**2. Methods**

#### *2.1. The Truncated Lévy Flight (TLF)*

We provide a brief overview of the Truncated Lévy Flight (TLF) model in this subsection. The most general representation of the Lévy stable distribution is denoted by the characteristic function:

$$\mathcal{K}(q,a) = \exp\{i\mu q \, - \, \sigma^a \, \vert \, q \, \vert^a \, [1 \, + \, i\beta \, sign(q) \, \phi(q,a)]\} \tag{1}$$

where,

$$\phi(q,\alpha) = \begin{cases} (2/\pi)\ln(q), & \alpha = 1 \\ -\tan(\pi\alpha/2), & \alpha \neq 1. \end{cases}$$

The stability exponent *α* ∈ (0, 2] defines the asymptotic decay of the pdf. *σ* ∈ (0, ∞) measures dispersion. Skewness parameter *β* ∈ [−1, 1] measures asymmetry of the distribution. *μ* ∈ (−∞, ∞) is a scalar which determines the "location" or shift of the distribution. The sign x is the signum function of *x* ∈ R defined as *sign*(*x*) = *<sup>x</sup>*/|*x*|. The problem is that the variance of the distribution in (1) is finite but is not stable. This is because, large cut-off *l* results in slow convergence and a smaller cut-off *l* may result in abrupt tail [8]. In [25], the author generated a TLF to address the convergence problem by using a decreasing exponential cut-off function. Thus, the process in Equation (1) is truncated to obtain the TLF given by:

$$\mathcal{T}(q,a) = \begin{cases} c \, \mathcal{K}(q,a), & |q| \, | \, \le |l| \\ 0, & |q| \, | \, > |l| \end{cases} \tag{2}$$

for some normalizing constant *c*, stability exponent *α* ∈ (0, 2] and cut-off length *l*. The characteristic function of the TLF in Equation (2) is given by

$$\ln\left[\mathcal{T}(q,a)\right] = \frac{2\pi A l^{-a} t \left[1 - \left((ql/\sigma)^2 + 1\right)^{a/2} \cos(a \arctan(ql/\sigma))\right]}{a \Gamma(a) \sin(\pi a)}.\tag{3}$$

To determine the best scaling exponent (*α*) from characteristic equation in (3), we adjust the values of *A*, the cut-off parameter *l* and the scaling exponent *α* simultaneously to fit the characteristic function to the data.

#### *2.2. Detrended Fluctuation Analysis (DFA)*

Given the noise-like time series *ψ*, we find the integrated series

$$Y = \sum\_{k} (\psi\_k - <\psi>). \tag{4}$$

to determine the Root Mean Squared Fluctuations (RMSF) from Equation (5) below

$$F(s) = \left\{ \frac{1}{N} \sum\_{\vec{j}} \left[ \mathbf{Y}\_{\vec{j}} - \mathbf{Y}\_{\vec{j}}^{s} \right]^2 \right\}^{1/2} \tag{5}$$

A log–log plot of the RMSF against the series length *s* produces a directly proportional relation given by

$$F(\mathbf{s}) \propto \mathbf{s}^H$$

$$\log F(\mathbf{s}) - H \log(\mathbf{s}) = \mathbf{K},\tag{6}$$

where *H* := Hurst exponent of the DFA and *Hmin* ≤ *H* ≤ *Hmax* [4].

#### *2.3. Cantor Detrended Fluctuation Analysis (CDFA)*

In this subsection, we prove that the subspace [*Hmin*, *Hmax*] of Hurst exponents is homeomorphic to [0, 1] of the Cantor set. We also present an illustration of the Cantor set and the algorithm for the CDFA.

**Theorem 1.** *A map f* : [*Hmin*, *Hmax*] → [0, 1] *between the topological spaces of Hurst exponents of noise-like time series and the Cantor set is a homeomorphism if it has the following properties:*


If two topological spaces admit a homeomorphism between them, we say they are homeomorphic: they are essentially the same topological space.

**Proof.** Let *Hmin* ≤ *H* ≤ *Hmax* and 0 ≤ *y* = *f*(*H*) ≤ 1, then the map *f* : [*Hmin*, *Hmax*] → [0, 1] gives

$$H\_{\min} - H\_{\min} \le H - H\_{\min} \le H\_{\max} - H\_{\min} \tag{7}$$

$$0 \le \frac{H - H\_{\rm min}}{H\_{\rm max} - H\_{\rm min}} \le 1. \tag{8}$$

Thus,

$$y = f(H) = \frac{H - H\_{\min}}{H\_{\max} - H\_{\min}}.\tag{9}$$

Now, we need to prove that the map *f* is homeomorphic to the Cantor set.

The map *f*(*H*) is said to be bijective if and only if *f*(*a*) = *f*(*b*) for all *a*, *b* implies that *a*= *b*. From

$$f(a) = \frac{a - H\_{\text{min}}}{H\_{\text{max}} - H\_{\text{min}}} \quad \text{and} \quad f(b) = \frac{b - H\_{\text{min}}}{H\_{\text{max}} - H\_{\text{min}}},$$

$$f(a) = f(b)$$

$$\implies a - H\_{\text{min}} = b - H\_{\text{min}}$$

$$\implies a = b.$$

Thus, the map *f*(*H*) is a bijection.

The map *f*(*H*) is continuous at some value *c* in its domain if *f*(*c*) is defined, the limit of *f* as *H* approaches *c* exists and the function value of *f* at *c* equals the limit of *f* as *H* approaches *c*. The function *f*(*c*) is defined as

$$f(c) = \frac{c - H\_{\text{min}}}{H\_{\text{max}} - H\_{\text{min}}}.\tag{10}$$

The limit of *f* as *H* approaches *c* equals

$$\lim\_{H \to \varepsilon^{+}} f(H) = \lim\_{H \to \varepsilon^{-}} f(H) = \frac{\varepsilon - H\_{\text{min}}}{H\_{\text{max}} - H\_{\text{min}}}.\tag{11}$$

The left- and right-sided limits are equal from (11). Therefore,

$$\lim\_{H \to c} f(H) = \frac{c - H\_{\text{min}}}{H\_{\text{max}} - H\_{\text{min}}}.\tag{12}$$

Hence we observe that the right hand side of Equation (10) is equal to right hand side of Equation (12). Thus, it follows that

$$\lim\_{H \to \mathcal{c}} f(H) = f(\mathcal{c}) = \frac{\mathcal{c} - H\_{\text{min}}}{H\_{\text{max}} - H\_{\text{min}}}.$$

Thus, the map *f* is continuous at some value *H* = *c* for a differentiable fractal. The inverse function of *f* (i.e., *f* −<sup>1</sup>(*H*)) exists.

$$y = f(H) = \frac{H - H\_{\text{min}}}{H\_{\text{max}} - H\_{\text{min}}} \tag{13}$$

$$(H\_{\max} - H\_{\min})y = H - H\_{\min} \tag{14}$$

$$H = H\_{\min} + (H\_{\max} - H\_{\min})y \tag{15}$$

Interchanging *H* and *y* gives

$$y = f^{-1}(H) = H\_{\text{min}} + (H\_{\text{max}} - H\_{\text{min}})H,\tag{16}$$

the inverse function of *f*(*H*). The inverse map *f* −1 is continuous at some value *s* in its domain if *f* <sup>−</sup><sup>1</sup>(*s*) is defined, the limit of *f* −1 as *H* approaches *s* exists and the function value of *f* −1 at *s* equals the limit of *f* −1 as *H* approaches *s*. *f* <sup>−</sup><sup>1</sup>(*s*) is defined as

$$f^{-1}(\mathbf{s}) = (1 - \mathbf{s})H\_{\text{min}} + \mathbf{s}H\_{\text{max}}.\tag{17}$$

The limit of *f* −1 as *H* approaches *s* equals

$$\lim\_{H \to s^{+}} f^{-1}(H) = \lim\_{H \to s^{-}} f^{-1}(H) = H\_{\min} + (H\_{\max} - H\_{\min})s.\tag{18}$$

$$\implies \lim\_{H \to s} f^{-1}(H) = H\_{\text{min}} + (H\_{\text{max}} - H\_{\text{min}})s. \tag{19}$$

Since the right hand side of Equation (17) equals the right hand side of Equation (19) it implies that,

$$\lim\_{H \to s} f^{-1}(H) = f^{-1}(s) = (1 - s)H\_{\min} + sH\_{\max}.$$

Thus, the inverse map *f* −1 exists and is continuous at some value *H* = *s*.

Therefore, the map *f*(*H*) is a homeomorphism and *H* ∈ [*Hmin*, *Hmax*] is homeomorphic to [0, 1] of the Cantor set for noise-like time series.

#### 2.3.1. Illustration of the Cantor Set

In this subsection, we take real-world noise-like time series and remove middle thirds up to four (4) levels so that it is similar to the Cantor set. This phenomenon is depicted in Figure 1 [26]. It shows that the segments appear the same at different scales in successive magnifications of the Cantor set from levels *C*0 to *C*6. *C*0 depicts the original time series with no missing parts and *C*6 represents the remaining time series after removing middle thirds for the sixth time. For the sake of experimentation, we limit our scope to levels from *C*0 to *C*3.

**Figure 1.** Fractal behavior of a ternary Cantor set.
