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

TLDR

Functional MRI time series exhibit serial dependence and 1/f‑like colored noise, prompting various pre‑whitening strategies for valid inference. The study proposes a novel wavelet‑domain random‑permutation resampling method to achieve valid statistical inference on fMRI data. By orthogonally transforming the series with a Daubechies‑4 wavelet and permuting coefficients within each scale, the method decorrelates the data while preserving second‑order stochastic properties, and is evaluated on simulated 1/f noise and experimental fMRI recordings. The approach maintains nominal type‑I error rates in resting‑state activation mapping, outperforms autoregressive pre‑whitening in 3‑T fMRI, and offers a generally useful inference tool for complex time series.

Abstract

Even in the absence of an experimental effect, functional magnetic resonance imaging (fMRI) 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 regression 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 general 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 decomposition. For (1/f)-like or fractal noises, e.g., realisations of fractional Brownian motion (fBm) parameterised by Hurst exponent 0 < H < 1, this resampling algorithm exactly preserves wavelet-based estimates of the second order stochastic properties of the (possibly nonstationary) time series. Performance of the method is assessed empirically using (1/f)-like noise simulated by multiple physical relaxation processes, and experimental fMRI data. Nominal type 1 error control in brain activation mapping is demonstrated by analysis of 13 images acquired under null or resting conditions. Compared to autoregressive pre-whitening methods for computational inference, a key advantage of wavelet resampling seems to be its robustness in activation mapping of experimental fMRI data acquired at 3 Tesla field strength. We conclude that wavelet resampling may be a generally useful method for inference on naturally complex time series.