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Nonparametric Inference of Hemodynamic
Response for fMRI data
Tingting ZhangUniversity of Virginia
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Joint work with Fan Li
Data from Duke Department of Psychology Lab of Ahmad Hariri
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Data fMRI data under a specific experimental design Neuroticism-Extroversion-Openness (NEO) Inventory
measurements: regarding individual personality, for example, Anxiety, Extraversion, Conscientiousness
Goal of the experiment Explore the differences of brain activities across
subjects to emotional stimuli
Understand the relationship between individual brain functions and their temperament and personality
Real Problem
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Participants completed a standardized protocol comprised of four blocks of a facial expression matching task interleaved with five blocks of a shape-matching control task.
Experiment
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Three stimuli: Fearful face matching, Angry face matching
and neutral shape matching Block Design
Experiment
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For every TR of 2s, a 3d brain image of dimension is acquired
The total experiment time is 390 s, so there are195 times points for each voxel
The fMRI Data
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Nonlinear ones such as the Balloon model (Buxon et al., 1998; Friston et al., 2000; Riera et al., 2004) in which differential equations are constructed to describe the brain hemodynamics
Linear Models: General Linear Model (GLM) (Friston et al., 1995a; Worsley and Friston, 1995; Goutte et al., 2000) in which fMRI time series are assumed to follow a linear regression of stimulus effects.
fMRI model
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Let ,I i=1,…N and t=1,…,T be one fMRI time series for subject i
Let v(t) be the stimulus function, v(t)=1, if the stimulus is evoked at time t, otherwise it equals zero.
GLM:
where m is some known constant, and is the hemodynamic response function of ith subject describing the underlying evoked brain activity due to the stimulus
GLM
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Extract important quantitative characteristics of individual HRF estimate to be regressed with individual NEO scores
HRF
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With K different stimuli, the fMRI is modeled as
Here, we would be interested in estimating
More than one Stimulus
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Parametric approaches usually assume parametric forms of HRF with
only one free parameter measuring the amplitude (Worsley and Friston, 1995)
Existing Methods for Estimation HRF
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Linear Fit: only magnitude is the free parameter
Nonlinear Fit, using Gauss-Newton algorithm to estimate six free parameters by minimizing MSE
Canonical HRF
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Inverse Logit Regression Model (Lindquist & Wager 2007)
Other Parametric Models
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The most flexible approach is to treat HRF at every time point as a free parameter (Glover, 1999; Goutte et al., 2000; Ollinger et al., 2001)
Nonparametric
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can be rewritten as
The most flexible approach
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Usually, the least square estimate has an unnatural high-frequency noise
The least square estimate
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Least square estimate of one voxel in ROI of one subject
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Inhomogeneous Variance
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Smoothing Finite Inverse Regression (SFIR) (Goutte et al. 2000)
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The smoothing parameters vary for different HRFs.
For easy computations, we consider do kernel smoothing on the least square estimate: use Nadaraya-Watson estimator
Kernel Smoothing
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For each stimulus k, we choose the optimal h that minimizes
Bandwidth Selection
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We
and . Then the kernel
estimate is linear of the least square estimate:
Estimate Bias and Variance
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Because for least square estimate, we have
Then
The variance can be estimated by plugging the OLS estimate of
Estimate the MSE
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In many situations, the matrix is ill-conditioned.
Even though is not ill-conditioned, due to the many parameters to be estimated, and the large variance involved, the kernel smoothing is not sufficient to reduce the estimation error.
We consider add Tikhonov regularization
HRF EstimationRidge Regression and Smoothing
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We select the parameters that minimize
The bias and variance of the estimate can be easily estimated
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With large , a large bias is incurred, so bias correction is necessary.
Because
The new estimator is defined as
Bias-Correction
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Histogram of SelectLamda2
SelectLamda2
Frequency
0 1 2 3 4 5 6 7
010
2030
40
TheoreticallyOptimalBandwidth
h
Frequency
2 4 6 8 100
510
15
TheoreticallyOptimalLambda
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Due to the large individual variance, all the existing nonparametric methods are
only feasible for magnitude estimation.
Comments
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Represent
B-spline
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Boundary
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Connecting HRF and other subject covariates with response variables
Interpretation
Future Research