Solving the missing value problem in PCA by Orthogonalized-Alternating Least Squares (O-ALS).

Dealing with missing data poses a challenge in Principal Component Analysis (PCA) since the most common algorithms are not designed to handle them. Several approaches have been proposed to solve the missing value problem in PCA, such as Imputation based on SVD (I-SVD), where missing entries are fill...

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Authors: Gómez Sánchez, Adrián, Vitale, Raffaele, Ruckebusch, Cyril, Juan Capdevila, Anna de
Format: article
Status:Published version
Publication Date:2024
Country:España
Institution:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
Repository:Recercat. Dipósit de la Recerca de Catalunya
OAI Identifier:oai:dnet:recercat____::689e32c15a46331e150b0f4e5da3a14c
Online Access:https://hdl.handle.net/2445/229355
Access Level:Open access
Keyword:Mínims quadrats
Polinomis ortogonals
Least squares
Orthogonal polynomials
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spelling Solving the missing value problem in PCA by Orthogonalized-Alternating Least Squares (O-ALS).Gómez Sánchez, AdriánVitale, RaffaeleRuckebusch, CyrilJuan Capdevila, Anna deMínims quadratsPolinomis ortogonalsLeast squaresOrthogonal polynomialsDealing with missing data poses a challenge in Principal Component Analysis (PCA) since the most common algorithms are not designed to handle them. Several approaches have been proposed to solve the missing value problem in PCA, such as Imputation based on SVD (I-SVD), where missing entries are filled by imputation and updated in every iteration until convergence of the PCA model, and the adaptation of the Nonlinear Iterative Partial Least Squares (NIPALS) algorithm, able to work skipping the missing entries during the least-squares estimation of scores and loadings. However, some limitations have been reported for both approaches. On the one hand, convergence of the I-SVD algorithm can be very slow for data sets with a high percentage of missing data. On the other hand, the orthogonality properties among scores and loadings might be lost when using NIPALS. To solve these issues and perform PCA of data sets with missing values without the need of imputation steps, a novel algorithm called Orthogonalized-Alternating Least Squares (O-ALS) is proposed. The O-ALS algorithm is an alternating least-squares algorithm that estimates the scores and loadings subject to the Gram-Schmidt orthogonalization constraint. The way to estimate scores and loadings is adapted to work only with the available information. In this study, the performance of O-ALS is tested and compared with NIPALS and I-SVD in simulated data sets and in a real case study. The results show that O-ALS is an accurate and fast algorithm to analyze data with any percentage and distribution pattern of missing entries, being able to provide correct scores and loadings in cases where I-SVD and NIPALS do not perform satisfactorily.Elsevier B.V.2026202620242026info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersion12 p.application/pdfhttps://hdl.handle.net/2445/229355https://hdl.handle.net/2445/229355reponame:Recercat. Dipósit de la Recerca de Catalunyainstname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)InglésReproducció del document publicat a: https://doi.org/10.1016/j.chemolab.2024.105153Chemometrics and Intelligent Laboratory Systems, 2024, vol. 250, num.2024, p. 1-12https://doi.org/10.1016/j.chemolab.2024.105153cc-by-nc (c) Elsevier B.V., 2024http://creativecommons.org/licenses/by-nc/4.0/info:eu-repo/semantics/openAccessoai:dnet:recercat____::689e32c15a46331e150b0f4e5da3a14c2026-05-29T05:05:01Z
dc.title.none.fl_str_mv Solving the missing value problem in PCA by Orthogonalized-Alternating Least Squares (O-ALS).
title Solving the missing value problem in PCA by Orthogonalized-Alternating Least Squares (O-ALS).
spellingShingle Solving the missing value problem in PCA by Orthogonalized-Alternating Least Squares (O-ALS).
Gómez Sánchez, Adrián
Mínims quadrats
Polinomis ortogonals
Least squares
Orthogonal polynomials
title_short Solving the missing value problem in PCA by Orthogonalized-Alternating Least Squares (O-ALS).
title_full Solving the missing value problem in PCA by Orthogonalized-Alternating Least Squares (O-ALS).
title_fullStr Solving the missing value problem in PCA by Orthogonalized-Alternating Least Squares (O-ALS).
title_full_unstemmed Solving the missing value problem in PCA by Orthogonalized-Alternating Least Squares (O-ALS).
title_sort Solving the missing value problem in PCA by Orthogonalized-Alternating Least Squares (O-ALS).
dc.creator.none.fl_str_mv Gómez Sánchez, Adrián
Vitale, Raffaele
Ruckebusch, Cyril
Juan Capdevila, Anna de
author Gómez Sánchez, Adrián
author_facet Gómez Sánchez, Adrián
Vitale, Raffaele
Ruckebusch, Cyril
Juan Capdevila, Anna de
author_role author
author2 Vitale, Raffaele
Ruckebusch, Cyril
Juan Capdevila, Anna de
author2_role author
author
author
dc.subject.none.fl_str_mv Mínims quadrats
Polinomis ortogonals
Least squares
Orthogonal polynomials
topic Mínims quadrats
Polinomis ortogonals
Least squares
Orthogonal polynomials
description Dealing with missing data poses a challenge in Principal Component Analysis (PCA) since the most common algorithms are not designed to handle them. Several approaches have been proposed to solve the missing value problem in PCA, such as Imputation based on SVD (I-SVD), where missing entries are filled by imputation and updated in every iteration until convergence of the PCA model, and the adaptation of the Nonlinear Iterative Partial Least Squares (NIPALS) algorithm, able to work skipping the missing entries during the least-squares estimation of scores and loadings. However, some limitations have been reported for both approaches. On the one hand, convergence of the I-SVD algorithm can be very slow for data sets with a high percentage of missing data. On the other hand, the orthogonality properties among scores and loadings might be lost when using NIPALS. To solve these issues and perform PCA of data sets with missing values without the need of imputation steps, a novel algorithm called Orthogonalized-Alternating Least Squares (O-ALS) is proposed. The O-ALS algorithm is an alternating least-squares algorithm that estimates the scores and loadings subject to the Gram-Schmidt orthogonalization constraint. The way to estimate scores and loadings is adapted to work only with the available information. In this study, the performance of O-ALS is tested and compared with NIPALS and I-SVD in simulated data sets and in a real case study. The results show that O-ALS is an accurate and fast algorithm to analyze data with any percentage and distribution pattern of missing entries, being able to provide correct scores and loadings in cases where I-SVD and NIPALS do not perform satisfactorily.
publishDate 2024
dc.date.none.fl_str_mv 2024
2026
2026
2026
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv https://hdl.handle.net/2445/229355
https://hdl.handle.net/2445/229355
url https://hdl.handle.net/2445/229355
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Reproducció del document publicat a: https://doi.org/10.1016/j.chemolab.2024.105153
Chemometrics and Intelligent Laboratory Systems, 2024, vol. 250, num.2024, p. 1-12
https://doi.org/10.1016/j.chemolab.2024.105153
dc.rights.none.fl_str_mv cc-by-nc (c) Elsevier B.V., 2024
http://creativecommons.org/licenses/by-nc/4.0/
info:eu-repo/semantics/openAccess
rights_invalid_str_mv cc-by-nc (c) Elsevier B.V., 2024
http://creativecommons.org/licenses/by-nc/4.0/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv 12 p.
application/pdf
dc.publisher.none.fl_str_mv Elsevier B.V.
publisher.none.fl_str_mv Elsevier B.V.
dc.source.none.fl_str_mv reponame:Recercat. Dipósit de la Recerca de Catalunya
instname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
instname_str Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
reponame_str Recercat. Dipósit de la Recerca de Catalunya
collection Recercat. Dipósit de la Recerca de Catalunya
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15.812429

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