**7. Box size**

The mDFA program generated by K. Tanaka was based on an article by Scafetta and Grigolini published in 2002 [20]. These authors pointed out that DFA does not reliably detect the distribution of the Lévy time series [20]. The mDFA program is intended to observe the scaling behavior over a wide range of SI values. DFA focuses on critical phenomena and an SI value near 1.0, which is a Gauss distribution time series. In the current study, there were significant differences in the computation results between the mDFA and Peng's DFA, especially when diseased hearts were examined. If mDFA was used instead of DFA, diseased hearts exhibited a high SI (SI=~1.4, Table 1).

The idea of using a box in mDFA (and DFA) computations is presented in Figure 11 for 2000 heartbeat interval time series obtained from a human subject. Seven clear irregular beats are evident, i.e., so-called premature ventral contractions (PVCs) or ectopic heartbeat. The four black bars indicate that the 2000-beat data set was divided into four boxes. Here, there are 500 beats in each box; therefore, the box size is 500. Every box always has an identical number of beats in any mDFA/DFA computation. In mDFA (not DFA) computations using a PC, the box size is changed cyclically from 10 to 1000 as shown in Figure 10.

**Figure 11.** Interval time series and boxes.

included box-ranges [30: 60], [70: 140], [130: 270], and [30: 270]. These four ranges were automatically and simultaneously computed. However, any box size range could be manually computed if necessary. Then, program B was added in 2006, which included box-ranges [30: 70], [70; 140], [130: 270], [51; 100], [30; 140], [30; 270] (Figure 9). These automatic ranges were arbitrarily determined. Until now, programs A and B have been used simultaneously. All

**Figure 10.** Box size (number of heartbeats) for the mDFA programs. Program A and Program B contained 53 and 136

Program A computed 53 box points and program B computed 136 box points (Figure 10). This increase in box number occurred because of an increase in the PC calculation speed due to a Windows software improvement. A device requires relatively rapid computation of the scaling exponent. An average scaling exponent was computed in each computation, calculated as the

mDFA calculations use recordings of approximately 2000 beats, digital recordings at a 1-KHz sampling ratio, and a box size range of [30: 270] in the following sections of this article, unless

The mDFA program generated by K. Tanaka was based on an article by Scafetta and Grigolini published in 2002 [20]. These authors pointed out that DFA does not reliably detect the distribution of the Lévy time series [20]. The mDFA program is intended to observe the scaling behavior over a wide range of SI values. DFA focuses on critical phenomena and an SI value near 1.0, which is a Gauss distribution time series. In the current study, there were significant differences in the computation results between the mDFA and Peng's DFA, especially when diseased hearts were examined. If mDFA was used instead of DFA, diseased hearts exhibited

average of the scaling exponents across the range of box sizes (see Figure 22 and 23).

existing EKG-mDFA results were computed using both programs.

otherwise mentioned.

boxes, respectively. Programming language C++.

368 Advances in Bioengineering

a high SI (SI=~1.4, Table 1).

**7. Box size**
