2.3.2. Definition

The subset of intervals of the Cantor set is defined recursively as:


The ternary Cantor set is defined as *C* = [0, 1] \ (∪<sup>∞</sup>*n*=<sup>1</sup>*Cn*). The level *C*0 indicates the interval we begin with. For *C*1, [0, 1] is divided into 3 sub-intervals and the middle sub-interval 13 , 23 is removed. For *C*2, each of the remaining intervals from *C*1 are divided into 3 sub-intervals and their middle sub-intervals 19 , 29 and - 79 , 89 are removed. This procedure can continue indefinitely by removing open middle third sub-interval of each interval obtained in the previous level. Due to issues with the dimension of the Cantor sets (i.e., dimension of 0.631 < 1), we rescale the integrated series *ψt* by dividing each observation by the maximum data point:

$$
\psi\_t = \frac{\Psi^t}{\max(\Psi\_t)}.
$$

s.t. *ψt* ∈ [0, 1].

#### 2.3.3. Algorithm of the CDFA

Here, we present a modification of the DFA algorithm called CDFA to generalize the segmen<sup>t</sup> division step of the DFA. The CDFA algorithm consists of four (4) main steps:

1. given the time series *ψt* of length *N*, find the integrated series shifted by the mean < *ψ* >,

$$\Upsilon\_{\vec{j}} = \sum\_{i=1}^{j} (\psi\_i - <\psi>).$$

2. the cumulatively summed series *Yj* is then segmented into equal non-overlapping segments of various sizes Δ*s*. Δ*s* is based on the Cantor set theory scale (Δ*s* = 3*<sup>n</sup>*, *n* ≥ 0). The number of non-overlapping segments is calculated as:

$$N\_{\Delta s} \equiv \operatorname{int} \left( \frac{N}{\Delta s} \right) = \operatorname{int} \left( \frac{N}{3^n} \right).$$

The Cantor set scaling function is computed for multiple segments to highlight both slow- and fast-evolving fluctuations that control the structure of the time series.

3. Root Mean Squared Fluctuation (RMSF) is computed for multiple scales of the integrated series:

$$F(\Lambda s) \equiv \left\{ \frac{1}{2N\_{\Lambda s}} \sum\_{j=1}^{2N\_{\Lambda s}} \left[ \Upsilon\_j - \Upsilon\_j^{\Lambda s} \right]^2 \right\}^{1/2}$$

where *j* denotes the sample size of segments *N*Δ*s*. We compute RMSF from *j* = 1 to 2*N*Δ*s* not *N*Δ*s*. We sum from beginning to end and from end to beginning, then an average of the values is calculated so that every data point is considered. Conversely, the large segments interweave many local periods with both small and large fluctuations and therefore average out their differences in magnitude.

4. the least squares regression fit of *<sup>F</sup>*(<sup>Δ</sup>*s*) versus the Cantor scales Δ*s* on a log–log scale produces the power-law notation computed for multiple scales:

$$F(\Delta s) \propto \left(\Delta s\right)^{H^c}$$

$$\log\left(F(\Delta s)\right) = H^c \log\left(\Delta s\right) + \log\left(C\right),$$

where *Hc* := Hurst exponent of the CDFA which measures memory behavior in the noise-like time series.

#### 2.3.4. Real Time Series

In Figure 2, the time series multifractal (upper panel), monofractal (middle panel) and white noise (lower panel) used in the experiment are noise-like biomedical time series with 8000 rescaled sample data points each [27]. The red trajectory depicts the random walk of the respective series. Observe that the fractal depicted by the multifractal time series at the peak looks very similar to the entire monofractal time series. Thus, comparing the series in the upper panel to the middle panel, the multifractal series has many fractals compared to the one for the monofractal series. We determine DFA's Hurst exponents for the remaining series after removing the middle thirds of each series at each level. It should be noted that the white noise time series has a structure independent of time with Hurst exponent close to *H* = 0.5 whereas noise-like monofractal and multifractal time series exhibit persistence behavior s.t. 0.5 < *H* ≤ 1.
