{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:51:44Z","timestamp":1760233904005,"version":"build-2065373602"},"reference-count":12,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2021,3,7]],"date-time":"2021-03-07T00:00:00Z","timestamp":1615075200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["41975064"],"award-info":[{"award-number":["41975064"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Jiangsu Program for Innovation and Entrepreneurship Groups","award":["2015"],"award-info":[{"award-number":["2015"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Recently, it has been shown that the information flow and causality between two time series can be inferred in a rigorous and quantitative sense, and, besides, the resulting causality can be normalized. A corollary that follows is, in the linear limit, causation implies correlation, while correlation does not imply causation. Now suppose there is an event A taking a harmonic form (sine\/cosine), and it generates through some process another event B so that B always lags A by a phase of \u03c0\/2. Here the causality is obviously seen, while by computation the correlation is, however, zero. This apparent contradiction is rooted in the fact that a harmonic system always leaves a single point on the Poincar\u00e9 section; it does not add information. That is to say, though the absolute information flow from A to B is zero, i.e., TA\u2192B=0, the total information increase of B is also zero, so the normalized TA\u2192B, denoted as \u03c4A\u2192B, takes the form of 00. By slightly perturbing the system with some noise, solving a stochastic differential equation, and letting the perturbation go to zero, it can be shown that \u03c4A\u2192B approaches 100%, just as one would have expected.<\/jats:p>","DOI":"10.3390\/e23030316","type":"journal-article","created":{"date-parts":[[2021,3,7]],"date-time":"2021-03-07T20:20:43Z","timestamp":1615148443000},"page":"316","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["A Note on Causation versus Correlation in an Extreme Situation"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8472-3211","authenticated-orcid":false,"given":"X. San","family":"Liang","sequence":"first","affiliation":[{"name":"Nanjing Institute of Meteorology, 219 Ningliu Blvd, Nanjing 210044, China"}]},{"given":"Xiu-Qun","family":"Yang","sequence":"additional","affiliation":[{"name":"School of Atmospheric Sciences, Nanjing University, 163 Xianlin Avenue, Nanjing 210023, China"}]}],"member":"1968","published-online":{"date-parts":[[2021,3,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"052201","DOI":"10.1103\/PhysRevE.94.052201","article-title":"Information flow and causality as rigorous notions ab initio","volume":"94","author":"Liang","year":"2016","journal-title":"Phys. Rev. E"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"052150","DOI":"10.1103\/PhysRevE.90.052150","article-title":"Unraveling the cause-effect relation between time series","volume":"90","author":"Liang","year":"2014","journal-title":"Phy. Rev. 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Fract."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/3\/316\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:34:18Z","timestamp":1760160858000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/3\/316"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,3,7]]},"references-count":12,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2021,3]]}},"alternative-id":["e23030316"],"URL":"https:\/\/doi.org\/10.3390\/e23030316","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2021,3,7]]}}}