[Paper Review] A maximum likelihood based technique for validating detrended fluctuation analysis (ML-DFA)

Maria Botcharova, Simon F. Farmer+1 more|arXiv (Cornell University)|Jun 21, 2013

Complex Systems and Time Series Analysis38 references16 citations

TL;DR

This paper introduces ML-DFA, a maximum likelihood-based method to validate the linearity of detrended fluctuation analysis (DFA) fluctuation plots, which is essential for ensuring the reliability of the DFA exponent as a measure of long-range temporal correlations. By comparing linear and alternative models (polynomial, spline, exponential, logarithmic) using AIC and BIC, ML-DFA identifies the best-fitting model while penalizing overfitting, thus determining whether the DFA exponent is statistically meaningful.

ABSTRACT

Detrended Fluctuation Analysis (DFA) is widely used to assess the presence of long-range temporal correlations in time series. Signals with long-range temporal correlations are typically defined as having a power law decay in their autocorrelation function. The output of DFA is an exponent, which is the slope obtained by linear regression of a log-log fluctuation plot against window size. However, if this fluctuation plot is not linear, then the underlying signal is not self-similar, and the exponent has no meaning. There is currently no method for assessing the linearity of a DFA fluctuation plot. Here we present such a technique, called ML-DFA. We scale the DFA fluctuation plot to construct a likelihood function for a set of alternative models including polynomial, root, exponential, logarithmic and spline functions. We use this likelihood function to determine the maximum likelihood and thus to calculate values of the Akaike and Bayesian information criteria, which identify the best fit model when the number of parameters involved is taken into account and over-fitting is penalised. This ensures that, of the models that fit well, the least complicated is selected as the best fit. We apply ML-DFA to synthetic data from FARIMA processes and sine curves with DFA fluctuation plots whose form has been analytically determined, and to experimentally collected neurophysiological data. ML-DFA assesses whether the hypothesis of a linear fluctuation plot should be rejected, and thus whether the exponent can be considered meaningful. We argue that ML-DFA is essential to obtaining trustworthy results from DFA.

Motivation & Objective

• To address the lack of a formal method to assess the linearity of DFA fluctuation plots, which is critical for the validity of the DFA exponent.
• To overcome the limitations of traditional R² and chi-squared assumptions, which are insensitive or invalid due to heteroscedasticity in DFA plots.
• To provide a statistically rigorous, model-selection-based approach to determine whether a fluctuation plot supports self-similarity and thus meaningful exponent estimation.
• To validate the method on synthetic FARIMA processes and real neurophysiological EEG data to demonstrate its reliability and practical utility.
• To establish ML-DFA as a necessary step before interpreting DFA exponents in neuroscience and other fields.

Proposed method

• Construct a likelihood function by normalizing DFA fluctuation magnitudes to form a probability density function over window sizes.
• Evaluate the log-likelihood for a linear model and five alternative models: polynomial, root, exponential, logarithmic, and spline functions.
• Use the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) to compare model fits while penalizing models with more parameters to avoid overfitting.
• Select the model with the lowest AIC or BIC as the best-fitting model; if the linear model performs best, the fluctuation plot is considered linear and the DFA exponent is deemed valid.
• Apply the method to synthetic data from FARIMA(0,d,0), FARIMA(1,d,0), and FARIMA(0,d,1) processes, where the expected fluctuation plot shapes are analytically known.
• Test the method on experimentally recorded human EEG data from 20 healthy subjects, using pre-processed, artifact-rejected signals with minimum 20-minute duration.

Experimental results

Research questions

• RQ1 Can ML-DFA reliably detect whether a DFA fluctuation plot is linear, thus validating the use of the DFA exponent as a measure of long-range temporal correlations?
• RQ2 How does ML-DFA perform in distinguishing between self-similar signals (e.g., fractional Gaussian noise) and non-self-similar signals with complex fluctuation patterns?
• RQ3 Does ML-DFA outperform traditional R²-based assessments in detecting non-linearity in fluctuation plots, especially when visual inspection is inconclusive?
• RQ4 Can ML-DFA be effectively applied to real neurophysiological EEG data to determine the validity of reported DFA exponents?
• RQ5 To what extent does ML-DFA reduce the risk of misinterpreting DFA results due to non-linear fluctuation plots in time series with mixed or short-range correlations?

Key findings

• ML-DFA successfully identified the correct underlying model in synthetic FARIMA processes, including cases with concave or convex fluctuation plots due to non-zero φ or θ parameters.
• For FARIMA(0,d,0) processes, which are theoretically self-similar and produce asymptotically linear DFA plots, ML-DFA consistently selected the linear model as the best fit.
• In cases with non-zero φ (e.g., FARIMA(1,d,0)), where the fluctuation plot shows concavity, ML-DFA correctly rejected the linear model in favor of a polynomial or spline fit.
• For FARIMA(0,d,1) processes with convex fluctuation plots due to non-zero θ, ML-DFA selected non-linear models (e.g., exponential or polynomial) over the linear model.
• In neurophysiological EEG data, ML-DFA revealed that a significant proportion of fluctuation plots were non-linear, indicating that many reported DFA exponents may not be statistically valid.
• The method demonstrated robustness in distinguishing between true self-similarity and spurious linearity, especially in noisy or multi-component signals where visual inspection fails.