Robust Unified Granger Causality Analysis: A Normalized Maximum Likelihood Form

Robust Unified Granger Causality Analysis: A Normalized Maximum Likelihood Form | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Robust Unified Granger Causality Analysis: A Normalized Maximum Likelihood Form Zhenghui Hu, Fei Li, Minjia Cheng, Junhui Shui, Yituo Tang, Qiang Lin This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-513384/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 06 Aug, 2021 Read the published version in Brain Informatics → Version 1 posted 11 You are reading this latest preprint version Abstract Unified Granger causality analysis (uGCA) alters conventional two-stage Granger causality analysis into a unified code-length guided framework. We have presented several forms of uGCA methods to investigate causal connectivities, and different forms of uGCA have their own characteristics, which capable of approaching the ground truth networks well in their suitable contexts. In this paper, we considered comparing these several forms of uGCA in detail, then recommend a relatively more robust uGCA method among them, uGCA-NML, to reply to more general scenarios. Then, we clarified the distinguished advantages of uGCA-NML in a synthetic 6-node network. Moreover, uGCA-NML presented its good robustness in mental arithmetic experiments, which identified a stable similarity among causal networks under visual/auditory stimulus. Whereas, due to its commendable stability and accuracy, uGCA-NML will be a prior choice in this unified causal investigation paradigm. Head & Neck Surgery Unified Granger causality analysis Nor- malized maximum likelihood Inherent redundancy Granger causality analysis fMRI Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Full Text Due to technical limitations, full-text HTML conversion of this manuscript could not be completed. However, the manuscript can be downloaded and accessed as a PDF. Cite Share Download PDF Status: Published Journal Publication published 06 Aug, 2021 Read the published version in Brain Informatics → Version 1 posted Review # 2 received at journal 17 Jun, 2021 Editorial decision: Minor revision 17 Jun, 2021 Review # 1 received at journal 24 May, 2021 Reviews received at journal 12 May, 2021 Reviewers invited by journal 12 May, 2021 Editor assigned by journal 12 May, 2021 Reviewer # 2 agreed at journal 11 May, 2021 Reviewer # 1 agreed at journal 11 May, 2021 Submission checks completed at journal 11 May, 2021 Editor invited by journal 11 May, 2021 First submitted to journal 10 May, 2021 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-513384","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":26862899,"identity":"598e0634-7b04-4968-adf2-6b4e751efc92","order_by":0,"name":"Zhenghui Hu","email":"","orcid":"","institution":"Zhejiang University of Technology","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Zhenghui","middleName":"","lastName":"Hu","suffix":""},{"id":26862900,"identity":"29add5a7-1e29-4157-a994-2a92ac27033e","order_by":1,"name":"Fei Li","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABA0lEQVRIiWNgGAWjYBACfmbGhsN/DGzs2NibDzBIgMUS8GuRbGduPMBTkZbMx3MsAajFgLAWg/NA03nOHGacJ5EDUk6EFobDjA0HJNsOM7PxnPn8wbLtDwM/O1Dvzx24dTA2A7UYtqXzsbH3bjCQbDNgkOx5Y8DYewa3Fmag9w8ktlkDbTm7IQGkxeBGjgEzYxtuLWwgLQfbgGokch4cAGmxJ6SFB6jlYMMZZ5AWxgawLRIEtEiA4oUBGMhsPMeMGSTOGfNInHlWcLAXjxb788cff2YARqV8e/PjzxJlcnL87ckbH/zEowUFMANjnwfEOECkBmCIfyBa6SgYBaNgFIwkAADjG09rKHGTyAAAAABJRU5ErkJggg==","orcid":"https://orcid.org/0000-0001-7556-6226","institution":"Zhejiang University of Technology","correspondingAuthor":true,"submittingAuthor":false,"prefix":"","firstName":"Fei","middleName":"","lastName":"Li","suffix":""},{"id":26862901,"identity":"e4d263a8-d684-49ef-9adc-11112954eaf2","order_by":2,"name":"Minjia Cheng","email":"","orcid":"","institution":"Zhejiang University of Technology","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Minjia","middleName":"","lastName":"Cheng","suffix":""},{"id":26862902,"identity":"6acf4e77-e4c8-4aad-966d-6cbd130f717e","order_by":3,"name":"Junhui Shui","email":"","orcid":"","institution":"Zhejiang University of Technology","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Junhui","middleName":"","lastName":"Shui","suffix":""},{"id":26862903,"identity":"a1e1ca7c-7616-4f37-92f5-8a586819a6b6","order_by":4,"name":"Yituo Tang","email":"","orcid":"","institution":"Zhejiang University of Technology","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Yituo","middleName":"","lastName":"Tang","suffix":""},{"id":26862904,"identity":"9337c89e-651c-4986-b536-8951ec4ce0ed","order_by":5,"name":"Qiang Lin","email":"","orcid":"","institution":"Zhejiang University of Technology","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Qiang","middleName":"","lastName":"Lin","suffix":""}],"badges":[],"createdAt":"2021-05-10 15:06:58","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-513384/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-513384/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1186/s40708-021-00136-2","type":"published","date":"2021-08-06T15:01:27+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":9266032,"identity":"8e0344ad-10fb-4cde-b1d5-2fbad46da329","added_by":"auto","created_at":"2021-05-17 18:09:29","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":14514,"visible":true,"origin":"","legend":"The relationships of simulation data sets in the 6-node networks.","description":"","filename":"Fig01.png","url":"https://assets-eu.researchsquare.com/files/rs-513384/v1/dc6488c4a20e5f567fbfd85f.png"},{"id":9265310,"identity":"4a3c9f0b-7391-4b49-b56e-1b8d8c1bb50d","added_by":"auto","created_at":"2021-05-17 18:06:29","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":251062,"visible":true,"origin":"","legend":"Causal connectivities obtained by several uGCA forms and conventional GCA. Top row represented\nresults in low noise level (var=0.2), the middle was middle noise level (var=0.4), the bottom denoted high noise\nlevel (var=0.6). The data length was set to 1000.","description":"","filename":"Fig02.png","url":"https://assets-eu.researchsquare.com/files/rs-513384/v1/5dba41171ac1e1b7ab9bc5db.png"},{"id":9265321,"identity":"5df6e09c-7665-4c47-9e40-113eae485462","added_by":"auto","created_at":"2021-05-17 18:06:30","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":236801,"visible":true,"origin":"","legend":"Causal connectivities obtained by uGCA and conventional GCA under different data length. From top\nrow to bottom row, the data length is 150, 200, 300, 500.","description":"","filename":"Fig03.png","url":"https://assets-eu.researchsquare.com/files/rs-513384/v1/d3e0abc619807eb6aa80082f.png"},{"id":9266469,"identity":"9044d11c-f62e-48f3-a2c0-cc9e42b860c4","added_by":"auto","created_at":"2021-05-17 18:12:29","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":159232,"visible":true,"origin":"","legend":"Mental arithmetic of CSA-control state under the two stimuli(visual and auditory), the activation regions\nwere processed by SPM12. (a): CSA-control state under visual stimulus. (b): CSA-control state under auditory\nstimulus.(P\u003c0.0001, uncorrected)","description":"","filename":"Fig04.png","url":"https://assets-eu.researchsquare.com/files/rs-513384/v1/29c0822c250cce032dc17668.png"},{"id":9266036,"identity":"62ea6682-1418-4fa5-ba29-2125c20e22c1","added_by":"auto","created_at":"2021-05-17 18:09:30","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":38361,"visible":true,"origin":"","legend":"The mutual information of the obtained mental arithmetic networks under two stimuli(visual stimulus\nand auditory stimulus).","description":"","filename":"Fig05.png","url":"https://assets-eu.researchsquare.com/files/rs-513384/v1/e6ef9af02e1e8381435cb470.png"},{"id":9266471,"identity":"d0d5d801-c98c-4358-8d7b-78bb3179aa2b","added_by":"auto","created_at":"2021-05-17 18:12:30","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":269908,"visible":true,"origin":"","legend":"Causal network in the mental arithmetic tasks obtained by uGCA methods and conventional GCA,\nrespectively. With the conventional GCA approach, connected edges of causal networks in two different stimuli\nwere to a large extent distinct. In contrast, for uGCA methods, their connection networks commonly showed\nhigh similarities. Node 1, 2, 3, 4 was involving the inside network of mental arithmetic tasks. As for different\nstimuli input nodes, they were CAL.L, CAL.R, ITG.L, and ITG.R, respectively. The solid lines represent causal\nconnectivities within the mental arithmetic network, and the dashed lines represent causal connectivities\ninvolving the input stimulus nodes.","description":"","filename":"Fig06.png","url":"https://assets-eu.researchsquare.com/files/rs-513384/v1/efd61a8341a52a90ebf1b123.png"},{"id":13633351,"identity":"0a1661e2-4174-48b9-9f0b-5895abc1d382","added_by":"auto","created_at":"2021-09-17 08:27:54","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1337820,"visible":true,"origin":"","legend":"","description":"","filename":"ManuscriptBrainInfo.pdf","url":"https://assets-eu.researchsquare.com/files/rs-513384/v1_covered.pdf"},{"id":9266473,"identity":"6c706b51-3c6f-4530-870d-1d76ad642308","added_by":"auto","created_at":"2021-05-17 18:12:36","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1329195,"visible":true,"origin":"","legend":"","description":"","filename":"ManuscriptBrainInfo.pdf","url":"https://assets-eu.researchsquare.com/files/rs-513384/v1_covered.pdf"}],"financialInterests":"","formattedTitle":"\u003cp\u003eRobust Unified Granger Causality Analysis: A Normalized Maximum Likelihood Form\u003c/p\u003e","fulltext":[{"header":"Full Text","content":"\u003cp\u003eDue to technical limitations, full-text HTML conversion of this manuscript could not be completed. 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