# Detrended Fluctuation Analysis (DFA): Tutorial > A hands-on tutorial for computing DFA on neuronal oscillations using MATLAB and the Neurophysiological Biomarker Toolbox (NBT). - URL: https://www.poil.dk/research-background/dfa-tutorial/ - Author: Simon-Shlomo Poil - First published: 2012-11-10 - Last updated: 2026-03-01 - Section: Research Background (https://www.poil.dk/research-background/) *This tutorial was originally published on nbtwiki.net as companion material to Hardstone et al. (2012). It has been migrated here with minor formatting updates. The MATLAB code uses functions from the [Neurophysiological Biomarker Toolbox (NBT)](https://www.poil.dk/projects/nbt/), available on [GitHub](https://github.com/NBT-Analytics/NBTpublic). For why long-range temporal correlations matter, see [The Brain as a Critical System](https://www.poil.dk/research-background/brain-criticality/).* *Please cite: R. Hardstone\*, S.-S. Poil\*, G. Schiavone, R. Jansen, V.V. Nikulin, H.D. Mansvelder, K. Linkenkaer-Hansen. Detrended fluctuation analysis: A scale-free view on neuronal oscillations. Frontiers in Physiology 3:450, 2012. [doi:10.3389/fphys.2012.00450](https://doi.org/10.3389/fphys.2012.00450)* --- **Approximate time to complete: 60 minutes.** ## What DFA measures The detrended fluctuation analysis (DFA) algorithm is a scaling analysis method used to estimate long-range temporal correlations of power-law form (Peng et al., 1995; Hardstone et al., 2012). If a sequence of events has a non-random temporal structure with slowly decaying auto-correlations, DFA quantifies how slowly these correlations decay, as indexed by the DFA power-law exponent α. Typical classes of correlations are reflected in DFA exponents of: - **Uncorrelated sequence:** α ≈ 0.5 - **Anti-correlated sequence:** 0 < α < 0.5 - **Long-range temporal correlations:** 0.5 < α < 1 - **Strong correlations (not power-law form):** α > 1 The interest in long-range temporal correlations (LRTC) is rooted in empirical and modelling results indicating that the dynamics of self-organising multi-unit systems is characterised by fluctuations that tend to appear in a specific temporal ordering, as indexed by power-law decaying correlations. That a system is "self-organising" means that, except for some fairly slow stochastic drive, most of the fluctuations are endogenous rather than determined by strong external inputs. The physical background is that activities make changes to the internal structure of these systems, and this structure in turn provides a bias for when and where future activity can flow (Poil et al., 2012). Think of water flowing down a mountain slope, carving a network of rivers that is ever changing but also sufficiently stable that it strongly influences where water can flow from day to day, month to month, year to year. The dynamical spatial structures in complex self-organising systems are the substrate of the memory that ensures a degree of correlation in activity over time. Long-range temporal correlations have been found in the amplitude modulation of human brain waves (Linkenkaer-Hansen et al., 2001), hippocampal fast network oscillations (Poil et al., 2011), and have been shown to be impaired in disorders such as Alzheimer's disease (Montez et al., 2009) and epilepsy (Monto et al., 2007). The interpretation of DFA crucially depends on the pre-processing of the data. In the case of EEG/MEG analysis, pre-processing consists of extracting the amplitude envelope of distinct oscillations — but DFA could equally be applied to, for example, time series of phase-locking values between brain areas, where it would quantify the possibly non-random temporal order of synchronisation (Montez et al., 2006). This tutorial has two sections: a basic introduction to **computing DFA using NBT**, and an **extended tutorial** that gives a deeper understanding of DFA and its practical pitfalls. --- ## Computing DFA using NBT We show how to perform detrended fluctuation analysis in MATLAB using NBT. In this example, we compute DFA in the alpha frequency band (8–13 Hz) with a sampling frequency of 250 Hz. If you do not have data, generate a test signal: ```matlab Signal = randn(300*250, 23); ``` This generates a 5-minute (300 seconds) signal with 23 channels. ### 1. Removing artefacts NBT has several functions for artefact removal. Large-amplitude transient artefacts will strongly influence the temporal structure of the signal and should be cut away. **Important:** the time series should be stitched together after cutting artefact segments. Signals in NBT have time along the 1st dimension and channels along the 2nd dimension. ### 2. Filter the signal and extract the amplitude envelope Use `nbt_GetAmplitudeEnvelope` to filter the signal and extract the amplitude envelope: ```matlab SignalInfo = nbt_Info; SignalInfo.converted_sample_frequency = 250; AmplitudeEnvelope = nbt_GetAmplitudeEnvelope(Signal, SignalInfo, 8, 13, 2/8); ``` The `nbt_Info` function defines an NBT info object containing information about your signal. The last parameter `2/8` is the filter order (in seconds), set so that at least two oscillation cycles are covered by the filter window. ### 3. Perform DFA Calculate the DFA exponent and plot the fluctuation function: ```matlab [DFAobject, DFA_exp] = nbt_doDFA(AmplitudeEnvelope, SignalInfo, [1 15], [0.8 25], 0.5, 1, 1, []); ``` Parameters: `Signal`, `SignalInfo`, `FitInterval` (time range for fitting), `CalcInterval` (time range for calculation), `DFA_Overlap` (window overlap, here 50%), `DFA_Plot`, `ChannelToPlot`, and `res_logbin` (resolution of logarithmic binning, default 10). After calculation, find the DFA values for all channels: ```matlab DFAobject.MarkerValues ``` --- ## Extended DFA tutorial ### Testing with random data First, verify that DFA yields α ≈ 0.5 for random data. Define a sampling frequency of 300 Hz and generate 20 minutes of random activity: ```matlab Fs = 300; Signal = randn(20*60*Fs, 1); SignalInfo = nbt_Info; SignalInfo.converted_sample_frequency = Fs; [DFAobject, DFA_exp] = nbt_doDFA(Signal, SignalInfo, [0.1 120], [0.08 1200], 0.5, 1, 1, []); ``` **Rule of thumb:** only estimate correlations up to time scales of 10% of the data length. Did you get a DFA exponent close to 0.5? ### The effect of filtering on correlations Long-range temporal correlations imply modulations on long time scales — but low-frequency fluctuations are typically removed during data acquisition with a high-pass filter around 0.1–1 Hz. Since we are interested in distinct oscillations, we use a bandpass filter and the Hilbert transform to extract the amplitude envelope of narrow-band oscillations. However, bandpass filtering introduces autocorrelations at short time scales, even in random data. To see this, filter the random data at 8–13 Hz and compute DFA on the amplitude envelope: ```matlab highpass = 8; lowpass = 13; fir_order = 0.25; % Filter order in seconds [Data_filtered] = nbt_filter_fir(Signal, highpass, lowpass, Fs, 2/8); ``` Plot the filtered signal and its amplitude envelope: ```matlab figure plot(Data_filtered, 'r') hold on plot(abs(hilbert(Data_filtered)), 'k') ``` You should see oscillations that fluctuate erratically in amplitude. Now compute DFA: ```matlab [DFAobject, DFA_exp] = nbt_doDFA(abs(hilbert(Data_filtered)), SignalInfo, [0.1 120], [0.08 1200], 0.5, 1, 1, []); ``` The DFA fluctuation function no longer increases linearly — the correlations do not decay according to a simple power law across all time scales. To find the time scales not affected by filter-induced correlations, fit separately to short and long ranges: **Short-range:** ```matlab [DFAobject, DFA_exp] = nbt_doDFA(abs(hilbert(Data_filtered)), SignalInfo, [0.1 1], [0.08 1200], 0.5, 1, 1, []); ``` **Long-range:** ```matlab [DFAobject, DFA_exp] = nbt_doDFA(abs(hilbert(Data_filtered)), SignalInfo, [3 120], [0.08 1200], 0.5, 1, 1, []); ``` At time scales from 100 ms to 1 s, the amplitude fluctuations are strongly correlated (α > 1) — this is the filter artefact. At time scales of 3–120 s, correlations are absent (α ≈ 0.5), as expected for random data. **For this specific 8–13 Hz filter, it is safe to interpret correlations on time scales longer than approximately 3 s as originating from the signal, not from filtering.** ### Applying DFA to real MEG data Now apply the same analysis to real MEG data. Download the [sample MEG recording](https://sourceforge.net/projects/nbtoolbox/files/Tutorials/MEG_data.mat/download) (`MEG_data.mat`, 2 MB). It contains 20 minutes of eyes-closed rest sampled at 300 Hz from a right and a left sensorimotor channel (`RSM`, `LSM`), and empty-room recordings from the same channels (`RSM_noise`, `LSM_noise`). The signals are stored as row vectors, so transpose them to put time along the first dimension. Use the fitting range you identified as having a proper null hypothesis, here 3–120 s: ```matlab load MEG_data.mat RSM = RSM'; % time along the 1st dimension [RSM_filtered] = nbt_filter_fir(RSM, highpass, lowpass, Fs, 2/8); [DFAobject, DFA_exp] = nbt_doDFA(abs(hilbert(RSM_filtered)), SignalInfo, [3 120], [0.08 1200], 0.5, 1, 1, []); ``` Long-range temporal correlations require a non-uniformity of fluctuations over time — a "patchy" time structure. Compare the filtered noise with the filtered MEG signal to see this visually. ### Signal-to-noise ratio Check whether LRTC are present in laboratory noise by analysing the empty-room recording `RSM_noise` in the same way. Narrow-band filtered MEG noise should have close to no long-range temporal autocorrelations. This raises a practical concern: if the signal-to-noise ratio is poor, can you reliably quantify the temporal correlations of the underlying physiological process? You can test this by adding amplified noise on top of neural data and checking whether the estimated DFA exponent changes. **This matters because:** different conditions, subjects, or patient groups may have different signal-to-noise ratios. A change in the DFA exponent could potentially reflect different oscillation amplitudes rather than a true effect on LRTC. This influence of signal-to-noise ratio on biomarker values is a generic problem that also applies to coherence, cross-frequency phase locking, graph analyses, and other measures. --- ## Summary - DFA quantifies long-range temporal correlations in time series - For uncorrelated data, α ≈ 0.5; for LRTC, 0.5 < α < 1 - Bandpass filtering introduces short-timescale correlations — always verify the safe fitting range using filtered random data - Signal-to-noise ratio can bias DFA estimates — this must be controlled for when comparing across conditions or groups ## References Hardstone R, Poil SS, Schiavone G, Jansen R, Nikulin VV, Mansvelder HD, Linkenkaer-Hansen K (2012) Detrended fluctuation analysis: A scale-free view on neuronal oscillations. *Front Physiol* 3:450. [doi:10.3389/fphys.2012.00450](https://doi.org/10.3389/fphys.2012.00450) Linkenkaer-Hansen K, Nikouline VV, Palva JM, Ilmoniemi RJ (2001) Long-range temporal correlations and scaling behavior in human brain oscillations. *J Neurosci* 21:1370. Montez T, Poil SS, Jones BF, et al. (2009) Altered temporal correlations in parietal alpha and prefrontal theta oscillations in early-stage Alzheimer disease. *PNAS* 106:1614. Montez T, Linkenkaer-Hansen K, van Dijk BW, Stam CJ (2006) Synchronization likelihood with explicit time-frequency priors. *NeuroImage* 33:1117–1125. Monto S, Vanhatalo S, Holmes MD, Palva JM (2007) Epileptogenic neocortical networks are revealed by abnormal temporal dynamics in seizure-free subdural EEG. *Cereb Cortex* 17:1386. Peng CK, Havlin S, Stanley HE, Goldberger AL (1995) Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series. *Chaos* 5:82–87. Poil SS, Jansen R, van Aerde K, et al. (2011) Fast network oscillations in vitro exhibit a slow decay of temporal auto-correlations. *Eur J Neurosci* 34:394–403. Poil SS, Hardstone R, Mansvelder HD, Linkenkaer-Hansen K (2012) Critical-state dynamics of avalanches and oscillations jointly emerge from balanced excitation/inhibition in neuronal networks. *J Neurosci* 32:9817–9823.