Bayesian Spatial Point Process Modeling of
Neuroimaging Data
Timothy D. Johnson
Department of Biostatistics
University of Michigan
Johnson (University of Michigan)
June 14, 2012
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Outline
1
Acknowledgements
2
Background/Motivation
3
Bayesian Spatial Point Process Models
4
MS Data Analysis
Johnson (University of Michigan)
June 14, 2012
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Acknowledgements
Jian Kang, Department of Biostatistics and Bioinformatics, Emory
University
Thomas E. Nichols, Department of Statistics, University of
Warwick
Ernst Wilhelm Radü, Department of Neurology, University
Hospital Basel
Tor D. Wager, Department of Psychology & Neuroscience,
University of Colorado, Boulder
Lisa Feldman Barrett, Department of Psychology, Northeastern
University
US NIH/NINDS grant: 1 R01 NS075066-01A1
Johnson (University of Michigan)
June 14, 2012
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Background/Motivation
Multiple Sclerosis
autoimmune disease affecting the central nervous system
affects women more than men
commonly diagnosed between 20 and 40 years of age.
caused by damage to the myelin sheath surrounding axons
slows nerve impulse
damage caused by inflammation
cause is unknown, theories include
viral infection
genetics
environmental factors
Johnson (University of Michigan)
June 14, 2012
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Background/Motivation
Multiple Sclerosis—symptoms
symptoms vary due to location and severity of attack
muscle symptoms
bowel/bladder problems
eye problems
brain/nerve problems
speech problems, etc
episodes can last from days to months
episodes alternate with periods of reduced or no symptoms
no known cure
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June 14, 2012
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Background/Motivation
Multiple Sclerosis Subtypes
CIS—clinically isolated syndrome
first neurological episode
PRL—progressive relapsing
gradual progression, short phases of remission
PRP—primary progressive
gradual progression, frequent phases of exacerbation
RLRM—relapsed-remitting
alternating phases of relapse followed by phases of remission
most common form (≈ 80% of cases)
SCP—secondary chronic progressive
progressive decline between acute attacks, no definite periods of
remission
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June 14, 2012
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Background/Motivation
MS imaging
Data are lesion’s centers-of-mass
T1 weighted MRI
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Background/Motivation
The Data
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Bayesian Spatial Point Process Models
Cox Processes
Point process (PP)
A point process is a probabilistic structure used to model a random
number of points in random locations
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June 14, 2012
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Bayesian Spatial Point Process Models
Cox Processes
Point process (PP)
A point process is a probabilistic structure used to model a random
number of points in random locations
Poisson point process
A PP, Y , is a Poisson PP (on brain B) with intensity function λ if
1
Rthe number of points, N(B), has a Poisson distribution with mean
B λ(y)dy
2
given the number of points,
N(B), the locations of the points are
R
i.i.d. with density λ(y)/ B λ(y)dy (normalized intensity)
Johnson (University of Michigan)
June 14, 2012
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Bayesian Spatial Point Process Models
Cox Processes
Point process (PP)
A point process is a probabilistic structure used to model a random
number of points in random locations
Poisson point process
A PP, Y , is a Poisson PP (on brain B) with intensity function λ if
1
Rthe number of points, N(B), has a Poisson distribution with mean
B λ(y)dy
2
given the number of points,
N(B), the locations of the points are
R
i.i.d. with density λ(y)/ B λ(y)dy (normalized intensity)
Cox process
Let λ be a random function. If conditional on λ, Y is a Poisson PP with
intensity λ, then Y is called a Cox process driven by λ.
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June 14, 2012
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Bayesian Spatial Point Process Models
Overdispersed Data
A Cox process has larger variance than a Poisson process
The data warrant this extra variability
Mean
Var
CIS
5.13
36.70
PRL
27.89 330.54
PRP
18.31 274.06
RLRM 15.52 229.13
SCP
22.00 328.91
# Subjects
8
10
13
165
45
Hence a Cox process is warranted
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June 14, 2012
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Bayesian Spatial Point Process Models
What is an intensity function?
Not the density of the point pattern
Integrates to the expected number of points (for a Poisson PP)
Integrating over any subregion results in the expected number of
points in that subregion
Probabilistically, determines the # of points and their locations
If we condition on the observed # of points, the normalized
intensity is the density function of the locations (Binomial process)
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June 14, 2012
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Bayesian Spatial Point Process Models
Poisson/gamma random field model
A (Bayesian) nonparametric representation of the intensity
function
Intensity function is the convolution of a Gaussian kernel and a
Gamma random field
A gamma random field can be represented as an infinite sum of its
jump heights νm at it jump locations θm
G(A) =
∞
X
νm I(θm ∈ A) ∼ GRF(U(A), β)
m=1
U(A) is the uniform distribution
β is the inverse scale
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June 14, 2012
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Bayesian Spatial Point Process Models
Gamma field—GRF(U(dx), β)
U(dx)—uniform distribution, β—inverse scale
Jump locations—θm i.i.d. uniformly over the brain
Arrival times—ζm follow a unit rate Poisson process
−1
Jump heights—ν
m = E1 {|B|ζm }/β,
R∞
where E1 (t) = t e−u u −1 du
Jump Heights
Johnson (University of Michigan)
Intensity function
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Bayesian Spatial Point Process Models
Naı̈ve Bayesian Classifier
Use the entire T1-black whole data
The point process models only use lesion’s centers-of-mass
Should give advantage to the Naı̈ve Bayesian classifier
Feature vector consists of 0’s, and 1’s
if voxels within lesion boundary, score it 1
Each element in vector treated independently
Binomial model at each vector for each subtype
continuity corrected
A priori, assume each subtype is equally probable
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June 14, 2012
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MS Data Analysis
Posterior Mean Intensity (Poisson Gamma random field model)
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June 14, 2012
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MS Data Analysis
Prediction Results
Naı̈ve Bayes
Table: LOOCV classification results
CIS
PRL
PRP RLRM
CIS
0.125 0.000 0.000 0.875
PRL
0.000 0.222 0.000 0.778
PRP
0.000 0.000 0.000 1.000
RLRM
0.000 0.000 0.000 0.994
SCP
0.000 0.000 0.000 1.000
Overall correct classification: 0.696
Average correct classification: 0.268
Johnson (University of Michigan)
SCP
0.000
0.000
0.000
0.006
0.000
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MS Data Analysis
Prediction Results
Poisson/gamma random field model
Table: LOOCV classification results
CIS
PRL
PRP RLRM
CIS
0.875 0.000 0.000 0.125
PRL
0.100 0.600 0.000 0.300
PRP
0.077 0.000 0.769 0.154
RLRM
0.182 0.000 0.018 0.782
SCP
0.044 0.000 0.000 0.222
Overall correct classification: 0.768
Average correct classification: 0.752
Johnson (University of Michigan)
SCP
0.000
0.000
0.000
0.018
0.734
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MS Data Analysis
Questions?
Johnson (University of Michigan)
June 14, 2012
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