Most of these data are multivariate, with multiple values at each time step (Wilms, Rombouts, & Croux, 2021; Lee, Kim & Sim, 2024). Consequently, utilizing multivariate time series data for knowledge extraction and application to societal issues is becoming increasingly prevalent, making multivariate time series prediction a challenging task widely regarded across most industries (Bidarkota, 1998). The traditional approach to addressing multivariate time series prediction problems involves using statistical methods, such as vector autoregression and autoregressive distributed lag (ARDL) models (Qu, Huang, She, Liad, & Lai, 2024). However, statistical methods may struggle to capture complex sequence patterns in the data because of irregularities and nonlinearities among the features (Salinas, Flunkert, Gasthaus, & Januschowski, 2020). Therefore, layer-based deep-learning models incorporating recurrent layers (Rumelhart, Hinton, & Williams, 1986), long short-term memory layers (Hochreiter & Schmidhuber, 1997), gated recurrent units (Chung, Gulcehre, Cho & Bengio, 2014), and attention mechanisms (Vaswani, Shazeer, Parmar, Uszkoreit, Jones, Gomez, & Polosukhin, 2017) have been widely explored in multivariate time series prediction (Catania, Grassi, & Ravazzolo, 2019). Deep learning-based models effectively learn sequential patterns in time series data and can outperform traditional statistical models, demonstrating superior prediction performance (Ortega, Otero, Solomon, Otero, & Fabregas, 2023). Another approach for improving the accuracy of multivariate time series prediction is to utilize models based on transformers, such as informers (Zhou, Zhang, Peng, Zhang, Li, Xiong, & Zhang, 2021) and autoformers (Wu, Xu, Wang, & Long, 2021). Transformer-based models effectively capture long-and short-term patterns in multivariate time series data and have surpassed traditional approaches in terms of prediction performance (Costa & Machado, 2023).