Effects of coarse-graining on the scaling behavior of long-range correlated and anti-correlated signals

We investigate how various coarse-graining methods affect the scaling properties of long-range power-law correlated and anti-correlated signals, quantified by the detrended fluctuation analysis. Specifically, for coarse-graining in the magnitude of a signal, we consider (i) the Floor, (ii) the Symmetry and (iii) the Centro-Symmetry coarse-graining methods. We find, that for anti-correlated signals coarse-graining in the magnitude leads to a crossover to random behavior at large scales, and that with increasing the width of the coarse-graining partition interval $\Delta$ this crossover moves to intermediate and small scales. In contrast, the scaling of positively correlated signals is less affected by the coarse-graining, with no observable changes when $\Delta<1$, while for $\Delta>1$ a crossover appears at small scales and moves to intermediate and large scales with increasing $\Delta$. For very rough coarse-graining ($\Delta>3$) based on the Floor and Symmetry methods, the position of the crossover stabilizes, in contrast to the Centro-Symmetry method where the crossover continuously moves across scales and leads to a random behavior at all scales, thus indicating a much stronger effect of the Centro-Symmetry compared to the Floor and the Symmetry methods. For coarse-graining in time, where data points are averaged in non-overlapping time windows, we find that the scaling for both anti-correlated and positively correlated signals is practically preserved. The results of our simulations are useful for the correct interpretation of the correlation and scaling properties of symbolic sequences.

💡 Research Summary

The paper investigates how different coarse‑graining (or “binning”) procedures influence the scaling behavior of long‑range correlated and anti‑correlated time series, as measured by detrended fluctuation analysis (DFA). The authors focus on two broad classes of coarse‑graining: (i) magnitude‑based discretization, where the raw signal values are mapped into bins of width Δ, and (ii) time‑based averaging, where non‑overlapping windows of length Δ are replaced by their mean.

For magnitude coarse‑graining three specific mapping rules are examined: the Floor method (each value is rounded down to the nearest multiple of Δ), the Symmetry method (bins are symmetric about zero and the sign of the original value is retained), and the Centro‑Symmetry method (bins are symmetric about zero but the sign is discarded, i.e., only the absolute value is binned). These transformations convert a continuous signal into a symbolic sequence, potentially altering its correlation structure.

Synthetic signals with known power‑law autocorrelation are generated: anti‑correlated series with DFA exponent α≈0.3 and positively correlated series with α≈0.7. For each series the authors vary Δ from 0.1 up to 5 and apply DFA to the coarse‑grained output. The main findings are:

1. Anti‑correlated signals are highly sensitive to magnitude coarse‑graining. When Δ≤0.5 the original exponent α≈0.3 is preserved. As soon as Δ exceeds 1, a crossover appears at large scales where the fluctuation function follows the random‑walk exponent α=0.5. Increasing Δ shifts this crossover toward smaller scales, eventually erasing the anti‑correlation across the whole range. The Centro‑Symmetry rule produces the most drastic effect: even for Δ>3 the crossover continues to move, and the series behaves randomly at all scales. This is because discarding the sign destroys the delicate balance of negative correlations.

2. Positively correlated signals are more robust. For Δ<1 the DFA exponent remains essentially unchanged (α≈0.7). When Δ>1 a crossover to α≈0.5 emerges first at the smallest scales, but as Δ grows the crossover moves to intermediate and then larger scales. For the Floor and Symmetry rules, the crossover stabilizes around Δ≈3–4, leaving a clear scaling regime at larger scales. The Centro‑Symmetry rule again shows a stronger impact, pushing the crossover further into the spectrum and reducing the range where the original correlation is observable.

3. Time‑based coarse‑graining (non‑overlapping averaging) has a negligible effect on both anti‑correlated and positively correlated series. Across a wide range of window sizes the DFA exponent remains within a few percent of its original value. This indicates that averaging primarily suppresses high‑frequency fluctuations while preserving the underlying long‑range dependence.

The authors interpret these results in terms of information loss. Magnitude coarse‑graining reduces the resolution of amplitude fluctuations; the larger the bin width, the more fine‑scale structure is lost. For anti‑correlated data, the sign of fluctuations is crucial, so any operation that blurs or eliminates sign information quickly destroys the correlation. Positively correlated data rely more on the persistence of large excursions, which are less affected by moderate binning.

The paper concludes that when symbolic or discretized representations of physiological, financial, or genomic data are analyzed for scaling properties, the choice of coarse‑graining method and bin width must be made carefully. In particular, researchers should avoid coarse‑graining schemes that discard sign information (e.g., Centro‑Symmetry) if anti‑correlations are of interest, and they should keep Δ below the threshold where a crossover to random behavior appears. Conversely, time‑averaging can be safely used for data compression or noise reduction without compromising DFA‑based scaling estimates. The study provides a quantitative guideline for interpreting correlation exponents in symbolic sequences and highlights the potential pitfalls of naïve discretization.