poster at HBM11

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.