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One limitation with CCM is that it uses both past and future values to predict the current value, which is inconsistent with the widely accepted definition of causality, where it is assumed that the future values of one process cannot influence the past of another. To overcome this obstacle, in our previous research, we introduced the concept of causalized convergent cross mapping (cCCM), where future values are no longer used to predict the current value. In this paper, we focus on the implementation of cCCM in causality analysis. More specifically, we demonstrate the effectiveness of cCCM in identifying both linear and nonlinear causal coupling in various settings through a large number of examples, including Gaussian random variables with additive noise, sinusoidal waveforms, autoregressive models, stochastic processes with a dominant spectral component embedded in noise, deterministic chaotic maps, and systems with memory, as well as experimental fMRI data. In particular, we analyze the impact of shadow manifold construction on the performance of cCCM and provide detailed guidelines on how to configure the key parameters of cCCM in different applications. Overall, our analysis indicates that cCCM is a promising and easy-to-implement tool for causality analysis in a wide spectrum of applications.<\/jats:p>","DOI":"10.3390\/e26070539","type":"journal-article","created":{"date-parts":[[2024,6,24]],"date-time":"2024-06-24T08:42:03Z","timestamp":1719218523000},"page":"539","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Causalized Convergent Cross Mapping and Its Implementation in Causality Analysis"],"prefix":"10.3390","volume":"26","author":[{"ORCID":"https:\/\/orcid.org\/0009-0007-9411-9975","authenticated-orcid":false,"given":"Boxin","family":"Sun","sequence":"first","affiliation":[{"name":"Department of Electrical and Computer Engineering, Michigan State University, East Lansing, MI 48824, USA"}]},{"ORCID":"https:\/\/orcid.org\/0009-0000-5517-1075","authenticated-orcid":false,"given":"Jinxian","family":"Deng","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, Michigan State University, East Lansing, MI 48824, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4641-7579","authenticated-orcid":false,"given":"Norman","family":"Scheel","sequence":"additional","affiliation":[{"name":"Department of Radiology, Michigan State University, East Lansing, MI 48824, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5264-0144","authenticated-orcid":false,"given":"David C.","family":"Zhu","sequence":"additional","affiliation":[{"name":"Department of Radiology, Albert Einstein College of Medicine, Bronx, NY 10461, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7796-8501","authenticated-orcid":false,"given":"Jian","family":"Ren","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, Michigan State University, East Lansing, MI 48824, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6534-5216","authenticated-orcid":false,"given":"Rong","family":"Zhang","sequence":"additional","affiliation":[{"name":"Institute for Exercise and Environmental Medicine, Texas Health Presbyterian Hospital Dallas, Dallas, TX 75231, USA"},{"name":"Department of Neurology and Internal Medicine, University of Texas Southwestern Medical Center, Dallas, TX 75390, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7506-5368","authenticated-orcid":false,"given":"Tongtong","family":"Li","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, Michigan State University, East Lansing, MI 48824, USA"},{"name":"Michigan Alzheimer\u2019s Disease Research Center, Ann Arbor, MI 48109, USA"}]}],"member":"1968","published-online":{"date-parts":[[2024,6,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Hua, J.C., Jin Kim, E., and He, F. 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