# Application of Wavelet Transform and Fractal Analysis for
## 2 Materials and Methods
### Methodology  
The schematic diagram presented in Figure 2 illustrates the research methodology employed by the authors in the study; it focuses on two methods of signal decomposition based on wavelet transforms to classify a patient undergoing pH-metric examination into one who is healthy or one with symptoms of reflux. First, a continuous wavelet transform (CWT) was used to determine the specific wavelet coefficients describing the correlation between the selected basis wavelet and the signal undergoing the decomposition process. Figure 2a illustrates the calculation methodology based on the application of the CWT, which led the authors to the calculation of the sums of the wavelet coefficients. Figure 1: Selected statistical parameters of healthy patients and patients with GERD. \begin{table}
\begin{tabular}{c c c} \hline
**Statistic** & **Healthy Patient Total DeMeester Parameter** & **GERD Patient Total DeMeester Parameter** \\ \hline max number of episodes & 79 & 235 \\ \hline min number of episodes & 4 & 35 \\ \hline mean number of episodes & 36.75 & 104.88 \\ \hline mean time of the longest reflux episode [s] & 620.230 & 5990.700 \\ \hline mean number of long reflux episodes & 3.250 & 11.813 \\ \hline \end{tabular}
\end{table} 
Table 3: Statistical analysis of total DeMeester count parameters for healthy and GERD patients. As the second method, the wavelet-based fractal analysis (WBFA) was chosen. The authors selected this approach due to the fact that the analysed signals can be interpreted as a time series (with some features of a stochastic process). At the same time, the signals manifesting the characteristics of a stochastic process can be well described by the power spectral density (PSD), which represents the signal power associated with particular frequencies, \(\omega\). The PSD calculations enable the study of the frequency content of the stochastic processes, as well as the identification of any periodic behaviour associated with them [40]. It should also be mentioned that many biosignals are characterised by PSD distributions corresponding to 1/f processes (pink noise), which in practice means that their amplitudes are inversely proportional to the frequency f [41; 42]. In the research presented in this paper, the authors used the WBFA method to determine the pH signal correlations in a manner similar to that of [PERSON] et al. [43]. They analysed the effect of different drugs on heart rate variability (HRV) in a group of 10 healthy subjects using fractal methods. This approach proved to be correct in a similar analysis [44]. Self-similarity of the tested signals was analysed using the spectral exponent \(\gamma\) determined by the WBFA method:  
\[\gamma=\frac{\Delta log_{2}var\ d_{m,n}}{\Delta m} \tag{3}\]  
where \(d_{m,n}\) denotes the wavelet coefficients calculated using the discrete wavelet transform (DWT) analysis [45]. The methodology employed based on the DWT analysis is schematically presented in Figure 2b. The Daubechies base wavelet (db3) and 8 decomposition levels were used to calculate the spectral exponent coefficients. Figure 2: A flow chart for the methodology of analysis for esophageal pH monitoring signal: (**a**) calculation of the sums of wavelet coefficients (CWT), (**b**) based on spectral exponent parameters. The essence of the method presented in this paper, based on the application of CWT, is the determination of the sum of the wavelet coefficients as a function of the scale parameter. In contrast, the application of the WBFA method provides spectral coefficients, including all the necessary resultant information. The differences between the results representing the 24-h pH measurements for the patients with different conditions may indicate the existence of some subtle but significant changes in the signal under investigation, which can help in making a proper diagnosis. The results obtained can also confirm the validity of the proposed methodology and the developed numerical tool for determining the patient's health status.