Optimal spatial regularisation of autocorrelation estimates in fMRI analysis Temujin Gautama * and Marc M. Van Hulle Laboratorium voor Neuro-en Psychofysiologie, B-3000 Leuven, Belgium Received 13 May 2004; revised 30 June 2004; accepted 12 July 2004 Available online 27 September 2004 In the General Linear Model (GLM) framework for the statistical analysis of fMRI data, the problem of temporal autocorrelations in the residual signal (after regression) has been frequently addressed in the open literature. There exist various methods for correcting the ensuing bias in the statistical testing, among which the prewhitening strategy, which uses a prewhitening matrix for rendering the residual signal white (i.e., without temporal autocorrelations). This correction is only exact when the autocorrelation structure of the noise-generating process is accurately known, and the estimates derived from the fMRI data are too noisy to be used for correction. Recently, Worsley and co- workers proposed to spatially smooth the noisy autocorrelation estimates, effectively reducing their variance and allowing for a better correction. In this article, a systematic study into the effect of the smoothing kernel width is performed and a method is introduced for choosing this bandwidth in an boptimalQ manner. Several aspects of the prewhitening strategy are investigated, namely the choice of the autocorrelation estimate (biased or unbiased), the accuracy of the estimates, the degree of spatial regularisation and the order of the autoregressive model used for characterising the noise. The proposed method is extensively evaluated on both synthetic and real fMRI data. D 2004 Elsevier Inc. All rights reserved. Keywords: Spatial regularisation; Autocorrelation; Prewhitening Introduction In the statistical analysis of fMRI data, the presence of intrinsic temporal autocorrelations in the noise-generating process is a well- studied topic. The time series of a given voxel is usually modelled as consisting of a possible haemodynamic response to the stimulus and coloured noise (containing temporal autocorrelations). Albeit the origin of this noise is still an open issue (see, e.g., Biswal et al., 1995; Woolrich et al., 2001; Zarahn et al., 1997), there is a general agreement that it needs to be taken into account in the analysis methods used for activation detection. The conventional statistical analysis techniques, such as SPM (Statistical Parametric Mapping, Wellcome Department of Cogni- tive Neurology, London), are based on the General Linear Model (GLM), which models an fMRI time series as a linear combination of paradigm-related responses, drift terms and an error term. The GLM analysis is only exact when the autocorrelation function of the (real) noise process, which generates the error term, is taken into account. In practice, however, the underlying process is unknown and alternative approaches have been devised, such as bprecolouringQ, which imposes a certain autocorrelation function on the noise term by temporal smoothing (Friston et al., 1995; Worsley and Friston, 1995), and bprewhiteningQ, which transforms the data such that the error term becomes white noise (Bullmore et al., 1996). Several studies have evaluated these (and other) approaches in combination with different statistical tests, both on synthetic and real-world fMRI data (Friston et al., 2000a; Purdon and Weisskoff, 1998; Wicker and Fonlupt, 2003; Woolrich et al., 2001). The prewhitening strategy yields the best (minimum variance) linear unbiased estimator, but only if the true autocorre- lation structure is known (Bullmore et al., 2001; Friston et al., 2000a; Woolrich et al., 2001), or at least if it can be accurately estimated. However, any mismatch between the true and the estimated autocorrelations will lead to a bias in the estimation of the parameter variance (Friston et al., 2000a), which is used for the statistical inference of effects. Therefore, an accurate model of the noise, from which the prewhitening matrix is computed, is essential to the efficacy of the prewhitening strategy, and various noise models have been proposed (Bullmore et al., 1996; Locascio et al., 1997; Purdon and Weisskoff, 1998; Zarahn et al., 1997). Additionally, the underlying noise process needs to be estimated from the residual signal after regression (the error term), which introduces a bias in the autocorrelation estimates (for an overview, see Marchini and Smith, 2003). Another important aspect of the autocorrelation is the spatial variability, which cannot be solely attributed to a difference 1053-8119/$ - see front matter D 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.neuroimage.2004.07.048 * Corresponding author. Laboratorium voor Neuro-en Psychofysiologie, K. U. Leuven, Campus Gasthuisberg, Herestraat 49, bus 801, B-3000 Leuven, Belgium. Fax: +32 16 34 59 60. E-mail addresses: temu@neuro.kuleuven.ac.be (T. Gautama)8 marc@neuro.kuleuven.ac.be (M.M. Van Hulle). Available online on ScienceDirect (www.sciencedirect.com.) www.elsevier.com/locate/ynimg NeuroImage 23 (2004) 1203 – 1216