods for classification and analysis of multivariate observations. In Proceedings of the fifth Berkeley symposium on mathematical statistics and probability, vol. 1 281–297 Oakland, CA USA.

  • McLachlan, G., & Peel, D. (2000). Finite mixture models. Hoboken: Wiley.

    Book  Google Scholar 

  • McNicholas, P.D. (2016). Mixture model-based classification. Boca Raton: Chapman and Hall/CRC Press.

    Book  Google Scholar 

  • Meilă, M. (2007). Comparing clusterings—an information based distance. Journal of multivariate analysis, 98(5), 873–895.

    Article  MathSciNet  Google Scholar 

  • Rand, W.M. (1971). Objective criteria for the evaluation of clustering methods. Journal of the American Statistical Association, 66, 846–850.

    Article  Google Scholar 

  • Scrucca, L., Fop, M., Murphy, T.B., & Raftery, A.E. (2016). mclust 5: clustering, classification and density estimation using gaussian finite mixture models. The R Journal, 8(1), 289.

    Article  Google Scholar 

  • Steinley, D. (2004). Properties of the hubert-arable adjusted rand index. Psychological Methods, 9(3), 386.

    Article  Google Scholar 

Download references

Acknowledgements

This research was supported by the Natural Sciences and Engineering Research Council of Canada through their Discovery Grants program (RGPIN-2018-04444 for Browne and RGPIN-2020-04646 for Andrews). Infrastructure support was provided by the Canada Foundation for Innovation through the John Evans Leaders Fund (Andrews, #35578).

Author information

Authors and Affiliations

  1. Department of Statistics, University of British Columbia – Okanagan Campus, Kelowna, BC, Canada

    Jeffrey L. Andrews & Chelsey D. Hvingelby

  2. Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON, Canada

    Ryan Browne

Authors
  1. Jeffrey L. Andrews
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Ryan Browne
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Chelsey D. Hvingelby
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Jeffrey L. Andrews.

Additional information

Publisher’s Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Andrews, J.L., Browne, R. & Hvingelby, C.D. On Assessments of Agreement Between Fuzzy Partitions. J Classif 39, 326–342 (2022). https://doi.org/10.1007/s00357-021-09407-3

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00357-021-09407-3

Keywords