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NeuroImage 12, 196–208 (2000)doi:10.1006/nimg.2000.0609, available online at http://www.idealibrary.com on

To Smooth or Not to Smooth?Bias and Efficiency in fMRI Time-Series Analysis

K. J. Friston,* O. Josephs,* E. Zarahn,† A. P. Holmes,* S. Rouquette,‡ and J.-B. Poline‡*The Wellcome Department of Cognitive Neurology, Institute of Neurology, Queen Square, London WC1N 3BG, United Kingdom;

†Department of Neurology, University of Pennsylvania, Philadelphia, Pennsylvania 19104; and‡CEA/DSV/DRM/SHFJ, 4 Place General Leclerc, 91406 Orsay, France

Received July 28, 1999

This paper concerns temporal filtering in fMRI time-series analysis. Whitening serially correlated data isthe most efficient approach to parameter estimation.However, if there is a discrepancy between the as-sumed and the actual correlations, whitening can ren-der the analysis exquisitely sensitive to bias when es-timating the standard error of the ensuing parameterestimates. This bias, although not expressed in termsof the estimated responses, has profound effects onany statistic used for inference. The special con-straints of fMRI analysis ensure that there will alwaysbe a misspecification of the assumed serial correla-tions. One resolution of this problem is to filter thedata to minimize bias, while maintaining a reasonabledegree of efficiency. In this paper we present expres-sions for efficiency (of parameter estimation) and bias(in estimating standard error) in terms of assumedand actual correlation structures in the context of thegeneral linear model. We show that: (i) Whiteningstrategies can result in profound bias and are there-fore probably precluded in parametric fMRI data anal-yses. (ii) Band-pass filtering, and implicitly smoothing,has an important role in protecting against inferentialbias. © 2000 Academic Press

Key Words: functional neuroimaging; fMRI; bias; ef-ciency; filtering; convolution; inference.

INTRODUCTION

This paper is about serial correlations in fMRI timeseries and their impact upon the estimations of, andinferences about, evoked hemodynamic responses. InFriston et al. (1994), we introduced the statistical com-

lications that arise, in the context of the general lin-ar model (or linear time invariant systems), due toemporal autocorrelations or “smoothness” in fMRIime series. Since that time a number of approaches tohese intrinsic serial correlations have been proposednd our own approach has changed substantially overhe years. In this paper we describe briefly the sources

1961053-8119/00 $35.00Copyright © 2000 by Academic PressAll rights of reproduction in any form reserved.

of correlations in fMRI error terms and consider differ-ent ways of dealing with them. In particular we con-sider the importance of filtering the data to conditionthe correlation structure despite the fact that removingcorrelations (i.e., whitening) would be more efficient.These issues are becoming increasingly important withthe advent of event-related fMRI that typically evokesresponses in the higher frequency range (Paradis et al.,1998).

This paper is divided into three sections. The firstdescribes the nature of, and background to, serial cor-relations in fMRI and the strategies that have beenadopted to accommodate them. The second sectioncomments briefly on the implications for optimum ex-perimental design and the third section deals, ingreater depth, with temporal filtering strategies andtheir impact on efficiency and robustness. In this sec-tion we deal first with efficiency and bias for a singleserially correlated time series and then consider theimplications of spatially varying serial correlationsover voxels.

SERIAL CORRELATIONS IN fMRI

fMRI time series can be viewed as a linear admixtureof signal and noise. Signal corresponds to neuronallymediated hemodynamic changes that can be modeledas a linear (Friston et al., 1994) or nonlinear (Friston etal., 1998) convolution of some underlying neuronal pro-cess, responding to changes in experimental factors.fMRI noise has many components that render it rathercomplicated in relation to other neurophysiologicalmeasurements. These include neuronal and nonneuro-nal sources. Neuronal noise refers to neurogenic signalnot modeled by the explanatory variables and occupiesthe same part of the frequency spectrum as the hemo-dynamic signal itself. These noise components may beunrelated to the experimental design or reflect varia-tions about evoked responses that are inadequatelymodeled in the design matrix. Nonneuronal compo-nents can have a physiological (e.g., Meyer’s waves) or

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197fMRI TIME-SERIES ANALYSIS

nonphysiological origin and comprise both white [e.g.,thermal (Johnston) noise] and colored components[e.g., pulsatile motion of the brain caused by cardiaccycles or local modulation of the static magnetic field(B0) by respiratory movement]. These effects are typi-cally low frequency (Holmes et al., 1997) or wide band(e.g., aliased cardiac-locked pulsatile motion). The su-perposition of these colored components creates serialcorrelations among the error terms in the statisticalmodel (denoted by Vi below) that can have a severeeffect on sensitivity when trying to detect experimentaleffects. Sensitivity depends upon (i) the relativeamounts of signal and noise and (ii) the efficiency of theexperimental design and analysis. Sensitivity also de-pends on the choice of the estimator (e.g., linear leastsquares vs Gauss–Markov) as well as the validity ofthe assumptions regarding the distribution of the er-rors (e.g., the Gauss–Markov estimator is the maxi-mum likelihood estimator only if the errors are multi-variate Gaussian as assumed in this paper).

There are three important considerations that arisefrom this signal processing perspective on fMRI timeseries: The first pertains to optimum experimental de-sign, the second to optimum filtering of the time seriesto obtain the most efficient parameter estimates, andthe third to the robustness of the statistical inferencesabout the parameter estimates that ensue. In whatfollows we will show that the conditions for both highefficiency and robustness imply a variance–bias trade-off that can be controlled by temporal filtering. Theparticular variance and bias considered in this paperpertain to the variance of the parameter estimates andthe bias in estimators of this variance.

The Background to Serial Correlations

Serial correlations were considered initially from thepoint of view of statistical inference. It was suggestedthat instead of using the number of scans as the de-grees of freedom for any statistical analysis, the effec-tive degrees of freedom should be used (Friston et al.,1994). The effective degrees of freedom were basedupon some estimate of the serial correlations and en-tered into the statistic so that its distribution underthe null hypothesis conformed more closely to thatexpected under parametric assumptions. The primaryconcern at this stage was the validity of the inferencesthat obtained. This heuristic approach was subse-quently corrected and refined, culminating in theframework described in Worsley and Friston (1995). InWorsley and Friston (1995) a general linear model wasdescribed that accommodated serial correlations, interms of both parameter estimation and inference us-ing the associated statistics. In order to avoid estimat-ing the intrinsic serial correlations, the data were con-volved with a Gaussian smoothing kernel to impose an
approximately known correlation structure. The criti- a
cal consideration at this stage was robustness in theface of errors in estimating the correlations. Bullmoreet al. (1996) then proposed an alternative approach,whereby the estimated temporal autocorrelation struc-ture was used to prewhiten the data, prior to fitting ageneral linear model with assumed identical and inde-pendently distributed error terms. This proposal wasmotivated by considerations of efficiency. Validity androbustness were ensured in the special case of theanalysis proposed by Bullmore et al. (1996) becausehey used randomization strategies for inference.1

Variations on an autoregressive characterization of se-rial correlations then appeared. For example Locascioet al. (1997) used autoregressive moving average(ARMA) models on a voxel-by-voxel basis. Purdon andWeisskoff (1998) suggested the use of an AR(1) pluswhite noise model. In the past years a number of em-pirical characterizations of the noise have been de-scribed. Among the more compelling is a modified 1/fcharacterization2 of Aguirre et al. (1997) and Zarahn etal. (1997) (see Appendix A for a more formal descrip-tion of autoregressive and modified 1/f models). Inhort, there have emerged a number of apparentlyisparate approaches to dealing with, and modeling,oise in fMRI time series. Which of these approaches orodels is best? The answer to this question can be

ramed in terms of efficiency and robustness.

Validity, Efficiency, and Robustness

Validity refers to the validity of the statistical infer-nce or the accuracy of the P values that are obtainedrom an analysis. A test is valid if the false-positiveate is less than the nominal specificity (usually a 5.05). An exact test has a false-positive rate that isqual to the specificity. The efficiency of a test relateso the estimation of the parameters of the statisticalodel employed. This estimation is more efficienthen the variability of the estimated parameters is

maller. A test that remains valid for a given departurerom an assumption is said to be robust to violation ofhat assumption. If a test becomes invalid, when thessumptions no longer hold, then it is not a robust test.n general, considerations of efficiency are subordinateo ensuring that the test is valid under the normalircumstances of its use. If these circumstances incurome deviation from the assumptions behind the test,obustness becomes the primary concern.

1 There is an argument that the permutation of scans implicit inthe randomization procedure is invalid if serial correlations persistafter whitening: Strictly speaking the scans are not interchangeable.The strategy adopted by Bullmore et al. (1996) further assumes thesame error variance for every voxel. This greatly reduces computa-tional load but is not an assumption that everyone accepts in fMRI.

2 This class of model should not be confused with conventional 1/fprocesses with fractal scaling properties, which are 1/f in power. Themodels referred to as “modified 1/f models” in this paper are 1/f in
mplitude.

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198 FRISTON ET AL.

OPTIMUM EXPERIMENTAL DESIGN

Any linear time invariant model (e.g., Friston et al.,1994; Boynton et al., 1996) of neuronally mediatedignals in fMRI suggests that only those experimentalffects whose frequency structure survives convolutionith the hemodynamic response function (HRF) can bestimated with any efficiency. Experimental variancehould therefore be elicited with reference to the trans-er function of the HRF. The corresponding frequencyrofile of a canonical transfer function is shown in Fig.(right). It is clear that frequencies around 1/32 Hzill be preserved, following convolution, relative tother frequencies. This frequency characterizes peri-dic designs with 32-s periods (i.e., 16-s epochs). Gen-rally the first objective of experimental design is toomply with the natural constraints imposed by theRF and to ensure that experimental variance occu-ies these intermediate frequencies.Clearly there are other important constraints. An

mportant one here is the frequency structure of noise,hich is much more prevalent at low frequencies (e.g.,/64 Hz and lower). This suggests that the experimen-al frequencies, which one can control by experimentalesign, should avoid these low-frequency ranges. Otheronstraints are purely experimental; for example, psy-hological constraints motivate the use of event-relatedesigns that evoke higher frequency signals (Paradis et

al., 1998) relative to equivalent block designs. Typicalregressors for an epoch- or block-related design and anevent-related design are shown in Fig. 2. The epoch-related design is shown at the top and comprises abox-car and its temporal derivative, convolved with acanonical HRF (shown on the left in Fig. 1). An event-related design, with the same number of trials pre-sented in a stochastic fashion, is shown at the bottom.Again the regressors correspond to an underlying set ofdelta functions (“stick” function) and their temporal

FIG. 1. Left: A canonical hemodynamic response function (HRF).The HRF in this instance comprises the sum of two gamma functionsmodeling a peak at 6 s and a subsequent undershoot. Right: Spectraldensity associated with the HRF expressed as a function of 1/fre-quency or period length. This spectral density is ul(v)u2 where l(v) isthe HRF transfer function.

derivatives convolved with a canonical HRF. The

event-related design has more high-frequency compo-nents and this renders it less efficient than the blockdesign from a statistical perspective (but more usefulfrom other perspectives). The regressors in Fig. 1 willbe used later to illustrate the role of temporal filtering.

TEMPORAL FILTERING

This section deals with temporal filtering and itseffect on efficiency and inferential bias. We start withthe general linear model and derive expressions forefficiency and bias in terms of assumed and actualcorrelations and some applied filter. The next subsec-tion shows that the most efficient filtering scheme (aminimum variance filter) introduces profound bias intothe estimates of standard error used to construct thetest statistic. This results in tests that are insensitive

FIG. 2. Regressors for an epoch-related or block design (top) andan event-related design (bottom) with 128 scans and a TR of 1.7 s.Both sets of regressors were constructed by convolving the appropri-ate stimulus function (box-car for the block design and a stick func-tion for the event-related design) and its temporal derivative withthe canonical HRF depicted in Fig. 1. These regressors have beenorthogonalized and Euclidean normalized.

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or potentially invalid (i.e., not robust). The positionadopted in this paper is that, for fMRI data analysis, aminimum variance filter is not appropriate. Instead wewould like to find a minimum bias filter. This is diffi-cult because one needs to know how the serial correla-tions that are likely to be encountered deviate from theassumed form. The next subsection presents a way ofestimating the expected bias and efficiency, given theprobability distribution of the intrinsic correlations.Using empirical estimates of this distribution it isshown that suppressing both high and low frequencieswith band-pass filtering is required to minimize bias.The expected values for bias and efficiency are thenused to compare three filtering strategies, (i) whiten-ing, (ii) high-pass, and (iii) band-pass (i.e., high passwith smoothing) filtering, under different models of thecorrelations. In brief it will be shown that supplement-ing high-pass filtering with smoothing has an impor-tant role in reducing bias whatever model is assumed.This section concludes by noting that minimizing biasover a range of deviations from the assumed form forthe correlations also renders bias less sensitive to spa-tial variations in serial correlations from voxel to voxel.

Efficiency and Bias

Here we provide expressions for the efficiency andbias for any experimental design, embodied in the ex-planatory variables or regressors that comprise thedesign matrix X and any contrast or compound of pa-rameter estimates specified with a vector of contrastweights. Consider the general linear model

Sy 5 SXb 1 SKiz, (1)

where y is a (n 3 1) response variable (measured fMRIignal at any voxel) and S is an extrinsic or appliedemporal filter matrix. If S has a Toeplitz form then itan be considered as an applied (de)convolution. How-ver, generally S can take any form. A distinction isade between the true intrinsic correlations and those

ssumed. These correlations are characterized by then 3 n) convolution matrices Ki and Ka, respectively,ith an ensuing noise process Kiz where z is an inde-

pendent innovation ;N(0, s2I).The corresponding autocorrelation matrices are Vi 5

KiKiT and Va 5 KaKa

T. The general least-squares esti-ator of the parameters is

bGLS 5 ~X TS TSX! 21~SX! TSy 5 ~SX! 1Sy, (2)

where 1 denotes the pseudoinverse. The efficiency ofestimating a particular contrast of parameters is in-versely proportional to the contrast variance,

T ˆ 2 T 1 T 1T
var$c bGLS% 5 s c ~SX! SViS ~SX! c, (3)
where c is a vector of contrast weights. One mightsimply proceed by choosing S to maximize efficiency or,equivalently, minimize contrast variance (see “Mini-mum Variance Filters”). However, there is anotherimportant consideration here: any statistic used tomake an inference about the significance of the con-trast is a function of that contrast and an estimate of itsvariance. This second estimate depends upon an esti-mate of s2 and an estimate of the intrinsic correlationsVi (i.e., Va). The estimator of the contrast variance willbe subject to bias if there is a mismatch between theassumed and the actual correlations. Such bias wouldinvalidate the use of theoretical distributions, of teststatistics derived from the contrast, used to controlfalse-positive rates.

Bias can be expressed in terms of the proportionaldifference between the true contrast variance and theexpectation of its estimator (see Appendix B for de-tails),

Bias$S, Vi%

5 1 2trace$RSViS T%c T~SX! 1SVaS T~SX! 1Tc

trace$RSVaS T%c T~SX! 1SViS T~SX! 1Tc,

(4)

where R 5 I 2 SX(SX)1 is a residual-forming matrix.When bias is less than zero the estimated standarderror is too small and the ensuing T or F statistic willbe too large, leading to capricious inferences (i.e., falsepositives). When bias is greater than zero the inferencewill be too conservative (but still valid). In short if Ki Þ

a then bias will depend on S. If Ki 5 Ka then bias 5but efficiency still depends upon S.

Minimum Variance Filters

Conventional signal processing approaches and esti-ation theory dictate that whitening the data engen-

ers the most efficient parameter estimation. This cor-esponds to filtering with a convolution matrix S that

is the inverse of the intrinsic convolution matrix Ki

(where Ki 5 Vi1/2). The resulting parameter estimates

are optimally efficient among all linear, unbiased esti-mators and correspond to the maximum likelihood es-timators under Gaussian assumptions. More formally,the general least-squares estimators bGLS are thenequivalent to the Gauss–Markov or linear minimumvariance estimators bGM (Lawson and Hanson, 1974).

In order to whiten the data one needs to know, orestimate, the intrinsic correlation structure. This re-duces to finding an appropriate model for the autocor-relation function or spectral density of the error termsand then estimating the parameters of that model.Clearly one can never know the true structure but wecan compare different models to characterize their rel-ative strengths and weaknesses. In this paper we will
take a high-order (16) autoregressive model as the

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200 FRISTON ET AL.

“gold standard” and evaluate simpler, but commonlyused, models in relation to it. In other words we willconsider the AR(16) model as an approximation to thetrue underlying correlations. The problem of estimat-ing serial correlations is highlighted in Fig. 3. Here theresiduals from a long (512-scan, TR 5 1.7 s) time serieswere used to estimate the spectral density and associ-

FIG. 3. Spectral densities (top) and corresponding autocorrela-ion functions (bottom) for the residual terms of a fMRI time seriesveraged over 512 voxels. Three cases are shown: (i) An AR(16)odel estimated using the Yule–Walker method (this is taken to begood approximation to the true correlations). The “bump” in the

pectrum at around 1/32 Hz may reflect harmonics of random vari-tions in trial-to-trial responses (every 16 s). (ii) An AR(1) modelstimate using the same method. (iii) For a model of the form (q1/f 1

q2) where f is frequency in Hz. These data came from a single-subject,event-related, single-word-presentation fMRI study acquired withmultislice EPI at 2 T with a TR of 1.7 s. The words were presentedevery 16 s. The data were globally normalized, and fitted event-related responses (Josephs et al., 1997) were removed. The 512voxels were selected on the basis of a nontrivial response to acousticstimulation, based on the F ratio in a conventional SPM analysis(P , 0.001 uncorrected). This ensured that the ensuing gray-mattervoxels represented a fairly homogeneous population in terms of theirfunctional specialization.

ated autocorrelation functions (where one is the

Fourier transform of the other) using the Yule–Walkermethod with an autoregression order of 16 (see Appen-dix A). The data came from a single-subject event-related study using sparse single-word presentationsevery 16 s. Evoked responses were removed followingglobal normalization. Estimates of the autocorrelationfunctions and spectral densities using a commonly as-sumed AR(1) model (Bullmore et al., 1996) and a mod-ified 1/f model (Zarahn et al., 1997) are also shown. TheAR(1) is inadequate in that it fails to model eitherlong-range (i.e., low frequencies) or intermediate cor-relations. The modified 1/f model shown here is a goodapproximation for the short-range and intermediatecorrelations but fails to model the long-range correla-tions as well as it could. Any discrepancy between theassumed and the actual correlation structure meansthat, when the data are whitened in accord with theassumed models, the standard error of the contrast isbiased. This leads directly to bias in the ensuing sta-tistics. For example the T statistic is simply the quo-tient of the contrast and its estimated standard error.It should be noted that the simple models would fitmuch better if drifts were first removed from the timeseries. However, this drift removal corresponds tohigh-pass filtering and we want to make the point thatfiltering is essential for reducing the discrepancy be-tween assumed and actual correlations (see below).

This bias is illustrated in Fig. 4 under a variety ofmodel-specific minimum-variance filters. Here the re-gressors from the epoch- and event-related designs inFig. 2 were used as the design matrix X, to calculatethe contrast variance and bias in the estimate of thisvariance according to Eq. (3) and Eq. (4), where

Vi 5 VAR(16),

Va 5 5ViVAR(1)V1/f1

, S 5 5K i

21 “correct” modelKAR(1)

21 AR~1! modelK1/f

21 1/f model1 “none”

.

The variances in Fig. 4 have been normalized by theminimum variance possible (i.e., that of the “correct”model). The biases are expressed in terms of the pro-portion of variance incorrectly estimated. Obviouslythe AR(16) model gives maximum efficiency and nobias (bias 5 0) because we have used the AR(16) esti-mates as an approximation to the actual correlations.Any deviation from this “correct” form reduces effi-ciency and inflates the contrast variance. Note thatmisspecifying the form for the serial correlations hasinflated the contrast variance more for the event-re-lated design (bottom) relative to the epoch-related de-sign (top). This is because the regressors in the epoch-related design correspond more closely to eigenvectors
of the intrinsic autocorrelation matrix (see Worsley


and Friston, 1995). Generally if the regressors conformto these eigenvectors then there is no loss of efficiency.

The bias incurred by mis-specification can be sub-stantial, resulting in an over- [AR(1)] or under- (1/f and“none”) estimation of the contrast variance leading, inturn, to inexact tests using the associated T statisticthat are unduly insensitive [AR(1)] or invalid (1/f and“none”). The effect is not trivial. For example the biasengendered by assuming an AR(1) form in Fig. 4 isabout 24%. This would result in about a 10% reductionof T values and could have profound effects on infer-ence.

In summary the use of minimum variance, or maxi-mum efficiency, filters can lead to invalid tests. Thissuggests that the use of whitening is inappropriate andthe more important objective is to adopt filtering strat-egies that minimize bias to ensure that the tests arerobust in the face of misspecified autocorrelation struc-tures (i.e., their validity is retained). The minimumbias approach is even more tenable given that sensi-

FIG. 4. Efficiencies and biases computed according to Eq. (3) andEq. (4) in the main text for the regressors in Fig. 2 and contrasts of[1 0] and [0 1] for three models of intrinsic correlations [AR(16),AR(1), and the modified 1/f models] and assuming they do not exist(“none”). The results are for the Gauss–Markov estimators (i.e.,using a whitening strategy based on the appropriate model in eachcase) using the first model [AR(16)] as an approximation to the truecorrelations. The contrast variances have been normalized to theminimum attainable. The bias and increased contrast variance in-duced result from adopting a whitening strategy when there is adiscrepancy between the actual and the assumed intrinsic correla-tions.

tivity in fMRI is not generally a great concern. This is

because a large number of scans enter into the estima-tion in fixed-effect analyses, and random-effects anal-yses (with fewer degrees of freedom) do not have tocontend with serial correlations.

Minimum Bias Filters

There are two fundamental problems when trying tomodel intrinsic correlations: (i) one generally does notknow the true intrinsic correlations and (ii) even if theywere known for any given voxel time series, adoptingthe same assumptions for all voxels will lead to biasand loss of efficiency because each voxel has a differentcorrelation structure (e.g., brain-stem voxels will besubject to pulsatile effects, ventricular voxels will besubject to CSF flow artifacts, white matter voxels willnot be subject to neurogenic noise). It should be notedthat very reasonable methods have been proposed forlocal estimates of spatially varying noise (e.g., Langeand Zeger, 1997). However, in this paper we assumethat computational, and other, constraints require usto use the same statistical model for all voxels.

One solution to the “bias problem” is described inWorsley and Friston (1995) and involves conditioningthe serial correlations by smoothing. This effectivelyimposes a structure on the intrinsic correlations thatrenders the difference between the assumed and theactual correlations less severe. Although generally lessefficient, the ensuing inferences are less biased andtherefore more robust. The loss of efficiency can beminimized by appropriate experimental design andchoosing a suitable filter S. In other words there arecertain forms of temporal filtering for which SViST 'SVaST even when Vi is not known [see Eq. (4)]. Thesefilters will minimize bias. The problem is to find asuitable form for S.

One approach to designing a minimum bias filter isto treat the intrinsic correlation Vi, not as an unknowndeterministic variable, but as a random variable,whose distributional properties are known or can beestimated. We can then choose a filter that minimizesthe expected square bias over all Vi,

SMB 5 min argS$j$S%%

j$S% 5 E bias$S, Vi%2p~Vi!dVi.

(5)

Equation (5) says that the ideal filter would minimizethe expected or mean square bias over all intrinsiccorrelation structures encountered. An expression formean square bias and the equivalent mean contrastvariance (i.e., 1/efficiency) is provided in Appendix C.This expression uses the first and second order mo-ments of the intrinsic autocorrelation function, param-
eterized in terms of the underlying autoregression co-

202 FRISTON ET AL.

efficients. More precisely, given the expectedcoefficients and their covariances, a simple second-order approximation to Eq. (5) obtains in terms of theeigenvectors and eigenvalues of the AR coefficient co-variance matrix. These characterize the principal vari-ations about the expected autocorrelation function.Empirical examples are shown in Fig. 5, based on thevariation over gray matter voxels in the data used inFig. 2. Here the principal variations, about the ex-pected autocorrelation function (lower right), are pre-sented (top) in terms of partial derivatives of the auto-correlation function with respect to the autoregressiveeigenvectors (see Appendix C).

The expressions for mean square bias and meancontrast variance will be used in the next subsection toevaluate the behavior of three filtering schemes inrelation to each other and a number of different corre-lation models. First however, we will use them to mo-

FIG. 5. Characterizing the variability in intrinsic correlations.Top: The 16 partial derivatives of the autocorrelation function withrespect to the eigenvectors of the covariance matrix of the underlyingautoregression coefficients. These represent changes to the autocor-relation function induced by the principal components of variationinherent in the coefficients. The covariances were evaluated over thevoxels described in Fig. 2. Lower left: The associated eigenvaluespectrum. Lower right: The autocorrelation function associated withthe mean of the autoregression coefficients. These characterizationsenter into Eq. (C.3) in Appendix C, to estimate the mean square biasfor a given filter.

tivate the use of band-pass filtering: Intuitively one

might posit band-pass filtering as a minimum biasfilter. In the limit of very narrow band-pass filteringthe spectral densities of the assumed and actual corre-lations, after filtering, would be identical and biaswould be negligible. Clearly this would be inefficientbut suggests that some band-pass filter might be anappropriate choice. Although there is no single univer-sal minimum bias filter, in the sense it will depend onthe design matrix and contrasts employed and otherdata acquisition parameters; one indication of whichfrequencies can be usefully attenuated derives fromexamining how bias depends on the upper and lowercutoff frequencies of a band-pass filter.

Figure 6 shows the mean square bias and meancontrast variance incurred by varying the upper andlower band-pass frequencies. These results are basedon the epoch-related regressors in Fig. 2 and assumethat the intrinsic correlations conform to the AR(16)estimate. This means that any bias is due to variation

FIG. 6. Top: Mean square bias (shown in image format in inset)as a function of high- and low-pass cutoff frequencies defining aband-pass filter. Darker areas correspond to lower bias. Note thatminimum bias attains when a substantial degree of smoothing orlow-pass filtering is used in conjunction with high-pass filtering(dark area on the middle left). Bottom: As for the top but nowdepicting efficiency. The gray scale is arbitrary.

h

ca


about that estimate and not due to specifying an inap-propriately simple form for the correlations.3 Theranges of upper and lower band-pass frequencies werechosen to avoid encroaching on frequencies that con-tain signal. This ensured that efficiency was not se-verely compromised. The critical thing to note fromFig. 6 is that the effects of low-pass and high-passfiltering are not linearly separable. In other wordslow-pass filtering or smoothing engenders minimumbias but only in the context of high-pass filtering. It isapparent that the minimum bias obtains with thegreatest degree of smoothing examined but only inconjunction with high-pass filtering, at around 1/96 persecond. In short, band-pass filtering (as opposed tohigh- or low-pass filtering on their own) minimizes biaswithout a profound effect on efficiency. Interestingly,in this example, although increasing the degree ofsmoothing or low-pass filtering increases contrast vari-ance (i.e., decreases efficiency), at high degrees ofsmoothing the minimum variance high-pass filter isvery similar to the minimum bias filter (see Fig. 6).

These results used filter matrices that have a simpleform in frequency space but are computationally ex-pensive to implement. Next we describe a band-passfilter used in practice (e.g., in SPM99) and for theremainder of this paper.

Computationally Efficient Band-Pass Filters

The filter S can be factorized into low- and high-pass4 components S 5 SLSH. The motivation for this ispartly practical and speaks to the special problems offMRI data analysis and the massive amount of datathat have to be filtered. The implementation of thefilter can be made computationally much more efficientif the high frequencies are removed by a sparseToeplitz convolution matrix and the high-pass compo-nent is implemented by regressing out low-frequencycomponents. In this paper we choose SL so that itstransfer function corresponds to that of the hemody-namic response function (the implied kernel is, how-ever, symmetrical and does not induce a delay). This isa principled choice because it is in these frequenciesthat the neurogenic signal resides. SH is, effectively,the residual-forming matrix associated with a discretecosine transform set (DCT) of regressors R up to afrequency specified in term of a minimum period, ex-pressed in seconds. In this paper we use a cutoff period

3 An AR(16) model stands in here for nearly every other possiblemodel of intrinsic correlations. For example an AR(16) model canemulate the AR plus white noise model of Purdon and Weisskoff(1998).

4 The terms low- and high-pass filtering are technically imprecisebecause the linear filter matrices are not generally convolution ma-trices (i.e., SH does not have a Toeplitz form). However, they remove

igh- and low-frequency components, respectively. m

of 64 s.5 Some people prefer the use of polynomial orspline models for drift removal because the DCT im-poses a zero slope at the ends of the time series and thisis not a plausible constraint. The (squared) transferfunctions and kernels associated with the high-passand combined high- and low-pass (i.e., band-pass) fil-ters are shown in Fig. 7.

The effect of filtering the data is to impose an auto-correlation structure or frequency profile on the errorterms. This reduces the discrepancy between the un-derlying serial correlations and those assumed by anyparticular model. This effect is illustrated in Fig. 8 inwhich the band-pass filter in Fig. 7 has been applied tothe empirical time series used above. Filtering mark-edly reduces the differences among the spectral densityand autocorrelation estimates using the various mod-els (compare Fig. 8 with Fig. 3). The question ad-dressed in the next subsection is whether this filterreduces mean square bias and, if so, at what cost interms of mean efficiency.

An Evaluation of Different Filtering Strategies

In this subsection we examine the effect of filteringon mean contrast variance (i.e., mean 1/efficiency) andmean square bias using the expressions in Appendix C.This is effected under three different models of serial

5 In practice filtering is implemented as Sy [ SLSHy [ SL(y 2R(RTy)). This regression scheme eschews the need to actually form alarge (nonsparse) residual-forming matrix associated with the DCTmatrix R. This form for S can be implemented in a way that gives abroad range of frequency modulation for relatively small numbers offloating point operations. Note that R is a unit orthogonal matrix. Rould comprise a Fourier basis set or indeed polynomials. We preferDCT set because of its high efficiency and its ability to remove

FIG. 7. The spectral density of a band-pass filter based on thehemodynamic response function in Fig. 1 (solid line) and a high-passcomponent with a cutoff at 1/64 Hz (broken line). The correspondingsymmetrical filter kernels are shown in the inset for the high-passfilter (broken line) and band-pass filter (solid line).

onotonic trends.

sF

aca

204 FRISTON ET AL.

correlations: (i) AR(1), (ii) 1/f, and (iii) AR(16). Againwe will assume that the AR(16) estimates are a goodapproximation to the true intrinsic correlations. Foreach model we compared the effect of filtering the datawith whitening. In order to demonstrate the interac-tion between high- and low-pass filtering we used ahigh-pass filter and a band-pass filter. The latter cor-responds to high-pass filtering with smoothing. Theaddition of smoothing is the critical issue here: Thehigh-pass component is motivated by considerations ofboth bias and efficiency. One might expect that high-pass filtering would decrease bias by removing lowfrequencies that are poorly modeled by simple models.Furthermore the effect on efficiency will be small be-cause the minimum variance filter is itself a high-passfilter. On the other hand smoothing is likely to reduceefficiency markedly and its application has to be justi-fied much more carefully.

The impact of filtering can be illustrated by paramet-

FIG. 8. The spectral density and autocorrelation functions pre-dicted on the basis of the three models shown in Fig. 3, after filteringwith the band-pass filter in Fig. 7. Compare these functions withthose in Fig. 3. The differences are now ameliorated.

rically varying the amount of filtering with a filter F, fi

where S 5 s z F 1 (1 2 s)1; 1 is the identity matrix andis varied between 0 (no filtering) and 1 (filtering with). The models and filters considered correspond to

Va 5 HVAR(1)V1/fVAR(16)

and F 5 HV a21/2 “whitening”

SH “high-pass”SLSH “band-pass”

.

Mean square bias and mean contrast variance areshown as functions of the filtering parameter s in Fig.9 for the various schema above. The effect on efficiencyis remarkably consistent over the models assumed forthe intrinsic correlations. The most efficient filter is awhitening filter that progressively decreases the ex-pected contrast variance. High-pass filtering does not

FIG. 9. The effect of filtering on mean square bias and meancontrast variance when applying three filters: the band-pass filter inFig. 8 (solid lines), the high-pass filter in Fig. 8 (dashed lines), and awhitening filter appropriate to the model assumed for the intrinsiccorrelations (dot–dash lines). Each of these filters was applied underthree models: an AR(1) model (top), a modified 1/f model (middle),nd an AR(16) model (bottom). The expected square biases (left) andontrast variances (right) were computed as described in Appendix Cnd plotted against a parameter s that determines the degree ofltering applied (see main text).


markedly change this mean contrast variance and ren-ders the estimation only slightly less efficient than nofiltering at all. The addition of smoothing to give band-pass filtering decreases efficiency by increasing themean contrast variance by about 15%. The effects onmean square bias are similarly consistent. Whiteningattenuates mean square bias slightly. High-pass filter-ing is more effective at attenuating bias but only forthe AR(1) model. This is because the whitening filtersfor the 1/f and AR(16) models more closely approximatethe high-pass filter. The addition of smoothing engen-ders a substantial and consistent reduction in bias,suggesting that, at least for the acquisition parametersimplicit in these data, smoothing has an important rolein minimizing bias. This is at the expense of reducedefficiency.

The profiles at the bottom in Fig. 9 are interestingbecause, as in the analysis presented in Fig. 6, themean intrinsic correlations and assumed autocorrela-tion structure are taken to be the same. This meansthat any effects on bias or efficiency are due solely tovariations about that mean (i.e., the second term in Eq.(C.3), Appendix C). The implication is that even a so-phisticated autocorrelation model will benefit, in termsof inferential bias, from the particular band-pass filter-ing considered here.

Spatially Varying Intrinsic Correlations

In the illustrative examples above we have assumedthat the variation within one voxel, over realizations,can be approximated by the variation over realizationsin different gray-matter voxels that show a degree offunctional homogeneity. The second problem, intro-duced at the beginning of this section, is that even if weassume the correct form for one voxel then we will benecessarily incorrect for every other voxel in the brain.This reflects the spatially dependent nature of tempo-ral autocorrelations (see also Locascio et al., 1997).Assuming the same form for all voxels is a specialconstraint under which analyses of fMRI data have tooperate. This is because we would like to use the samestatistical model for every voxel. There are both com-putational and theoretical reasons for this, which de-rive from the later use of Gaussian Field theory whenmaking inferences that are corrected for the volumeanalyzed. The theoretical reasons are that differentintrinsic correlations would lead to statistics with dif-ferent (effective) degrees of freedom at each voxel.Whether Gaussian Field theory is robust to this effectremains to be addressed.

Not only does appropriate temporal filtering reducebias engendered by misspecification of the intrinsiccorrelations at any one voxel, and stochastic variationsabout that specification, but it also addresses the prob-
lem of spatially varying serial correlations over voxels.
The top of Fig. 10 shows bias, computed using Eq. (3)over 512 voxels using an AR(16) voxel-specific estimatefor the intrinsic correlations Vi and an AR(1) modelaveraged over all voxels for the assumed correlationsVa. With whitening the biases (solid bars) range from250 to 250%. With band-pass filtering (open bars) theyare reduced substantially. The effect on efficiency isshown at the bottom. Here filtering increases the con-trast variance in a nontrivial way (by as much as 50%in some voxels). It is interesting to note that spatiallydependent temporal autocorrelations render the effi-ciency very variable over voxels (by nearly an order ofmagnitude). This is important because it means thatfMRI is not homogeneous in its sensitivity to evokedresponses from voxel to voxel, simply because of differ-

FIG. 10. The distribution of bias (top) and contrast variance(bottom) over voxels using an AR(16) model to estimate intrinsiccorrelations at 512 voxels and assuming the same AR(1) autocorre-lation structure over voxels. Distributions are shown with band-passfiltering (open bars) and with whitening (filled bars). Note howband-pass filtering reduces bias (at the expense of reduced efficien-cy). Efficiency is proportional to the inverse of the contrast variance.

ences in serial correlations.

t

[ft

206 FRISTON ET AL.

DISCUSSION

This paper has addressed temporal filtering in fMRItime-series analysis. Whitening serially correlateddata is the most efficient approach to parameter esti-mation. However, whitening can render the analysissensitive to inferential bias, if there is a discrepancybetween the assumed and the actual autocorrelations.This bias, although not expressed in terms of the esti-mated model parameters, has profound effects on anystatistic used for inference. The special constraints offMRI analysis ensure that there will always be a mis-specification of the intrinsic autocorrelations becauseof their spatially varying nature over voxels. One res-olution of this problem is to filter the data to ensurebias is small while maintaining a reasonable degree ofefficiency.

Filtering can be chosen in a principled way to main-tain efficiency while minimizing bias. Efficiency can beretained by band-pass filtering to preserve frequencycomponents that correspond to signal (i.e., the frequen-cies of the HRF) while suppressing high- and low-frequency components. By estimating the mean squarebias over the range of intrinsic autocorrelation func-tions that are likely to be encountered, it can be shownthat supplementing a high-pass filter with smoothinghas an important role in reducing bias.

The various strategies that can be adopted can besummarized in terms of two choices: (i) the assumedform for the intrinsic correlations Va and (ii) the filterapplied to the data S. Permutations include

Va 5 511VAR(p)VAR(p)

,

S 5 51 ordinary least squaresSLSH conventional “filtering” ~SPM97!KAR(p)

21 conventional “whitening”SLSH band-pass filtering ~SPM99!

,

where the assumed form is an AR(p) model. The laststrategy, adopted in SPM99, is motivated by a balancebetween efficiency and computational expediency. Be-cause the assumed correlations Va do not enter explic-itly into the computation of the parameter estimatesbut only into the subsequent estimation of their stan-dard error (and ensuing T or F statistics), the autocor-relation structure and parameter estimation can beimplemented in a single pass through the data. SPM99has the facility to use an AR(1) estimate of intrinsiccorrelations, in conjunction with (separately) specifiedhigh- and low-pass filtering.

A further motivation for filtering the data, to exertsome control over the variance–bias trade-off, is that
the exact form of serial correlations will vary with a
scanner, acquisition parameters, and experiment. Forexample the relative contribution of aliased bio-rhythms will change with TR and field strength. Byexplicitly acknowledging a discrepancy between theassumed approximation to underlying correlations aconsistent approach to data analysis can be adoptedover a range of experimental parameters and acquisi-tion systems.

The issues considered in this paper apply even whenthe repetition time (TR) or interscan interval is long.This is because serial correlations can enter as low-frequency components, which are characteristic of allfMRI time series, irrespective of the TR. Clearly as theTR gets longer, higher frequencies cease to be a con-sideration and the role of low-pass filtering or smooth-ing in reducing bias will be less relevant.

The conclusions presented in this paper arise underthe special constraints of estimating serial correlationsin a linear framework and, more specifically, usingestimators that can be applied to all voxels. The inher-ent trade-off between the efficiency of parameter esti-mation and bias in variance estimation is shaped bythese constraints. Other approaches using, for exam-ple, nonlinear observation models, may provide moreefficient and unbiased estimators than those we haveconsidered.

It should be noted that the main aim of this paper isto provide an analytical framework within which theeffects of various filtering strategies on bias and effi-ciency can be evaluated. We have demonstrated theuse of this framework using only one data set and donot anticipate that all the conclusions will necessarilygeneralize to other acquisition parameters or statisti-cal models. What has been shown here is, however,sufficient to assert that, for short repetition times,band-pass filtering can have an important role in ame-liorating inferential bias and consequently in ensuringthe relative robustness of the resulting statistical tests.

APPENDIX A

Forms of Intrinsic and Assumed Autocorrelations

A distinction is made between the true intrinsic au-tocorrelations in a fMRI time series of length n andthose assumed. These correlations are characterized bythe (n 3 n) convolution matrices Ki and Ka, respec-ively, with an ensuing noise process e 5 Kiz, where z,

is an independent innovation ;N(0, s2I) anddiag{KiKi

T} 5 1. The corresponding autocorrelation ma-trices are Vi 5 KiKi

T and Va 5 KaKaT. In this appendix

we describe some models of intrinsic autocorrelations,specifically pth order autoregressive models AR(p)e.g., AR(1); Bullmore et al., 1996)] or those that deriverom characterizations of noise autocovariance struc-ures or equivalently their spectral density (Zarahn et
l., 1997). For any process the spectral density g(v) and

1Tmnfi

dta

ott


autocorrelation function r(t) are related by the Fouriertransform pair

r~t! 5 FT$g~v!%, g~v! 5 IFT$r~t!%,

where V 5 Toeplitz$r~t!% and K 5 V 1/2.(A.1)

Here t 5 [0, 1, . . . , n 2 1] is the lag in scans with r(0) 5and v 5 2pi (i 5 [0, . . . , n 2 1]) denotes frequency.he Toeplitz operator returns a Toeplitz matrix (i.e., aatrix that is symmetrical about the leading diago-al). The transfer function associated with the linearlter K is l(v), where g(v) 5 ul(v)u2.

Autoregressive Models

Autoregressive models have the following form:

e 5 Ae 1 z N e 5 ~1 2 A! 21z,

giving Ka } ~1 2 A! 21 where A

5 30 0 0 0 . . .a1 0 0 0a2 a1 0 0a3 a2 a1 0···

· · ·4 .

(A.2)

The autoregression coefficients in the triangular ma-trix A can be estimated using

a 5 @a1, . . . , ap#

5 31 r~1! · · · r~p!

r~1! 1 ······· · · r~1!

r~p! · · · r~1! 14

21

3 r~1!r~2!

···r~p 1 1!4 .(A.3)

The corresponding spectral density over n frequenciesis given by

g~v! 5 uFTn$@1, 2a#%u 22, (A.4)

where FTn{ z } is a n-point Fourier transform with zeropadding (cf. the Yule–Walker method of spectral den-sity estimation). With these relationships [(A.1) to(A.3)] one can take any empirical estimate of the auto-correlation function r(t) and estimate the autoregres-sion model-specific convolution Ka and autocorrelationVa 5 KaKa

T matrices.

Modified 1/f Models

Ka and Va can obviously be estimated using spectralensity through Eq. (A.1) if the model of autocorrela-ions is expressed in frequency space; here we use thenalytic form suggested by the work of Zarahn et al.

(1997),

us~v!u 5q1

v1 q2, where g~v! 5 us~v!u 2. (A.5)

Note that Eq. (A.5) is linear in the parameters whichcan therefore be estimated in an unbiased mannerusing ordinary least squares.

APPENDIX B

Efficiency and Bias

In this section we provide expressions for the effi-ciency and bias for any design matrix X and any con-trast of parameter estimates specified with a vector ofcontrast weights c (this framework serves all the fil-tering schemes outlined in the main text). These ex-pressions are developed in the context of the generallinear model and parametric statistical inference (Fris-ton et al., 1995). The bias here is in terms of theestimated standard error associated with any ensuingstatistic. Consider the general linear model,

Sy 5 SXb 1 SKiz, (B.1)

where y is the response variable and S is an appliedfilter matrix. The efficiency of the general least squarescontrast estimator cTbGLS is inversely proportional toits variance,

Efficiency } Var$c TbGLS% 21

Var$c TbGLS% 5 s 2c T~SX! 1SViS T~SX! 1Tc,(B.2)

where 1 denotes the pseudoinverse. Efficiency is max-imized with the Gauss–Markov estimator, when S 5Ki

21 if the latter were known. However, there is an-ther important consideration: The variance of the con-rast has to be estimated in order to provide for statis-ical inference. This estimator and its expectation are

Var$c Tb%

5y TS TR TRSy

trace$RSVaS T%c T~SX! 1SVaS T~SX! 1Tc

^Var$c Tb%& (B.3)

5s 2trace$RSViS T%

trace$RSVaS T%c T~SX! 1SVaS T~SX! 1Tc,

where R 5 I 2 SX(SX)1 is a residual-forming matrix.Bias can be expressed in terms of the (normalized)difference between the actual and the expected con-
trast variance estimators,

T

v

B

F

F

F

H

J

L

L

L

P

P

W

Z

208 FRISTON ET AL.

Bias$S, Vi%

5Var$c TbGLS% 2 ^Var$c TbGLS%&

Var$c TbGLS%(B.4)

5 1 2trace$RSViS T%c T~SX! 1SVaS T~SX! 1Tc

trace$RSVaS T%c T~SX! 1SViS T~SX! 1Tc.

The critical thing about Eq. (B.4) is that bias is reducedas S induces more correlations relative to Vi. In otherwords there are certain forms of temporal filtering forwhich SViST ' SVaST even if Vi Þ Va. In extreme casesthis allows one to essentially ignore intrinsic correlationswith temporal filtering (i.e., Va 5 I) although this strat-egy may be inefficient. Generally bias becomes sensitiveto discrepancies between the actual and the assumedcorrelations when no filtering is employed and exquis-itely so when S is a deconvolution matrix (e.g., whiteningunder the wrong assumptions about Vi).

APPENDIX C

Mean Square Bias

In Appendix B the expression for bias treated Vi as afixed deterministic variable. In this appendix we derivean approximation for the expected squared bias whenVi is a random stochastic variable parameterized interms of its AR coefficients a. Let

f~r, S! 5 Bias$V$a# 1 Qr%, S% 2. (C.1)

he operator V{a} returns the autocorrelation matrixgiven the underlying AR coefficients according to A.2. a#are the expected coefficients and Q contains the eigen-vectors of Cov{a} 5 ^(a 2 a# ) z (a 2 a# )T& such that

r 5 Q T~a 2 a# ! f H ^r& 5 0^r z r T& 5 l (C.2)

and l is a leading diagonal matrix of associated eigen-alues. Mean square bias is approximated with

j~S! 5 ^f~r, S!&r < f~0, S! 1 Oi

li

2

2f~0, S!

r i2

. (C.3)

This follows from taking the expectation of the second-order approximation of the Taylor expansion of Eq. (C.1)around r 5 0 and substituting the expectations in Eq.(C.2). Note that, apart from the second-order approxima-tion, no distributional assumptions have been madeabout the intrinsic AR coefficients. If these coefficientshad a multivariate Gaussian distribution then (C.3)would be exactly right. The first term in (C.3) reflects the
mean square bias attributable to the deviation between
the assumed and the expected autocorrelation function.The second term reflects the contribution to bias due tovariation about that expectation. An equivalent expres-sion for mean contrast estimate variance obtains by sub-stituting Eq. (B.2) into Eq. (C.1).

In this paper the expectation and covariances of theAR coefficients were determined empirically usingmultiple realizations over gray-matter voxels and anAR(16) model. The partial derivatives in C.3 were es-timated numerically.

ACKNOWLEDGMENTS

K.J.F., O.J., and A.P.H., were funded by the Wellcome trust. Wethank the two anonymous reviewers for guidance in the developmentand presentation of this work.

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