{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,22]],"date-time":"2026-03-22T05:32:49Z","timestamp":1774157569766,"version":"3.50.1"},"reference-count":43,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2019,12,15]],"date-time":"2019-12-15T00:00:00Z","timestamp":1576368000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"LABEX IMUST","award":["ANR-10-LABX-0064"],"award-info":[{"award-number":["ANR-10-LABX-0064"]}]},{"DOI":"10.13039\/501100001665","name":"Agence Nationale de la Recherche","doi-asserted-by":"publisher","award":["ANR-11-IDEX-0007"],"award-info":[{"award-number":["ANR-11-IDEX-0007"]}],"id":[{"id":"10.13039\/501100001665","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We describe how to analyze the wide class of non-stationary processes with stationary centered increments using Shannon information theory. To do so, we use a practical viewpoint and define ersatz quantities from time-averaged probability distributions. These ersatz versions of entropy, mutual information, and entropy rate can be estimated when only a single realization of the process is available. We abundantly illustrate our approach by analyzing Gaussian and non-Gaussian self-similar signals, as well as multi-fractal signals. Using Gaussian signals allows us to check that our approach is robust in the sense that all quantities behave as expected from analytical derivations. Using the stationarity (independence on the integration time) of the ersatz entropy rate, we show that this quantity is not only able to fine probe the self-similarity of the process, but also offers a new way to quantify the multi-fractality.<\/jats:p>","DOI":"10.3390\/e21121223","type":"journal-article","created":{"date-parts":[[2019,12,17]],"date-time":"2019-12-17T02:59:01Z","timestamp":1576551541000},"page":"1223","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":20,"title":["Information Theory for Non-Stationary Processes with Stationary Increments"],"prefix":"10.3390","volume":"21","author":[{"given":"Carlos","family":"Granero-Belinch\u00f3n","sequence":"first","affiliation":[{"name":"Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique, F-69342 Lyon, France"},{"name":"ONERA-DOTA, University of Toulouse, FR-31055 Toulouse, France"}]},{"given":"St\u00e9phane G.","family":"Roux","sequence":"additional","affiliation":[{"name":"Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique, F-69342 Lyon, France"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1094-7201","authenticated-orcid":false,"given":"Nicolas B.","family":"Garnier","sequence":"additional","affiliation":[{"name":"Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique, F-69342 Lyon, France"}]}],"member":"1968","published-online":{"date-parts":[[2019,12,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1007\/s10546-007-9252-z","article-title":"Identifying nonstationarity in turbulence series","volume":"127","author":"Andreas","year":"2008","journal-title":"Bound. Layer Meteorol."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2777","DOI":"10.5194\/hess-21-2777-2017","article-title":"A non-stationary stochastic ensemble generator for radar rainfall fields based on the short-space Fourier transform","volume":"21","author":"Nerini","year":"2017","journal-title":"Hydrol. Earth Syst. Sci."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"108","DOI":"10.1109\/MSP.2013.2265914","article-title":"Time-frequence processing of nonstationary signals","volume":"30","author":"Boashash","year":"2013","journal-title":"IEEE Signal Process. Mag."},{"key":"ref_4","first-page":"1","article-title":"Forecasting non-stationary economic time series","volume":"18","author":"Couts","year":"1966","journal-title":"Manag. Sci."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1002\/for.3980130210","article-title":"Time-variable parameter and trend estimation in non-stationary economic time series","volume":"13","author":"Young","year":"1994","journal-title":"J. Forecast."},{"key":"ref_6","unstructured":"Yang, K., and Shahabi, C. (2005, January 27\u201330). On the stationarity of multivariate time series for correlation-based data analysis. Proceedings of the Fifth IEEE International Conference on Data Mining (ICDM\u201905), Houston, TX, USA."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"752","DOI":"10.1016\/j.spl.2006.11.012","article-title":"On processes with summable partial autocorrelations","volume":"77","year":"2007","journal-title":"Stat. Probab. Lett."},{"key":"ref_8","first-page":"141","article-title":"Correlation theory of processes with random stationary nth increments","volume":"37","author":"Yaglom","year":"1955","journal-title":"Mat. Sb."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Ibe, O. (2013). 11-Levy processes. Markov Processes for Stochastic Modeling, Elsevier. [2nd ed.].","DOI":"10.1016\/B978-0-12-407795-9.00011-6"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Frisch, U. (1995). Turbulence: The Legacy of A.N. Kolmogorov, Cambridge University Press.","DOI":"10.1017\/CBO9781139170666"},{"key":"ref_11","first-page":"388","article-title":"A Mathematical Theory of Communication","volume":"XXVII","author":"Shannon","year":"1948","journal-title":"Bell Syst. Tech. J."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Kantz, H., and Schreiber, T. (2003). Nonlinear Time Series Analysis, Cambridge University Press.","DOI":"10.1017\/CBO9780511755798"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"688","DOI":"10.1162\/neco.2008.01-08-700","article-title":"Information in the Non-Stationary Case","volume":"21","author":"Vu","year":"2009","journal-title":"Neural Comput."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"5077","DOI":"10.1016\/j.physa.2010.07.010","article-title":"On the characterization of non-stationary chaotic systems: Autonomous and non-autonomous cases","volume":"389","author":"Ray","year":"2010","journal-title":"Phys. A"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1958","DOI":"10.3390\/e17041958","article-title":"Assessing coupling dynamics from an ensemble of time series","volume":"17","author":"Wu","year":"2015","journal-title":"Entropy"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"3796","DOI":"10.1109\/TSP.2019.2920472","article-title":"Probing high-order dependencies with information theory","volume":"67","author":"Roux","year":"2019","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"422","DOI":"10.1137\/1010093","article-title":"Fractional brownian motions fractional noises and applications","volume":"10","author":"Mandelbrot","year":"1968","journal-title":"SIAM Rev."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Takens, F. (1981). Detecting strange attractors in turbulence. Dynamical Systems and Turbulence, Warwick: Proceedings of a Symposium Held at the University of Warwick 1979\/80, Springer.","DOI":"10.1007\/BFb0091924"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Granero-Belinchon, C., Roux, S., Abry, P., Doret, M., and Garnier, N. (2017). Information Theory to Probe Intrapartum Fetal Heart Rate Dynamics. Entropy, 19.","DOI":"10.3390\/e19120640"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1063\/1.1530990","article-title":"Regularities unseen, randomness observed: The entropy convergence hierarchy","volume":"15","author":"Crutchfield","year":"2003","journal-title":"Chaos"},{"key":"ref_21","unstructured":"Mandelbrot, B. (1982). The Fractal Geometry of Nature, W.H. Freeman and Co."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"4483","DOI":"10.1021\/ma00202a018","article-title":"Dielectric relaxation studies of ion motions in electrolyte-containing perfluorosulfonate ionomers: 4. long-range ion transport","volume":"22","author":"Mauritz","year":"1989","journal-title":"Macromolecules"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"899","DOI":"10.1016\/j.crhy.2012.09.002","article-title":"A phenomenological theory of Eulerian and Lagrangian velocity fluctuations in turbulent flows","volume":"13","author":"Chevillard","year":"2012","journal-title":"C. R. Phys."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"075201","DOI":"10.1063\/1.4927301","article-title":"Introduction to the focus issue: physics of scaling and self-similarity in hydrologic dynamics, hydrodynamics and climate","volume":"25","author":"Kavvas","year":"2015","journal-title":"Chaos"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"3367","DOI":"10.1029\/96WR02397","article-title":"On Hack\u2019s law","volume":"32","author":"Rigon","year":"1996","journal-title":"Water Resour. Res."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1283","DOI":"10.1175\/1520-0450(1998)037<1283:AFDAOT>2.0.CO;2","article-title":"A fractal dimensional analysis on the cloud shape parameters of cumulus over land","volume":"37","author":"Gotoh","year":"1998","journal-title":"J. Appl. Meteorol."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2128","DOI":"10.1029\/2001JB001651","article-title":"Bath\u2019s law and the self-similarity of earthquakes","volume":"108","author":"Console","year":"2003","journal-title":"J. Geophys. Res. Solid Earth"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"041920","DOI":"10.1103\/PhysRevE.79.041920","article-title":"Levels of complexity in scale-invariant neural signals","volume":"79","author":"Ivanov","year":"2009","journal-title":"Phys. Rev. E"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"589","DOI":"10.1007\/s100510050890","article-title":"Imprints of log-periodic self-similarity in the stock market","volume":"10","author":"Drozdz","year":"1999","journal-title":"Eur. Phys. J. B Condens. Matter Complex Syst."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1007\/978-3-662-09799-1_5","article-title":"Scaling in stock market data: stable laws and beyond","volume":"Volume 7","author":"Cont","year":"1997","journal-title":"Scale Invariance and Beyond"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"945","DOI":"10.1007\/s10044-014-0435-1","article-title":"A systematic evaluation of the scale invariance of texture recognition methods","volume":"18","author":"Uhl","year":"2015","journal-title":"Pattern Anal. Appl."},{"key":"ref_32","first-page":"119","article-title":"Self-similar and fractal nature of internet traffic","volume":"14","author":"Chakraborty","year":"2004","journal-title":"Netw. Manag."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"910","DOI":"10.1109\/18.119751","article-title":"Wavelet analysis and synthesis of fractional Brownian motion","volume":"38","author":"Flandrin","year":"1992","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1016\/j.spl.2004.10.023","article-title":"Expressions for R\u00e9nyi and Shannon entropies for multivariate distributions","volume":"71","author":"Zografos","year":"2005","journal-title":"Stat. Probab. Lett."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"58003","DOI":"10.1209\/0295-5075\/115\/58003","article-title":"Scaling of information in turbulence","volume":"115","author":"Roux","year":"2016","journal-title":"EPL"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1741","DOI":"10.1016\/j.sigpro.2011.01.020","article-title":"Synthesis of multivariate stationary series with prescribed marginal distributions and covariance using circulant matrix embedding","volume":"91","author":"Helgason","year":"2011","journal-title":"Signal Process."},{"key":"ref_37","first-page":"95","article-title":"Sample estimate of entropy of a random vector","volume":"23","author":"Kozachenko","year":"1987","journal-title":"Probl. Inf. Transm."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"066138","DOI":"10.1103\/PhysRevE.69.066138","article-title":"Estimating mutual information","volume":"69","author":"Kraskov","year":"2004","journal-title":"Phys. Rev. E"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"5629","DOI":"10.1109\/TIT.2018.2807481","article-title":"Demystifying Fixed k-Nearest Neighbor Information Estimators","volume":"64","author":"Gao","year":"2018","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"026103","DOI":"10.1103\/PhysRevE.64.026103","article-title":"Multifractal random walk","volume":"64","author":"Bacry","year":"2001","journal-title":"Phys. Rev. E"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"056121","DOI":"10.1103\/PhysRevE.66.056121","article-title":"Multifractal stationary random measures and multifractal random walk with log-infinitely divisible scaling laws","volume":"66","author":"Bacry","year":"2002","journal-title":"Phys. Rev. E"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1007\/s100510170074","article-title":"Intermittency of 1D velocity spatial profiles in turbulence: A magnitude cumulant analysis","volume":"23","author":"Delour","year":"2001","journal-title":"Eur. Phys. J. B"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"013107","DOI":"10.1103\/PhysRevE.97.013107","article-title":"Kullback-Leibler divergence measure of intermittency: Application to turbulence","volume":"97","author":"Roux","year":"2018","journal-title":"Phys. Rev. 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