{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T03:23:40Z","timestamp":1740108220390,"version":"3.37.3"},"reference-count":44,"publisher":"Springer Science and Business Media LLC","license":[{"start":{"date-parts":[[2024,9,12]],"date-time":"2024-09-12T00:00:00Z","timestamp":1726099200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,9,12]],"date-time":"2024-09-12T00:00:00Z","timestamp":1726099200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100004359","name":"Vetenskapsr\u00e5det","doi-asserted-by":"publisher","award":["2021-04810"],"award-info":[{"award-number":["2021-04810"]}],"id":[{"id":"10.13039\/501100004359","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Umea University"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comput Stat"],"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Partial least squares regression (PLS-R) has been an important regression method in the life sciences and many other fields for decades. However, PLS-R is typically solved using an opaque algorithmic approach, rather than through an optimisation formulation and procedure. There is a clear optimisation formulation of the PLS-R problem based on a Krylov subspace formulation, but it is only rarely considered. The popularity of PLS-R is attributed to the ability to interpret the data through the model components, but the model components are not available when solving the PLS-R problem using the Krylov subspace formulation. We therefore highlight a simple reformulation of the PLS-R problem using the Krylov subspace formulation as a promising modelling framework for PLS-R, and illustrate one of the main benefits of this reformulation\u2014that it allows arbitrary penalties of the regression coefficients in the PLS-R model. Further, we propose an approach to estimate the PLS-R model components for the solution found through the Krylov subspace formulation, that are those we would have obtained had we been able to use the common algorithms for estimating the PLS-R model. We illustrate the utility of the proposed method on simulated and real data.<\/jats:p>","DOI":"10.1007\/s00180-024-01545-7","type":"journal-article","created":{"date-parts":[[2024,9,12]],"date-time":"2024-09-12T12:02:19Z","timestamp":1726142539000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Using the Krylov subspace formulation to improve regularisation and interpretation in partial least squares regression"],"prefix":"10.1007","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7119-7646","authenticated-orcid":false,"given":"Tommy","family":"L\u00f6fstedt","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,9,12]]},"reference":[{"issue":"1","key":"1545_CR1","doi-asserted-by":"publisher","first-page":"97","DOI":"10.1002\/wics.51","volume":"2","author":"H Abdi","year":"2010","unstructured":"Abdi H (2010) Partial least squares regression and projection on latent structure regression (PLS regression). WIREs Comput Stat 2(1):97\u2013106","journal-title":"WIREs Comput Stat"},{"issue":"4","key":"1545_CR2","doi-asserted-by":"publisher","first-page":"302","DOI":"10.1002\/sam.11169","volume":"6","author":"GI Allen","year":"2013","unstructured":"Allen GI, Peterson C, Vannucci M, Maleti\u0107-Savati\u0107 M (2013) Regularized partial least squares with an application to NMR spectroscopy. Stat Anal Data Min 6(4):302\u2013314","journal-title":"Stat Anal Data Min"},{"key":"1545_CR3","doi-asserted-by":"publisher","first-page":"166","DOI":"10.1002\/cem.785","volume":"17","author":"M Barker","year":"2003","unstructured":"Barker M, Rayens W (2003) Partial least squares for discrimination. J Chemom 17:166\u2013173","journal-title":"J Chemom"},{"issue":"3","key":"1545_CR4","doi-asserted-by":"publisher","first-page":"503","DOI":"10.1023\/B:JOTA.0000025708.31430.22","volume":"120","author":"HH Bauschke","year":"2004","unstructured":"Bauschke HH, Kruk SG (2004) Reflection-projection method for convex feasibility problems with an obtuse cone. J Optim Theory Appl 120(3):503\u2013531","journal-title":"J Optim Theory Appl"},{"key":"1545_CR5","volume-title":"Nonlinear programming","author":"DP Bertsekas","year":"1999","unstructured":"Bertsekas DP (1999) Nonlinear programming. Athena Scientific, Belmont"},{"key":"1545_CR6","doi-asserted-by":"publisher","first-page":"129","DOI":"10.1016\/S0003-2670(00)85465-3","volume":"150","author":"ML Bisani","year":"1983","unstructured":"Bisani ML, Faraone D, Clementi S, Esbensen KH, Wold S (1983) Principal components and partial least-squares analysis of the geochemistry of volcanic rocks from the aeolian archipelago. Anal Chim Acta 150:129\u2013143","journal-title":"Anal Chim Acta"},{"key":"1545_CR7","doi-asserted-by":"crossref","unstructured":"Bj\u00f6rck A, Indahl UG (2017) Fast and stable partial least squares modelling: a benchmark study with theoretical comments. J Chemomet 31","DOI":"10.1002\/cem.2898"},{"issue":"1","key":"1545_CR8","doi-asserted-by":"publisher","first-page":"32","DOI":"10.1093\/bib\/bbl016","volume":"8","author":"A-L Boulesteix","year":"2007","unstructured":"Boulesteix A-L, Strimmer K (2007) Partial least squares: a versatile tool for the analysis of high-dimensional genomic data. Brief Bioinform 8(1):32\u201344","journal-title":"Brief Bioinform"},{"issue":"1","key":"1545_CR9","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1561\/2200000016","volume":"3","author":"S Boyd","year":"2010","unstructured":"Boyd S, Parikh N, Chu E, Peleato B, Eckstein J (2010) Distributed optimization and statistical learning via the alternating direction method of multipliers. Found Trends Machine Learn 3(1):1\u2013122","journal-title":"Found Trends Machine Learn"},{"issue":"3","key":"1545_CR10","doi-asserted-by":"publisher","first-page":"585","DOI":"10.1111\/1467-9868.00252","volume":"62","author":"NA Butler","year":"2000","unstructured":"Butler NA, Denham MC (2000) The peculiar shrinkage properties of partial least squares regression. J Roy Stat Soc 62(3):585\u2013593","journal-title":"J Roy Stat Soc"},{"issue":"1","key":"1545_CR11","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1111\/j.1467-9868.2009.00723.x","volume":"72","author":"H Chun","year":"2010","unstructured":"Chun H, Kele\u015f S (2010) Sparse partial least squares regression for simultaneous dimension reduction and variable selection. J Roy Stat Soc 72(1):3\u201325","journal-title":"J Roy Stat Soc"},{"key":"1545_CR12","doi-asserted-by":"publisher","first-page":"185","DOI":"10.1007\/978-1-4419-9569-8_10","volume-title":"Fixed-Point Algorithms for Inverse Problems in Science and Engineering","author":"PL Combettes","year":"2011","unstructured":"Combettes PL, Pesquet J-C (2011) Proximal splitting methods in signal processing. In: Bauschke HH, Burachik RS, Combettes PL, Elser V, Luke DR, Wolkowicz H (eds) Fixed-Point Algorithms for Inverse Problems in Science and Engineering. Springer, New York, U.S.A., pp 185\u2013212"},{"key":"1545_CR13","doi-asserted-by":"publisher","first-page":"11","DOI":"10.1016\/S0167-9473(03)00138-5","volume":"46","author":"L Eld\u00e9n","year":"2004","unstructured":"Eld\u00e9n L (2004) Partial least-squares vs. Lanczos bidiagonalization\u2013I: analysis of a projection method for multiple regression. Comput Stat Data Anal 46:11\u201331","journal-title":"Comput Stat Data Anal"},{"issue":"1","key":"1545_CR14","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1002\/cem.899","volume":"19","author":"R Ergon","year":"2005","unstructured":"Ergon R (2005) PLS post-processing by similarity transformation (PLS+ST): a simple alternative to OPLS. J Chemom 19(1):1\u20134","journal-title":"J Chemom"},{"issue":"2","key":"1545_CR15","doi-asserted-by":"publisher","first-page":"109","DOI":"10.1080\/00401706.1993.10485033","volume":"35","author":"IE Frank","year":"1993","unstructured":"Frank IE, Friedman JH (1993) A statistical view of some chemometrics regression tools. Technometrics 35(2):109\u2013135","journal-title":"Technometrics"},{"issue":"1","key":"1545_CR16","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1016\/0898-1221(76)90003-1","volume":"2","author":"D Gabay","year":"1976","unstructured":"Gabay D, Mercier B (1976) A dual algorithm for the solution of nonlinear variational problems via finite element approximation. Comput Math Appl 2(1):17\u201340","journal-title":"Comput Math Appl"},{"issue":"16","key":"1545_CR17","doi-asserted-by":"publisher","first-page":"10","DOI":"10.1016\/j.aca.2015.02.012","volume":"879","author":"PS Gromski","year":"2015","unstructured":"Gromski PS, Muhamadali H, Ellis DI, Xu Y, Correa E, Turner ML, Goodacre R (2015) A tutorial review: metabolomics and partial least squares-discriminant analysis - a marriage of convenience or a shotgun wedding. Anal Chim Acta 879(16):10\u201323","journal-title":"Anal Chim Acta"},{"issue":"2","key":"1545_CR18","doi-asserted-by":"publisher","first-page":"581","DOI":"10.1080\/03610918808812681","volume":"17","author":"IS Helland","year":"1988","unstructured":"Helland IS (1988) On the structure of partial least squares regression. Commun Stat Simul Comput 17(2):581\u2013607","journal-title":"Commun Stat Simul Comput"},{"key":"1545_CR19","first-page":"27","volume":"8","author":"AE Hoerl","year":"1970","unstructured":"Hoerl AE, Kennard RW (1970) Ridge regression: biased estimation for nonorthogonal problems. Technometrics 8:27\u201351","journal-title":"Technometrics"},{"key":"1545_CR20","doi-asserted-by":"publisher","first-page":"211","DOI":"10.1002\/cem.1180020306","volume":"2","author":"A H\u00f6skuldsson","year":"1988","unstructured":"H\u00f6skuldsson A (1988) PLS regression methods. J Chemom 2:211\u2013228","journal-title":"J Chemom"},{"key":"1545_CR21","doi-asserted-by":"publisher","first-page":"630","DOI":"10.1002\/cem.831","volume":"17","author":"A H\u00f6skuldsson","year":"2003","unstructured":"H\u00f6skuldsson A (2003) Analysis of latent structures in linear models. J Chemom 17:630\u2013645","journal-title":"J Chemom"},{"key":"1545_CR22","doi-asserted-by":"publisher","first-page":"251","DOI":"10.1016\/0169-7439(93)85002-X","volume":"18","author":"S Jong","year":"1993","unstructured":"Jong S (1993) SIMPLS: an alternative approach to partial least squares regression. Chemom Intell Lab Syst 18:251\u2013263","journal-title":"Chemom Intell Lab Syst"},{"key":"1545_CR23","doi-asserted-by":"crossref","unstructured":"Kloss RB, Cirne MVM, Silva S, Pedrini H, Schwartz WR (2015) Partial least squares image clustering. In: Oliveira LR, Apolin\u00e1rio\u00a0Junior AL, Lemes RP (eds.) Proceedings of the 28th conference on graphics, patterns and images (SIBGRAPI 2015), pp. 41\u201348","DOI":"10.1109\/SIBGRAPI.2015.25"},{"issue":"2","key":"1545_CR24","doi-asserted-by":"publisher","first-page":"455","DOI":"10.1016\/j.neuroimage.2010.07.034","volume":"56","author":"A Krishnana","year":"2011","unstructured":"Krishnana A, Williams LJ, McIntosh AR, Abdi H (2011) Partial least squares (PLS) methods for neuroimaging: a tutorial and review. Neuroimage 56(2):455\u2013475","journal-title":"Neuroimage"},{"key":"1545_CR25","doi-asserted-by":"publisher","first-page":"249","DOI":"10.1007\/s00180-007-0038-z","volume":"22","author":"N Kr\u00e4mer","year":"2007","unstructured":"Kr\u00e4mer N (2007) An overview on the shrinkage properties of partial least squares regression. Comput Stat 22:249\u2013273","journal-title":"Comput Stat"},{"issue":"1","key":"1545_CR26","doi-asserted-by":"publisher","first-page":"49","DOI":"10.1002\/cem.1193","volume":"23","author":"OM Kvalheim","year":"2009","unstructured":"Kvalheim OM, Rajalahti T, Arneberg R (2009) X-tended target projection (XTP)\u00e2\u20ac\u201dcomparison with orthogonal partial least squares (OPLS) and PLS post-processing by similarity transformation (PLS+ST). J Chemom 23(1):49\u201355","journal-title":"J Chemom"},{"key":"1545_CR27","doi-asserted-by":"publisher","first-page":"1","DOI":"10.3390\/e23010018","volume":"23","author":"P Linardatos","year":"2020","unstructured":"Linardatos P, Papastefanopoulos V, Kotsiantis S (2020) Explainable AI: a review of machine learning interpretability methods. Entropy (Basel) 23:1","journal-title":"Entropy (Basel)"},{"key":"1545_CR28","doi-asserted-by":"crossref","unstructured":"L\u00ea Cao K-A, Rossouw D, Robert-Grani\u00e9 C, Besse P (2008) A sparse PLS for variable selection when integrating omics data. Stat Appl Genet Mol Biol 7(1)","DOI":"10.2202\/1544-6115.1390"},{"key":"1545_CR29","doi-asserted-by":"publisher","first-page":"187","DOI":"10.1016\/0169-7439(87)80096-5","volume":"2","author":"R Manne","year":"1987","unstructured":"Manne R (1987) Analysis of two partial-least-squares algorithms for multivariate calibration. Chemom Intell Lab Syst 2:187\u2013197","journal-title":"Chemom Intell Lab Syst"},{"key":"1545_CR30","doi-asserted-by":"publisher","first-page":"234","DOI":"10.1080\/01621459.1965.10480787","volume":"60","author":"WF Massy","year":"1965","unstructured":"Massy WF (1965) Principal components regression in exploratory statistical research. J Am Stat Assoc 60:234\u2013246","journal-title":"J Am Stat Assoc"},{"key":"1545_CR31","doi-asserted-by":"publisher","first-page":"2305","DOI":"10.1080\/03610921003778225","volume":"40","author":"G Mateos-Aparicio","year":"2011","unstructured":"Mateos-Aparicio G (2011) Partial least squares (PLS) methods: origins, evolution, and application to social sciences. Commun Stat Theor Methods 40:2305\u20132317","journal-title":"Commun Stat Theor Methods"},{"key":"1545_CR32","unstructured":"Rabbani T, Jain A, Rajkumar A, Huang F (2021) Practical and fast momentum-based power methods. In: Proceedings of the 2nd annual conference on mathematical and scientific machine learning. Proceedings of machine learning research, vol. 145, pp. 1\u201336. PMLR, Cambridge, Ma., U.S.A"},{"issue":"7","key":"1545_CR33","doi-asserted-by":"publisher","first-page":"1664","DOI":"10.1016\/j.soilbio.2007.01.022","volume":"39","author":"R Rinnan","year":"2007","unstructured":"Rinnan R, Rinnan \u00c5 (2007) Application of near infrared reflectance (NIR) and fluorescence spectroscopy to analysis of microbiological and chemical properties of arctic soil. Soil Biol Biochem 39(7):1664\u20131673","journal-title":"Soil Biol Biochem"},{"key":"1545_CR34","doi-asserted-by":"publisher","first-page":"34","DOI":"10.1007\/11752790_2","volume-title":"Subspace, latent structure and feature selection","author":"R Rosipal","year":"2006","unstructured":"Rosipal R, Kr\u00e4mer N (2006) Overview and recent advances in partial least squares. In: Saunders C, Grobelnik M, Gunn S, Shawe-Taylor J (eds) Subspace, latent structure and feature selection. Springer, Berlin Heidelberg, pp 34\u201351"},{"key":"1545_CR35","doi-asserted-by":"publisher","first-page":"461","DOI":"10.1016\/B978-0-444-87877-9.50042-X","volume-title":"Pattern recognition in practice","author":"M Sj\u00f6str\u00f6m","year":"1986","unstructured":"Sj\u00f6str\u00f6m M, Wold S, S\u00f6derstr\u00f6m B (1986) PLS discriminant plots. In: Gelsema ES, Kanal LN (eds) Pattern recognition in practice. Elsevier Science Publishers BV, North-Holland, pp 461\u2013470"},{"key":"1545_CR36","doi-asserted-by":"publisher","first-page":"185","DOI":"10.1002\/cem.1180010306","volume":"1","author":"L St\u00e5hle","year":"1987","unstructured":"St\u00e5hle L, Wold S (1987) Partial least squares analysis with cross-validation for the two-class problem: a Monte Carlo study. J Chemom 1:185\u2013196","journal-title":"J Chemom"},{"key":"1545_CR37","doi-asserted-by":"publisher","first-page":"119","DOI":"10.1002\/cem.695","volume":"16","author":"J Trygg","year":"2002","unstructured":"Trygg J, Wold S (2002) Orthogonal projections to latent structures (O-PLS). J Chemom 16:119\u2013128","journal-title":"J Chemom"},{"issue":"21","key":"1545_CR38","doi-asserted-by":"publisher","first-page":"8331","DOI":"10.1021\/jf071538s","volume":"55","author":"P Valderrama","year":"2007","unstructured":"Valderrama P, Braga JWB, Poppi RJ (2007) Variable selection, outlier detection, and figures of merit estimation in a partial least-squares regression multivariate calibration model. A case study for the determination of quality parameters in the alcohol industry by near-infrared spectroscopy. J Agric Food Chem 55(21):8331\u20138338","journal-title":"J Agric Food Chem"},{"key":"1545_CR39","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898717808","volume-title":"The matrix Eigenvalue problem","author":"DS Watkins","year":"2007","unstructured":"Watkins DS (2007) The matrix Eigenvalue problem. Society for Industrial and Applied Mathematics, Philadelphia, Pa., U.S.A"},{"key":"1545_CR40","doi-asserted-by":"publisher","first-page":"286","DOI":"10.1007\/BFb0062108","volume":"973","author":"S Wold","year":"1983","unstructured":"Wold S, Martens H, Wold H (1983) The multivariate calibration problem in chemistry solved by the PLS method. Lect Notes Math 973:286\u2013293","journal-title":"Lect Notes Math"},{"issue":"3","key":"1545_CR41","doi-asserted-by":"publisher","first-page":"735","DOI":"10.1137\/0905052","volume":"5","author":"S Wold","year":"1984","unstructured":"Wold S, Ruhe A, Wold H, Dunn WJI (1984) The collinearity problem in linear regression. The partial least squares (PLS) approach to generalized inverses. SIAM J Sci Stat Comput 5(3):735\u2013743","journal-title":"SIAM J Sci Stat Comput"},{"issue":"2","key":"1545_CR42","doi-asserted-by":"publisher","first-page":"109","DOI":"10.1016\/S0169-7439(01)00155-1","volume":"58","author":"S Wold","year":"2001","unstructured":"Wold S, Sj\u00f6str\u00f6m M, Eriksson L (2001) PLS-regression: a basic tool of chemometrics. Chemom Intell Lab Syst 58(2):109\u2013130","journal-title":"Chemom Intell Lab Syst"},{"key":"1545_CR43","unstructured":"Xu P, He B, De\u00a0Sa C, Mitliagkas I, Re C (2018) Accelerated stochastic power iteration. In: Storkey A, Perez-Cruz F (eds.) Proceedings of the Twenty-First International Conference on Artificial Intelligence and Statistics. Proceedings of Machine Learning Research, vol. 84, pp. 58\u201367. PMLR, Cambridge, Ma., U.S.A"},{"issue":"2","key":"1545_CR44","doi-asserted-by":"publisher","first-page":"301","DOI":"10.1111\/j.1467-9868.2005.00503.x","volume":"67","author":"H Zou","year":"2005","unstructured":"Zou H, Hastie T (2005) Regularization and variable selection via the elastic net. J Roy Stat Soc 67(2):301\u2013320","journal-title":"J Roy Stat Soc"}],"container-title":["Computational Statistics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00180-024-01545-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00180-024-01545-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00180-024-01545-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,9,12]],"date-time":"2024-09-12T12:03:39Z","timestamp":1726142619000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00180-024-01545-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,9,12]]},"references-count":44,"alternative-id":["1545"],"URL":"https:\/\/doi.org\/10.1007\/s00180-024-01545-7","relation":{},"ISSN":["0943-4062","1613-9658"],"issn-type":[{"type":"print","value":"0943-4062"},{"type":"electronic","value":"1613-9658"}],"subject":[],"published":{"date-parts":[[2024,9,12]]},"assertion":[{"value":"20 December 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 August 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 September 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}