Wavelet methods for characterising mono- and poly-fractal noise structures in shortish time series: an application to functional MRI

Functional magnetic resonance imaging (fMRI) time series generally demonstrate serial dependence. This endogenous auto-correlation typically exhibits long-range dependence described by a 1/f-like power law. We present a novel wavelet-based methodology for characterising the noise structure in short-medium length (shortish) fMRI time series. Mono-fractality is assessed in terms of the Hurst exponent and the noise variance. We then investigate potential local stationarity of the Hurst exponent in MM data and present a uniformly most powerful test for its time constancy. A novel bootstrap approach is presented as an alternative to the normal assumption and its advantages are discussed. From several datasets investigated, we specifically show that the 1/f model is particularly suited to describe color in MM nose. We also demonstrate that even if most of the brain voxels are mono-fractal, there are many locations in the brain where time constancy of the Hurst exponent is violated, ie, the noise structure is poly-fractal