{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:56:19Z","timestamp":1760144179896,"version":"build-2065373602"},"reference-count":39,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2024,3,29]],"date-time":"2024-03-29T00:00:00Z","timestamp":1711670400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>Clustering algorithms are usually iterative procedures. In particular, when the clustering algorithm aims to optimise an objective function like in k-means clustering or Gaussian mixture models, iterative heuristics are required due to the high non-linearity of the objective function. This implies higher computational costs and the risk of finding only a local optimum and not the global optimum of the objective function. In this paper, we demonstrate that in the case of one-dimensional clustering with one main and one noise cluster, one can formulate an objective function, which permits a closed-form solution with no need for an iteration scheme and the guarantee of finding the global optimum. We demonstrate how such an algorithm can be applied in the context of laboratory medicine as a method to estimate reference intervals that represent the range of \u201cnormal\u201d values.<\/jats:p>","DOI":"10.3390\/a17040143","type":"journal-article","created":{"date-parts":[[2024,3,29]],"date-time":"2024-03-29T06:33:16Z","timestamp":1711693996000},"page":"143","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["An Objective Function-Based Clustering Algorithm with a Closed-Form Solution and Application to Reference Interval Estimation in Laboratory Medicine"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9613-182X","authenticated-orcid":false,"given":"Frank","family":"Klawonn","sequence":"first","affiliation":[{"name":"Institute for Information Engineering, Ostfalia University, 38302 Braunschweig, Germany"},{"name":"Biostatistics Group, Helmholtz Centre for Infection Research, 38124 Braunschweig, Germany"}]},{"given":"Georg","family":"Hoffmann","sequence":"additional","affiliation":[{"name":"Medizinischer Fachverlag Trillium GmbH, 82284 Grafrath, Germany"}]}],"member":"1968","published-online":{"date-parts":[[2024,3,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"James, G., Witten, D., Hastie, T., and Tibshirani, R. (2021). An Introduction to Statistical Learning, Springer. [2nd ed.].","DOI":"10.1007\/978-1-0716-1418-1"},{"key":"ref_2","unstructured":"Duda, R.O., Hart, P.E., and Stork, D.G. (2000). Pattern Classification, Wiley. [2nd ed.]."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Giordani, P., Brigida Ferraro, M., and Martella, F. (2020). An Introduction to Clustering with R, Springer.","DOI":"10.1007\/978-981-13-0553-5"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"\u017bytkow, J.M., and Rauch, J. (1999). Principles of Data Mining and Knowledge Discovery, Springer.","DOI":"10.1007\/b72280"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"6439","DOI":"10.1007\/s10462-022-10325-y","article-title":"Data clustering: Application and trends","volume":"56","author":"Oyewole","year":"2023","journal-title":"Artif. Intell. Rev."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1445","DOI":"10.1243\/09544062JMES508","article-title":"Clustering techniques and their applications in engineering","volume":"221","author":"Pham","year":"2007","journal-title":"Proc. Inst. Mech. Eng. Part J. Mech. Eng. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1117","DOI":"10.1007\/s11192-020-03717-w","article-title":"Studying the heterogeneity of European higher education institutions","volume":"125","author":"Bruni","year":"2020","journal-title":"Scientometrics"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1016\/j.patcog.2012.07.021","article-title":"An extensive comparative study of cluster validity indices","volume":"46","author":"Arbelaitz","year":"2013","journal-title":"Pattern Recognit."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Liu, Y., Li, Z., Xiong, H., Gao, X., and Wu, J. (2010, January 13\u201317). Understanding of Internal Clustering Validation Measures. Proceedings of the IEEE International Conference on Data Mining, Sydney, NSW, Australia.","DOI":"10.1109\/ICDM.2010.35"},{"key":"ref_10","unstructured":"Rend\u00f3n, E., Abundez, I.M., Gutierrez, C., Zagal, S.D., Arizmendi, A., Quiroz, E.M., and Arzate, H.E. (2011, January 29\u201331). A comparison of internal and external cluster validation indexes. Proceedings of the 2011 American Conference on Applied Mathematics and the 5th WSEAS International Conference on Computer Engineering and Applications, Puerto Morelos, Mexico."},{"key":"ref_11","unstructured":"Ester, M., Kriegel, H.-P., Sander, J., and Xu, X. (1996, January 2\u20134). A density-based algorithm for discovering clusters in large spatial databases with noise. Proceedings of the Second International Conference on Knowledge Discovery and Data Mining (KDD-96), Portland, OR, USA."},{"key":"ref_12","unstructured":"Ankerst, M., Breunig, M.M., Kriegel, H.-P., and Sander, J. (June, January 31). OPTICS: Ordering points to identify the clustering structure. Proceedings of the ACM SIGMOD International Conference on Management of Data, Philadelphia, PA, USA."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1007\/BF00337288","article-title":"Self-organized formation of topologically correct feature maps","volume":"43","author":"Kohonen","year":"1982","journal-title":"Biol. Cybern."},{"key":"ref_14","unstructured":"Arthur, D., and Vassilvitskii, S. (2007, January 7\u20139). k-means++: The advantages of careful seeding. Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms, New Orleans, LA, USA."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"511","DOI":"10.1111\/1467-9868.00082","article-title":"The EM algorithm\u2014An old folk-song sung to a fast new tune","volume":"59","author":"Meng","year":"1997","journal-title":"J. R. Stat. Soc. Ser. Stat. Methodol."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1016\/j.tcs.2010.05.034","article-title":"The planar k-means problem is NP-hard","volume":"442","author":"Mahajan","year":"2012","journal-title":"Theor. Comput. Sci."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.is.2016.02.007","article-title":"Single-pass and linear-time k-means clustering based on MapReduce","volume":"60","author":"Shahrivari","year":"2016","journal-title":"Inf. Syst."},{"key":"ref_18","unstructured":"Yi, J., Zhang, L., Wang, J., Jin, R., and Jain, A.K. (2014, January 22\u201324). A single-pass algorithm for efficiently recovering sparse cluster centers of high-dimensional data. Proceedings of the 31st International Conference on Machine Learning, Beijing, China."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Behnezhad, S., Charikar, M., Ma, W., and Tan, L.-Y. (2023, January 22\u201325). Single-Pass Streaming Algorithms for Correlation Clustering. Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), Florence, Italy.","DOI":"10.1137\/1.9781611977554.ch33"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Moewes, C., and N\u00fcrnberger, A. (2013). Computational Intelligence in Intelligent Data Analysis, Springer.","DOI":"10.1007\/978-3-642-32378-2"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1080\/01969727308546046","article-title":"A fuzzy relative of the ISODATA process and its use in detecting compact well-separated clusters","volume":"3","author":"Dunn","year":"1973","journal-title":"J. Cybern."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Bezdek, J.C. (1981). Pattern Recognition with Fuzzy Objective Function Algorithms, Plenum Press.","DOI":"10.1007\/978-1-4757-0450-1"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1016\/S0019-9958(70)80040-7","article-title":"Generalized information functions","volume":"16","year":"1970","journal-title":"Inf. Control"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"508","DOI":"10.1109\/TFUZZ.2004.840104","article-title":"Regularized linear fuzzy clustering and probabilistic PCA mixture models","volume":"13","author":"Honda","year":"2005","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_25","first-page":"406","article-title":"Characterization and detection of noise in clustering","volume":"12","year":"1991","journal-title":"Pattern Recognit. Lett."},{"key":"ref_26","unstructured":"Berthold, M.R., Lenz, H.-J., Bradley, E., Kruse, R., and Borgelt, C. (2003). Advances in Intelligent Data Analysis V, Springer."},{"key":"ref_27","first-page":"14","article-title":"Cluster analysis via the dynamic data assigning assessment algorithm","volume":"2","author":"Georgieva","year":"2006","journal-title":"Inf. Technol. Control"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1305","DOI":"10.1016\/j.asoc.2007.11.006","article-title":"Dynamic data assigning assessment clustering of streaming data","volume":"8","author":"Georgieva","year":"2008","journal-title":"Appl. Soft Comput."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1381","DOI":"10.1016\/j.procs.2016.08.183","article-title":"Exploring data sets for clusters and validating single clusters","volume":"96","author":"Klawonn","year":"2016","journal-title":"Procedia Comput. Sci."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1515\/cclm-2018-0073","article-title":"Indirect methods for reference interval determination: Review and recommendations","volume":"57","author":"Jones","year":"2019","journal-title":"Clin. Chem. Lab. Med."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"16023","DOI":"10.1038\/s41598-021-95301-2","article-title":"refineR: A novel algorithm for reference interval estimation from real-world data","volume":"11","author":"Ammer","year":"2021","journal-title":"Sci. Rep."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1043","DOI":"10.1515\/CCLM.2007.250","article-title":"A plea for intra-laboratory reference limits. Part 2. A bimodal retrospective concept for determining reference limits from intra-laboratory databases demonstrated by catalytic activity concentrations of enzymes","volume":"45","author":"Arzideh","year":"2007","journal-title":"Clin. Chem. Lab. Med."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1016\/j.cca.2009.03.057","article-title":"A new approach for the determination of reference intervals from hospital-based data","volume":"405","author":"Concordet","year":"2009","journal-title":"Clin. Chim. Acta"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1933","DOI":"10.1515\/cclm-2018-1341","article-title":"A new indirect estimation of reference intervals: Truncated minimum chi-square (TMC) approach","volume":"57","author":"Wosniok","year":"2019","journal-title":"Clin. Chem. Lab. Med."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1537","DOI":"10.1515\/CCLM.2010.319","article-title":"An appraisal of statistical procedures used in derivation of reference intervals","volume":"48","author":"Ichihara","year":"2010","journal-title":"Clin. Chem. Lab. Med."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"172","DOI":"10.3390\/stats7010011","article-title":"Importance and uncertainty of \u03bb-estimation for Box-Cox transformations to compute and verify reference intervals in laboratory medicine","volume":"7","author":"Klawonn","year":"2024","journal-title":"Stats"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1515\/labmed-2020-0005","article-title":"Quantitative laboratory results: Normal or lognormal distribution","volume":"44","author":"Klawonn","year":"2020","journal-title":"J. Lab. Med."},{"key":"ref_38","unstructured":"Misra, G. (2019). Data Processing Handbook for Complex Biological Data Sources, Academic Press."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"18308","DOI":"10.1038\/srep18308","article-title":"Antibody induced CD4 down-modulation of T cells is site-specifically mediated by CD64+ cells","volume":"5","author":"Vogel","year":"2015","journal-title":"Sci. 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