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I'm trying to get an intuitive understanding of the different coefficients in detrended fluctuation analysis (DFA).

It is used to detect fractal patterns in time series and it yields a coefficient, Alpha, which, as I understand is a Hurst coefficient mapped for different scales — the slope of which is the alpha.

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What I don't understand is if alpha shows a correlation between the frequency fluctuation and the scale (i.e. the bigger is the scale, the higher are the frequency fluctuations at that scale) — why is it that a time series is scale-free when alpha is around 1 or random when alpha is around 0.5?

We have a correlation in both cases, so where do these precise values come from? Why 0.5 and not 0.3? Why 1 and not 0.9?

Maybe somebody has an explanation for this as I haven't found anything precise in scientific literature.

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The application of the DFA method starts with a conversion of a bounded time series under test into an unbounded process, the elements of the unbounded process being partial sums of the bounded time series sequence.

Let we have a (discrete-time) realization of WGN of length N. It is a bounded time series. The starting conversion procedure of the DFA method transforms this WGN time series sequence into a realization of the random-walk process by the definition of the random walk process. For the RMS deviation's log-log plot of this random-walk process realization (this RMS deviation named the fluctuation in the DFA parlance), we draw a straight trend line using the least squares method. The slope of this trend line approaches 1/2 when N tends to infinity.

Hence the exact equality of DFA's alpha to 1/2 for the random process. You may have failed to find an explanation in scientific literature, because it should be read from textbooks, for example, read the Wikipedia article on DFA.

UPDATE

For an uncorrelated random walk of length N, fluctuation F(N), introduced in the DFA Wiki article, grows like a square root of N, i.e., its Hurst exponent is 1/2. Recall that the "fluctuation" of the uncorrelated time sequence such as a white noise realization does not depend on the sequence length by definition, its Hurst exponent is 0, and a white noise realization has its self-affinity at negative maximum -- the stretches of steady increase/decrease must be compensated by the stretches of the opposite direction, because the mean is zero. The alphas between zero and 1/2 correspond to an "anti-correlated" behavior from the POV of self-similarity. Likewise, the alphas between 1/2 and 1 correspond to correlation, and the alphas greater than one bear witness of the presence of non-stationary, unbounded process. The alpha=1 process is the stationary process, correlated at the maximum amount of self-affinity, it is the "1/f noise", because its PSD(f) is proportional to the inverse of frequency. The variance of an uncorrelated random walk is proportional to the square root of the series length; for the time series characterized by the Hurst exponent H, the variance and higher order moments are proportional to the series length raised to power H. This statistics results in the scaling behavior $$ x(at) \sim a^Hx(t) $$ where x is a random variable of the time series generating distribution, a is the scaling coefficient, H is the Hurst coefficient.

With H=1, the scaling behavior is closest to the concept of pure self-similarity, x(at) ~ ax(t), reproducing the system behavior at every time range.

These explanations do not prove the self-similarity claims, as mathematics requires it. But it would greatly help you learn the signal processing theory and master developer skills, if you compute and simulate the problems and processes. You can create an array of data and initialize it with a WGN samples in Matlab or Python; transform this array into a random walk process realization; compute trend lines with least squares and plot log-log graphs. Then, you can generate 1/f noise samples (or whatever noise spectrum you want) and plot trend lines for these distributions. Changing time scales, you will see for youself the claimed self-similarity properties. And in the process, you will master the necessary math.

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  • $\begingroup$ Thank you! What about the alpha value between 0.5 to 1? How do we know that 1 is scale-free and, for instance, 0.75, is not? $\endgroup$ Jan 10, 2021 at 9:20
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    $\begingroup$ I would say alpha=1 places the system behavior closest to an ideal self-similarity; see UPDATE in my answer. With alpha=1, x(at) ~ ax(t), where a is the scaling coefficient; the system demonstrates conformance invariance. With alpha=.75, x(at) ~ a^.75·x(t). With repeating scaling, the power exponent dwindles (.75·.75.·75·...) and the similarity is blurred. $\endgroup$
    – V.V.T
    Jan 10, 2021 at 10:50
  • $\begingroup$ Thank you for your update, @V.V.T! I just don't get one thing from your update: "For an uncorrelated random walk of length N, fluctuation F(N), introduced in the DFA Wiki article, grows like a square root of N, i.e., its Hurst exponent is 1/2". But then you say that the "fluctuation of the uncorrelated time sequence such as a white noise realization does not depend on the sequence length by definition, its Hurst exponent is 0," — so why is it 1/2 in one case and 0 in the other, when the two cases are uncorrelated white noise? $\endgroup$ Jan 11, 2021 at 15:55
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    $\begingroup$ Yes, "for an uncorrelated random walk ... fluctuation F(N) ... grows like a square root of N, its Hurst exponent is 1/2", and then I say "the fluctuation of .. a white noise realization does not depend on the sequence length by definition, its Hurst exponent is 0", In the first phrase, I talk about a random walk, in the second, about a white noise. Maybe we should blame my verbosity that caused your confusion... $\endgroup$
    – V.V.T
    Jan 12, 2021 at 10:01
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    $\begingroup$ In simulation, one can construct a random walk realization from a white noise realization; still, these two are different random processes. For example, besides they have different fluctuations $F(N)$, the corresponding noises have different power spectral density (PSD) functions: $PSD_{brownian}(f)$ ~ $1\over{f^2}$, $PSD_{whitenoise}(f)$ ~ $const$ etc. ( see enlightening graphs in oreilly.com/library/view/think-dsp/9781491938508/ch04.html ) $\endgroup$
    – V.V.T
    Jan 12, 2021 at 13:08

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