**3. Results**

Below are tables of results of power-law exponents from the implementation of the DFA algorithm on the respective time series. Note from Figure 1 that


It should be noted that the same data points from partitioning correspond to the white noise, monofractal, and multifractal time series. The Hurst exponents also follow respectively.

**Table 1.** DFA's Hurst Exponents of White noise time series.


From Table 1, we observe closeness of the Hurst exponents of the white noise series to *H* = 0.5 for all levels from *C*0 to *C*3. This confirms the phenomena that are exhibited in the fractal nature of the Cantor set in white noise time series. No matter how many sections of a white noise series are removed, the left-over series still exhibits similar characteristics as the whole white noise series.

**Table 2.** DFA's Hurst exponents of monofractal time series.


Table 2 present Hurst exponents between 0.5 and 1 (0.5 < *H* ≤ 1) for long memory monofractal time series for levels *C*0, *C*1, *C*2 and *C*3. The phenomena exhibited in the monofractal time series from the table above are similar to the fractal nature of the Cantor set. The series left behind after removing the middle thirds of the monofractal time series exhibits similar statistical properties as the whole.

**Table 3.** DFA's Hurst exponents of multifractal time series.


Hurst exponents of the multifractal time series lie within the range 0.5 < *H* ≤ 1 for all levels *C*0, *C*1, *C*2 and *C*3 from Table 3. This illustrates the fractal phenomena depicted by the Cantor set where successive magnification of the Cantor produces a copy of itself. This can be seen in Figure 1. Thus, self-similar behavior persists after removing the middle thirds of the whole series up to the level *C*3. Results from Tables 1–3 confirm that successive magnification of noise-like time series shows a similar pattern at increasingly smaller scales. Thus, the statistical characteristics of part of noise-like series are similar to that of the whole. This phenomenon is commonly known in fractals as self-similarity.

Figures 3–8 shows the log–log fits of RMSF and scales of the white noise, monofracal and multifractal bio-medical series using the DFA and the CDFA. The first two (2) plots (i.e., Figures 3 and 4) present fits of the white noise using the DFA and CDFA. The next two (2) plots (i.e., Figures 5 and 6) illustrate the fit of monofractal series using the DFA and CDFA. The last two(2) plots (i.e., Figures 7 and 8) show fits of the multifractal series using the DFA and the CDFA.

Table 4 above has six (6) columns of results in total. The first column (*H*) represents the Hurst exponents of the DFA, the second column (*Hc*) denotes the Hurst exponents of the CDFA and the difference between the exponents in the first two columns are found in the third column. The column for *α* denotes the scaling exponents of the TLF. The last two columns represent the multiplication of the Hurst exponents of the DFA (*H*) and the scaling exponent of the TLF (*α*), as well as the multiplication of the Hurst exponents of the CDFA (*Hc*) and the scaling exponents of the TLF. Upon investigating Hurst exponents of white noise, monofractal and multifractal time series using the DFA and CDFA, we observe differences in their exponents, as shown in Table 4. Hurst exponent of white noise time series changes slightly but that of the monofractal and multifractal time series changes about 1%. The slight changes in the exponents are a result of subdividing the time series as multiples of 3 (ternary base) at each level using the CDFA. This helps to curb the problem of overestimation associated with DFA. Notwithstanding the differences between the exponents, they still depict the same processes modeled herein (i.e., noise-like time series). The exponent of the white noise is close to 0.5 whereas that of the noise-like monofractal and multifractal series lie within the range 0.5 < *H* ≤ 1, depicting long-memory behavior.

**Table 4.** Comparison of scaling exponents of DFA(*H*) & CDFA(*Hc*) & TLF (*α*) on noise-like time series.


**Figure 3.** Log–log fit of white noise time series using DFA.

**Figure 4.** Log–log fit of white noise time series using CDFA.

**Figure 5.** Log–log fit of monofractal time series using DFA.

**Figure 6.** Log–log fit of monofractal time series using CDFA.

**Figure 7.** Log–log fit of multifractal time series using DFA.

**Figure 8.** Log–log fit of multifractal time series using CDFA.
