Accepted Manuscript
fMRI hemodynamic response function estimation in autoregressive noise by avoiding the
drift
Abd-Krim Seghouane, Adnan Shah, Chee-Ming Ting
PII:
DOI:
Reference:
S1051-2004(17)30075-1
http://dx.doi.org/10.1016/j.dsp.2017.04.006
YDSPR 2102
To appear in:
Digital Signal Processing
Please cite this article in press as: A.-K. Seghouane et al., fMRI hemodynamic response function estimation in autoregressive noise by
avoiding the drift, Digit. Signal Process. (2017), http://dx.doi.org/10.1016/j.dsp.2017.04.006
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fMRI Hemodynamic Response Function Estimation in
Autoregressive Noise by Avoiding the Drift
Abd-Krim Seghouane*a , Adnan Shaha , Chee-Ming Tingb
a
Department of Electrical and Electronic Engineering, Melbourne School of Engineering,
The University of Melbourne, Melbourne, VIC 3010, Australia.
(e-mail: [email protected]; [email protected])
b
Center for Biomedical Engineering, Universiti Teknologi Malaysia, 81310 Skudai,
Johor, Malaysia. (e-mail: [email protected])
Abstract
The measured functional magnetic resonance imaging (fMRI) time series is
typically corrupted by instrumental drift and physiological noise due to respiration and heartbeat giving rise to temporal correlation in the signals. Most
methods proposed so far for nonparametric hemodynamic response function
(HRF) estimation in fMRI data do not account for these confounding effects, and thus produce biased and inefficient estimates. The aim of this
paper is to address this issue by modeling the noise in fMRI time series using
an autoregressive model of order p (AR(p)). Making use of a semiparametric model to characterize the fMRI time series and the AR(p) to model the
temporally correlated noise, a generalized least squares (GLS) estimator for
voxelwise consistent nonparametric HRF estimation is derived. The proposed estimation method is a three-stage procedure that relies on first-order
differencing to remove drift and a novel structured covariance estimator for
the AR noise based on Cholesky decomposition to derive the best linear
unbiased estimator (BLUE) of the HRF. We also establish the asymptotic
consistency of the proposed estimator. Simulation results show that the proposed method generates more accurate HRF estimates compared to existing
methods. When applied to real fMRI data, it demonstrates the effectiveness
in uncovering the brain response temporal dynamics for both event-related
and block-design paradigms. Our approach removes the two types of noise in
fMRI data simultaneously, thus providing efficient estimation of brain hemodynamic responses, while allowing for flexible characterization of the shape
and timing of the voxelwise HRF.
Preprint submitted to Digital Signal Processing
April 18, 2017
Keywords: functional MRI, hemodynamic response function, drift,
autoregressive noise, generalized least squares.
1. Introduction
Functional magnetic resonance imaging (fMRI) has been widely used in
neuroscience studies to analyze cognitive functions in human brain. Besides
the use in detecting activated brain areas in response to a specific stimulus or
task, finding the temporal dynamics of the brain response from fMRI during
activations is also an important problem to address. The blood oxygen leveldependent (BOLD) fMRI relies on the coupling between increases in neuronal
activity and increases in blood flow and volume that accompany the local increase in oxygen demand to offer a proxy of the underlying local neuronal
activity [1]. Identifying the temporal dynamics of brain responses during
activation is among reasons why development of HRF estimation methods
for the BOLD fMRI signal in a particular voxel has attracted a lot of attention. The BOLD response is usually, as a first approximation, modeled as
a convolution of a stimulus function based on the experimental paradigm,
with a hemodynamic response function (HRF) characterizing the temporal
dynamics of the brain response. The key assumptions related to the convolution model are the stationarity and linearity of the neurovascular system for
which some evidences have been provided in [2, 3, 4]. Despite the flexibilities
of non-linear modeling [5], the ability to assume linearity is important to
allow for the use of a linear time invariant model, which can provide more
robust estimation and interpretable characterizations in noisy systems [6].
This is related to critical issues that HRF estimation is typically complicated by the confounding effect of various noise sources in the fMRI data
which cause bias in the estimates, mainly from (1) low-frequency drift due
to instrumental instabilities, and (2) oscillatory noise due to respiration and
cardiac pulsation. In this paper, we address the important question of how
to obtain the most efficient estimates of the HRF with the least amount of
bias and misspecification, particularly in the presence of baseline drift and
temporally-correlated physiological noise in fMRI data.
Common approach to estimating the HRF relies on parametric modeling
of the HRF and the drift. This approach imposes a priori shapes on the
HRF with some families of statistical distributions such as Gamma function,
and on the drift with slowly-varying parametric models such as weighted discrete cosine transform (DCT) basis functions [7, 8, 9, 10], polynomials [11]
2
or splines [12], which often assumes unrealistically the same model to fit at
each voxel, under all conditions and subjects. While parametric models are
very useful in providing a parsimonious description, they have drawbacks of
introducing modeling biases [13] and lack of flexibility. Moreover, many fitting procedures for parametric models are computationally expensive. The
alternative nonparametric approach uses a set of basis functions without the
constraints of any arbitrary modeling structures, and thus allowing for a
more flexible representation of a broader class of HRF and drift signals compared to using parametric models in HRF estimation [7, 8, 9, 10, 14, 15].
Allowing flexibility in both the HRF and the drift across different regions,
conditions and subjects, will help reduce the bias due to model misspecification and hence more accurate HRF estimates. Moreover, this approach
avoids the selection of nuisance covariates and the estimation of their coefficients, and may also capture physiologically-meaningful parameters in the
HRF. We adopt a non-parametric approach proposed in our recent works
[16, 17, 18], which estimates the HRF directly as unknown parameters in
the linear model, by least-square fitting of signals with first-order differencing to remove the drift effect, without using any basis functions. The use
of nonparametric component for the drift leads to a flexible semi-parametric
estimation of the underlying HRF [19, 20].
Another key ingredient of fMRI time series models besides the drift is
specification of the additive noise. The convenient assumption of temporally
independent noise is inappropriate, as the residuals of fMRI signals were
shown to be non-white, but colored noise exhibiting temporal correlation
arising from the aliased physiological artifacts [21, 22, 23]. Ignoring this autocorrelation in fMRI data renders the ordinary least squares (LS) estimate
of HRF parameters asymptotically inefficient, and worse introduces negative
bias in the estimated parameter standard errors, resulting in invalid statistical inference. To overcome this problem, most fMRI analyses adopted the
pre-whitening strategy which first estimates the autocorrelations by assuming parametric models for the colored noise, e.g. autoregressive process of
order one, AR(1) [24, 25, 26, 27] or higher orders, AR(p) [11, 21, 28, 29], and
then use the estimated parameters to ‘pre-whiten’ or decorrelate the errors
to produce an efficient HRF estimate. However, these studies still relied on
the parametric estimation of HRF and drift with the drawbacks discussed
above.
In this paper, we propose a unified nonparametric framework for consistent estimation of hemodynamic response by simultaneously dealing with the
3
temporally correlated noise and avoiding drift present in fMRI data. More
precisely, we develop a novel nonparametric estimator of HRF by first-order
differencing of the fMRI signals to remove the drift, and then derive a generalized LS (GLS) estimator to account for the temporal autocorrelations in
the noise. The proposed GLS estimator is an extension of the LS fitting of
differenced fMRI signals for the white noise case considered in our previous
approach [16, 17, 18]. It is obtained by incorporating the covariance structure
of the colored noise to further improve the HRF estimation. The GLS estimator is a best linear unbiased estimator (BLUE) (with minimum variance)
given a known noise covariance. However, it is typically unknown and current
methods are unsatisfactory in providing an accurate estimate to reduce the
bias. In order to construct the BLUE of HRF, we further derive a consistent
estimator for the noise covariance based on drift-free residuals and a parameterization of the covariance structure by Cholesky decomposition assuming
an AR(p) noise model. The entries in the Cholesky decomposition of the
covariance matrix can be interpreted as AR parameters and innovation variances. Therefore, the proposed estimation procedure consists of three steps:
[Step 1.] The drift-corrected, unbiased residuals are generated from the LS
fitting of the non-parametric fMRI model to the differenced signals. [Step 2.]
The covariance estimator is then constructed by substituting in the Cholesky
factors with the AR parameters estimated based on the differenced residuals
in Step 1. [Step 3.] A novel GLS estimator of HRF with improved asymptotic efficiency (lower bias and variance) can be computed from the necessary
statistics obtained in the first two steps. To our knowledge, there were limited studies on the asymptotic properties of HRF estimation. We provide
the asymptotic consistency of the proposed estimator which is important for
theoretical inferences.
Our proposed GLS estimator which integrates the estimated noise covariance directly in the LS fitting, is more general than using it separately in
an initial stage for pre-whitening. A similar GLS method was proposed for
non-parametric HRF estimation in [30] which, however, has not taken into
account the drift effect and only used an AR(1) model for noise. In contrast,
our approach utilizes the drift-corrected signals and a consistent noise covariance estimate based on the Cholesky factorization of AR(p) noise structure,
thus producing more efficient HRF estimates. It is also computationally more
efficient than the voxel-wise expectation maximization (EM) estimation in
[31, 27], since the estimates of the large covariance matrix and its inverse are
simple constructs of the AR Cholesky factors. Besides, the derivation of our
4
GLS estimator is not straightforward. It relies on the covariance of a differenced AR(p) noise process equivalent to an AR-moving average (ARM A)
process, instead of an AR(p) process in the standard GLS.
The rest of the paper is organized as follows. The proposed voxel-wise
semiparametric fMRI time series model is presented in Section II. The threestep procedure to obtain the novel GLS HRF estimator is described in Section
III. The performance evaluation of the proposed HRF estimator via simulations and real fMRI data for both block-design and event-related paradigms
are given in Sections IV and V respectively. The conclusion is given in Section
VI.
2. fMRI Signal Model
Let yi = (yi (t1 ), ..., yi (tN )) be the discrete time BOLD fMRI signal measured over the time course of N scanned volumes during an fMRI experiment,
at a particular voxel location Vi , i = 1, ..., I with I the total number of vovels.
yi (tj ) is the signal sample for voxel i at discrete time tj , j = 1, ..., N . We
assume yi can be modeled as a sum of three components: an experimentally
induced controlled activation response in voxel i, an uncontrolled confound
part or low-frequency drift (including possible unknown nuisance effects) and
a noise term. In vector form, this model is given by [19, 20, 18]
yi = Xθ i + fi + i ,
i ∼ N (0, σ2 IN ),
(1)
where X is a known (N × q) experimental design matrix consisting of the
lagged stimulus covariates. The parameter vector θ i is an unknown q-dimensional
vector representing the unknown HRF samples to be estimated. fi = (fi (t1 ), . . . , fi (tN ))
is a discrete time sequence which is independent of X and represents the uncontrolled baseline drift including other unknown nuisance effects. Using the
semiparametric model in (1) for hemodynamic response estimation in the
presence of unknown smooth drift has the advantage of not assuming any
particular parametric form for both the HRF and the drift function. It offers
more flexibility in approximating the drift, which helps reduce the bias due to
model miss-specification. The noise i = (i (t1 ), . . . , i (tN )) is modeled as a
stationary multivariate Gaussian process with variance σ2 and autocorrelation matrix Σ. In practice however, Σ is unknown and varies considerably
over both voxels and subjects. Despite this fact, a widely used assumption
in fMRI is that the noise is white with Σ reduced to an identity matrix. For
5
simplicity of notation, we use only the discrete time index j to indicate tj ,
and drop the voxel index i since our approach performs in a voxel-specific
basis, i.e. y = (y1 , . . . , yN ) , f = (f1 , . . . , fN ) and = (1 , . . . , N ) .
fMRI time series are known to contain temporally correlated noise arising
from both physical and physiological processes [23]. A way to model the noise
correlation in fMRI is to use a stationary stochastic process, for instance, an
autoregressive (AR) process of order p, AR(p). For the general case of AR(p),
the model of the fMRI time series in a voxel is given by
y = Xθ + f + with
j = α1 j−1 + . . . + αp j−p + ηj
j = 1, . . . , N
(2)
where αk , k = 1, . . . , p are AR coefficients, and ηj are independent identically
distributed (i.i.d.) Gaussian variables with mean zero and variance ση2 . The
process j is assumed stationary
and causal, i.e. all the roots z of the autoregressive polynomial φ(z) = 1 − pk=1 αk z k = 0 lie within the unit circle, i.e.
|z|≤ 1 for all z ∈ C. Our aim in model (2) is to estimate the parameter vector θ representing the HRF. In the following, we shall propose a consistent
HRF estimation procedure which relies on a first-order differencing approach
to generate the necessary component for the best linear unbiased estimator
(BLUE), i.e., the noise covariance matrix.
3. HRF estimation in autoregressive noise
In practice, the optimal estimation of HRF is complicated by the presence of the drift in (1) as well as the unknown noise covariance matrix Σ
in (2). The proposed estimation method below is a three-stage procedure to
generate a BLUE of the HRF, which includes (a) First-order signal differencing and initial HRF estimation based on least squares (LS), (b) Consistent
AR noise covariance estimation based on the differenced residuals and the
modified Cholesky decomposition, and (c) HRF estimation based on GLS
incorporating the estimated noise covariance. The proposed procedure for
HRF estimation in AR noise is detailed in Algorithm 1 in Table 1.
3.1. Initial HRF estimation
In this section, we describe how to obtain the preliminary estimate of
HRF based on the LS fitting of the non-parametric fMRI model (2) to the
difference signals. Applying a first order difference to the fMRI time series
6
Table 1: The proposed algorithm for HRF estimation in autoregressive noise
Algorithm 1
Given: The fMRI time series y = (y1 , ..., yN ) , design matrix X, and AR order p
Step 1: Initial LS Estimation
1.1: Generate the first-order difference time series z = (y2 − y1 , ..., yN − yN −1 )
1.2: Compute the least squares estimator θ̂ LS according to (5)
Step
2.1:
2.2:
2.3:
2: Noise Covariance Estimation
Generate the estimate of unobservable error ν̂ according to (13)
Generate the estimate of autoregressive noise ˆ according to (14)
Compute the estimate of AR(p) parameters α̂ and σ̂η2 based on ˆ
2.4: Compute the estimate Σ̂ = L̂D̂L̂ by substituting α̂ and σ̂η2 into (12)
−1
2.5: Compute the estimate Ω̂ using (11)
Step 3: GLS Estimation
3.1: Compute the generalized least squares estimator θ̂ GLS according to (8)
end.
Output: θ̂ GLS
helps eliminate the drift [32]. Under the assumption of low-frequency smooth
drift signal f
fj − fj−1 O N −1
(3)
where fj and fj−1 are the j th and (j − 1)th drift samples, it follows that
yj − yj−1 +
+
(xj − xj−1 )θ + j − j−1
(xj − xj−1 )θ + α1 (j−1 − j−2 )
α2 (j−2 − j−3 ) + . . . + αp (j−p − j−p−1 )
ηj − ηj−1
(xj − xj−1 )θ + νj
(4)
where yj and yj−1 are the j th and (j − 1)th fMRI time series samples and xj
and xj−1 are the j th and (j − 1)th rows of the design matrix X. For large N ,
7
the amplitude of the first-order difference of the drift component is negligible
as in (3), and can be eliminated in the HRF estimation using the differenced
signal in (4) [16, 18, 17].
Define z = (y2 − y1 , ..., yN − yN −1 ) as the (N − 1) × 1 vector of the
differenced fMRI signals, and R = [(x2 − x1 ) , ..., (xN − xN −1 ) ] as an
(N −1)×q matrix of the differenced regressors. We developed a LS estimator
for HRF based on the differenced signals in [16]
−1 θ̂ LS = R R
R z
(5)
The error term of the differenced signal in (4) is given by
νj = j − j−1 =
p
αk (j−k − j−k−1 ) + ηj − ηj−1
k=1
p
=
αk νj−k + ej ,
j = 2, . . . , N
(6)
k=1
where ej = ηj − ηj−1 is distributed as N (0, σe2 = 2ση2 ), and has the form of an
ARM A(p, 1) with AR coefficients (α1 , . . . , αp ) and M A coefficient θ1 = −1
and ej is not white but serially correlated
E (ej ej−1 ) = E ((ηj − ηj−1 )(ηj−1 − ηj−2 ))
= E (ηj−1 ηj−1 )
= ση2
3.2. GLS estimation
One approach to accommodating the temporal correlation in the fMRI
noise is to estimate the fMRI model in (2) using the GLS method which
requires an estimate of the noise covariance matrix Σ. We propose a novel
GLS estimator of HRF based on the differenced signals in (4) which are driftcorrected. It incorporates the covariance structure of the differenced noise,
instead of Σ of the original noise . The proposed estimator is defined by
−1 −1
θ̂ GLS = R Ω−1 R
R Ω z
(7)
where Ω is the unknown covariance matrix of the differenced noise ν =
(ν2 , ..., νN ) in (6). It is a BLUE for the HRF θ given a known Ω. When Ω is
unknown, computing θ̂ GLS of (7) is infeasible, but an appropriate estimate
Ω̂ can be substituted for Ω to obtain
−1
−1
−1
θ̂ EGLS = R Ω̂ R
R Ω̂ z.
(8)
8
3.3. Covariance estimation
In this section, we derive a consistent parametric estimator for Ω, and
then integrate it in the GLS estimator (8) to obtain the BLUE of HRF in
(7). The estimator is based on the differenced residuals from the initial LS
fitting on the differenced signals, and a modified Cholesky decomposition of
Σ with an AR(p) structure.
Estimation of the covariance Ω and its inverse Ω−1 can be derived based
on the following relation between the noise terms
ν = (I − U)
where U is an N × N matrix defined by
⎛
0 0 0 0
⎜ 1 0 0 0
⎜
⎜ 0 1 0 0
⎜
U=⎜
⎜ 0 0 1 0
⎜ . . . .
⎝ .. .. .. . .
0 0 0 0
Therefore,
(9)
⎞
··· 0
··· 0 ⎟
⎟
··· 0 ⎟
. ⎟
⎟
· · · .. ⎟
⎟
..
. 0 ⎠
1 0
Ω = (I − U)Σ(I − U )
(10)
where Σ is the temporal covariance matrix of the AR(p) process, j =
p
j = 2, ..., N and the inverse is given by
k=1 αk j−k + ηj
Ω−1 = (I − U )−1 Σ−1 (I − U)−1
(11)
where Σ−1 is band diagonal with p bands above and below the main diagonal
[33] fully defined by the parameters of the AR(p) noise j , i.e., αk , k = 1, ..., p
and ση2 . For estimating Σ and its inverse Σ−1 , we propose to use the modified
Cholesky decomposition
Σ = LDL
and Σ−1 = T D−1 T
(12)
where T = L−1 , D = diag(ση2 , ..., ση2 ) is a diagonal covariance matrix of noise
η = (η2 , ..., ηN ) of the AR(p) process in (2) and T is a unit lower triangular
9
matrix of AR coefficients [34]
⎛
1
0
0
0
0
···
0
⎜ −α1
1
0
0
0
···
0
⎜
⎜ −α2 −α1
1
0
0
·
·
·
0
⎜
⎜ ..
⎜ .
−α2 −α1
1
0
···
0
⎜
..
T=⎜
1
0
.
−α2 −α1
⎜ −αp
⎜
.
.
.
.
.
⎜ 0 −α
..
..
..
..
..
p
⎜
⎜ ..
.
..
..
..
⎝ .
.
.
1
−α2 −α1
0
···
0 −αp · · · −α2 −α1
0
0
0
0
0
0
0
1
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
In order to compute a consistent estimate of Ω and then θ̂ EGLS , the
estimate of the unobservable differenced errors is first obtained based on
the estimated residuals from LS-fitted fMRI model based on the differenced
signals in (5)
ν̂ = z − Rθ̂ LS ,
(13)
where ν̂ = (ν̂1 , ν̂2 , ..., ν̂N −1 ) . Since (I − U ) is invertible, the relation between ν and is bijective. The noise samples of can then be computed
as
ˆ = (I − U)−1 ν̂.
(14)
The estimates of α = (α1 , ..., αp ) and ση2 can be obtained by fitting the
AR(p) to ˆ using least squares for example, denoted here by α̂ and σ̂η2 . By
substituting α̂ and σ̂η2 into (12), and then using (10), we obtain the noise
covariance estimator as
Σ̂ = L̂D̂L̂
and Ω̂ = (I − U)Σ̂(I − U )
(15)
By replacing Ω in (7) with the consistent estimate Ω̂, we compute the θ̂ EGLS
according to (8).
3.4. Asymptotic Theory
In this section, we establish the consistency of the proposed noise covariance estimator Ω̂ and the subsequent GLS HRF estimator θ̂ EGLS .
Asymptotic distribution of θ̂ LS : We derive the limiting distribution of the
LS estimator obtained based on the differenced signals, as in the following
theorem.
10
Theorem 1: The LS estimate θ̂ LS as defined in (5) is consistent and has
an asymptotic normal distribution, as N → ∞
√
d
N (θ̂ LS − θ) → N (0, cov(θ̂ LS ))
(16)
−1 −1
R ΩR R R . The proof is provided in
where cov(θ̂ LS ) = R R
Appendix A. However, the LS estimator is less efficient relative to θ̂ GLS
since [35]
| cov(θ̂ GLS ) |≤| cov(θ̂ LS ) |
−1
−1
.
where cov(θ̂ GLS ) = R Ω̂ R
Consistency of Ω̂: Under ν̂−ν = Op (N −1/2 ) and hence ˆ− = Op (N −1/2 ),
it follows that the LS estimates of the AR parameters are consistent and
asymptotically normal-distributed as (Proposition 8.10.1 in [36])
√
d
N (α̂ − α) → N (0, ση2 Σ−1 )
√
d
N (σ̂η2 − ση2 ) → N (0, 2ση4 )
which implies that α̂ − α = Op (N −1/2 ) and σ̂η2 − ση2 = Op (N −1/2 ). The
consistency of the substitution noise covariance estimators are then directly
follows from the consistency of the AR parameters in the entries of the decomposition (12).
Proposition 1: Let Σ̂ and Ω̂ be estimators obtained by substituting the
LS estimates α̂ and σ̂η2 into the Cholesky decomposition (12). We have
Σ̂ − Σ = Op (N −1/2 ) and Ω̂ − Ω = Op (N −1/2 ).
Consistency of θ̂ EGLS : The following theorem studies the properties of
the GLS estimator for the HRF θ̂ EGLS , when Ω̂ is used to replace Ω in θ̂ GLS .
Theorem 2: Let θ̂ EGLS be the GLS estimator defined in (8) where Ω̂ is
obtained according to (15). Then, as N → ∞, we have
θ̂ EGLS = θ̂ GLS + Op (N −1/2 )
which implies
√
d
N (θ̂ EGLS − θ) →
√
N (θ̂ GLS − θ).
The proof relies on the results in Proposition 1, and is given in Appendix B.
Theorem 2 states that the proposed GLS estimator with the consistent noise
covariance√estimate based on the AR Cholesky decomposition, converges with
a rate of N to the BLUE of HRF θ̂ GLS where Ω is known.
11
4. Simulation results
In this section, we assess the performance of our method for estimating
HRF from fMRI data with correlated noise via two simulation studies. The
first compares the estimation errors obtained using various estimation procedures for the semi-parametric fMRI model (i.e. pre-whitening, LS with
and without differencing and the proposed GLS with differencing) for two
main fMRI experimental designs. The second compares the proposed semiparametric model with various widely used HRF models for estimation under
different noise structures.
4.1. Simulation I
Simulated fMRI time series were generated according to
y(tj ) = x(tj ) θ(t) + f (tj ) + (tj )
(17)
with experimental stimulus x(t) based on event-related and block-design
fMRI paradigms. 200 fMRI time series of N = 260 time points each for
TR = 1 second were simulated based on the true HRF θ 0 generated according to [37] with q = 20 whose
exact shape is depicted in Figure 2. The
drift function f (tj ) = sin(π Nj − 0.21 ), j = 1, ..., 260 was used to simulate the drift. For the variability of the drift, each realization of the drift
was randomly scaled with scaling parameter drawn from normal distribution
with standard deviation of 0.5. We simulated AR noise with different orders
[21, 11]: AR(1) [α1 = 0.9], AR(2) [α1 = 0.9, α2 = -0.01], AR(3) [α1 = 0.9, α2
= -0.01, α3 = -0.005], and i.i.d. samples from a Gaussian distribution with
mean zero and variance ση2 were used for ηj .
To simulate each time series in the event-related paradigm, the stimulus
sequence which was a realization of a random event-related stimulus generated from independent Bernoulli events with p(x(tj ) = 1) = 0.85, was
convolved with the true HRF θ 0 . Simulated drift f (tj ) and AR noise of
different orders with variance varied by ση2 = 1.00, 0.75, 0.5, 0.1 were superimposed on each time series. For block-design fMRI time series generation,
block-design stimuli were convolved with the true HRF θ 0 and the function
f (tj ) was used for the drift. The HRF estimation performance was also investigated for AR noise of different orders with variance fixed ση2 = 0.5. We
also examined the influence of experimental conditions of the block design
including the number of blocks and the variations in the block-stimuli and
rest durations. The number of blocks was varied from 3 to 7 with increments
12
5
5
4
4
3
3
2
2
1
1
amplitude
amplitude
of two-blocks alternating between the stimuli-ON block (with duration varied from 23 to 45 time-points) and stimuli-OFF block (with duration varied
from 13 to 35 time-points). One generated realization of fMRI time series
with AR(2) noise with ση2 = 0.50 based on the block design with three blocks
is shown in Figure 1(a), and randomized event-related design in Figure 1(b),
with the corresponding stimulus sequence.
0
−1
0
−1
−2
−2
−3
−3
−4
−4
−5
1
50
100
150
200
−5
1
250
time points, TR
50
100
150
200
250
scan−points, TR
(a) 3-Blocks design.
(b) Event-related design.
Figure 1: Simulation I: One realization of simulated fMRI time series corrupted by AR(2)
noise with ση2 = 0.5 and drift, for (a) 3-blocks design and (b) randomized event-related
design. Red pulse train represents the stimulus function of the experimental design.
Although block design paradigms are considered not optimal for HRF
parameter estimation, a recent study [38] investigated block-design fMRI for
82 adult twins and established the test-retest reliability of HRF parameters
from block-design paradigms. Therefore, we performed this simulation to
report on the performance of the proposed GLS method for HRF estimation
in block-design fMRI data as well.
Performance Analysis
We compare the performance of the proposed HRF estimator based on
GLS with the pre-whitening, LS estimator without differencing and LS estimator based on first-order signal differencing [16] defined in (5), using the
semi-parametric model in (2). The ordinary LS estimator based on the
−1 original fMRI signals is defined by θ̂ OLS = X X
X y. In the prewhitening approach, both the data and the design matrix are pre-multiplied
by a whitening matrix, before the LS fitting of the transformed model which
13
gives [21]
θ̂ OLS−W = (WX)+ Wy
where W = Σ−1/2 can be computed based on the modified Cholesky decomposition Σ = LDL of the noise covariance matrix Σ in (12), and (WX)+ de
−1
notes the pseudo-inverse of (WX) given by (WX)+ = (WX) WX
(WX) .
The whitening matrix W can be estimated by replacing Σ with an estimate
Σ̂ from the residuals y − Xθ̂ OLS .
The various HRF estimators are assessed using the squared error as a
measure of unbiasedness and efficiency:
ηθ̂ =
1
θ̂ − θ 0 2
q−1
where θ 0 and θ̂ respectively represent the true unknown and the estimated
HRF with q number of samples.
Table II reports the HRF estimation results in squared error averaged
over the 200 fMRI time series generated based on (17) using randomized
event-related paradigm, under AR noise of different orders with varying noise
variances ση2 . Figure 2 shows the corresponding estimated HRF with the
proposed method for (a) AR(1) noise and (b) AR(3) noise both with ση2 =
0.5. The central line and lower and upper box boundaries of the boxplots
represent respectively the median, 25th and 75th percentiles of the 200 simulations. Table III reports the HRF estimation results for the block-design
paradigm under AR noise of different orders and varying experimental conditions. Figure 3 shows the estimated HRF with the proposed method under
(a) 3-blocks design and (b) 7-blocks design using AR(3) noise with ση2 = 0.5.
The results from both Table II and III show that both the proposed estimators based on the differencing, significantly outperform the pre-whitening
method and the LS without differencing for both paradigms. This is evident
from the substantial reduction of mean squared errors indicating improved
efficiency of our estimators. The lower performance of the both conventional
LS methods is possibly due to the neglects of drift effects in the estimation, despite that pre-whitening has mitigated some autocorrelation effects
in the noise. Moreover, the pre-whitening tends to produce less efficient
HRF estimates when relying on an ill-conditioned sample estimate for the
high-dimensional covariance matrix, compared to our structured estimator
parameterized by the AR(p) Cholesky decomposition. Among the differencing estimators, the proposed GLS estimator offers further improvements over
14
the LS estimator, by integrating the autocorrelation of the AR noise at different lags. This can be reflected by the low variance of the estimated HRF
functions relative to the ground-truth in Figure 2 and 3. As expected, the
use of event-related paradigms leads to more accurate HRF estimates from
the fMRI data, compared to the block-designs.
Table 2: Simulation I: Squared error (×10−3 ) for the estimated HRF from randomized
event-related paradigm by pre-whitening, LS method without differencing [15], LS method
with differencing [16] and the proposed GLS method, under AR noise of different orders
with varying noise variances ση 2 .
AR noise
AR(1)
AR(2)
AR(3)
[α1 = 0.9]
[α1 = 0.9, α2 = -0.01]
[α1 = 0.9, α2 = -0.01, α3 = -0.005]
ση 2
0.10
0.5
0.75
1.00
0.10
0.5
0.75
1.00
0.10
0.5
0.75
1.00
LS method with pre-whitening
8.81
35.24
51.48
73.93
9.03
35.99
58.22
72.75
7.68
38.22
53.97
74.88
LS method without differencing
8.80
35.25
51.47
73.92
9.02
35.96
58.22
72.75
7.67
38.21
53.97
74.87
LS method with differencing [16]
6.92
33.37
50.56
70.37
7.06
33.84
52.73
67.05
6.47
37.31
51.98
42.91
GLS method with differencing [proposed]
6.17
27.01
40.72
57.31
6.29
27.45
42.49
55.97
5.88
29.02
43.31
32.61
Table 3: Simulation I: Squared error (×10−3 ) for the estimated HRF from block-design
paradigm by pre-whitening, LS method without differencing [15], LS method with differencing [16] and the proposed GLS method, under AR noise (ση 2 = 0.5) of different orders
and varying experimental conditions.
AR noise, ση 2 = 0.5
Number of Blocks
AR(1)
AR(2)
AR(3)
[α1 = 0.9]
[α1 = 0.9, α2 = -0.01]
[α1 = 0.9, α2 = -0.01, α3 = -0.005]
3-Blocks
5-Blocks
7-Blocks
3-Blocks
5-Blocks
7-Blocks
3-Blocks
5-Blocks
7-Blocks
LS method with pre-whitening
130.80
87.57
79.28
131.01
90.70
71.55
128.21
93.49
71.49
LS method without differencing
130.80
87.57
79.28
131.00
90.71
71.56
128.21
93.49
71.49
LS method with differencing [16]
90.18
58.05
52.12
93.22
59.06
50.83
90.51
60.30
51.51
GLS method with differencing [proposed]
89.40
57.64
51.26
92.44
58.92
50.53
90.17
59.99
51.40
15
1.5
1
1
0.5
0.5
amplitude
amplitude
1.5
0
−0.5
−1
0
−0.5
1
2
3
4
5
6
7
8
9
−1
10 11 12 13 14 15 16 17 18 19 20
time points
1
(a) AR(1) noise ση2 = 0.5.
Ground truth (in black).
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
time points
(b) AR(3) noise ση2 = 0.5.
Ground truth (in black).
1.5
1.5
1
1
0.5
0.5
amplitude
amplitude
Figure 2: Simulation I: Boxplot of HRF estimated by the proposed method from fMRI
time series generated with event-related paradigm, under (a) AR(1) noise with ση2 = 0.5
and (b) AR(3) noise with ση2 = 0.5. The true HRF is represented by the solid curve.
0
−0.5
−1
0
−0.5
1
2
3
4
5
6
7
8
9
−1
10 11 12 13 14 15 16 17 18 19 20
time points
(a) 3-Blocks design. Ground
truth (in black).
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
time points
(b) 7-Blocks design. Ground
truth (in black).
Figure 3: Simulation I: Boxplot of HRF estimated by the proposed method from fMRI
time series generated with block design paradigm, under AR(3) noise with ση2 = 0.5 for
(a) 3-Blocks and (b) 7-Blocks. The true HRF is represented by the solid curve.
16
4.2. Simulation II
In this simulation, we further compare our method with four generallyused HRF models in fMRI data analysis: i) the inverse logit (IL) model [39]
with 3 logistic functions, ii) standard finite impulse response (FIR) basis set
model [37], iii) semi-parametric smoothed FIR (sFIR) model [40], iv) and
the canonical HRF model plus its temporal and dispersion derivative (DD)
model [41, 42]. See [6, 38] for a review of these methods. The proposed
HRF estimation approach based on semi-parametric models fitted by GLS
is also compared with the LS method (DIFF) derived in [16], both based on
first-order signal differencing. Moreover, the presence of structured noise in
fMRI data arising from either an AR noise process or an 1/f noise process
[43] is also taken into account in separate investigations. Furthermore, a
ground truth HRF with an initial dip component representing the picking
up of the oxygen consumption in the BOLD signal was used in this simulation. fMRI time series were generated using this known HRF θ 0 , which
was obtained based on the superposition of the SPM canonical HRF with
its time and dispersion derivatives generating an initial-dip followed by rise
to peak and subsequently an undershoot that settles and recovers back to
the zero-baseline. Random event-related stimuli generated from independent
Bernoulli events with p(x(tj ) = 1) = 0.25 were convolved with the ground
truth (GT) HRF θ 0 to generate simulated fMRI time series of 320 time
points for a scanning repetition time TR = 0.5 seconds based on (17). 100
realizations were generated where each time series was superimposed with
the same drift function f (tj ) as described in Simulation I with j = 1, ..., 320
and scaling parameter drawn from normal distribution with standard deviation of 2.5, and then added with structured noise derived based on i) an AR
process, and ii) an 1/f process. The GT HRF exhibiting an initial dip was
estimated from these simulated fMRI time series by fitting the various HRF
models.
In this simulation, we evaluated the performance of the different methods
for estimating five important HRF paramters: (1) time-to-initial-dip (T2iD)
defined as the local minimum of the signal change in 1/6th of the 30 seconds peristimulus time window, (2) time-to-peak (T2P) for the HRF signal
to reach its peak value, (3) full-width at half maximum (FWHM) of the
HRF, (4) amplitude (A): the maximum signal change in the peristimulus
time window, and (5) onset (O): the index of the first time-sample with
signal intensity > 0.1 × A after the stimulus presentation. Moreover, the
squared error (SE) of estimation as reported in Simulation I was also used
17
as a measure of trueness and precision.
Table-IV reports the mean values of these parameters of interest for the
estimated HRFs by the different models, over the 100 simulated realizations
under AR(1) noise with α1 = 0.2 for ση2 = 0.05 and 1/f noise with Hurst
component of 0.6. The HRF estimates for AR(1) noise and 1/f noise are
shown respectively in Figures 4(a) and 5(a). Figure 4(b) and 5(b) show
the corresponding variations in the estimated parameters of interest for the
different models. These results clearly demonstrate the suitability of the
proposed method for HRF estimation, providing remarkable improvement
over the available methods for HRF modelling in fMRI data.
Table 4: Simulation II: Mean parameters of interest for estimated HRFs from 100 realizations. Last row represents the ground truth (GT) values.
AR(1)
1/f
[α1 = 0.2]
[H = 0.6]
Noise Structure
T2iD
T2P
FWHM
A
O
SE (x 10−3 )
T2iD
T2P
FWHM
A
O
IL
0.50
7.40
3.98
0.82
7.51
33.42
0.50
7.19
3.87
0.84
7.48
29.4
FIR
3.48
6.54
3.26
1.00
7.87
39.20
3.49
6.69
3.28
1.00
8.1
34.8
sFIR
3.44
6.50
3.31
0.97
8.08
36.58
3.49
6.75
3.32
0.94
8.16
33.0
DD
2.58
6.15
3.82
0.88
8.02
20.09
2.63
6.08
3.72
0.83
8.22
20.41
DIFF
3.50
6.50
3.17
1.00
9.02
0.85
3.5
6.5
3.17
1.00
9.00
0.84
Proposed
3.50
6.50
3.14
1.00
9.00
0.33
3.5
6.5
3.12
1.00
9.00
0.19
GT
3.5
6.50
3.00
1.00
9.00
0
3.5
6.50
3.00
1.00
9.00
0
Parameters
18
SE (x 10−3 )
IL
GT
1
0.5
FIR
GT
1
0.5
0
0
−0.5
−0.5
−1
−1
0
5
10
15
20
25
30
35
40
45
50
55
0
sFIR
GT
1
0.5
5
10
15
20
25
30
35
40
45
50
55
DD
GT
1
0.5
0
0
−0.5
−0.5
−1
−1
0
5
10
15
20
25
30
35
40
45
50
55
0
DIFF
GT
1
0.5
5
10
15
20
25
30
35
40
45
50
55
Proposed
GT
1
0.5
0
0
−0.5
−0.5
−1
−1
0
5
10
15
20
25
30
35
time−points
40
45
50
55
0
5
10
15
20
25
30
35
time−points
40
45
50
55
(a) HRF Estimates, AR(1) noise
20
T2P
T2iD
3
2
15
10
1
5
IL
FIR
sFIR
DD
DIFF
Proposed
GT
FIR
sFIR
DD
DIFF
Proposed
GT
IL
FIR
sFIR
DD
DIFF
Proposed
GT
IL
FIR
sFIR
DD
DIFF
Proposed
GT
1.5
10
8
1
6
0.5
A
FWHM
IL
4
0
2
−0.5
0
IL
FIR
sFIR
DD
DIFF
Proposed
GT
10
0.25
0.2
6
QE
O
8
0.15
0.1
4
0.05
2
0
IL
FIR
sFIR
DD
DIFF
Proposed
GT
(b) Variations in Parameters-of-Interest, AR(1) noise
Figure 4: Simulation II with AR noise: (a) HRF estimates using different models used
in fMRI data analysis: IL (top-left), FIR (top-right), sFIR (middle-left), DD (middle-right)
and non-parametric methods: DIFF [16] (bottom-left), and the proposed method (bottomright) for 100 realizations of simulated event-related fMRI data under AR(1) noise with
α1 = 0.2 and ση2 = 0.05. (b) Variations of the estimated parameters relative to the GT,
indicating mis-modeling by these methods.
19
IL
GT
1.5
1
0.5
FIR
GT
1.5
1
0.5
0
0
−0.5
−0.5
−1.0
−1.0
0
5
10
15
20
25
30
35
40
45
50
55
0
sFIR
GT
1.5
1
0.5
5
10
15
20
25
30
35
40
45
50
55
DD
GT
1.5
1
0.5
0
0
−0.5
−0.5
−1.0
−1.0
0
5
10
15
20
25
30
35
40
45
50
55
0
DIFF
GT
1.5
1
0.5
5
10
15
20
25
30
35
40
45
50
55
Proposed
GT
1.5
1
0.5
0
0
−0.5
−0.5
−1.0
−1.0
0
5
10
15
20
25
30
35
time−points
40
45
50
55
0
5
10
15
20
25
30
35
time−points
40
45
50
55
(a) HRF Estimates, 1/f noise
20
T2P
T2iD
3
2
15
10
1
5
IL
FIR
sFIR
DD
DIFF
Proposed
GT
IL
FIR
sFIR
DD
DIFF
Proposed
GT
IL
FIR
sFIR
DD
DIFF
Proposed
GT
IL
FIR
sFIR
DD
DIFF
Proposed
GT
1.5
1
6
0.5
A
FWHM
8
4
0
2
−0.5
0
−1
IL
FIR
sFIR
DD
DIFF
Proposed
GT
0.3
10
0.2
QE
O
8
6
0.1
4
2
0
IL
FIR
sFIR
DD
DIFF
Proposed
GT
(b) Variations in Parameters-of-Interest, 1/f noise
Figure 5: Simulation II with 1/f noise: (a) HRF estimates using different models used
in fMRI data analysis: IL (top-left), FIR (top-right), sFIR (middle-left), DD (middleright) and non-parametric methods: DIFF [16] (bottom-left), and the proposed method
(bottom-right) for 100 realizations of simulated event-related fMRI data under 1/f noise
of H = 0.6. (b) Variations of the estimated parameters relative to the GT, indicating
mis-modeling by these methods.
20
5. Applications To Real fMRI data
The proposed method was tested on event-related and block-deign experimental fMRI data sets of finger tapping task. Using a 3.0 T functional
MRI system (ISOL, Republic of Korea), EPI sequences were obtained with
TR/TE = 2000/35 ms for event-related design, and TR/TE = 3000/35 ms
for block-design. For both experiments, the flip angle = 80◦ and slice thickness = 4mm. In the block-design experiment, a 15 s task period alternated
with a 72 s resting period was repeated 4 times for each subject followed by
an additional 30 s of rest. During task period, subjects performed a right
finger flexion whereas during rest-period subjects focused on a fixed point
to minimize eye movement. In the event-related experiment, the right finger tapping task and resting periods were repeated 40 times followed by an
additional 30 s of rest with an average interstimulus interval (ISI) period of
12 s and ISI ranged between 4 and 20 s. Further details about acquisition
parameters and experimental protocols are described in [44, 45].
Image processing, statistical analysis and activation detection were carried out using SPM8 [46] and Matlab. The data were pre-processed by first
realignment of all volumes to the first volume to correct changes in signal intensity over time arising from uncontrolled head motion within the
fMRI scanner. A structural MRI image, acquired using a standard threedimensional T1 weighted sequence was then co-registered to the mean T2
image. This was followed by spatial normalization to a standard Tailarach
template, re-sampling to 2 mm × 2 mm × 2 mm voxels, and smoothing using a 8mm full width at half maximum (FWHM) isotropic Gaussian kernel.
The right finger tapping task stimulates the regions of the brain responsible for motor activity. The intensity values of the pre-processed images at
each voxel location were collected to form individual voxel time series. For
inference on activation maps, we used the cross-correlation method [47] to
detect the activity in the brain. The level of activation at a voxel is measured
by the correlation between the voxel’s time series and the predicted BOLD
response (by convolution of the stimulus function with a canonical HRF).
Strong positive correlation indicates activated voxels, and when there is no
correlation, voxels are inactive. The deactivated voxels indicated by negative
correlation were not of interest. The activation detection was performed for a
random field corrected p-value p < 0.005. The detected activations obtained
in the motor areas are similar when compared to the results in the study
[44], where this fMRI data has been discussed in detail.
21
50 most-significant voxels time series with highest positive correlation values from the activated region of interest (ROI) in primary motor area were
extracted as in [45] for two different subjects one from each experimental
design. These voxels were then approached for HRF estimation with the
proposed method with the AR(p) noise structure where the order of the estimated noise in (14) is determined for each voxel using approximations to
Bayesian information criterion as implemented in the ARFIT package [48].
These results by the proposed method are shown in Figure 6 (bottom) for
the event-related (Figure 6(a)) and block-design (Figure 6(b)) real fMRI data
sets. To compare with a standard parametric HRF, the estimates using the
canonical models plus its time and dispersion derivatives are also shown in
Figure 6 (top). Note that the canonical model was fitted after removing
the drift effects. As revealed in these figures, there is a clear difference in
HRF shapes obtained with different experimental designs. The HRF estimated with event-related design show an initial dip usually believed to arise
from an increase in oxygen consumption, however, the block-design estimates
miss this information. This may be due to that the block-design fMRI experiments, despite its high detection power, have a lower ability to estimate
the shape of the HRF appropriately, compared to the event-related designs
[38]. As predicted by simulation results, the estimates by the canonical models have larger variance than our estimator as shown by the boxplots. The
shape of the HRF obtained with the proposed method allows more variability
than the parametric form because it offers more flexibility in the estimation
(which is the aim of nonparametric estimation [7, 8, 9]). Both methods are
able to identify the peaks of the hemodynamic response function in a similar
time position. However, we can observe that the proposed method performs
better than the canonical models in capturing the ‘post-stimulus undershoot’
located at 13s, as particularly evident for the block-design estimates. This
undershoot is a main feature of BOLD response possibly due to elevation
of venous volume or cerebral metabolic rate of oxygen (CMRO2) [49]. The
canonical estimates also give a wider FWHM compared to our method. These
results consistent with other findings of strong departure of the HRF from a
canonical shape in the motor cortex [50].
22
0.8
0.3
0.6
0.2
amplitude
amplitude
0.4
0.2
0
0.1
1
−0.2
−0.1
−0.4
5
10
15
time [seconds]
20
25
−0.2
30
5
10
15
time [seconds]
20
25
30
0.5
0.4
0.4
0.3
amplitude
amplitude
0.3
0.2
0.2
0.1
0.1
0
−0.1
0
−0.2
−0.1
5
10
15
20
time [seconds]
25
30
5
(a) HRF in Event-related fingertapping task.
10
15
20
time [seconds]
25
30
(b) HRF in Block-design fingertapping task.
Figure 6: Real fMRI data: Estimates of HRF for activated 50 voxels from primary
motor area using the canonical models plus its time and dispersion derivatives (top) and
proposed method (bottom), for finger-tapping task-related fMRI data obtained based on
(a) an event-related design, and (a) a block-design. Solid line represents the mean of the
HRF estimates over the 50 voxels.
23
6. Conclusion
We have developed a novel nonparametric method based on the generalized least squares for estimating HRF from fMRI data based on signal differencing while taking into account the AR(p) noise structure. The proposed
method exploits the semiparametric modeling to describe the BOLD response
with a temporally correlated noise. It is a three-stage voxel-wise approach
that produces consistent HRF estimates based on an error structure estimation method that bypasses any nonparametric estimation. The effectiveness
and performance of the proposed method is illustrated on simulated data
and tested on real fMRI data obtained using both block-design and eventrelated paradigms. The results are in accordance with previous study in
[38], revealing the difference in HRF estimates between the block-design and
event-related experiments. These results suggest that the proposed method
is a good alternative to existing methods for HRF estimation in fMRI data
as it takes into account the temporal correlation nature of the noise.
There are two potential limitations of this work. First, we use a voxelwise approach which essentially relies on fitting an ensemble of univariate
models separately to individual voxels assumed to be independent. Despite
the benefit of its scalability of the voxel-specific modeling, it can only capture the autocorrelation of noise in each individual voxel, but neglects the
cross-correlations between voxels. The proposed method can be extended in
the future to multivariate modeling which incorporates both the auto- and
cross-covariance noise structure of multiple voxels to improve the HRF estimation for a specific brain region. Though this will invite new challenges in
estimating reliably the resultant high-dimensional covariance matrix and in
handling the increased computational complexity. The second shortcoming
is that the non-parametric estimation might lead to possible over-fitting of
the signals. This can be possibly overcome by introducing sparse estimation
of HRF in our current framework, as recently developed in [51, 52], which
can data-adaptively select the dominant representative structure of the signals and prevent over-fitting. It also avoids assuming prior knowledge of the
dimension of the HRF parameter vector as in many existing HRF estimation
methods. Future works will also consider extensions based on mixed-effect
models to handle multi-subject analysis, and to account for variability in the
drift and noise characteristics.
24
Appendix A: Proof of Theorem 1
√
Derivation of the N consistency of θ̂ LS given in (5).
For j = 2, ..., N , the vector of differences between two consecutive samples
of the fMRI time series takes the form
z = Rθ + g + ν
(18)
where z = (y2 − y1 , ..., yN − yN −1 ) is an (N − 1) × 1 vector, R = [(x2 −
x1 ) , ..., (xN −xN −1 ) ] is an (N −1)×q matrix, g = (f2 −f1 , ..., fN −fN −1 )
is an (N − 1) × 1 vector, ν = (2 − 1 , ..., N − N −1 ) is an (N − 1) × 1 vector.
From (5), we have
(R R)−1 R z = θ + (R R)−1 R g + (R R)−1 R ν.
(19)
Assuming that f is Lipschitz continuous and that the matrix
N −1
1 1
(xj − xj−1 ) (xj − xj−1 ) = R R
CN =
N j=1
N
converge toward a positive definite matrix C as N → ∞, we have
√
1
N (R R)−1 R g O
E
N 3/2
1
−1 −1
E N (R R) R gg R(R R)
=O
.
N3
and with the central limit theorem
√
N (R R)−1 R ν = 0
E
and the variance is
E N (R R)−1 R νν R(R R)−1
−1 −1
R ΩR R R
= R R
Therefore for sufficiently large N , we have
25
√
−1 −1
N (θ̂ LS −θ) −→ N (0, R R
R ΩR R R ).
Appendix B: Proof of Theorem 2
√
Derivation of the N consistency of θ̂ EGLS given in (8).
−1
−1 −1
−1
−1
R Ω̂ z − R Ω−1 R
R Ω z
θ̂ EGLS − θ̂ GLS = R Ω̂ R
−1
−1
−1
−1
(20)
=
R Ω̂ R
− R Ω−1 R
R Ω̂ z
−1 −1
R Ω̂ − Ω−1 z.
+ R Ω−1 R
The second term of (20) can be approximated as
−1 −1 −1
R Ω̂ − Ω−1 z
R Ω R
−1 −1 I − Ω̂Ω−1 z
R Ω̂
= R Ω−1 R
−1 −1 R Ω̂
= R Ω−1 R
Ω − Ω̂ Ω−1 z
−1 −1 = R Ω−1 R
R Ω
Ω − Ω̂ Ω−1 z
−1 −1
+ R Ω−1 R
R Ω̂ − Ω−1 Ω − Ω̂ Ω−1 z
−1 −1 Ω − Ω̂ Ω−1 z
= R Ω−1 R
R Ω
−1 −1 Ω − Ω̂ Ω−1 Ω − Ω̂ Ω−1 z
+ R Ω−1 R
R Ω̂
∝ Op N −1/2 + Op N −1
∝ Op N −1/2
(21)
where the last two lines of (21) are obtained from the fact that Ω̂ − Ω =
Op (N −1/2 ).
26
The approximation of the first term of (20) relies on the approximation of
−1 −1 −1 −1 −1
−1
R Ω R
R Ω̂ R
− R Ω−1 R
= R Ω̂ R
−1
−1
− R Ω̂ R R Ω−1 R
−1
−1
= R Ω̂ R
R Ω−1 Ω̂
−1 −1
− Ω Ω̂ R R Ω−1 R
−1
−1
= R Ω̂ R
R Ω−1 Ω̂
−1
− Ω Ω−1 R R Ω−1 R
−1
−1
−1
+ R Ω̂ R
R Ω−1 Ω̂ − Ω Ω̂
−1
− Ω−1 R R Ω−1 R
(22)
The first term of (22) is of order Op (N −1/2 ) whereas the second term can be
approximated by
−1 −1
−1
−1
R Ω−1 Ω̂ − Ω Ω−1 Ω̂ − Ω Ω̂ R R Ω−1 R
− R Ω̂ R
(23)
which is of order Op (N −1 ).
Furthermore,
−1
−1
−1
−1
−Ω
z
= R Ω̂ − Ω−1 z + R Ω−1 z
−1
Ω − Ω̂ Ω−1 z + R Ω−1 z
= R Ω̂
R Ω̂ z = R
Ω̂
+Ω
−1
(24)
= R Ω−1 z + Op (N −1/2 )
Combining (22) and (24) we conclude that the first term of (20) is also of
order Op (N −1/2 ). Therefore
θ̂ EGLS = θ̂ GLS + Op (N −1/2 )
which implies
√
d
N (θ̂ EGLS − θ) →
27
√
N (θ̂ GLS − θ).
[1] N. K. Logothetis, J. Pauls, M. Augath, T. Trinath, A. Oeltermann,
Neurophysiological investigation of the basis of the fMRI signal, Nature
412 (2001) 150–157.
[2] G. M. Boynton, S. A. Engel, G. H. Glover, D. J. Heeger, Linear systems analysis of functional magnetic resonance imaging in human V1,
Neuroscience 16 (1996) 4207–4221.
[3] A. M. Dale, R. L. Buckner, Selective averaging of rapidly presented
individual trials using fMRI, Human Brain Mapping 5 (1997) 329–340.
[4] R. L. Buckner, Event-related fMRI and the hemodynamic response, Human Brain Mapping 6 (1998) 373–377.
[5] R. B. Buxton, K. Uludağ, D. J. Dubowitz, T. T. Liu, Modeling the
hemodynamic response to brain activation, NeuroImage 23 (2004) 220–
233.
[6] M. A. Lindquist, J. M. Loh, L. Y. Atlas, T. D. Wager, Modeling
the hemodynamic response function in fMRI: Efficiency, bias and mismodeling, NeuroImage 45 (1) (2009) 187–198.
[7] G. Marrelec, H. Benali, P. Ciuciu, M. Pelegrini-Issac, J. B. Poline,
Robust Bayesian estimation of the hemodynamic response function in
event-related BOLD fMRI using basic physiological information, Human
Brain Mapping 19 (2003) 1–17.
[8] G. Marrelec, P. Ciuciu, , M. Pelegrini-Issac, H. Benali, Estimation of
the hemodynamic response in event related functional MRI: Bayesian
networks as a framework for efficient Bayesian modeling and inference,
IEEE Transactions on Medical Imaging 23 (2004) 959–967.
[9] P. Ciuciu, J. B. Poline, G. Marrelec, J. Idier, C. Pallier, H. Benali, Unsupervised robust nonparametric estimation of the hemodynamic response
function for any fMRI experiment, IEEE Transactions on Medical Imaging 22 (2003) 1235–1251.
[10] S. Makni, J. Idier, J. B. Poline, Joint detection-estimation of brain activity in functional MRI: A multichannel deconvolution solution, IEEE
Transactions on Signal Processing 53 (2005) 3488–3502.
28
[11] K. J. Worlsey, C. H. Liao, J. Aston, V. Petre, G. H. Duncan, F. Morales,
A. C. Evans, A general statistical analysis for fMRI, NeuroImage 15
(2002) 1–15.
[12] J. Tanabe, D. Miller, J. Tregellas, R. Freedman, F. G. Meyer, Comparison of detrending methods for optimal fMRI preprocessing, NeuroImage
15 (2002) 902–907.
[13] M. A. Lindquist, T. D. Wager, Validity and power in hemodynamic
response modeling: A comparison study and a new approach, Human
Brain Mapping 28 (2007) 764–784.
[14] A.-K. Seghouane, A Kullback-Leibler methodology for HRF estimation
in fMRI data, in: Proceedings of Annual International Conference of
the IEEE Engineering in Medicine and Biology Society, Buenos Aires,
Argentina, 2010, pp. 2910–2913.
[15] A. K. Seghouane, A. Shah, HRF estimation in fMRI data with unknown drift matrix by iterative minimization of the Kullback-Leibler
divergence, IEEE Transactions on Medical Imaging 31 (2) (2012) 192–
206.
[16] A. K. Seghouane, A. Shah, Consistent hemodynamic response function
estimation in functional MRI by first order differencing, in: Proceedings of the 10th IEEE International Symposium on Biomedical Imaging
(ISBI), San Francisco, US, 2013, pp. 282–285.
[17] A. Shah, A. K. Seghouane, Consistent estimation of hemodynamic response function in fNIRS, in: Proceedings of the 2013 IEEE International Conference on Acoustics, Speech and signal Processing (ICASSP),
Vancouver, Canada, 2013, pp. 1281–1285.
[18] A. Shah, A. K. Seghouane, An integrated framework for joint HRF and
drift estimation and HbO/HbR signal improvement in fNIRS data, IEEE
Transactions on Medical Imaging 33 (2014) 2086–2097.
[19] F. G. Meyer, Wavelet-based estimation of a semi-parametric generalized
linear model of fMRI time series, IEEE Transactions on Medical Imaging
22 (2003) 315–322.
29
[20] J. Fadili, E. Bullmore, Penalized partially linear models using sparse
representations with an application to fMRI time series, IEEE Transactions on Signal Processing 53 (2005) 3436–3448.
[21] M. W. Woolrich, B. D. Ripley, M. Brady, S. M. Smith, Temporal autocorrelation in univariate linear modeling of fMRI data, Neuroimage 14
(2001) 1370–1386.
[22] E. Bullmore, C. Long, J. Suckling, J. Fadili, G. Calvert, F. Zelaya, T. A.
Carpenter, M. Brammer, Colored noise and computational inference in
neurophysiological (fMRI) time series analysis: Resampling methods in
time and wavelet domains, Human Brain Mapping 12 (2) (2001) 61–78.
[23] T. E. Lund, K. H. Madsen, K. Sidaros, W.-L. Luo, T. E. Nichols, Nonwhite noise in fMRI: Does modelling have an impact?, NeuroImage
29 (1) (2006) 54–66.
[24] E. Bullmore, M. Brammer, S. C. Williams, S. Rabe-Hesketh, N. Janot,
A. David, J. Mellers, P. Sham, Statistical methods of estimation and
inference for functional MR image analysis, Magnetic Resonance in
Medicine 35 (1996) 261–277.
[25] P. L. Purdon, V. Solo, R. M. Weisskoff, E. N. Brown, Locally regularized spatiotemporal modeling and model comparison for functional
MRI, NeuroImage 14 (2001) 912–923.
[26] S. Makni, P. Ciuciu, J. Idier, J. B. Poline, Joint detection-estimation of
brain activity in fMRI using an autoregressive model, in: Proceedings
of the IEEE International Symposium on Biomedical Imaging (ISBI),
Arlington, USA, 2006, pp. 1048–1051.
[27] K. J. Friston, J. T. Ashburner, S. J. Kiebel, T. E. Nichols, W. D. Penny,
Statistical Parametric Mapping: The Analysis of Functional Brain Images., Academic press, London, 2006.
[28] L. Harrison, W. D. Penny, K. Friston, Multivariate autoregresive modeling of fMRI time series., NeuroImage 19 (2003) 1477–1491.
[29] S. M. Smith et al., Advances in functional and structural MR image
analysis and implementation as FSL, NeuroImage 23 (2004) 208–219.
30
[30] M. A. Burock, A. M. Dale, Estimation and detection of event-related
fMRI signals with temporally correlated noise: A statistically efficient
and unbiased approach., Human Brain Mapping 11 (4) (2000) 249–260.
[31] K. J. Friston, W. Penny, C. Phillips, S. Kiebel, G. Hinton, J. Ashburner,
Classical and Bayesian inference in neuroimaging: Theory., NeuroImage
16 (2) (2002) 484–512.
[32] A. Yatchew, An elementary estimator of the partial linear model, Economics Letters 57 (1997) 135–143.
[33] J. Wise, The autocorrelation function and the spectral density function,
Biometrika 42 (1955) 151–159.
[34] M. Pourahmadi, Joint mean covariance models with applications to longitudinal data: Unconstrained parametrisation, Biometrika 86 (1999)
677–690.
[35] H. Theil, Principles of Econometrics, New York: Wiley, 1971.
[36] P. J. Brockwell, R. A. Davis, Time Series: Theory and Methods,
Springer-Verlag, New York, 1991.
[37] G. H. Glover, Deconvolution of impulse response in event-related bold
fMRI, NeuroImage 9 (4) (1999) 416–429.
[38] Z. Y. Shan, M. J. Wright, P. M. Thompson, K. L. McMahon, G. G. Blokland, G. I. Zubicaray, N. G. Martin, A. A. Vinkhuyzen, D. C. Reutens,
Modeling of the hemodynamic responses in block design fMRI studies,
Journal of Cerebral Blood Flow and Metabolism 34 (2014) 316–324.
[39] M. A. Lindquist, T. D. Wager, Validity and power in hemodynamic
response modeling: A comparison study and a new approach, Human
brain mapping 28 (8) (2007) 764–784.
[40] C. Goutte, F. A. Nielsen, K. H. Hansen, Modeling the hemodynamic response in fMRI using smooth FIR filters, IEEE Transactions on Medical
Imaging 19 (12) (2000) 1188–1201.
[41] K. J. Friston, O. Josephs, G. Rees, R. Turner, Nonlinear event-related
responses in fMRI, Magnetic Resonance in Medicine 39 (1) (1998) 41–52.
31
[42] K. J. Friston, D. E. Glaser, R. N. Henson, S. Kiebel, C. Phillips, J. Ashburner, Classical and Bayesian inference in neuroimaging: Applications,
NeuroImage 16 (2) (2002) 484–512.
[43] E. Zarahn, G. K. Aguirre, M. D’Esposito, Empirical analyses of BOLD
fMRI statistics. I. Spatially unsmoothed data collected under nullhypothesis conditions., NeuroImage 5 (1997) 179–97.
[44] K. Lee, S. Tak, J. C. Ye, A data-driven sparse GLM for fMRI analysis
using sparse dictionary learning with MDL criterion, IEEE Trans. Med.
Imaging 30 (5) (2011) 1076–89.
[45] A. Shah, A model-free de-drifting approach for detecting BOLD activities in fMRI data, Journal of Signal Processing Systems 79 (2) (2015)
133–143.
[46] K. J. Friston, A. P. Holmes, J. Ashburner, J. B. Poline, SPM8,
http://www.fil.ion.ucl.ac.uk/spm/.
[47] P. A. Bandettini, A. Jesmanowicz, E. C. Wong, J. S. Hyde, Processing
strategies for time-course data sets in functional MRI of the human
brain, Magnetic Resonance in Medicine 30 (2) (1993) 161–173.
[48] T. Schneider, A. Neumaier, ARFIT: A Matlab package for the estimation of parameters and eigenmodes of multivariate autoregressive models, ACM Transactions on Mathematical Software 27 (2001) 58–65.
[49] P. C. M. van Zijl, J. Hua, H. Lu, The BOLD post-stimulus undershoot,
one of the most debated issues in fMRI, NeuroImage 62 (2) (2012) 1092–
1102.
[50] S. Badillo, T. Vincent, P. Ciuciu, Group-level impacts of withinand between-subject hemodynamic variability in fMRI, NeuroImage 82
(2013) 433–448.
[51] A. K. Seghouane, L. A. Johnston, Consistent hemodynamic response estimation function in fMRI using sparse prior information, in: Proceedings of the 11th IEEE International Symposium on Biomedical Imaging
(ISBI), Beijing, China, 2014, pp. 596–599.
32
[52] A. K. Seghouane, A. Shah, Sparse estimation of the hemodynamic response function in near infrared spectroscopy, in: Proceedings of the
2014 IEEE International Conference on Acoustics, Speech and signal
Processing (ICASSP), Florence, Italy, 2014, pp. 2074–2078.
33
Authors' biographies
Article reference: YDSPR_DSP-D-16-00801
Article title: fMRI Hemodynamic Response Function Estimation in Autoregressive Noise by
Avoiding the Drift
To be published in: Digital Signal Processing
Abd-Krim Seghouane received his PhD in Signal Processing and Control from Université
Paris-sud XI, France in 2003. After one year postdoc at INRIA Roquencourt, France he joined
The Australian National University and National ICT Australia in 2005. Since 2013, he has been
with the Department of Electrical and Electronic Engineering, The University of Melbourne.
Adnan Shah received the B.S. (BSc/BEng) Degree in Electronics Engineering from GIK
Institute of Engineering Sciences & Technology, Pakistan, in 2003, and M.Sc. Degree in
Information and Communication engineering (ICE) from TU Darmstadt, Germany, in 2008, and
Ph.D. Degree in Control and Signal Processing Group from the Australian National University,
Canberra, Australia, in 2014. His interests include biomedical signal and image processing,
embedded systems engineering, and design of RF/mixed-signal circuits.
Chee-Ming Ting received the B.Eng. and M.Eng. degrees both in electrical engineering, and
the Ph.D. degree in mathematics from the Universiti Teknologi Malaysia (UTM), Johor,
Malaysia, in 2005, 2007 and 2012, respectively. He is currently a Senior Lecturer in the Faculty
of Biosciences and Medical Engineering, UTM. From 2012 to 2013, he was a Postdoctoral
Fellow with the Center for Biomedical Engineering, UTM, where he is currently a Research
Fellow. He was a Visiting Researcher at the University of Melbourne, Australia, and the
University of California, Irvine, in 2014 and 2015, respectively. In 2016 and 2017, he was a
Visiting Lecturer at the King Abdullah University of Science and Technology, Saudi Arabia. His
research interests include statistical signal processing, state-space methods, time series
analysis, and high-dimensional statistics with applications to biomedical signal processing.