Bayesian modelling of brain network data via latent space models Emanuele Aliverti , University of Padova, Department of Statistical Sciences � emanuelealiverti.github.io � aliverti@stat.unipd.it 22 th November 2019
Brain Networks ◮ Non-invasive imaging technologies provide accurate data on brain activity and structure at increasing resolution for multiple subjects ◮ Neuro–imaging study comprising ● data for m = 21 individuals ● (Landman et al., 2011). ● ● ● ● ● ● ● ● ◮ n = 68 brain regions (nodes), ● ● ● ● ● ● ● ● spatially located ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 2
Brain Networks ◮ Non-invasive imaging technologies provide accurate data on brain activity and structure at increasing resolution for multiple subjects ◮ Neuro–imaging study comprising ● data for m = 21 individuals ● (Landman et al., 2011). ● ● ● ● ● ● ● ● ◮ n = 68 brain regions (nodes), ● ● ● ● ● ● ● ● spatially located ● ● ● ● ● ● ◮ Lobe and hemisphere information ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 2
Brain Networks ◮ Non-invasive imaging technologies provide accurate data on brain activity and structure at increasing resolution for multiple subjects ◮ Neuro–imaging study comprising ● data for m = 21 individuals ● (Landman et al., 2011). ● ● ● ● ● ● ● ● ◮ n = 68 brain regions (nodes), ● ● ● ● ● ● ● ● spatially located ● ● ● ● ● ● ◮ Lobe and hemisphere information ● ● ● ● ● ● ● ● ● ● ◮ For each individual, brain network ● ● ● ● ● ● ● ● ● ● ● ● connections (edges) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 2
Brain Networks ◮ Non-invasive imaging technologies provide accurate data on brain activity and structure at increasing resolution for multiple subjects ◮ Neuro–imaging study comprising data for m = 21 individuals (Landman et al., 2011). ◮ n = 68 brain regions (nodes), spatially located ◮ Lobe and hemisphere information ◮ For each individual, brain network connections (edges) ◮ High-resolution scans with n = 998 for m = 5 subjects (Hagmann et al., 2008) 2
Brain Networks ◮ Non-invasive imaging technologies provide accurate data on brain activity and structure at increasing resolution for multiple subjects ◮ Neuro–imaging study comprising data for m = 21 individuals (Landman et al., 2011). ◮ n = 68 brain regions (nodes), spatially located ◮ Lobe and hemisphere information ◮ For each individual, brain network connections (edges) ◮ High-resolution scans with n = 998 for m = 5 subjects (Hagmann et al., 2008) 2
Brain Networks ◮ Non-invasive imaging technologies provide accurate data on brain activity and structure at increasing resolution for multiple subjects ◮ Neuro–imaging study comprising data for m = 21 individuals (Landman et al., 2011). ◮ n = 68 brain regions (nodes), spatially located ◮ Lobe and hemisphere information ◮ For each individual, brain network connections (edges) ◮ High-resolution scans with n = 998 for m = 5 subjects (Hagmann et al., 2008) ◮ Goal: investigate network connectivity patterns, accounting for anatomical constraints and unobservable patters (e.g. shapes, functionalities) 2
Brain Network Data ◮ For each subject, data can be represented as an ( n × n ) symmetric adjacency matrix A ( k ) , k = 1 , . . . , m Subject 37 3
Brain Network Data ◮ For each subject, data can be represented as an ( n × n ) symmetric adjacency matrix A ( k ) , k = 1 , . . . , m Subject 37 ◮ a ( k ) = a ( k ) = 1 if at least one white ij ji matter fiber has been observed between regions i = 2 , . . . , n and j = 1 , . . . , i − 1 ◮ a ( k ) = a ( k ) = 0 otherwise. ij ji 3
Brain Network Data ◮ For each subject, data can be represented as an ( n × n ) symmetric adjacency matrix A ( k ) , k = 1 , . . . , m Subject 37 ◮ a ( k ) = a ( k ) = 1 if at least one white ij ji matter fiber has been observed between regions i = 2 , . . . , n and j = 1 , . . . , i − 1 ◮ a ( k ) = a ( k ) = 0 otherwise. ij ji Anatomical information ◮ Spatial coordinates for the i -th region ( x i , y i , z i ) ◮ Lobes and hemisphere membership → lobe ij = 1 region i and region j are in the same lobe → hemi ij = 1 region i and region j are in the same hemisphere 3
Latent Space Models - intuition ◮ Developed in social sciences (e.g. Hoff et al., 2002; Hoff, 2008) ◮ Edges a ( k ) are conditionally ij Beregond independent given their own Eorl probability π ij Edoras Éomer Gorbag Éowyn Shelob Wormtongue Mount Doom Shadowfax Orthanc Helm Ents Osgiliath Denethor Morgul Faramir Mordor Númenor Rohan Gollum Isengard Elendil Treebeard Théoden Saruman Gondor Company Orcs Sauron Anduin Boromir Legolas Gandalf Frodo Pippin Merry Gimli Sam Aragorn Lórien Elves Elrond Arwen Ring Shire Dwarves Moria Galadriel Isildur Balin Arathorn Rivendell Nine Riders Bilbo Durin Thráin Glóin Gildor Celeborn Thorin Glorfindel Bree Lothlórien Bombadil Haldir Bag End Hobbiton Bill 4
Latent Space Models - intuition ◮ Developed in social sciences (e.g. Hoff et al., 2002; Hoff, 2008) ◮ Edges a ( k ) are conditionally ij Beregond independent given their own Eorl probability π ij Edoras Éomer Gorbag Éowyn Shelob Wormtongue Mount Doom ◮ The probability π ij of Shadowfax Orthanc Helm Ents Osgiliath Denethor Morgul Faramir connection between i and j , is a Mordor Númenor Rohan Gollum Isengard Elendil Treebeard function of their positions in a Théoden Saruman Gondor Company Orcs Sauron Anduin H -dimensional latent space Boromir Legolas Gandalf Frodo Pippin Merry Gimli Sam Aragorn Lórien Elves Elrond Arwen Ring Shire Dwarves Moria Galadriel Isildur Balin Arathorn Rivendell Nine Riders Bilbo Durin Thráin Glóin Gildor Celeborn Thorin Glorfindel Bree Lothlórien Bombadil Haldir Bag End Hobbiton Bill 4
Latent Space Models - intuition ◮ Developed in social sciences (e.g. Hoff et al., 2002; Hoff, 2008) ◮ Edges a ( k ) are conditionally ij Beregond independent given their own Eorl probability π ij Edoras Éomer Gorbag Éowyn Shelob Wormtongue Mount Doom ◮ The probability π ij of Shadowfax Orthanc Helm Ents Osgiliath Denethor Morgul Faramir connection between i and j , is a Mordor Númenor Rohan Gollum Isengard Elendil Treebeard function of their positions in a Théoden Saruman Gondor Company Orcs Sauron Anduin H -dimensional latent space Boromir Legolas Gandalf Frodo Pippin Merry Gimli Sam Aragorn Lórien Elves Elrond Arwen Ring Shire Dwarves Moria Galadriel Isildur Benefits Balin Arathorn Rivendell Nine Riders Bilbo Durin Thráin Glóin Gildor Celeborn ◮ Reduce dimensionality from Thorin Glorfindel Bree Lothlórien n × ( n − 1 ) / 2 to n × H Bombadil Haldir ◮ Takes into account network Bag End properties (e.g. transitivity, Hobbiton Bill homophily) 4
Latent spaces and beyond ◮ Each latent coordinate may be interpreted as measure of its “propensity” for different functions / metabolic processes ◮ Regions with similar propensities are more likely to be connected 5
Latent spaces and beyond ◮ Each latent coordinate may be interpreted as measure of its “propensity” for different functions / metabolic processes ◮ Regions with similar propensities are more likely to be connected Desiderata ◮ Allow modelling of replicated networks, multiple subjects → Joint modelling of m networks 5
Latent spaces and beyond ◮ Each latent coordinate may be interpreted as measure of its “propensity” for different functions / metabolic processes ◮ Regions with similar propensities are more likely to be connected Desiderata ◮ Allow modelling of replicated networks, multiple subjects → Joint modelling of m networks ◮ Include covariates → Connectivity as a function of anatomical constraints (distance, lobes) 5
Latent spaces and beyond ◮ Each latent coordinate may be interpreted as measure of its “propensity” for different functions / metabolic processes ◮ Regions with similar propensities are more likely to be connected Desiderata ◮ Allow modelling of replicated networks, multiple subjects → Joint modelling of m networks ◮ Include covariates → Connectivity as a function of anatomical constraints (distance, lobes) ◮ Estimate local clusters of brain regions → Some regions might be similar only wrt subset of latent features. 5
Specification ◮ Focus on modelling A = � m k = 1 A ( k ) ( a ij | π ij ) ∼ Binom ( m , π ij ) logit ( π ij ) = β 0 + β 1 hem ij + β 2 lobe ij + β 3 d ij − ¯ d ij 6
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