Colored noise and computational inference in neurophysiological (fMRI) time series analysis: Resampling methods in time and wavelet domains

Even in the absence of an experimental effect, functional magnetic resonanc e imaging (fMRT) time series generally demonstrate serial dependence. This colored noise or endogenous autocorrelation typically has disproportionate spectral power at low frequencies, i.e., its spectrum is 1/f-like. Various pre-whitening and pre-coloring strategies have been proposed to make valid inference on standardised test statistics estimated by time series regressi on in this context of residually autocorrelated errors. Here we introduce a new method based on random permutation after orthogonal transformation of the observed time series to the wavelet domain. This scheme exploits the ge neral whitening or decorrelating property of the discrete wavelet transform and is implemented using a Daubechies wavelet with four vanishing moments to ensure exchangeability of wavelet coefficients within each scale of deco mposition. For 1/f-like or fractal noises, e.g., realisations of fractional Brownian motion (fBm) parameterised by Hurst exponent 0 < H < 1, this resa mpling algorithm exactly preserves wavelet-based estimates of the second or der stochastic properties of the (possibly nonstationary) time series. Perf ormance of the method is assessed empirically using 1/f-like noise simulate d by multiple physical relaxation processes, and experimental fMRI data. No minal type 1 error control in brain activation mapping is demonstrated by a nalysis of 13 images acquired under null or resting conditions. Compared to autoregressive pre-whitening methods for computational inference, a key ad vantage of wavelet resampling seems to be its robustness in activation mapp ing of experimental fMRI data acquired at 3 Tesla field strength. We conclu de that wavelet resampling may be a generally useful method for inference o n naturally complex time series. (C) 2001 Wiley-Liss, Inc.