Pitfalls in fractal time series analysis: fMRI BOLD as an exemplary case

Abstract

This article will be positioned on our previous work demonstrating the importance of adhering to a carefully selected set of criteria when choosing the suitable method from those available ensuring its adequate performance when applied to real temporal signals, such as fMRI BOLD, to evaluate one important facet of their behavior, fractality. Earlier, we have reviewed on a range of monofractal tools and evaluated their performance. Given the advance in the fractal field, in this article we will discuss the most widely used implementations of multifractal analyses, too. Our recommended flowchart for the fractal characterization of spontaneous, low frequency fluctuations in fMRI BOLD will be used as the framework for this article to make certain that it will provide a hands-on experience for the reader in handling the perplexed issues of fractal analysis. The reason why this particular signal modality and its fractal analysis has been chosen was due to its high impact on today’s neuroscience given it had powerfully emerged as a new way of interpreting the complex functioning of the brain (see “intrinsic activity”). The reader will first be presented with the basic concepts of mono and multifractal time series analyses, followed by some of the most relevant implementations, characterization by numerical approaches. The notion of the dichotomy of fractional Gaussian noise and fractional Brownian motion signal classes and their impact on fractal time series analyses will be thoroughly discussed as the central theme of our application strategy. Sources of pitfalls and way how to avoid them will be identified followed by a demonstration on fractal studies of fMRI BOLD taken from the literature and that of our own in an attempt to consolidate the best practice in fractal analysis of empirical fMRI BOLD signals mapped throughout the brain as an exemplary case of potentially wide interest.

Decomposing Multifractal Crossovers

“…Rodent fMRI-BOLD imaging data of limited definition (Eke et al, 2012) was chosen to demonstrate the performance of our SFD method on scans of bimodal BOLD time series data obtained across a section at the level of the bregma (Herman et al, 2011; Figure 9A). Criteria of 0.5 < H [ f X i ](2) < 2, s x within the full available SR, and 0.1 < s b / s x < 10 yielded valid estimates of crossovers with MSE:SFD-FMF < MSE:qSRA-FMF at a rate of 68%.…”

“…These endpoints are h max [the Hölder exponent, h , at the peak of D ( h )] and fwhm [the full width of D ( h ) at half maximum] (Eke et al, 2012). h max captures the degree of correlation, and fwhm can be taken as the measure of the supporting base of the singularity spectrum (see Figure 3 of Mukli et al, 2015).…”

Multifractal detrended fluctuation analysis of human gait diseases

“…Fractional Gaussian noise also mimics typical shape expected from long range correlations and hence can lead to erroneous results (Eke et al, 2012; Delignières and Marmelat, 2013). Bashan et al (2008) have observed that MFDFA1 systematically overestimates the scale scaling exponents for small scale s. In this respect Centered Moving Average (CMA) (Alvarez-Ramirez et al, 2005) will be more fruitful in analyzing the scaling properties in short data sets without trends.…”

Using Continuous Glucose Monitoring Data and Detrended Fluctuation Analysis to Determine Patient Condition

“…33,[35][36][37] This means the Hurst coefficient must remain constant across all q-order statistical moments. 35,37 More simply, the signal must be simple enough to be described by one fractal dimension. When the selfsimilarity of a time series changes with spatial and temporal variations, the signal is deemed to be multifractal.…”