{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:21:36Z","timestamp":1760145696798,"version":"build-2065373602"},"reference-count":71,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2024,8,17]],"date-time":"2024-08-17T00:00:00Z","timestamp":1723852800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Innovation Fund Denmark","award":["10599","10577","NNF23OC0083524","10.46540\/2035-00294B"],"award-info":[{"award-number":["10599","10577","NNF23OC0083524","10.46540\/2035-00294B"]}]},{"name":"Novo Nordisk Foundation","award":["10599","10577","NNF23OC0083524","10.46540\/2035-00294B"],"award-info":[{"award-number":["10599","10577","NNF23OC0083524","10.46540\/2035-00294B"]}]},{"name":"Independent Research Fund Denmark","award":["10599","10577","NNF23OC0083524","10.46540\/2035-00294B"],"award-info":[{"award-number":["10599","10577","NNF23OC0083524","10.46540\/2035-00294B"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The Parallel Factor Analysis 2 (PARAFAC2) is a multimodal factor analysis model suitable for analyzing multi-way data when one of the modes has incomparable observation units, for example, because of differences in signal sampling or batch sizes. A fully probabilistic treatment of the PARAFAC2 is desirable to improve robustness to noise and provide a principled approach for determining the number of factors, but challenging because direct model fitting requires that factor loadings be decomposed into a shared matrix specifying how the components are consistently co-expressed across samples and sample-specific orthogonality-constrained component profiles. We develop two probabilistic formulations of the PARAFAC2 model along with variational Bayesian procedures for inference: In the first approach, the mean values of the factor loadings are orthogonal leading to closed form variational updates, and in the second, the factor loadings themselves are orthogonal using a matrix Von Mises\u2013Fisher distribution. We contrast our probabilistic formulations to the conventional direct fitting algorithm based on maximum likelihood on synthetic data and real fluorescence spectroscopy and gas chromatography\u2013mass spectrometry data showing that the probabilistic formulations are more robust to noise and model order misspecification. The probabilistic PARAFAC2, thus, forms a promising framework for modeling multi-way data accounting for uncertainty.<\/jats:p>","DOI":"10.3390\/e26080697","type":"journal-article","created":{"date-parts":[[2024,8,19]],"date-time":"2024-08-19T07:37:21Z","timestamp":1724053041000},"page":"697","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Probabilistic PARAFAC2"],"prefix":"10.3390","volume":"26","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1642-7450","authenticated-orcid":false,"given":"Philip J. H.","family":"J\u00f8rgensen","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics and Computer Science, Technical University of Denmark, 2800 Kongens Lyngby, Denmark"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6713-9144","authenticated-orcid":false,"given":"S\u00f8ren F.","family":"Nielsen","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics and Computer Science, Technical University of Denmark, 2800 Kongens Lyngby, Denmark"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0258-7151","authenticated-orcid":false,"given":"Jesper L.","family":"Hinrich","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics and Computer Science, Technical University of Denmark, 2800 Kongens Lyngby, Denmark"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6927-8869","authenticated-orcid":false,"given":"Mikkel N.","family":"Schmidt","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics and Computer Science, Technical University of Denmark, 2800 Kongens Lyngby, Denmark"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8606-7641","authenticated-orcid":false,"given":"Kristoffer H.","family":"Madsen","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics and Computer Science, Technical University of Denmark, 2800 Kongens Lyngby, Denmark"},{"name":"Danish Research Centre for Magnetic Resonance, Centre for Functional and Diagnostic Imaging and Research, Copenhagen University Hospital Amager and Hvidovre, 2650 Hvidovre, Denmark"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4985-4368","authenticated-orcid":false,"given":"Morten","family":"M\u00f8rup","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics and Computer Science, Technical University of Denmark, 2800 Kongens Lyngby, Denmark"}]}],"member":"1968","published-online":{"date-parts":[[2024,8,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1007\/BF02310791","article-title":"Analysis of individual differences in multidimensional scaling via an n-way generalization of \u201cEckart-Young\u201d decomposition","volume":"35","author":"Carroll","year":"1970","journal-title":"Psychometrika"},{"key":"ref_2","first-page":"84","article-title":"Foundations of the PARAFAC procedure: Models and conditions for an \u201cexplanatory\u201d multimodal factor analysis","volume":"16","author":"Harshman","year":"1970","journal-title":"UCLA Work. 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