A Brief Overview of Multifractal Time Series

Part 1: Fractal behavior in time series

The functions f(t) typically studied in mathematical analysis are continuous and have continuous derivatives. Hence, they can be approximated in the vicinity of some time ti by a so-called Taylor series or power series

      (eqn. 1)
For small regions around ti, just a few terms of the expansion (eqn. 1) are necessary to approximate the function f(t). In contrast, most time series f(t) found in "real-life" applications appear quite noisy (Fig. 1). Therefore, at almost every point in time, they cannot be approximated either by Taylor series (or by Fourier series) of just a few terms. Moreover, many experimental or empirical time series have fractal features--i.e., for some times ti, the series f(t) displays singular behavior. By this, we mean that at those times ti, the signal has components with non-integer powers of time which appear as step-like or cusp-like features, the so-called singularities, in the signal (see Figs. 1b,c).

Noisy signals
Figure 1: (a) A common example of numerically-generated "noise" with long-range power-law correlations. This signal has a power-law distributed power spectrum which increases as (frequency)0.6. (b) Cardiac interbeat intervals (in arbitrary units) for a healthy subject under ambulatory conditions. Note that the interbeat intervals are plotted against beat number. It is known that heart rate variability has long-range correlations characterized by a power spectrum that decreases as (frequency)-1.0. (c) Another example of numerically-generated noise also with long-range correlations but of a different type. In this case, the power spectrum decreases as (frequency)-1.4. Note how the high-frequency features of the signal decrease from (a) to (c). Note also: plotted in green are two instances of cusp-singularities (c1) and plotted in blue (b1) is one example of a step-singularity. Panel (b) also illustrates another complicating factor in many instances, singularities are not isolated, but may instead appear very close to one another, making their characterization rather more complex.

Formally, one can write (2, 4):
  (eqn. 2)
where t is inside a small vicinity of ti, and hi is a non-integer number quantifying the local singularity of f(t) at t = ti.
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