{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:53:02Z","timestamp":1760237582815,"version":"build-2065373602"},"reference-count":29,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2022,8,24]],"date-time":"2022-08-24T00:00:00Z","timestamp":1661299200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["EXC-2046\/1, project ID 390685689"],"award-info":[{"award-number":["EXC-2046\/1, project ID 390685689"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>This work addresses the problem of determining the number of components from sequential spectroscopic data analyzed by non-negative matrix factorization without separability assumption (SepFree NMF). These data are stored in a matrix M of dimension \u201cmeasured times\u201d versus \u201cmeasured wavenumbers\u201d and can be decomposed to obtain the spectral fingerprints of the states and their evolution over time. SepFree NMF assumes a memoryless (Markovian) process to underline the dynamics and decomposes M so that M=WH, with W representing the components\u2019 fingerprints and H their kinetics. However, the rank of this decomposition (i.e., the number of physical states in the process) has to be guessed from pre-existing knowledge on the observed process. We propose a measure for determining the number of components with the computation of the minimal memory effect resulting from the decomposition; by quantifying how much the obtained factorization is deviating from the Markovian property, we are able to score factorizations of a different number of components. In this way, we estimate the number of different entities which contribute to the observed system, and we can extract kinetic information without knowing the characteristic spectra of the single components. This manuscript provides the mathematical background as well as an analysis of computer generated and experimental sequentially measured Raman spectra.<\/jats:p>","DOI":"10.3390\/a15090297","type":"journal-article","created":{"date-parts":[[2022,8,24]],"date-time":"2022-08-24T21:03:51Z","timestamp":1661375031000},"page":"297","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["SepFree NMF: A Toolbox for Analyzing the Kinetics of Sequential Spectroscopic Data"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9836-8297","authenticated-orcid":false,"given":"Renata","family":"Sechi","sequence":"first","affiliation":[{"name":"Furukawa Electric Institute of Technology, 1158 Budapest, Hungary"},{"name":"Zuse Institute Berlin (ZIB), Takustr. 7, 14195 Berlin, Germany"},{"name":"Department of Inorganic and Analytical Chemistry, Budapest University of Technology and Economics, 1111 Budapest, Hungary"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4373-3672","authenticated-orcid":false,"given":"Konstantin","family":"Fackeldey","sequence":"additional","affiliation":[{"name":"Zuse Institute Berlin (ZIB), Takustr. 7, 14195 Berlin, Germany"},{"name":"Institute for Mathematics, TU Berlin, Strasse des 17. Juni 135, 10623 Berlin, Germany"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1428-349X","authenticated-orcid":false,"given":"Surahit","family":"Chewle","sequence":"additional","affiliation":[{"name":"Zuse Institute Berlin (ZIB), Takustr. 7, 14195 Berlin, Germany"},{"name":"Federal Institute for Materials Research and Testing (BAM), Richard-Willst\u00e4tter-Stra\u00dfe 11, 12489 Berlin, Germany"}]},{"given":"Marcus","family":"Weber","sequence":"additional","affiliation":[{"name":"Zuse Institute Berlin (ZIB), Takustr. 7, 14195 Berlin, Germany"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Risken, H. (1996). The Fokker-Planck Equation, Springer.","DOI":"10.1007\/978-3-642-61544-3"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"983","DOI":"10.1103\/PhysRev.124.983","article-title":"Memory Effects in Irreversible Thermodynamics","volume":"124","author":"Zwanzig","year":"1961","journal-title":"Phys. Rev."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"530","DOI":"10.1111\/fcp.12052","article-title":"Avidity and positive allosteric modulation\/cooperativity act hand in hand to increase the residence time of bivalent receptor ligands","volume":"28","author":"Vauquelin","year":"2014","journal-title":"Fundam. Clin. Pharmacol."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"152707","DOI":"10.1063\/1.4993228","article-title":"Transition path times reveal memory effects and anomalous diffusion in the dynamics of protein folding","volume":"147","author":"Satija","year":"2017","journal-title":"J. Chem. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"318","DOI":"10.1137\/13091124X","article-title":"Computing the Minimal Rebinding Effect Included in a Given Kinetics","volume":"12","author":"Weber","year":"2014","journal-title":"Multiscale Model. Simul."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"575","DOI":"10.1007\/s10910-020-01201-7","article-title":"Analyzing Raman spectral data without separabiliy assumption","volume":"59","author":"Fackeldey","year":"2021","journal-title":"J. Math. Chem."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Schrader, B. (1995). Infrared and Raman Spectroscopy, Wiley-VCH.","DOI":"10.1002\/9783527615438"},{"key":"ref_8","unstructured":"Ferraro, J.R., Nakamotot, K., and Brown, C.W. (2003). Introductory Raman Spectroscopy, Academic Press. [2nd ed.]."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Smith, E., and Dent, G. (2005). Modern Raman Spectroscopy, Wiley-VCH.","DOI":"10.1002\/0470011831"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1016\/j.aca.2012.12.028","article-title":"Multivariate curve resolution: A review of advanced and tailored applications and challenges","volume":"765","author":"Ruckebusch","year":"2013","journal-title":"Anal. Chim. Acta"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"11701","DOI":"10.1021\/ja2038084","article-title":"Using Covalent Dimers of Human Carbonic Anhydrase II To Model Bivalency in Immunoglobulins","volume":"133","author":"Mack","year":"2011","journal-title":"J. Am. Chem. Soc."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2698","DOI":"10.1021\/ja992648l","article-title":"Design, Synthesis, and Characterization of a High-Affinity Trivalent System Derived from Vancomycin and l-Lys-d-Ala-d-Ala","volume":"122","author":"Rao","year":"2000","journal-title":"J. Am. Chem. Soc."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"25659","DOI":"10.1073\/pnas.1902909116","article-title":"Mechanisms of noncanonical binding dynamics in multivalent protein\u2013protein interactions","volume":"116","author":"Errington","year":"2019","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1464","DOI":"10.1177\/0003702816662600","article-title":"Using separable nonnegative matrix factorization techniques for the analysis of time-resolved raman spectra","volume":"70","author":"Luce","year":"2016","journal-title":"Appl. Spectrosc."},{"key":"ref_15","first-page":"231","article-title":"Fast Conical Hull Algorithms for Near-separable Non-negative Matrix Factorization","volume":"Volume 28","author":"Dasgupta","year":"2013","journal-title":"Proceedings of the 30th International Conference on Machine Learning"},{"key":"ref_16","unstructured":"Kijima, M. (2013). Markov Processes for Stochastic Modeling, Springer."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1364","DOI":"10.1137\/070709967","article-title":"On the Complexity of Nonnegative Matrix Factorization","volume":"20","author":"Vavasis","year":"2010","journal-title":"SIAM J. Optim."},{"key":"ref_18","unstructured":"Sechi, R. (2021). Unravelling the Kinetics of Time-Resolved Spectra by Matrix-Factorization without Separability Assumption and by Markov State Modeling with PCCA+ Projection. [Master\u2019s Thesis, FU Berlin]."},{"key":"ref_19","unstructured":"R\u00f6blitz, S. (2009). Statistical Error Estimation and Grid-Free Hierarchical Refinement in Conformation Dynamics. [Ph.D. Thesis, FU Berlin]."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"024103","DOI":"10.1063\/1.2404953","article-title":"A coarse graining method for the identification of transition rates between molecular conformations","volume":"126","author":"Kube","year":"2007","journal-title":"J. Chem. Phys."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1307","DOI":"10.1007\/s00332-015-9258-5","article-title":"A Data\u2013Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition","volume":"25","author":"Williams","year":"2015","journal-title":"J. Nonlinear Sci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"A1482","DOI":"10.1137\/17M115414X","article-title":"Scalable Extended Dynamic Mode Decomposition Using Random Kernel Approximation","volume":"41","author":"Degennaro","year":"2019","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_23","unstructured":"Klus, S. (2020). Data-Driven Analysis of Complex Dynamical Systems. [Ph.D. Thesis, FU Berlin]."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Heida, M., Kantner, M., and Stephan, A. (2021). Consistency and convergence for a family of finite volume discretizations of the Fokker-Planck operator. arXiv.","DOI":"10.1051\/m2an\/2021078"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"9029","DOI":"10.1039\/C5CE01583A","article-title":"Direct evidence of polyamorphism in paracetamol","volume":"17","author":"Rademann","year":"2015","journal-title":"CrystEngComm"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1023\/A:1010184805966","article-title":"In situ characterization of polymorphic forms: The potential of Raman techniques","volume":"57","author":"Szelagiewicz","year":"1999","journal-title":"J. Therm. Anal. Calorim."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"3181","DOI":"10.1039\/b822882e","article-title":"Crystal structure determination of the elusive paracetamol Form III","volume":"22","author":"Perrin","year":"2009","journal-title":"Chem. Commun."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1391","DOI":"10.1021\/je990124v","article-title":"Solubility of paracetamol in pure solvents","volume":"44","author":"Granberg","year":"1999","journal-title":"J. Chem. Eng. Data"},{"key":"ref_29","unstructured":"R\u00f6hm, J. (2017). Non-Negative Matrix Factorization for Raman Data Spectral Analysis. [Master\u2019s Thesis, Technische Universitat Berlin]."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/15\/9\/297\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:14:22Z","timestamp":1760141662000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/15\/9\/297"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,8,24]]},"references-count":29,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2022,9]]}},"alternative-id":["a15090297"],"URL":"https:\/\/doi.org\/10.3390\/a15090297","relation":{},"ISSN":["1999-4893"],"issn-type":[{"type":"electronic","value":"1999-4893"}],"subject":[],"published":{"date-parts":[[2022,8,24]]}}}