https://doi.org/10.1140/epjp/s13360-025-07077-3
Regular Article
Multivariate time series prediction using clustered echo state network
Department of Physics, Centre for Nonlinear Science and Engineering, School of Electrical and Electronics Engineering, SASTRA Deemed University, 613401, Thanjavur, India
a
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Received:
20
September
2025
Accepted:
14
November
2025
Published online:
26
November
2025
Abstract
Many natural and physical processes can be understood by analyzing multiple system variables evolving, forming a multivariate time series. Predicting such time series is challenging due to the inherent noise and interdependencies among variables. Echo state networks (ESNs), a class of Reservoir Computing (RC) models, offer an efficient alternative to conventional recurrent neural networks by training only the output weights while keeping the reservoir dynamics fixed, reducing computational complexity. We propose a clustered ESNs (CESNs) that enhances the ability to model and predict multivariate time series by organizing the reservoir nodes into clusters, each corresponding to a distinct input variable. Input signals are directly mapped to their associated clusters, and intra-cluster connections remain dense, while inter-cluster connections are sparse, mimicking the modular architecture of biological neural networks. This architecture improves information processing by limiting cross-variable interference and enhances computational efficiency through independent cluster-wise training via ridge regression. We further explore different reservoir topologies, including ring, Erdős–Rényi (ER), and scale-free (SF) networks, to evaluate their impact predictive performance. Our algorithm works well across diverse real-world datasets such as the stock market, solar wind, and chaotic Rössler system, demonstrating that CESNs consistently outperform conventional ESNs in terms of predictive accuracy and robustness to noise, particularly when using ER and SF topologies. These findings highlight the adaptability of CESNs for complex, multivariate time series forecasting.
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© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2025
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
