{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,15]],"date-time":"2026-01-15T12:39:41Z","timestamp":1768480781890,"version":"3.49.0"},"reference-count":40,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2023,11,15]],"date-time":"2023-11-15T00:00:00Z","timestamp":1700006400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Puglia Region (Italy)\u2014Project TANDEM \u2018digiTAl twiN green aDvancEd Manufacturing\u2019","award":["P.O. PUGLIA FESR 2014-2020"],"award-info":[{"award-number":["P.O. PUGLIA FESR 2014-2020"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>Considering the issue of energy consumption reduction in industrial plants, we investigated a clustering method for mining the time-series data related to energy consumption. The industrial case study considered in our work is one of the most energy-intensive processes in the plastics industry: the plastic injection molding process. Concerning the industrial setting, the energy consumption of the injection molding machine was monitored across multiple injection molding cycles. The collected data were then analyzed to establish patterns and trends in the energy consumption of the injection molding process. To this end, we considered mixtures of regression models given their flexibility in modeling heterogeneous time series and clustering time series in an unsupervised machine learning framework. Given the assumption of autocorrelated data and exogenous variables in the mixture model, we implemented an algorithm for model fitting that combined autocorrelated observations with spline and polynomial regressions. Our results demonstrate an accurate grouping of energy-consumption profiles, where each cluster is related to a specific production schedule. The clustering method also provides a unique profile of energy consumption for each cluster, depending on the production schedule and regression approach (i.e., spline and polynomial). According to these profiles, information related to the shape of energy consumption was identified, providing insights into reducing the electrical demand of the plant.<\/jats:p>","DOI":"10.3390\/a16110524","type":"journal-article","created":{"date-parts":[[2023,11,15]],"date-time":"2023-11-15T10:57:46Z","timestamp":1700045866000},"page":"524","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Utilizing Mixture Regression Models for Clustering Time-Series Energy Consumption of a Plastic Injection Molding Process"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3712-7932","authenticated-orcid":false,"given":"Massimo","family":"Pacella","sequence":"first","affiliation":[{"name":"Department of Engineering for Innovation, University of Salento, Piazza Tancredi 7, 73100 Lecce, Italy"}]},{"given":"Matteo","family":"Mangini","sequence":"additional","affiliation":[{"name":"DGS SPA, Via Paolo di Dono 73, 00142 Roma, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1507-8188","authenticated-orcid":false,"given":"Gabriele","family":"Papadia","sequence":"additional","affiliation":[{"name":"Department of Engineering for Innovation, University of Salento, Piazza Tancredi 7, 73100 Lecce, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2023,11,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1016\/j.jclepro.2015.07.119","article-title":"Environmental impact analysis of the injection molding process: Analysis of the processing of high-density polyethylene parts","volume":"108","author":"Elduque","year":"2015","journal-title":"J. Clean. Prod."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"943","DOI":"10.1002\/pen.26256","article-title":"Strategic cost and sustainability analyses of injection molding and material extrusion additive manufacturing","volume":"63","author":"Kazmer","year":"2023","journal-title":"Polym. Eng. Sci."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"117784","DOI":"10.1016\/j.jclepro.2019.117784","article-title":"Classification and clustering of the German plastic industry with a special focus on the implementation of low and high temperature waste heat","volume":"238","author":"Dunkelberg","year":"2019","journal-title":"J. Clean. Prod."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"121375","DOI":"10.1016\/j.jclepro.2020.121375","article-title":"Mold cooling in thermoplastics injection molding: Effectiveness and energy efficiency","volume":"264","author":"Rashid","year":"2020","journal-title":"J. Clean. Prod."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"857","DOI":"10.1002\/pen.20335","article-title":"Comparison of injection molding machine performance","volume":"45","author":"Kelly","year":"2005","journal-title":"Polym. Eng. Sci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"118355","DOI":"10.1016\/j.jclepro.2019.118355","article-title":"Research on energy consumption of injection molding machine driven by five different types of electro-hydraulic power units","volume":"242","author":"Liu","year":"2020","journal-title":"J. Clean. Prod."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Elduque, A., Elduque, D., Pina, C., Claver\u00eda, I., and Javierre, C. (2018). Electricity consumption estimation of the polymer material injection-molding manufacturing process: Empirical model and application. Materials, 11.","DOI":"10.3390\/ma11091740"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"342","DOI":"10.1016\/j.procir.2017.11.042","article-title":"Analysis of process parameters affecting energy consumption in plastic injection moulding","volume":"69","author":"Meekers","year":"2018","journal-title":"Procedia CIRP"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"JAMDSM0055","DOI":"10.1299\/jamdsm.2020jamdsm0055","article-title":"Simulated annealing based simulation method for minimizing electricity cost considering production line scheduling including injection molding machines","volume":"14","author":"Ishihara","year":"2020","journal-title":"J. Adv. Mech. Des. Syst. Manuf."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"51518","DOI":"10.1007\/s11356-023-26007-3","article-title":"Generative machine learning-based multi-objective process parameter optimization towards energy and quality of injection molding","volume":"30","author":"Wu","year":"2023","journal-title":"Environ. Sci. Pollut. Res."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"804","DOI":"10.1016\/j.jclepro.2017.02.053","article-title":"Energy monitoring of plastic injection molding process running with hydraulic injection molding machines","volume":"148","author":"Mianehrow","year":"2017","journal-title":"J. Clean. Prod."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"100771","DOI":"10.1016\/j.segan.2022.100771","article-title":"A novel cluster-specific analysis framework for demand-side management and net metering using smart meter data","volume":"31","author":"Ahir","year":"2022","journal-title":"Sustain. Energy Grids Netw."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1016\/j.is.2015.04.007","article-title":"Time-series clustering\u2014A decade review","volume":"53","author":"Aghabozorgi","year":"2015","journal-title":"Inf. Syst."},{"key":"ref_14","first-page":"100001","article-title":"A benchmark study on time series clustering","volume":"1","author":"Javed","year":"2020","journal-title":"Mach. Learn. Appl."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"611","DOI":"10.1198\/016214502760047131","article-title":"Model-based clustering, discriminant analysis, and density estimation","volume":"97","author":"Fraley","year":"2002","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_16","first-page":"1","article-title":"K-means clustering of electricity consumers using time-domain features from smart meter data","volume":"10","author":"Okereke","year":"2023","journal-title":"J. Electr. Syst. Inf. Technol."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"121607","DOI":"10.1016\/j.apenergy.2023.121607","article-title":"Interpretable building energy consumption forecasting using spectral clustering algorithm and temporal fusion transformers architecture","volume":"349","author":"Zheng","year":"2023","journal-title":"Appl. Energy"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Pacella, M., and Papadia, G. (2022). Finite Mixture Models for Clustering Auto-Correlated Sales Series Data Influenced by Promotions. Computation, 10.","DOI":"10.3390\/computation10020023"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Czepiel, M., Ba\u0144kosz, M., and Sobczak-Kupiec, A. (2023). Advanced Injection Molding Methods. Materials, 16.","DOI":"10.3390\/ma16175802"},{"key":"ref_20","first-page":"101","article-title":"Curve Clustering with Random Effects Regression Mixtures","volume":"Volume R4","author":"Bishop","year":"2003","journal-title":"Proceedings of the Ninth International Workshop on Artificial Intelligence and Statistics"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1198\/016214503000189","article-title":"Clustering for Sparsely Sampled Functional Data","volume":"98","author":"James","year":"2003","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1361","DOI":"10.1016\/j.csda.2008.11.019","article-title":"Simultaneous curve registration and clustering for functional data","volume":"53","author":"Liu","year":"2009","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"2308","DOI":"10.1080\/00949655.2015.1109096","article-title":"Unsupervised learning of regression mixture models with unknown number of components","volume":"86","author":"Chamroukhi","year":"2016","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"567","DOI":"10.1002\/sam.10143","article-title":"Model-based clustering of regression time series data via APECM\u2014an AECM algorithm sung to an even faster beat","volume":"4","author":"Chen","year":"2011","journal-title":"Stat. Anal. Data Mining ASA Data Sci. J."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"513","DOI":"10.1080\/00401706.2013.826146","article-title":"A parallel EM algorithm for model-based clustering applied to the exploration of large spatio-temporal data","volume":"55","author":"Chen","year":"2013","journal-title":"Technometrics"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1214\/aos\/1176344136","article-title":"Estimating the Dimension of a Model","volume":"6","author":"Schwarz","year":"1978","journal-title":"Ann. Stat."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"716","DOI":"10.1109\/TAC.1974.1100705","article-title":"A new look at the statistical model identification","volume":"19","author":"Akaike","year":"1974","journal-title":"IEEE Trans. Autom. Control."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2523","DOI":"10.1214\/08-AOS651","article-title":"Hypothesis test for normal mixture models: The EM approach","volume":"37","author":"Chen","year":"2009","journal-title":"Ann. Stat."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"180","DOI":"10.1016\/j.csda.2018.05.005","article-title":"Hypothesis testing for finite mixture models","volume":"132","author":"Wichitchan","year":"2019","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Ahmed, M., Seraj, R., and Islam, S.M. (2020). The k-means algorithm: A comprehensive survey and performance evaluation. Electronics, 9.","DOI":"10.3390\/electronics9081295"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Holder, C., Middlehurst, M., and Bagnall, A. (2023). A review and evaluation of elastic distance functions for time series clustering. Knowl. Inf. Syst., 1\u201345.","DOI":"10.1007\/s10115-023-01952-0"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1477","DOI":"10.1007\/s00521-013-1439-2","article-title":"The latest research progress on spectral clustering","volume":"24","author":"Jia","year":"2014","journal-title":"Neural Comput. Appl."},{"key":"ref_33","unstructured":"Zhang, J., and Shen, Y. (2015, January 28\u201330). Review on spectral methods for clustering. Proceedings of the 2015 34th Chinese Control Conference (CCC), Hangzhou, China."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Pacella, M., and Papadia, G. (2020). Fault diagnosis by multisensor data: A data-driven approach based on spectral clustering and pairwise constraints. Sensors, 20.","DOI":"10.3390\/s20247065"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1212","DOI":"10.1109\/TKDE.2019.2903410","article-title":"Ultra-scalable spectral clustering and ensemble clustering","volume":"32","author":"Huang","year":"2019","journal-title":"IEEE Trans. Knowl. Data Eng."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/03610927408827101","article-title":"A dendrite method for cluster analysis","volume":"3","author":"Harabasz","year":"1974","journal-title":"Commun.-Stat.-Theory Methods"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"224","DOI":"10.1109\/TPAMI.1979.4766909","article-title":"A cluster separation measure","volume":"PAMI-1","author":"Davies","year":"1979","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Kolluri, J., Kotte, V.K., Phridviraj, M.S.B., and Razia, S. (2020, January 15\u201317). Reducing overfitting problem in machine learning using novel L1\/4 regularization method. Proceedings of the 2020 4th International Conference on Trends in Electronics and Informatics (ICOEI)(48184), Tirunelveli, India.","DOI":"10.1109\/ICOEI48184.2020.9142992"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/j.csda.2017.09.003","article-title":"A globally convergent algorithm for lasso-penalized mixture of linear regression models","volume":"119","author":"Nguyen","year":"2018","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"450","DOI":"10.1109\/TNNLS.2020.2978755","article-title":"Data clustering via uncorrelated ridge regression","volume":"32","author":"Zhang","year":"2020","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/16\/11\/524\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T21:23:40Z","timestamp":1760131420000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/16\/11\/524"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,11,15]]},"references-count":40,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2023,11]]}},"alternative-id":["a16110524"],"URL":"https:\/\/doi.org\/10.3390\/a16110524","relation":{},"ISSN":["1999-4893"],"issn-type":[{"value":"1999-4893","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,11,15]]}}}