{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,6]],"date-time":"2026-05-06T22:49:30Z","timestamp":1778107770124,"version":"3.51.4"},"reference-count":19,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2021,1,24]],"date-time":"2021-01-24T00:00:00Z","timestamp":1611446400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004663","name":"Ministry of Science and Technology, Taiwan","doi-asserted-by":"publisher","award":["MOST 108-2221-E-155-013"],"award-info":[{"award-number":["MOST 108-2221-E-155-013"]}],"id":[{"id":"10.13039\/501100004663","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Discovering densely-populated regions in a dataset of data points is an essential task for density-based clustering. To do so, it is often necessary to calculate each data point\u2019s local density in the dataset. Various definitions for the local density have been proposed in the literature. These definitions can be divided into two categories: Radius-based and k Nearest Neighbors-based. In this study, we find the commonality between these two types of definitions and propose a canonical form for the local density. With the canonical form, the pros and cons of the existing definitions can be better explored, and new definitions for the local density can be derived and investigated.<\/jats:p>","DOI":"10.3390\/sym13020185","type":"journal-article","created":{"date-parts":[[2021,1,25]],"date-time":"2021-01-25T12:28:31Z","timestamp":1611577711000},"page":"185","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Generalizing Local Density for Density-Based Clustering"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6844-1182","authenticated-orcid":false,"given":"Jun-Lin","family":"Lin","sequence":"first","affiliation":[{"name":"Department of Information Management, Yuan Ze University, Taoyuan 32003, Taiwan"},{"name":"Innovation Center for Big Data and Digital Convergence, Yuan Ze University, Taoyuan 32003, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,1,24]]},"reference":[{"key":"ref_1","unstructured":"Han, J., Kamber, M., and Pei, J. (2011). Data Mining: Concepts and Techniques, Morgan Kaufmann Publishers Inc.. [3rd ed.]."},{"key":"ref_2","unstructured":"Ester, M., Kriegel, H.-P., Sander, J., and Xu, X. (1996, January 2\u20134). A density-based algorithm for discovering clusters 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, Portland, OR, USA."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Ankerst, M., Breunig, M.M., Kriegel, H.-P., and Sander, J. (1999, January 1\u20133). OPTICS: Ordering points to identify the clustering structure. Proceedings of the 1999 ACM SIGMOD International Conference on Management of Data, Philadelphia, PA, USA.","DOI":"10.1145\/304182.304187"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1492","DOI":"10.1126\/science.1242072","article-title":"Clustering by fast search and find of density peaks","volume":"344","author":"Rodriguez","year":"2014","journal-title":"Science"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"208","DOI":"10.1016\/j.knosys.2017.07.010","article-title":"Adaptive density peak clustering based on K-nearest neighbors with aggregating strategy","volume":"133","author":"Liu","year":"2017","journal-title":"Knowl. Based Syst."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/j.ins.2016.03.011","article-title":"Robust clustering by detecting density peaks and assigning points based on fuzzy weighted K-nearest neighbors","volume":"354","author":"Xie","year":"2016","journal-title":"Inf. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1016\/j.knosys.2016.02.001","article-title":"Study on density peaks clustering based on k-nearest neighbors and principal component analysis","volume":"99","author":"Du","year":"2016","journal-title":"Knowl. Based Syst."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Liu, Y., Liu, D., Yu, F., and Ma, Z. (2020). A Double-Density Clustering Method Based on \u201cNearest to First in\u201d Strategy. Symmetry, 12.","DOI":"10.3390\/sym12050747"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Lin, J.-L., Kuo, J.-C., and Chuang, H.-W. (2020). Improving Density Peak Clustering by Automatic Peak Selection and Single Linkage Clustering. Symmetry, 12.","DOI":"10.3390\/sym12071168"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Lv, Y., Liu, M., and Xiang, Y. (2020). Fast Searching Density Peak Clustering Algorithm Based on Shared Nearest Neighbor and Adaptive Clustering Center. Symmetry, 12.","DOI":"10.3390\/sym12122014"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1016\/j.patcog.2007.04.010","article-title":"Robust path-based spectral clustering","volume":"41","author":"Chang","year":"2008","journal-title":"Pattern Recognit."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Fu, L., and Medico, E. (2007). FLAME, a novel fuzzy clustering method for the analysis of DNA microarray data. BMC Bioinform., 8.","DOI":"10.1186\/1471-2105-8-3"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1145\/1217299.1217303","article-title":"Clustering aggregation","volume":"1","author":"Gionis","year":"2007","journal-title":"ACM Trans. Knowl. Discov. Data"},{"key":"ref_14","unstructured":"Jain, A.K., and Law, M.H. (2005, January 20\u201322). Data clustering: A user\u2019s dilemma. Proceedings of the 2005 International Conference on Pattern Recognition and Machine Intelligence, Kolkata, India."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1273","DOI":"10.1109\/TPAMI.2002.1033218","article-title":"A maximum variance cluster algorithm","volume":"24","author":"Veenman","year":"2002","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"68","DOI":"10.1109\/T-C.1971.223083","article-title":"Graph-Theoretical Methods for Detecting and Describing Gestalt Clusters","volume":"100","author":"Zahn","year":"1971","journal-title":"IEEE Trans. Comput."},{"key":"ref_17","unstructured":"K\u00e4rkk\u00e4inen, I., and Fr\u00e4nti, P. (2002). Dynamic Local Search Algorithm for the Clustering Problem, University of Joensuu. A-2002-6."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"761","DOI":"10.1016\/j.patcog.2005.09.012","article-title":"Iterative shrinking method for clustering problems","volume":"39","author":"Virmajoki","year":"2006","journal-title":"Pattern Recognit."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2173","DOI":"10.1109\/TKDE.2016.2551240","article-title":"Set Matching Measures for External Cluster Validity","volume":"28","author":"Rezaei","year":"2016","journal-title":"IEEE Trans. Knowl. Data Eng."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/2\/185\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:14:48Z","timestamp":1760159688000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/2\/185"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,24]]},"references-count":19,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2021,2]]}},"alternative-id":["sym13020185"],"URL":"https:\/\/doi.org\/10.3390\/sym13020185","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1,24]]}}}