Citation: Isah, A.; Shin, H.; Oh, S.; Oh, S.; Aliyu, I.; Um, T.-w.; Kim, J. Digital Twins Temporal Dependencies-Based on Time Series Using Multivariate Long Short-Term Memory. Electronics 2023, 12, 4187. https://doi.org/10.3390/ electronics12194187 Academic Editors: Martin Reisslein and Franco Cicirelli Received: 31 July 2023 Revised: 5 October 2023 Accepted: 7 October 2023 Published: 9 October 2023 Copyright: © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). electronics Article Digital Twins TemporalDependencies-Based on Time Series Using Multivariate Long Short-Term Memory Abubakar Isah 1 , Hyeju Shin 1 , Seungmin Oh 1 , Sangwon Oh 1 , Ibrahim Aliyu 1 , Tai-won Um 2, * and Jinsul Kim 1, * 1 Department of ICT Convergence System Engineering, Chonnam National University, Gwangju 61186, Republic of Korea; abubakarisah@jnu.ac.kr (A.I.); sinhye102@jnu.ac.kr (H.S.); 216655@jnu.ac.kr (S.O.); osw0782@naver.com (S.O.); aliyu@jnu.ac.kr (I.A.) 2 Graduate School of Data Science, Chonnam National University, Gwangju 61186, Republic of Korea * Correspondence: stwum@jnu.ac.kr (T.-w.U.); jsworld@jnu.ac.kr (J.K.) Abstract: Digital Twins, which are virtual representations of physical systems mirroring their be- havior, enable real-time monitoring, analysis, and optimization. Understanding and identifying the temporal dependencies included in the multivariate time series data that characterize the behavior of the system are crucial for improving the effectiveness of Digital Twins. Long Short-Term Memory (LSTM) networks have been used to represent complex temporal dependencies and identify long- term links in the Industrial Internet of Things (IIoT). This paper proposed a Digital Twin temporal dependency technique using LSTM to capture the long-term dependencies in IIoT time series data, estimate the lag between the input and intended output, and handle missing data. Autocorrelation analysis showed the lagged links between variables, aiding in the discovery of temporal dependen- cies. The system evaluated the LSTM model by providing it with a set of previous observations and asking it to forecast the value at future time steps. We conducted a comparison between our model and six baseline models, utilizing both the Smart Water Treatment (SWaT) and Building Automation Transaction (BATADAL) datasets. Our model’s effectiveness in capturing temporal dependencies was assessed through the analysis of the Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF). The results of our experiments demonstrate that our enhanced model achieved a better long-term prediction performance. Keywords: temporal dependency; Digital Twins; LSTM; multivariate time series 1. Introduction Digital Twins connect the real and virtual worlds [1,2], offering simulations, projec- tions, and insights that can be applied to decision-making, optimization, and maintenance tasks [3]. The Digital Twin can learn and capture the underlying patterns and dependencies of the dynamic system by examining the historical and real-time data of the multivariate time series and training an LSTM network with this data [4]. This enables it to generate precise simulations and predictions. Digital Twin is a technology that is still under devel- opment, but it has the potential to revolutionize the way we manage assets. Digital Twins are virtual representations of physical assets that can be used to simulate, monitor, and optimize the performance of those assets [5]. Temporal dependencies are the patterns and relationships that develop over time between the variables in multivariate time series data. These dependencies can include seasonality, trends, lagged relationships, sequential patterns, and other temporal structures. Temporal dependencies and multivariate time series analysis are crucial in many areas, including weather forecasting, industrial operations, and others [6]. To make effective decisions and maximize system performance, it is essential to have the ability to accurately analyze and forecast [7] the behavior of complex dynamics. Time series data can be utilized Electronics 2023, 12, 4187. https://doi.org/10.3390/electronics12194187 https://www.mdpi.com/journal/electronics