Adachi, Kohei and Trendafilov, Nickolay T.
(2016).
*Computational Statistics*, 31(4) pp. 1403–1427.

DOI (Digital Object Identifier) Link: | https://doi.org/10.1007/s00180-015-0608-4 |
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Google Scholar: | Look up in Google Scholar |

## Abstract

Most of the existing procedures for sparse principal component analysis (PCA) use a penalty function to obtain a sparse matrix of weights by which a data matrix is post-multiplied to produce PC scores. In this paper, we propose a new sparse PCA procedure which differs from the existing ones in two ways. First, the new procedure does not sparsify the weight matrix. Instead, the so-called loadings matrix is sparsified by which the score matrix is post-multiplied to approximate the data matrix. Second, the cardinality of the loading matrix i.e., the total number of nonzero loadings, is pre-specified to be an integer without using penalty functions. The procedure is called unpenalized sparse loading PCA (USLPCA). A desirable property of USLPCA is that the indices for the percentages of explained variances can be defined in the same form as in the standard PCA. We develop an alternate least squares algorithm for USLPCA which uses the fact that the PCA loss function can be decomposed as a sum of a term irrelevant to the loadings, and another one being easily minimized under cardinality constraints. A procedure is also presented for selecting the best cardinality using information criteria. The procedures are assessed in a simulation study and illustrated with real data examples.

Item Type: | Journal Item |
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Copyright Holders: | 2015 Springer-Verlag |

ISSN: | 1613-9658 |

Keywords: | penalty-free approach; sparse loadings; alternating least squares; percentages of explained variances |

Academic Unit/School: | Faculty of Science, Technology, Engineering and Mathematics (STEM) > Mathematics and Statistics Faculty of Science, Technology, Engineering and Mathematics (STEM) |

Item ID: | 49277 |

Depositing User: | Nickolay Trendafilov |

Date Deposited: | 02 May 2017 15:19 |

Last Modified: | 03 May 2019 03:43 |

URI: | http://oro.open.ac.uk/id/eprint/49277 |

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