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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, Maryland



Abstract                                                                                                           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 as
clustering 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., do
classification 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% accuracy
achieves only 75% accuracy even upon using sophisticated multivariate machine learning
methods [2]
• Extensions and further applications aplenty.

                                                                                                                                                                                     incoherent estimator                                                                     coherent estimator
Methods                                                                                                                                                                                                                                                                                                           0.5




                                                                                                                                         misclassification rate




                                                                                                                                                                                                                                           # signal−vertices
                                                                                                                                                                    0.5 L π
                                                                                                                                                                        ˆˆ                                            ˆ
                                                                                                                                                                                                                      L n b = 0. 41                                                   ˆ
Connectome 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)∈ES

Classifier
• 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
                                                                                                                   precipitously
Signal 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                                                                                                                                                       0.5
Fisherʼs 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                                                    0.3
s 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 classification
FigureFigure 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 performance
Connections 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, and
terms 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 improvements
matrices 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. The
Each 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, computed
region 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 lower
index, 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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Connectome Classification: Statistical Graph Theoretic Methods for Analysis of MR-Connectome Data

  • 1. 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, Maryland Abstract 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 as clustering 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., do classification 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% accuracy achieves only 75% accuracy even upon using sophisticated multivariate machine learning methods [2] • Extensions and further applications aplenty. incoherent estimator coherent estimator Methods 0.5 misclassification rate # signal−vertices 0.5 L π ˆˆ ˆ L n b = 0. 41 ˆ Connectome 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)∈ES Classifier • 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 precipitously Signal 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 0.5 Fisherʼs 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 0.3 s 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 classification FigureFigure 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 performance Connections 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, and terms 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 improvements matrices 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. The Each 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, computed region 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 lower index, 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.