Some notes on the $k$-means clustering for missing data
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866914961456889856 |
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| author | Terada, Yoshikazu Guan, Xin |
| author_facet | Terada, Yoshikazu Guan, Xin |
| contents | The classical $k$-means clustering requires a complete data matrix without missing entries. As a natural extension of the $k$-means clustering for missing data, the $k$-POD clustering has been proposed, which ignores the missing entries in the $k$-means clustering. This paper shows the inconsistency of the $k$-POD clustering even under the missing completely at random mechanism. More specifically, the expected loss of the $k$-POD clustering can be represented as the weighted sum of the expected $k$-means losses with parts of variables. Thus, the $k$-POD clustering converges to the different clustering from the $k$-means clustering as the sample size goes to infinity. This result indicates that although the $k$-means clustering works well, the $k$-POD clustering may fail to capture the hidden cluster structure. On the other hand, for high-dimensional data, the $k$-POD clustering could be a suitable choice when the missing rate in each variable is low. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_00546 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Some notes on the $k$-means clustering for missing data Terada, Yoshikazu Guan, Xin Statistics Theory The classical $k$-means clustering requires a complete data matrix without missing entries. As a natural extension of the $k$-means clustering for missing data, the $k$-POD clustering has been proposed, which ignores the missing entries in the $k$-means clustering. This paper shows the inconsistency of the $k$-POD clustering even under the missing completely at random mechanism. More specifically, the expected loss of the $k$-POD clustering can be represented as the weighted sum of the expected $k$-means losses with parts of variables. Thus, the $k$-POD clustering converges to the different clustering from the $k$-means clustering as the sample size goes to infinity. This result indicates that although the $k$-means clustering works well, the $k$-POD clustering may fail to capture the hidden cluster structure. On the other hand, for high-dimensional data, the $k$-POD clustering could be a suitable choice when the missing rate in each variable is low. |
| title | Some notes on the $k$-means clustering for missing data |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2410.00546 |