# Application of Wavelet Transform and Fractal Analysis for
## 3 Results
To confirm the validity of these conclusions, the calculations were carried out, and the results were then compared for all the patients in the measurement programme. A summary of the results (Figure 7) shows a clear clustering of the curves for the patients with diagnosed GERD (red line) above the curve taken as the cut-off value (DMS = 15.9, blue line) and a clustering of healthy patients below the cut-off value. Only in one case was a healthy person assigned to the group of patients with GERD (DMS = 12.4, green dashed line), and one patient with diagnosed GERD (DMS = 17.0, red dashed line) was marked as healthy. In order to more clearly differentiate the group of patients with diagnosed reflux from healthy patients, a transformation using a scaling function specially designed for this purpose was applied. The scaling function is shown in Figure 8. The parameters of the exponential scaling function were determined from the results of the calculations for the marginal cases of a healthy patient (DeMeester score = 0.8) and a patient with reflux (DeMeester score = 107.8). The patient case with a calculated DeMeester score equal to 15.9 was taken as the borderline between the sick and healthy cases. Figure 5: Scalograms of pH-metry signals obtained for Haar wavelet related to the 24-h measurement procedure. Scalograms show the result of the continuous wavelet transform, based on pH signals for the marginal cases of calculated DeMeester score. The result of the rescaling process for the marginal cases is shown in Figure 9. Figure 10a shows the results of the classification process for all the healthy patients (DMS! 15.9), while Figure 10b shows the results of the classification process for all the patients diagnosed with GERD (DMS? 15.9). The main aim of this procedure was to obtain curves with values ranging from \(-1\) (completely healthy patient--lowest DeMeester Score) to \(+1\) (patient with the highest DeMeester Score). The curve drawn from the pH signal of the patient whose measurement data were taken as the boundary between the sick and healthy group corresponds to the zero value on the graph. Figure 8: Rescaling function designed based on selected patient cases. Figure 6: Curves representing sums of wavelet coefficients. Each curve was obtained as a result of summing up wavelet coefficients after the application of the continuous wavelet transform of pH signals for a selected patient during 24 h of measurement procedure. Different colours indicate marginal cases of calculated DeMeester score: blue line—base line (DeMeester 15.9); red—a patient diagnosed with reflux (DeMeester 107.8); green line—healthy patient (DeMeester 0.8). Figure 7: Curves representing sums of wavelet coefficients. Each curve was obtained as a result of summing up wavelet coefficients after the application of the continuous wavelet transform of pH signals for a selected patient during 24 h of measurement procedure. Different colours indicate marginal cases of calculated DeMeester score: blue line—base line (DeMeester 15.9); red—a patient diagnosed with reflux (DeMeester > 15.9); green lines—healthy patient (DeMeester < 15.9); black dashed line—misclassified patients. A summary of the results shows a clear clustering of the curves for the patients diagnosed with GERD above the curve taken as the cut-off (scale value greater than zero at point 2000 in Figure 10) and a clustering of healthy patients below the cut-off (scale value less than zero at point 2000 in Figure 10). Based on the Formulas (1) and (2), the sensitivity and specificity for the results of the calculations with the use of CWT were determined. As the patient's data with a DeMeester score of 15.9 were adopted as the borderline for the distinction between sick and healthy, this case was not taken into account when determining the values of sensitivity and specificity. The results of the calculation of these parameters are summarised in Figure 4.0 The calculations were repeated several times--each time obtaining the same assignment result.