{"id":473,"date":"2022-04-21T15:10:50","date_gmt":"2022-04-21T13:10:50","guid":{"rendered":"https:\/\/benediktehinger.de\/blog\/science\/?p=473"},"modified":"2022-04-21T15:10:51","modified_gmt":"2022-04-21T13:10:51","slug":"modelling-circular-effects-using-splines","status":"publish","type":"post","link":"https:\/\/benediktehinger.de\/blog\/science\/modelling-circular-effects-using-splines\/","title":{"rendered":"Modelling Circular Effects using Splines"},"content":{"rendered":"\n
\"\"<\/a>
ERP effect of absolute angle between two saccades. Dimigen & Ehinger 2021<\/a> <\/figcaption><\/figure><\/div>\n\n\n\n

Note: This blog is just explaining the basis sets, not how to actually fit models \/ get parameters etc. <\/em><\/p>\n\n\n\n

Recently, we used the unfoldtoolbox<\/a> (Matlab or Julia; access it from python!) to analyze some fixation-related ERPs. The approach we used (multiple regression with deconvolution) allowed us to include this circular-predictor: absolute saccade-angle. <\/p>\n\n\n\n

Why can’t we model saccade-angle using a linear predictor? The issue is straightforward: Look at this plot.<\/p>\n\n\n\n

\"\"<\/a>
Predicting circular data<\/mark> with a straight line<\/mark> is difficult.<\/figcaption><\/figure><\/div>\n\n\n\n

Ok, why can’t we use a standard non-linear spline regression?<\/p>\n\n\n\n

Wait – what even is a standard non-linear spline regression? Great that you asked. Instead of fitting a straight line with parameters slope & intercept (think y = m*x + c<\/em>), we can split up the x-axis regressor in multiple “local” regressors. <\/p>\n\n\n\n

\"\"
Taken from Ehinger & Dimigen 2019<\/figcaption><\/figure><\/div>\n\n\n\n

But: The angle starts at x=0, and goes to x=360 – but we all know, x=0 and x=360 are actually identical! The orange line added to the plot would have 0 & 360 at different values, except for a slope of 0.<\/p>\n\n\n\n

How do we fix this, how do we “wrap around” the predictor axis?<\/p>\n\n\n\n

Circular Splines!<\/strong><\/p>\n\n\n\n

\"\"<\/a>
We use a basis function that wraps around 0 \/ 360\u00b0<\/figcaption><\/figure><\/div>\n\n\n\n

Instead of having basis functions that have bounds at 0 \/ 360\u00b0, we are using a basis function set that wraps the circular space. If the idea didn’t become clear, here is an alternative visualization:<\/p>\n\n\n\n

\"\"<\/a>
“default” splines, spanning 0 to 1 or 0 to 360\u00b0<\/figcaption><\/figure><\/div>\n\n\n\n
\"\"<\/a>
Circular splines – spanning 0 – 1 in a circular fashion. Note the first spline (second column) that is “active” both at 0 and at 1 (aka 0\u00b0 and 360\u00b0).<\/figcaption><\/figure><\/div>\n\n\n\n

Hopefully, these visualizations help some of you to understand circular splines better!<\/p>\n\n\n\n

Disclaimer: Thanks to Judith Schepers for discussing this with me; sorry for the typos etc. I am a bit in a rush<\/p>\n","protected":false},"excerpt":{"rendered":"

Note: This blog is just explaining the basis sets, not how to actually fit models \/ get parameters etc. Recently, we used the unfoldtoolbox (Matlab or Julia; access it from python!) to analyze some fixation-related ERPs. The approach we used (multiple regression with deconvolution) allowed us to include this circular-predictor: absolute saccade-angle. Why can’t we model saccade-angle using a linear predictor? The issue is straightforward: Look at this plot. Ok, why can’t we use a standard non-linear spline regression? Wait – what even is a standard non-linear spline regression? Great that you asked. Instead of fitting a straight line with…<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5],"tags":[],"class_list":["post-473","post","type-post","status-publish","format-standard","hentry","category-blog"],"_links":{"self":[{"href":"https:\/\/benediktehinger.de\/blog\/science\/wp-json\/wp\/v2\/posts\/473","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/benediktehinger.de\/blog\/science\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/benediktehinger.de\/blog\/science\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/benediktehinger.de\/blog\/science\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/benediktehinger.de\/blog\/science\/wp-json\/wp\/v2\/comments?post=473"}],"version-history":[{"count":0,"href":"https:\/\/benediktehinger.de\/blog\/science\/wp-json\/wp\/v2\/posts\/473\/revisions"}],"wp:attachment":[{"href":"https:\/\/benediktehinger.de\/blog\/science\/wp-json\/wp\/v2\/media?parent=473"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/benediktehinger.de\/blog\/science\/wp-json\/wp\/v2\/categories?post=473"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/benediktehinger.de\/blog\/science\/wp-json\/wp\/v2\/tags?post=473"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}