Acceleration of the alternating least squares algorithm for principal components analysis
Principal components analysis (PCA) is a popular descriptive multivariate method for handling quantitative data and it can be extended to deal with qualitative data and mixed measurement level data. The existing algorithms for extended PCA are PRINCIPALS of Young et al. (1978) and PRINCALS of Gifi (...
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| Vydané v: | Computational statistics & data analysis Ročník 55; číslo 1; s. 143 - 153 |
|---|---|
| Hlavní autori: | , , , |
| Médium: | Journal Article |
| Jazyk: | English |
| Vydavateľské údaje: |
Amsterdam
Elsevier B.V
2011
Elsevier |
| Edícia: | Computational Statistics & Data Analysis |
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| ISSN: | 0167-9473, 1872-7352 |
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| Abstract | Principal components analysis (PCA) is a popular descriptive multivariate method for handling quantitative data and it can be extended to deal with qualitative data and mixed measurement level data. The existing algorithms for extended PCA are PRINCIPALS of
Young et al. (1978) and PRINCALS of
Gifi (1989) in which the alternating least squares algorithm is utilized. These algorithms based on the least squares estimation may require many iterations in their application to very large data sets and variable selection problems and may take a long time to converge. In this paper, we derive a new iterative algorithm for accelerating the convergence of PRINCIPALS and PRINCALS by using the vector
ε
algorithm of
Wynn (1962). The proposed acceleration algorithm speeds up the convergence of the sequence of the parameter estimates obtained from PRINCIPALS or PRINCALS. Numerical experiments illustrate the potential of the proposed acceleration algorithm. |
|---|---|
| AbstractList | Principal components analysis (PCA) is a popular descriptive multivariate method for handling quantitative data and it can be extended to deal with qualitative data and mixed measurement level data. The existing algorithms for extended PCA are PRINCIPALS of
Young et al. (1978) and PRINCALS of
Gifi (1989) in which the alternating least squares algorithm is utilized. These algorithms based on the least squares estimation may require many iterations in their application to very large data sets and variable selection problems and may take a long time to converge. In this paper, we derive a new iterative algorithm for accelerating the convergence of PRINCIPALS and PRINCALS by using the vector
ε
algorithm of
Wynn (1962). The proposed acceleration algorithm speeds up the convergence of the sequence of the parameter estimates obtained from PRINCIPALS or PRINCALS. Numerical experiments illustrate the potential of the proposed acceleration algorithm. Principal components analysis (PCA) is a popular descriptive multivariate method for handling quantitative data and it can be extended to deal with qualitative data and mixed measurement level data. The existing algorithms for extended PCA are PRINCIPALS of and PRINCALS of in which the alternating least squares algorithm is utilized. These algorithms based on the least squares estimation may require many iterations in their application to very large data sets and variable selection problems and may take a long time to converge. In this paper, we derive a new iterative algorithm for accelerating the convergence of PRINCIPALS and PRINCALS by using the vector e algorithm of . The proposed acceleration algorithm speeds up the convergence of the sequence of the parameter estimates obtained from PRINCIPALS or PRINCALS. Numerical experiments illustrate the potential of the proposed acceleration algorithm. Principal components analysis (PCA) is a popular descriptive multivariate method for handling quantitative data and it can be extended to deal with qualitative data and mixed measurement level data. The existing algorithms for extended PCA are PRINCIPALS of Young et al. (1978) and PRINCALS of Gifi (1989) in which the alternating least squares algorithm is utilized. These algorithms based on the least squares estimation may require many iterations in their application to very large data sets and variable selection problems and may take a long time to converge. In this paper, we derive a new iterative algorithm for accelerating the convergence of PRINCIPALS and PRINCALS by using the vector [epsilon] algorithm of Wynn (1962). The proposed acceleration algorithm speeds up the convergence of the sequence of the parameter estimates obtained from PRINCIPALS or PRINCALS. Numerical experiments illustrate the potential of the proposed acceleration algorithm. |
| Author | Iizuka, Masaya Kuroda, Masahiro Sakakihara, Michio Mori, Yuichi |
| Author_xml | – sequence: 1 givenname: Masahiro surname: Kuroda fullname: Kuroda, Masahiro email: kuroda@soci.ous.ac.jp organization: Department of Socio-Information, Okayama University of Science, 1-1 Ridaicho, Kita-ku, Okayama 700-0005, Japan – sequence: 2 givenname: Yuichi surname: Mori fullname: Mori, Yuichi organization: Department of Socio-Information, Okayama University of Science, 1-1 Ridaicho, Kita-ku, Okayama 700-0005, Japan – sequence: 3 givenname: Masaya surname: Iizuka fullname: Iizuka, Masaya organization: Graduate School of Environmental Science, Okayama University 1-1-1 Tsushima-naka, Kita-ku, Okayama 700-8530, Japan – sequence: 4 givenname: Michio surname: Sakakihara fullname: Sakakihara, Michio organization: Department of Information Science, Okayama University of Science, 1-1 Ridaicho, Kita-ku, Okayama 700-0005, Japan |
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| Cites_doi | 10.1016/S0167-9473(02)00142-1 10.1007/BF02293871 10.1016/j.csda.2005.09.003 10.1090/S0025-5718-1962-0145647-X 10.1016/j.csda.2006.05.004 10.1007/s00180-007-0089-1 10.1007/BF02293796 10.1214/ss/1028905828 |
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| Keywords | Vector ε algorithm Alternating least squares algorithm PRINCIPALS PRINCALS Acceleration of convergence Data analysis Convergence acceleration Estimation Iterative method Iteration Multivariate analysis Algorithm Convergence Convergence speed Selection problem Numerical analysis Statistical computation Least squares method Algorithm analysis Principal component analysis Variable selection |
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| References | Mori, Tanaka, Tarumi (b7) 1997 Wynn (b14) 1962; 16 Michailidis, de Leeuw (b6) 1998; 13 Young (b11) 1981; 46 Kuroda, Sakakihara (b5) 2006; 51 Young, Takane, de Leeuw (b12) 1978; 43 Kiers (b3) 2002; 41 R Development Core Team (b9) 2008 Brezinski, Zaglia (b1) 1991 Wang, Kuroda, Sakakihara, Geng (b13) 2008; 23 Mori, Y., Matsumoto, Y., Iizuka, M., Tanaka, Y., 2007. A variable selection in modified principal component analysis for qualitative data. In: The 56th Session of the International Statistical Institute. Abstract Book, 337, Proceedings CD-ROM. Gifi, A., 1989. Algorithm descriptions for ANACOR, HOMALS, PRINCIPALS, and OVERALS. Report RR 89-01. Department of Data Theory, University of Leiden, Leiden. Krijnen (b4) 2006; 51 Tanaka, Mori (b10) 1997; 17 10.1016/j.csda.2010.06.001_b2 R Development Core Team (10.1016/j.csda.2010.06.001_b9) 2008 Kiers (10.1016/j.csda.2010.06.001_b3) 2002; 41 Tanaka (10.1016/j.csda.2010.06.001_b10) 1997; 17 10.1016/j.csda.2010.06.001_b8 Michailidis (10.1016/j.csda.2010.06.001_b6) 1998; 13 Mori (10.1016/j.csda.2010.06.001_b7) 1997 Young (10.1016/j.csda.2010.06.001_b11) 1981; 46 Wynn (10.1016/j.csda.2010.06.001_b14) 1962; 16 Kuroda (10.1016/j.csda.2010.06.001_b5) 2006; 51 Brezinski (10.1016/j.csda.2010.06.001_b1) 1991 Krijnen (10.1016/j.csda.2010.06.001_b4) 2006; 51 Wang (10.1016/j.csda.2010.06.001_b13) 2008; 23 Young (10.1016/j.csda.2010.06.001_b12) 1978; 43 |
| References_xml | – start-page: 547 year: 1997 end-page: 554 ident: b7 article-title: Principal component analysis based on a subset of variables for qualitative data publication-title: Data Science, Classification, and Related Methods (Proceedings of IFCS-96) – volume: 41 start-page: 157 year: 2002 end-page: 170 ident: b3 article-title: Setting up alternating least squares and iterative majorization algorithm for solving various matrix optimization problems publication-title: Computational Statistics and Data Analysis – year: 1991 ident: b1 article-title: Extrapolation Methods: Theory and Practice – volume: 16 start-page: 301 year: 1962 end-page: 322 ident: b14 article-title: Acceleration techniques for iterated vector and matrix problems publication-title: Mathematics of Computation – volume: 51 start-page: 1549 year: 2006 end-page: 1561 ident: b5 article-title: Accelerating the convergence of the EM algorithm using the vector epsilon algorithm publication-title: Computational Statistics and Data Analysis – year: 2008 ident: b9 article-title: R: A language and environment for statistical computing – volume: 17 start-page: 61 year: 1997 end-page: 89 ident: b10 article-title: Principal component analysis based on a subset of variables: variable selection and sensitivity analysis publication-title: The American Journal of Mathematical and Management Sciences – volume: 13 start-page: 307 year: 1998 end-page: 336 ident: b6 article-title: The Gifi system of descriptive multivariate analysis publication-title: Statistical Science – reference: Mori, Y., Matsumoto, Y., Iizuka, M., Tanaka, Y., 2007. A variable selection in modified principal component analysis for qualitative data. In: The 56th Session of the International Statistical Institute. Abstract Book, 337, Proceedings CD-ROM. – volume: 46 start-page: 357 year: 1981 end-page: 388 ident: b11 article-title: Quantitative analysis of qualitative data publication-title: Psychometrika – volume: 23 start-page: 469 year: 2008 end-page: 486 ident: b13 article-title: Acceleration of the EM algorithm using the vector epsilon algorithm publication-title: Computational Statistics – reference: Gifi, A., 1989. Algorithm descriptions for ANACOR, HOMALS, PRINCIPALS, and OVERALS. Report RR 89-01. Department of Data Theory, University of Leiden, Leiden. – volume: 51 start-page: 481 year: 2006 end-page: 489 ident: b4 article-title: Convergence of the sequence of parameters generated by alternating least squares algorithms publication-title: Computational Statistics and Data Analysis – volume: 43 start-page: 279 year: 1978 end-page: 281 ident: b12 article-title: Principal components of mixed measurement level multivariate data: an alternating least squares method with optimal scaling features publication-title: Psychometrika – volume: 41 start-page: 157 year: 2002 ident: 10.1016/j.csda.2010.06.001_b3 article-title: Setting up alternating least squares and iterative majorization algorithm for solving various matrix optimization problems publication-title: Computational Statistics and Data Analysis doi: 10.1016/S0167-9473(02)00142-1 – volume: 43 start-page: 279 year: 1978 ident: 10.1016/j.csda.2010.06.001_b12 article-title: Principal components of mixed measurement level multivariate data: an alternating least squares method with optimal scaling features publication-title: Psychometrika doi: 10.1007/BF02293871 – ident: 10.1016/j.csda.2010.06.001_b8 – volume: 17 start-page: 61 year: 1997 ident: 10.1016/j.csda.2010.06.001_b10 article-title: Principal component analysis based on a subset of variables: variable selection and sensitivity analysis publication-title: The American Journal of Mathematical and Management Sciences – volume: 51 start-page: 481 year: 2006 ident: 10.1016/j.csda.2010.06.001_b4 article-title: Convergence of the sequence of parameters generated by alternating least squares algorithms publication-title: Computational Statistics and Data Analysis doi: 10.1016/j.csda.2005.09.003 – year: 2008 ident: 10.1016/j.csda.2010.06.001_b9 – year: 1991 ident: 10.1016/j.csda.2010.06.001_b1 – ident: 10.1016/j.csda.2010.06.001_b2 – volume: 16 start-page: 301 year: 1962 ident: 10.1016/j.csda.2010.06.001_b14 article-title: Acceleration techniques for iterated vector and matrix problems publication-title: Mathematics of Computation doi: 10.1090/S0025-5718-1962-0145647-X – start-page: 547 year: 1997 ident: 10.1016/j.csda.2010.06.001_b7 article-title: Principal component analysis based on a subset of variables for qualitative data – volume: 51 start-page: 1549 year: 2006 ident: 10.1016/j.csda.2010.06.001_b5 article-title: Accelerating the convergence of the EM algorithm using the vector epsilon algorithm publication-title: Computational Statistics and Data Analysis doi: 10.1016/j.csda.2006.05.004 – volume: 23 start-page: 469 year: 2008 ident: 10.1016/j.csda.2010.06.001_b13 article-title: Acceleration of the EM algorithm using the vector epsilon algorithm publication-title: Computational Statistics doi: 10.1007/s00180-007-0089-1 – volume: 46 start-page: 357 year: 1981 ident: 10.1016/j.csda.2010.06.001_b11 article-title: Quantitative analysis of qualitative data publication-title: Psychometrika doi: 10.1007/BF02293796 – volume: 13 start-page: 307 year: 1998 ident: 10.1016/j.csda.2010.06.001_b6 article-title: The Gifi system of descriptive multivariate analysis publication-title: Statistical Science doi: 10.1214/ss/1028905828 |
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| SubjectTerms | Acceleration Acceleration of convergence Algorithms Alternating least squares algorithm Alternating least squares algorithm Vector [epsilon] algorithm Acceleration of convergence PRINCIPALS PRINCALS Convergence Exact sciences and technology General topics Least squares method Mathematical analysis Mathematical models Mathematics Multivariate analysis Numerical analysis Numerical analysis. Scientific computation Numerical methods in probability and statistics PRINCALS Principal component analysis PRINCIPALS Probability and statistics Sciences and techniques of general use Statistics Vector [formula omitted] algorithm |
| Title | Acceleration of the alternating least squares algorithm for principal components analysis |
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