On uniqueness of the set of k-means
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916442834731008 |
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| author | Cárcamo, Javier Cuevas, Antonio Rodríguez, Luis A. |
| author_facet | Cárcamo, Javier Cuevas, Antonio Rodríguez, Luis A. |
| contents | We provide necessary and sufficient conditions for the uniqueness of the k-means set of a probability distribution. This uniqueness problem is related to the choice of k: depending on the underlying distribution, some values of this parameter could lead to multiple sets of k-means, which hampers the interpretation of the results and/or the stability of the algorithms. We give a general assessment on consistency of the empirical k-means adapted to the setting of non-uniqueness and determine the asymptotic distribution of the within cluster sum of squares (WCSS). We also provide statistical characterizations of k-means uniqueness in terms of the asymptotic behavior of the empirical WCSS. As a consequence, we derive a bootstrap test for uniqueness of the set of k-means. The results are illustrated with examples of different types of non-uniqueness and we check by simulations the performance of the proposed methodology. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_13495 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On uniqueness of the set of k-means Cárcamo, Javier Cuevas, Antonio Rodríguez, Luis A. Statistics Theory Methodology Machine Learning 62H30, 62E20 (primary), 62G20 (secondary) We provide necessary and sufficient conditions for the uniqueness of the k-means set of a probability distribution. This uniqueness problem is related to the choice of k: depending on the underlying distribution, some values of this parameter could lead to multiple sets of k-means, which hampers the interpretation of the results and/or the stability of the algorithms. We give a general assessment on consistency of the empirical k-means adapted to the setting of non-uniqueness and determine the asymptotic distribution of the within cluster sum of squares (WCSS). We also provide statistical characterizations of k-means uniqueness in terms of the asymptotic behavior of the empirical WCSS. As a consequence, we derive a bootstrap test for uniqueness of the set of k-means. The results are illustrated with examples of different types of non-uniqueness and we check by simulations the performance of the proposed methodology. |
| title | On uniqueness of the set of k-means |
| topic | Statistics Theory Methodology Machine Learning 62H30, 62E20 (primary), 62G20 (secondary) |
| url | https://arxiv.org/abs/2410.13495 |