A robust and sparse K-means clustering algorithm

In many situations where the interest lies in identifying clusters one might expect that not all available variables carry information about these groups. Furthermore, data quality (e.g. outliers or missing entries) might present a serious and sometimes hard-to-assess problem for large and complex datasets. In this paper we show that a small proportion of atypical observations might have serious adverse effects on the solutions found by the sparse clustering algorithm of Witten and Tibshirani (2010). We propose a robustification of their sparse K-means algorithm based on the trimmed K-means algorithm of Cuesta-Albertos et al. (1997) Our proposal is also able to handle datasets with missing values. We illustrate the use of our method on microarray data for cancer patients where we are able to identify strong biological clusters with a much reduced number of genes. Our simulation studies show that, when there are outliers in the data, our robust sparse K-means algorithm performs better than other competing methods both in terms of the selection of features and also the identified clusters. This robust sparse K-means algorithm is implemented in the R package RSKC which is publicly available from the CRAN repository.

On the stability of bootstrap estimators

It is shown that bootstrap approximations of an estimator which is based on a continuous operator from the set of Borel probability measures defined on a compact metric space into a complete separable metric space is stable in the sense of qualitative robustness. Support vector machines based on shifted loss functions are treated as special cases.