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Automatic detection of outliers and the number of clusters in k-means clustering via Chebyshev-type inequalities

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Abstract

We address two key challenges of k-means clustering. In the first part of the paper, we show that: when a dataset is partitioned with an appropriate number of clusters (k), not more than 1/9 of D will exceed twice its standard deviation (2 s.d.), and not more than 4/9 of D will exceed its standard deviation (1 s.d.) (σ), where D is a vector comprising the distance of each point to its cluster centroid. Our bounds assume unimodal symmetrical clusters (a generalization of k-means’ Gaussian assumption). In the second part of the paper, we show that a non-outlier will not be further from its cluster centroid than 14.826 times the median of absolute deviations from the median of D. Interestingly, D is already available from the k-means process. The first insight leads to an enhanced k-means algorithm (named Automatic k-means) which  efficiently estimates k. Unlike popular techniques, ours eliminates the need to supply a search range for k. Meanwhile, since practical datasets may deviate from the ideal distribution, the 1 and 2 s.d. tests may yield different k estimates. Both estimates constitute effective lower and upper bounds. Thus, our algorithm also provides a general way to speed up and automate existing techniques, via automatically determined narrow search range. We demonstrate this by presenting enhanced versions of the popular silhouette and gap statistics techniques (Auto-Silhouette and Auto-Gap). We apply the second theoretical insight to incorporate automatic outlier detection into k-means. Our outlier-aware algorithm (named k-means#) is identical to the standard k-means in the absence of outliers. In the presence of outliers, it is identical to a known outlier-aware algorithm, named k-means-−, except for the crucial difference that k-means−− relies on the user to supply the number of outliers, while our algorithm is automated. Our technique solves a puzzle described by the authors of k-means−− regarding the difficulty of complete automation, which was considered an open problem.

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Acknowledgements

This work was funded by the University of Johannesburg Global Excellence and Stature Doctoral Scholarship.

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Authors and Affiliations

  1. Department of Electrical and Electronics Engineering, Institute for Intelligent Systems, University of Johannesburg, Johannesburg, South Africa

    Peter Olukanmi, Fulufhelo Nelwamondo & Tshilidzi Marwala

  2. Durban University of Technology, Durban, South Africa

    Bhekisipho Twala

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  1. Peter Olukanmi
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  2. Fulufhelo Nelwamondo
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  3. Tshilidzi Marwala
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  4. Bhekisipho Twala
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Correspondence to Peter Olukanmi.

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Olukanmi, P., Nelwamondo, F., Marwala, T. et al. Automatic detection of outliers and the number of clusters in k-means clustering via Chebyshev-type inequalities. Neural Comput & Applic 34, 5939–5958 (2022). https://doi.org/10.1007/s00521-021-06689-x

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