## 5 Conclusions  
This study delineates a method and associated algorithms for tracking the energy consumption in injection molding operations, which are the most energy-intensive processes in the plastics industry. The energy profiles collected from a real industrial plant were analyzed using data mining algorithms to identify distinct clusters that represent the specific energy consumption profiles of various production variants. We utilized mixtures of regression models due to their flexibility in modeling heterogeneous time series and clustering time series in an unsupervised machine-learning framework. We opted to analyze the mixture models because of their robust statistical foundation and the interpretability of their results. Furthermore, because clusters correspond to the model components, the number of clusters can be easily determined by utilizing the likelihood function for the components in the mixture model. Mixture regression models assume that observations are independent and identically distributed, which may not be true for time-series data. Additionally, the model assumes that the errors are normally distributed, which may not be the case for the energy consumption data. Therefore, we developed a method for clustering autocorrelated time-series data that combines the APECM algorithm with B-spline and polynomial regressions within a mixture regression framework. This approach has been shown to outperform popular techniques such as the \(K\)-means and SC algorithms through its application to real-world data. Our analysis led to the accurate grouping of the energy consumption profiles, where each cluster was associated with 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. Through these profiles, we identified information related to the shape of energy consumption, which provided insights into reducing the electrical demand of the plant. Our research presents a parametric model-based approach that provides a precise representation of energy consumption profiles. This method is particularly well suited for practical industrial applications, as it is an automated, data-driven approach within an unsupervised training framework. The selection of an appropriate model structure, including the types and orders of the regressors and the number of clusters, is a critical task that must be performed by the analyst. To address this challenge, we employed a BIC-based rule for model selection. Figure 11: Graphical depiction of the centroids for each of the six clusters (1–6) resulting from the implementation of polynomial regression mixtures. Varying colors differentiate profiles. The primary limitation of the mixture regression models presented in this study is their vulnerability to overfitting, particularly when the number of components is large or when the data are noisy. Overfitting can result in poor generalization performance and inaccurate cluster labels. In future work, we will focus on implementing techniques to mitigate overfitting in mixture regression models for clustering time-series data. These techniques may include the application of regularization methods, such as \(L1\) or \(L2\) regularization [38], and feature selection methods, such as backward elimination or LASSO [39]. Additionally, shrinkage methods, such as ridge regression, may be employed to penalize the coefficients and reduce overfitting [40]. Further work in this area includes the incorporation of additional variables into the mixture models to assess their impact on the clustering performance and the exploration of supervised learning techniques, such as support vector machines (SVMs) or neural networks, to determine their potential to improve the accuracy of clustering. Conceptualization, M.P. ; methodology, M.P. ; software, M.P. ; validation, M.P. and G.P. ; resources, M.M. and G.P. ; writing--original draft preparation, M.P. ; writing--review and editing, M.P. and M.M. ; supervision, M.M. and G.P.