Functional Bayesian point process model for
neuroimaging meta-analysis data
Silvia Montagna
Joint work with T. D. Johnson and T. E. Nichols
November 27, 2014
Functional neuroimaging: an overview
Functional neuroimaging has become an essential tool for
non-invasively studying the brain of normal and clinical
populations
Ex: compare activations between two similar but different
types of experiments
Increasing interest in inverse inference
Work supported by the simplicity and computational efficiency of
the mass univariate approach
Limitations: low power, low reliability, etc.
Need meta-analyis!
Coordinate-based meta-analysis
Often, peak activation coordinates (foci) only are reported
Coordinate-based meta-analysis
Often, peak activation coordinates (foci) only are reported
Adopt spatial point processes
Goal: build a Bayesian model for coordinate meta-analysis data that
allows inference on the most likely domain for any new experiment
Spatial point processes
A spatial point process is a stochastic process where we observe
random occurrences of events, possibly with a value or mark
attached to each occurrence.
Examples:
Locations of foci in a brain (mark: z- or t-score)
Locations of trees in a forest (mark: type of tree, diameter, etc.)
Occurrences of disease (mark: type of disease)
Poisson processes
A Poisson process X is defined by a non-negative finite intensity
measure M which defines, for any A ⊆ B, the number of expected
points in A:
Z
N(A) ∼ Poisson(M(A)), where M(A) ≡
µ(ν)dν < ∞
A
We write X ∼ Poisson(B, µ), where B is a common brain template
B ⊂ 3 with finite volume |B|, and µ is the intensity function.
R
The density of a realisation x of X is
π(x) = exp{|B| − M(B)}
Y
µ(x)
x∈x
A Cox process is a Poisson process with random intensity µ(x).
Assumptions and notation
N independent observations, i.e., N independent point patterns
arising from N studies
i
from study i = 1, 2, . . . , N
Foci xi = {xij }nj=1
Each xi is a realisation from a Cox process Xi driven by a
random intensity function µi
Given that observations are independent, the sampling distribution is
( N
) N
X
Y Y
N
N
π({xi }i=1 |{µi }i=1 ) ∝ exp −
Mi (B)
µi (xij )
i=1
i=1 xij ∈xi
Z
Mi (B)
µi (ν)dν < ∞
=
B
B is a common brain template B ⊂
R3 with finite volume |B|.
Log-Gaussian Cox process (LGCP)
One model for Cox processes is an LGCP. Writing the intensity as
µ(ν) = exp{z> (ν)β}µ0 (ν), with z denoting a set of covariates, one
models
µ0 (ν) = exp{Ψ(ν)},
Ψ(ν) ∼ GP(ϕ, C(ν, ν 0 )),
with C(ν, ν 0 ) = σ 2 exp{−φ||ν − ν 0 ||}.
Eµ0 (ν) = exp{ϕ + σ2 /2}, so a useful specification is to set
ϕ = −σ 2 /2 so that Eµ0 (ν) = 1.
Many decent options for fitting LGCPs: INLA, MCMC, MALA,
Hamiltonian MC.
Alternatives to LGCPs
Look into functional representation of the log intensity function
Natural computational advantage over a GP
Functional representations use a linear mean structure to
describe the intensity’s behaviour whereas GPs use the
covariance matrix
Functional Latent Factor Regression Model (LFRM)
We write the (log) intensity as
log µi (ν) =
p
X
bm (ν)θim = b(ν)> θ i
(1)
m=1
bm (ν) corresponds to the mth basis function evaluated at voxel ν
Potential choices for b(·)> : B-splines, 3D Gaussian kernels
θ i is the vector of study-specific basis function coefficients
(scores)
Variations between intensities are reflected through the variations in
the score vectors.
LFRM
We specify a sparse factor model for the basis coefficients as
θ i = Λη i + ζ i ,
ζ i ∼ Np (0, Σ)
(2)
Λ is a p × k factor loading matrix
η i is a k × 1 vector of latent factors
ζ i is a residual vector that is uncorrelated with other variables in
the model, with Σ = diag(σ12 , . . . , σp2 )
This structure induces a low-dimensional representation of log µi .
LFRM
Information from study-specific covariates zi can be incorporated
through a simple linear model
η i = ι> zi + ∆i ,
∆i ∼ Nk (0, I)
(2)
where ι is a r × k matrix of unknown coefficients, and r denotes the
dimension of ιi .
LFRM
Despite the simplicity of this linear model, the resulting model on
log µ1 , . . . , log µn allows a very flexible accommodation of covariate
information.
Conditionally on ({bm }pm=1 , Λ, Σ, ι, {zi }ni=1 ), the log intensities are
independent (finite rank) Gaussian processes with covariate
dependent mean functions
E[log µi (ν)] =
k
X
ι>
l zi φl (ν)
l=1
and a common covariance function
Cov{log µi (ν), log µi (ν )} =
0
k
X
l=1
where φl (ν) =
Pp
m=1
λlm bm (ν).
0
φl (ν)φl (ν ) +
p
X
m=1
2
σm
bm (ν)bm (ν 0 ),
Bayesian analysis: the MGPS prior
We adopt the multiplicative gamma process shrinkage (MGPS) prior
on the loadings [1]:
λjh |ψjh , τh
δ1
−1 −1
∼ N(0, ψjh
τh ),
∼ Gamma(a1 , 1),
ψjh ∼ Gamma
ρ ρ
,
,
2 2
δl ∼ Gamma(a2 , 1),
τh =
h
Y
δl
l=1
l ≥2
τh is a global shrinkage parameter for the hth column
τh ’s are stochastically increasing under the restriction a2 > 1
Favors more shrinkage overall as the column index increases
ψjh ’s are local shrinkage parameters for the elements in the hth
column
Avoids over-shrinking the non-zero loadings
MGPS prior
The MGPS prior let Λ be q × ∞ with priors increasingly
concentrated at zero as the column index increases
The number of factors is automatically selected using adaptive
Gibbs sampler
The MGPS prior allows many of the loadings to be close to zero,
thus inducing effective basis selection
Bhattacharya and Dunson approach - computationally very
efficient (block updating)
Posterior computation
Update the model parameters sampling from their full conditional
posterior distributions when available in closed form
Resort to Hamiltonian MC to update the basis coefficients θ i
Adapt the number of factors as the sampler progresses
Emotion meta-analysis dataset
164 publications
219 studies and 1393 foci
Five emotions: sad, happy, anger, fear, and disgust
(Picture from [2])
Some estimated intensities
Est int. 148 − Exp. Foci = 1
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Est int. 64 − Exp. Foci = 2
Foci
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Est int. 164 − Exp. Foci = 4
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Est int. 142 − Exp. Foci = 2
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Summary
Foci & mean posterior intensity
Posterior stand dev
Foci
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Probit model for valence
Consider a meta-analysis data set of emotion studies, which can be
split into positive and negative studies by valence. Let
(
1 if study i is positive
yi =
0 if study i is negative
with P(yi = 1|α, γ, η i ) = Φ(α + γ > η i ).
The same set of latent factors impacts on the intensity function via
the basis coefficients θ i and on the response variable via the
probability of a positive study.
The model is extendable to different types of outcomes.
ROC – Emotion dataset
0.6
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True positive rate
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ROC
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False positive rate
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Simulations
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Est intensities
Est intensities
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SIM 3: Data
●
1.5
1.5
Positive
Negative
Kernels
1.5
SIM 2: Data
●
1.5
SIM 1: Data
POS foci
NEG foci
Kernels
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POS Intens
NEG Intens
−1.5
−1.0
−0.5
0.0
ROC
ROC
ROC
0.8
1.0
0.2
0.4
0.6
False positive rate
0.8
1.0
1.5
0.8
0.6
0.2
0.2
0.0
1.0
0.4
True positive rate
0.8
0.6
True positive rate
0.8
0.6
0.4
True positive rate
0.2
0.6
False positive rate
0.5
1.0
ROC
1.0
x
0.8
0.4
0.0
x
0.6
0.2
−0.5
x
0.4
0.0
−1.0
x
0.2
True positive rate
−1.0
POS Intens
NEG Intens
0.4
−0.5
POS Intens
NEG Intens
1.0
−1.0
1.0
−1.5
−1.5
−1.5
−1.5
●
POS Intens
NEG Intens
0.0
0.2
0.4
0.6
False positive rate
0.8
1.0
0.0
0.2
0.4
0.6
False positive rate
0.8
1.0
Summary and Discussion
Functional representation of the log intensity function of a Cox
process with inclusion of a high-dimensional set of pre-specified
basis functions
Allow for automatic shrinkage and effective removal of basis
coefficients that are not needed to characterise the intensities
Covariates allowed to impact on the latent factor scores
Easy modifications for joint modelling of disparate data of many
different types
θ i replaceable with concatenated coefficients within component
models for different types of objects, including images, movies,
text, etc.
Extendable to a semiparametric case that allows the latent
variables densities to be unknown via nonparametric Bayes
priors
References
Anirban Bhattacharya and David B Dunson.
Sparse Bayesian infinite factor models.
Biometrika, 98(2):291–306, June 2011.
Jian Kang, Timothy D Johnson, and Thomas E Nichols.
A Bayesian hierarchical spatial point process model for multi-type neuroimaging meta-analysis.
Annals of Applied Statistics, 8(3):1800–1824, 2014.