{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,2]],"date-time":"2025-11-02T09:47:29Z","timestamp":1762076849395,"version":"build-2065373602"},"reference-count":40,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2022,11,23]],"date-time":"2022-11-23T00:00:00Z","timestamp":1669161600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Science and Technology Project of Hebei Education Department","award":["ZD2021088","2017YFE0135700"],"award-info":[{"award-number":["ZD2021088","2017YFE0135700"]}]},{"name":"S&amp;T Major Project of the Science and Technology Ministry of China","award":["ZD2021088","2017YFE0135700"],"award-info":[{"award-number":["ZD2021088","2017YFE0135700"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>An echo state network (ESN) is an efficient recurrent neural network (RNN) that is widely used in time series prediction tasks due to its simplicity and low training cost. However, the \u201cblack-box\u201d nature of reservoirs hinders the development of ESN. Although a large number of studies have concentrated on reservoir interpretability, the perspective of reservoir modeling is relatively single, and the relationship between reservoir richness and reservoir projection capacity has not been effectively established. To tackle this problem, a novel reservoir interpretability framework based on permutation entropy (PE) theory is proposed in this paper. In structure, this framework consists of reservoir state extraction, PE modeling, and PE analysis. Based on these, the instantaneous reservoir states and neuronal time-varying states are extracted, which are followed by phase space reconstruction, sorting, and entropy calculation. Firstly, the obtained instantaneous state entropy (ISE) and global state entropy (GSE) can measure reservoir richness for interpreting good reservoir projection capacity. On the other hand, the multiscale complexity\u2013entropy analysis of global and neuron-level reservoir states is performed to reveal more detailed dynamics. Finally, the relationships between ESN performance and reservoir dynamic are investigated via Pearson correlation, considering different prediction steps and time scales. Experimental evaluations on several benchmarks and real-world datasets demonstrate the effectiveness and superiority of the proposed reservoir interpretability framework.<\/jats:p>","DOI":"10.3390\/e24121709","type":"journal-article","created":{"date-parts":[[2022,11,23]],"date-time":"2022-11-23T03:15:24Z","timestamp":1669173324000},"page":"1709","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Reservoir Dynamic Interpretability for Time Series Prediction: A Permutation Entropy View"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5101-5953","authenticated-orcid":false,"given":"Xiaochuan","family":"Sun","sequence":"first","affiliation":[{"name":"College of Artificial Intelligence, North China University of Science and Technology, Bohai Road, Tangshan 063210, China"},{"name":"Hebei Key Laboratory of Industrial Perception, Tangshan 063210, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mingxiang","family":"Hao","sequence":"additional","affiliation":[{"name":"College of Artificial Intelligence, North China University of Science and Technology, Bohai Road, Tangshan 063210, China"},{"name":"Hebei Key Laboratory of Industrial Perception, Tangshan 063210, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yutong","family":"Wang","sequence":"additional","affiliation":[{"name":"College of Artificial Intelligence, North China University of Science and Technology, Bohai Road, Tangshan 063210, China"},{"name":"Hebei Key Laboratory of Industrial Perception, Tangshan 063210, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yu","family":"Wang","sequence":"additional","affiliation":[{"name":"College of Artificial Intelligence, North China University of Science and Technology, Bohai Road, Tangshan 063210, China"},{"name":"Hebei Key Laboratory of Industrial Perception, Tangshan 063210, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhigang","family":"Li","sequence":"additional","affiliation":[{"name":"College of Artificial Intelligence, North China University of Science and Technology, Bohai Road, Tangshan 063210, China"},{"name":"Hebei Key Laboratory of Industrial Perception, Tangshan 063210, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yingqi","family":"Li","sequence":"additional","affiliation":[{"name":"School of Computer and Communication Engineering, University of Science and Technology Beijing, Beijing 100083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,11,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"5564","DOI":"10.1038\/s41467-021-25801-2","article-title":"Next generation reservoir computing","volume":"12","author":"Gauthier","year":"2021","journal-title":"Nat. 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