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Connectome Classification: Statistical Graph Theoretic Methods for Analysis of MR-Connectome Data Joshua T. Vogelstein , William R. Gray , John A. Bogovic , 1 1,2 1 3 1 1 Susan M. Resnick , Jerry L. Prince , Carey E. Priebe , R. Jacob Vogelstein1,2 1 2 Johns Hopkins University, Baltimore, Maryland, Johns Hopkins University Applied Physics Laboratory, Laurel, Maryland 3 National Institutes of Health, Bethesda, MarylandAbstract Results• Methods for high-throughput MR connectome inference are available [1] Gender Classiﬁer• Previous analyses of connectome data relied on classical graph theoretic tools, such asclustering coefﬁcient • Coherent and incoherent classiﬁers perform better than chance and the naive Bayes• We develop a statistical graph theoretic framework to apply to generic connectome classiﬁer (coherent classiﬁer is signiﬁcant with p-value < 0.0001).classiﬁcation problems • Best classiﬁer achieves 83% accuracy using 12 signal vertices and 360 signal edges• Applying the tools to 49 senior individuals from the BLSA data set resulted in connectome • Classical graph theoretic tools, such as clustering coefﬁecient, number of triangles, etc., doclassiﬁcation accuracy of up to 85% not use vertex labels, which contain useful classiﬁcation signal.• Using standard graph theoretic measures, like clustering coefﬁcient, ignores vertex labels, and • SOA Machine learning techniques [2] using classical graph theory yield only 75% accuracyachieves only 75% accuracy even upon using sophisticated multivariate machine learningmethods [2]• Extensions and further applications aplenty. incoherent estimator coherent estimatorMethods 0.5 misclassification rate # signal−vertices 0.5 L π ˆˆ ˆ L n b = 0. 41 ˆConnectome Inference = 0. 5 10 L c o h= 0. 16 0.4• MR Connectome Automated Pipeline (MRCAP) [1] to infer connectomes 0.25 20 0.3 ˆ L i n c= 0. 27• Vertices are neuroanatomical gyral regions [3], edges are estimated tracts using FACT [4]• 49 subjects from the Baltimore Longitudinal Study on Aging; 25 male, 24 female 30 0 0 1 2 3 0.16 10 10 10 10 200 400 600 800 1000 log size of signal subgraph size of signal subgraph some coherent estimators zoomed in coherent estimator misclassification rate 0.5 0.5 # star−vertices 15 0.4 18 0.25 0.3 0.16 21 0 0 1 2 3 0.16 10 10 10 10 400 500 600 log size of signal subgraph size of signal subgraph coherent signal subgraph estimate coherogram 30 20 20 vertex 20 40 40 10 60 60 0 20 40 60 0.04 0.14 0.29 0.55 vertex threshold Figure Legend (above): The top two panels depict the relative performances of the incoherent (left) and coherent (right) classiﬁers as a function of their hyper-parameters. The middle two depict misclassiﬁcation rate (left) for a few different choices of # of signal vertices and (right) a zoomed in depiction of the top right panel. The bottom left panel shows the estimated signal subgraph, and the bottom right shows the coherogram. Together, these bottom panels suggest that the signal subgraph for these data is neither particularly coherent or incoherent. (below): The figure below visualizes the twelve signal subgraph nodes. Each subplot shows the signal subgraph induced by one of the 12 signal vertices estimated using the coherent classifier. There are 360 edges in the signal subgraph. MRCAP is available at: http://www.nitrc.org/projects/mrcap/Model• Joint graph/class model• Each edge is an independent binary random variable• A subset of edges comprise the signal subgraph FGY = FG|Y FY = Bern(auv ; puv|y )πy (u,v)∈S Bern(auv ; puv ) (u,v)∈ESClassiﬁer• Bayes plug-in classiﬁer is asymptotically optimal• Robust estimators have better convergence properties than the MLE Synthetic Data Analysis y= ˆ Bern(auv ; puv|y )ˆy ˆ π • Simulations as true to real data as possible suggest model is not wholly unreasonable • Even under true model, we only expect about 50% of the identiﬁed edges are true signal ˆ (u,v)∈S edges with 50 samples • With only a few more samples, both misclassiﬁcation rate and missed-edge rate drop precipitouslySignal Subgraph Estimator incoherent estimator coherent estimator 1 misclassification rate• The signal subgraph could be all edges, an incoherent subset, or a coherent subset # star−vertices 0.75 0.7• We devise a different estimator for the two special cases 10• For each edge, we compute the signiﬁcance of the difference between the two clases, using a 0.5 0.5Fisherʼs exact test, which is optimal under the model 20• The incoherent signal subgraph estimator chooses the s most signiﬁcant edges 0.25• The coherent signal subgraph estimator chooses the m most signiﬁcant vertices, and then the 30 0.3s most signiﬁcant edges incident to those vertices 0 0.18 0 1 2 3 200 400 600 800 1000 10 10 10 10 log size of signal subgraph size of signal subgraph 1 0.5 misclassification rate missed−edge rate coh 0.4 inc 0.3 nb 0.5 0.2 0.1 0 0 20 40 60 80 100 0 20 40 60 80 100 # training samples # training samples Assumption Checking • Correlation matrix is signiﬁcantly correlated, suggesting independent edge assumption is poor (data not shown) Discussion • MRCAP is an effective tool for high-throughput connectome inference •Signal subgraph classiﬁers signiﬁcantly improve performance over standard classiﬁcationFigureFigure 2: (Top) Gyral labelslabels and associated numeric indicesRef. 5). Connections Legend: (Top) Gyral and associated numeric indices (adapted from (adapted from [3]). results in both real and synthetic data between these regions, as revealed through the DTI tensor data, are quantified in terms of the mean • Synthetic data suggests a few additional datapoints could yield vastly improved performanceConnections between these regions, as revealed through the DTI tensor data, are quantified in fractional anisotropy (FA) of the estimated fibers. (Bottom) Adjacency matrices illustrating connections • Assumption suggests performance improvements are despite some model inaccuracies, andterms of the mean regions (vertices) in female(FA)male brains. Each entry in these adjacencyAdjacency between gyral fractional anisotropy and of the estimated fibers. (Bottom) matrices generalized models might yield further improvementsmatrices illustrating connections between gyral gyral region indicated by the row index and terminating represents the mean FA of fibers originating in the regions (vertices) in female and male brains. • Standard graph theoretical tools are less effective and do not suggest a signal subgraph in the gyral region indicated by the column index, averaged across all subjects from each sex. TheEach entry in these adjacency matrices represents the mean FA of fibers originating in the gyral significance of the difference (uncorrected, exact p-values) between female and male brains, computedregion with Fisher’sby the row also shown. In all plots, lighter the gyralimplies higher values.by the column indicated exact test, is index and terminating in coloration region indicated Only the lowerindex, triangle is shown becausesubjects from each sex.and therefore the adjacency matrices are averaged across all these graphs are undirected The significance of the difference References(uncorrected, exact p-values) assigned to the left hemisphere; 36–70 are assigned to the right symmetric. Labels 1–35 are between female and male brains, computed with Fisher’s exact hemisphere. [1] Gray et al, submitted and available at: http://www.nitrc.org/projects/mrcap/. .test, is also shown. In all plots, lighter coloration implies higher values. Only the lower triangle [2] Drezde et al, 2008.is shown because these graphs are undirected and therefore the adjacency matrices are [3] Desikan et al, 2006.symmetric. Labels 1–35 are assigned to the left hemisphere; 36–70 are assigned to the right [4] Mori,et al. 1999.hemisphere.
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