{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,6]],"date-time":"2026-03-06T01:27:05Z","timestamp":1772760425007,"version":"3.50.1"},"reference-count":51,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2023,5,16]],"date-time":"2023-05-16T00:00:00Z","timestamp":1684195200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100006769","name":"Russian Science Foundation","doi-asserted-by":"publisher","award":["22-11-00055"],"award-info":[{"award-number":["22-11-00055"]}],"id":[{"id":"10.13039\/501100006769","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>Entropy measures are effective features for time series classification problems. Traditional entropy measures, such as Shannon entropy, use probability distribution function. However, for the effective separation of time series, new entropy estimation methods are required to characterize the chaotic dynamic of the system. Our concept of Neural Network Entropy (NNetEn) is based on the classification of special datasets in relation to the entropy of the time series recorded in the reservoir of the neural network. NNetEn estimates the chaotic dynamics of time series in an original way and does not take into account probability distribution functions. We propose two new classification metrics: R2 Efficiency and Pearson Efficiency. The efficiency of NNetEn is verified on separation of two chaotic time series of sine mapping using dispersion analysis. For two close dynamic time series (r = 1.1918 and r = 1.2243), the F-ratio has reached the value of 124 and reflects high efficiency of the introduced method in classification problems. The electroencephalography signal classification for healthy persons and patients with Alzheimer disease illustrates the practical application of the NNetEn features. Our computations demonstrate the synergistic effect of increasing classification accuracy when applying traditional entropy measures and the NNetEn concept conjointly. An implementation of the algorithms in Python is presented.<\/jats:p>","DOI":"10.3390\/a16050255","type":"journal-article","created":{"date-parts":[[2023,5,17]],"date-time":"2023-05-17T01:58:06Z","timestamp":1684288686000},"page":"255","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["Neural Network Entropy (NNetEn): Entropy-Based EEG Signal and Chaotic Time Series Classification, Python Package for NNetEn Calculation"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9341-1831","authenticated-orcid":false,"given":"Andrei","family":"Velichko","sequence":"first","affiliation":[{"name":"Institute of Physics and Technology, Petrozavodsk State University, 185910 Petrozavodsk, Russia"}]},{"given":"Maksim","family":"Belyaev","sequence":"additional","affiliation":[{"name":"Institute of Physics and Technology, Petrozavodsk State University, 185910 Petrozavodsk, Russia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4217-7969","authenticated-orcid":false,"given":"Yuriy","family":"Izotov","sequence":"additional","affiliation":[{"name":"Institute of Physics and Technology, Petrozavodsk State University, 185910 Petrozavodsk, Russia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5839-4589","authenticated-orcid":false,"given":"Murugappan","family":"Murugappan","sequence":"additional","affiliation":[{"name":"Intelligent Signal Processing (ISP) Research Lab, Department of Electronics and Communication Engineering, Kuwait College of Science and Technology, Block 4, Doha 13133, Kuwait"},{"name":"Department of Electronics and Communication Engineering, Faculty of Engineering, Vels Institute of Sciences, Technology, and Advanced Studies, Chennai 600117, India"},{"name":"Centre of Excellence for Unmanned Aerial Systems (CoEUAS), Universiti Malaysia Perlis, Arau 02600, Perlis, Malaysia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6321-3295","authenticated-orcid":false,"given":"Hanif","family":"Heidari","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Damghan University, Damghan 36716-41167, Iran"}]}],"member":"1968","published-online":{"date-parts":[[2023,5,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Ribeiro, M., Henriques, T., Castro, L., Souto, A., Antunes, L., Costa-Santos, C., and Teixeira, A. (2021). The Entropy Universe. Entropy, 23.","DOI":"10.3390\/e23020222"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1023\/A:1023785123428","article-title":"Horizon Entropy","volume":"33","author":"Jacobson","year":"2003","journal-title":"Found. Phys."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Bejan, A. (2020). Discipline in thermodynamics. Energies, 13.","DOI":"10.3390\/en13102487"},{"key":"ref_4","unstructured":"Bagnoli, F. (2016). Thermodynamics, entropy and waterwheels. arXiv, 1\u201318."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"720","DOI":"10.3389\/fphys.2017.00720","article-title":"Stability, consistency and performance of distribution entropy in analysing short length heart rate variability (HRV) signal","volume":"8","author":"Karmakar","year":"2017","journal-title":"Front. Physiol."},{"key":"ref_6","first-page":"72","article-title":"Approximate Entropy and Its Application to Biosignal Analysis","volume":"22","author":"Yang","year":"2000","journal-title":"Nonlinear Biomed. Signal Process."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Bakhchina, A.V., Arutyunova, K.R., Sozinov, A.A., Demidovsky, A.V., and Alexandrov, Y.I. (2018). Sample entropy of the heart rate reflects properties of the system organization of behaviour. Entropy, 20.","DOI":"10.3390\/e20060449"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1650005","DOI":"10.1142\/S0129065716500052","article-title":"Discriminating multiple emotional states from EEG using a data-adaptive, multiscale information-theoretic approach","volume":"26","author":"Tonoyan","year":"2016","journal-title":"Int. J. Neural Syst."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"700","DOI":"10.3389\/fnins.2020.00700","article-title":"Functional MRI signal complexity analysis using sample entropy","volume":"14","author":"Nezafati","year":"2020","journal-title":"Front. Neurosci."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Chanwimalueang, T., and Mandic, D.P. (2017). Cosine Similarity Entropy: Self-Correlation-Based Complexity Analysis of Dynamical Systems. Entropy, 19.","DOI":"10.3390\/e19120652"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Simons, S., Espino, P., and Ab\u00e1solo, D. (2018). Fuzzy Entropy analysis of the electroencephalogram in patients with Alzheimer\u2019s disease: Is the method superior to Sample Entropy?. Entropy, 20.","DOI":"10.3390\/e20010021"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2871","DOI":"10.1016\/j.asoc.2010.11.020","article-title":"Complexity analysis of the biomedical signal using fuzzy entropy measurement","volume":"11","author":"Xie","year":"2011","journal-title":"Appl. Soft Comput."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"103255","DOI":"10.1109\/ACCESS.2019.2929266","article-title":"Wavelet-Based EEG Processing for Epilepsy Detection Using Fuzzy Entropy and Associative Petri Net","volume":"7","author":"Chiang","year":"2019","journal-title":"IEEE Access"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1186\/s40708-021-00141-5","article-title":"EEG-based human emotion recognition using entropy as a feature extraction measure","volume":"8","author":"Patel","year":"2021","journal-title":"Brain Inform."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"64704","DOI":"10.1109\/ACCESS.2019.2917303","article-title":"Analyzing the Dynamics of Lung Cancer Imaging Data Using Refined Fuzzy Entropy Methods by Extracting Different Features","volume":"7","author":"Hussain","year":"2019","journal-title":"IEEE Access"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1007\/s11517-014-1216-0","article-title":"Assessing the complexity of short-term heartbeat interval series by distribution entropy","volume":"53","author":"Li","year":"2015","journal-title":"Med. Biol. Eng. Comput."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1553","DOI":"10.3390\/e14081553","article-title":"Permutation entropy and its main biomedical and econophysics applications: A review","volume":"14","author":"Zanin","year":"2012","journal-title":"Entropy"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1140\/epjst\/e2013-01862-7","article-title":"Practical considerations of permutation entropy","volume":"222","author":"Riedl","year":"2013","journal-title":"Eur. Phys. J. Spec. Top."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2711","DOI":"10.1109\/TBME.2017.2664105","article-title":"Bubble Entropy: An entropy almost free of parameters","volume":"64","author":"Manis","year":"2017","journal-title":"IEEE Trans. Biomed. Eng."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Liu, X., Jiang, A., Xu, N., and Xue, J. (2016). Increment entropy as a measure of complexity for time series. Entropy, 18.","DOI":"10.3390\/e18010022"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"118","DOI":"10.1016\/j.ins.2013.12.029","article-title":"Feature selection with SVD entropy: Some modification and extension","volume":"264","author":"Banerjee","year":"2014","journal-title":"Inf. Sci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"433","DOI":"10.1007\/s11741-008-0511-3","article-title":"Analysis of heart rate variability based on singular value decomposition entropy","volume":"12","author":"Li","year":"2008","journal-title":"J. Shanghai Univ. Engl. Ed."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"280","DOI":"10.1016\/j.compbiomed.2019.04.015","article-title":"Novel gridded descriptors of poincar\u00e9 plot for analyzing heartbeat interval time-series","volume":"109","author":"Yan","year":"2019","journal-title":"Comput. Biol. Med."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"105006","DOI":"10.1088\/1361-6579\/ab499e","article-title":"Phase entropy: A new complexity measure for heart rate variability","volume":"40","author":"Rohila","year":"2019","journal-title":"Physiol. Meas."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1109\/TAFFC.2020.3031004","article-title":"Classification of interbeat interval time-series using attention entropy","volume":"14","author":"Yang","year":"2023","journal-title":"IEEE Trans. Affect. Comput."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Velichko, A., and Heidari, H. (2021). A Method for Estimating the Entropy of Time Series Using Artificial Neural Networks. Entropy, 23.","DOI":"10.3390\/e23111432"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Velichko, A. (2020). Neural network for low-memory IoT devices and MNIST image recognition using kernels based on logistic map. Electronics, 9.","DOI":"10.3390\/electronics9091432"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"32015","DOI":"10.1088\/1742-6596\/2094\/3\/032015","article-title":"An improved LogNNet classifier for IoT applications","volume":"2094","author":"Heidari","year":"2021","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"9305","DOI":"10.1007\/s11071-023-08298-w","article-title":"Novel techniques for improving NNetEn entropy calculation for short and noisy time series","volume":"111","author":"Heidari","year":"2023","journal-title":"Nonlinear Dyn."},{"key":"ref_30","unstructured":"LeCun, Y., Cortes, C., and Burges, C. (2018, November 09). MNIST Handwritten Digit Database. Available online: http:\/\/yann.lecun.com\/exdb\/mnist\/."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"111446","DOI":"10.1016\/j.measurement.2022.111446","article-title":"Research on feature extraction method of ship radiated noise with K-nearest neighbor mutual information variational mode decomposition, neural network estimation time entropy and self-organizing map neural network","volume":"199","author":"Li","year":"2022","journal-title":"Measurement"},{"key":"ref_32","unstructured":"Murugappan, M., and Rajamanickam, Y. (2022). Biomedical Signal Analysis Using Entropy Measures: A Case Study of Motor Imaginary BCI in End Users with Disability BT\u2014Biomedical Signals Based Computer-Aided Diagnosis for Neurological Disorders, Springer International Publishing."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Velichko, A., Wagner, M.P., Taravat, A., Hobbs, B., and Ord, A. (2022). NNetEn2D: Two-Dimensional Neural Network Entropy in Remote Sensing Imagery and Geophysical Mapping. Remote Sens., 14.","DOI":"10.3390\/rs14092166"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Boriskov, P., Velichko, A., Shilovsky, N., and Belyaev, M. (2022). Bifurcation and Entropy Analysis of a Chaotic Spike Oscillator Circuit Based on the S-Switch. Entropy, 24.","DOI":"10.3390\/e24111693"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"e2022JA030630","DOI":"10.1029\/2022JA030630","article-title":"Dynamical complexity response in Traveling Ionospheric Disturbances across Eastern Africa sector during geomagnetic storms using Neural Network Entropy","volume":"127","author":"Oludehinwa","year":"2022","journal-title":"J. Geophys. Res. Space Phys."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Huyut, M.T., and Velichko, A. (2022). Diagnosis and Prognosis of COVID-19 Disease Using Routine Blood Values and LogNNet Neural Network. Sensors, 22.","DOI":"10.3390\/s22134820"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Miltiadous, A., Tzimourta, K.D., Giannakeas, N., Tsipouras, M.G., Afrantou, T., Ioannidis, P., and Tzallas, A.T. (2021). Alzheimer\u2019s disease and frontotemporal dementia: A robust classification method of EEG signals and a comparison of validation methods. Diagnostics, 11.","DOI":"10.3390\/diagnostics11081437"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"369","DOI":"10.1007\/s11571-020-09626-1","article-title":"Complex networks and deep learning for EEG signal analysis","volume":"15","author":"Gao","year":"2021","journal-title":"Cogn. Neurodyn."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Murugappan, M., and Murugappan, S. (2013, January 8\u201310). Human emotion recognition through short time Electroencephalogram (EEG) signals using Fast Fourier Transform (FFT). Proceedings of the 2013 IEEE 9th International Colloquium on Signal Processing and its Applications, Kuala Lumpur, Malaysia.","DOI":"10.1109\/CSPA.2013.6530058"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1007\/s13246-015-0333-x","article-title":"Feature extraction and classification for EEG signals using wavelet transform and machine learning techniques","volume":"38","author":"Amin","year":"2015","journal-title":"Australas. Phys. Eng. Sci. Med."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1016\/j.knosys.2015.08.004","article-title":"Application of entropies for automated diagnosis of epilepsy using EEG signals: A review","volume":"88","author":"Acharya","year":"2015","journal-title":"Knowl. Based Syst."},{"key":"ref_42","unstructured":"Gopika Gopan, K., Neelam, S., and Dinesh Babu, J. (2016, January 22\u201325). Statistical feature analysis for EEG baseline classification: Eyes Open vs Eyes Closed. Proceedings of the 2016 IEEE Region 10 Conference (TENCON), Singapore."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"204","DOI":"10.1109\/RBME.2020.2969915","article-title":"A Review on Machine Learning for EEG Signal Processing in Bioengineering","volume":"14","author":"Hosseini","year":"2021","journal-title":"IEEE Rev. Biomed. Eng."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Markoulidakis, I., Rallis, I., Georgoulas, I., Kopsiaftis, G., Doulamis, A., and Doulamis, N. (2021). Multiclass Confusion Matrix Reduction Method and Its Application on Net Promoter Score Classification Problem. Technologies, 9.","DOI":"10.3390\/technologies9040081"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"513","DOI":"10.1007\/s11265-020-01611-5","article-title":"Machine learning based stress monitoring in older adults using wearable sensors and cortisol as stress biomarker","volume":"94","author":"Nath","year":"2022","journal-title":"J. Signal Process. Syst."},{"key":"ref_46","unstructured":"Miltiadous, A., Tzimourta, K.D., Afrantou, T., Ioannidis, P., Grigoriadis, N., Tsalikakis, D.G., Angelidis, P., Tsipouras, M.G., Glavas, E., and Giannakeas, N. (2023, May 01). Available online: https:\/\/openneuro.org\/datasets\/ds004504\/versions\/1.0.4."},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Flood, M.W., and Grimm, B. (2021). EntropyHub: An open-source toolkit for entropic time series analysis. PLoS ONE, 16.","DOI":"10.1371\/journal.pone.0259448"},{"key":"ref_48","unstructured":"Vallat, R. (2023, April 26). AntroPy: Entropy and Complexity of (EEG) Time-Series in Python. Available online: https:\/\/github.com\/raphaelvallat\/antropy."},{"key":"ref_49","unstructured":"(2023, April 26). Numba: A High Performance Python Compiler. Available online: https:\/\/numba.pydata.org\/."},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Obukhov, Y.V., Kershner, I.A., Tolmacheva, R.A., Sinkin, M.V., and Zhavoronkova, L.A. (2021). Wavelet ridges in EEG diagnostic features extraction: Epilepsy long-time monitoring and rehabilitation after traumatic brain injury. Sensors, 21.","DOI":"10.3390\/s21185989"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"838","DOI":"10.1016\/j.bspc.2013.08.001","article-title":"Respiratory cycle related EEG changes: Modified respiratory cycle segmentation","volume":"8","author":"Hill","year":"2013","journal-title":"Biomed. Signal Process. 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