In stochastic processes, chaos theory and time series analysis, detrended fluctuation analysis (DFA) is a method for determining the statistical self-affinity of a signal. It is useful for analysing time series that appear to be long-memory processes (diverging correlation time, e.g. power-law decaying autocorrelation function) or 1/f noise.

The obtained exponent is similar to the Hurst exponent, except that DFA may also be applied to signals whose underlying statistics (such as mean and variance) or dynamics are non-stationary (changing with time). It is related to measures based upon spectral techniques such as autocorrelation and Fourier transform.

Peng et al. introduced DFA in 1994 in a paper that has been cited over 3,000 times as of 2022 and represents an extension of the (ordinary) fluctuation analysis (FA), which is affected by non-stationarities.

Systematic studies of the advantages and limitations of the DFA method were performed by PCh Ivanov et al. in a series of papers focusing on the effects of different types of nonstationarities in real-world signals: (1) types of trends; (2) random outliers/spikes, noisy segments, signals composed of parts with different correlation; (3) nonlinear filters; (4) missing data; (5) signal coarse-graining procedures and comparing DFA performance with moving average techniques (cumulative citations > 4,000).  Datasets generated to test DFA are available on PhysioNet.

Given: a time series ${\displaystyle x_{1},x_{2},...,x_{N}}$.

Compute its average value ${\displaystyle \langle x\rangle ={\frac {1}{N}}\sum _{t=1}^{N}x_{t}}$.

Sum it into a process ${\displaystyle X_{t}=\sum _{i=1}^{t}(x_{i}-\langle x\rangle )}$. This is the cumulative sum, or profile, of the original time series. For example, the profile of an i.i.d. white noise is a standard random walk.

Select a set ${\displaystyle T=\{n_{1},...,n_{k}\}}$ of integers, such that ${\displaystyle n_{1}<n_{2}<\cdots <n_{k}}$, the smallest ${\displaystyle n_{1}\approx 4}$, the largest ${\displaystyle n_{k}\approx N/4<ref></ref>}$, and the sequence is roughly distributed evenly in log-scale: ${\displaystyle \log(n_{2})-\log(n_{1})\approx \log(n_{3})-\log(n_{2})\approx \cdots }$. In other words, it is approximately a geometric progression.