zenodoopen

FEX3-ECG/Charts02: Least Squares Approximation of ECG Signals with Rational Functions

We introduce a new algorithm for "Least Squares Approximation of ECG Signals with Rational Functions". Detailed description here: <a href="https://doi.org/10.5281/zenodo.7628747">https://doi.org/10.5281/zenodo.7628747</a> . The following figures show the results of some of the approximations. The original ECG signals:   DOI:     <a href="https://doi.org/10.13026/C28C71">https://doi.org/10.13026/C28C71</a>   License: Open Data Commons Attribution License v1.0 Notes: - We refer to the database records as follows:  s.... = serial number, p... = patient. For example: s0508_p269. If we want to specify the location more precisely within the record, we also indicate the lead and the time of the QRS. For example: s0508_p269, vy, 6689 (Where "vy" is one of the Frank leads.). - The figures were created with Microsoft Excel (gif, non-animated). In addition to the figures, data files of the results are usually also included. In the text files, we used a decimal point (for GNU Octave) or a decimal comma (for Excel). Any text editor can be used to transform them, replacing all commas with points or vice versa. - In some ppaarraamm.txt files, we have written the location and values of the parameters of each approximation (in Hungarian). The starting point of the time coordinate is now the first point of the ECG signal drawing. For the quadratic parameters of the numerator, the values of the two real roots or the conjugate complex root are placed one after the other. Similarly, the quadratic parameters of the denominator have the values of the conjugate complex root one after the other. The data from example_1a.m in  <a href="https://doi.org/10.5281/zenodo.6479410">https://doi.org/10.5281/zenodo.6479410</a>  is from From_series\s0035_p009\vy\ppaarraamm.txt also found below. Mini Hu==>En dictionary: ha==>if; nincs==>there is none; számláló==>numerator; nevező==>denominator; a pontok száma==>the number of points; nullad- és elsőfokú==>zeroth and first degree; másodfokú==>second degree; hely_bazis==>location base (the origin of the drawings). - Below, the meaning of the two numbers after the wave letter is related to the number of approximation parameters. For example: P35 means: P wave, 3 is the number of parameters of the numerator (including the main coefficient d), while 5 is the number of pole pairs (factors) of the denominator. So the number of approximation parameters is 3 + 2×5=13. The meaning of Ta here: the approximation includes the Ta wave. - Below is the meaning of the 2×6 digit number, for example in filename:       yymmdd hhmmss:  year month day   hour minute second (time stamp). Notes on pole-zero representation: - In the figures, the poles are marked with x and the zeros with o. - In the figures, only one of the two conjugated complex roots was shown, namely always the one falling towards the momentary value of the signal. - The pole of the Ta wave was not marked. - In the older figures, only the poles are visible, the zeros are not indicated. When the poles are drawn in the diagram shared with the signs, their location (coordinate t) is good, but the imaginary part was taken into account in a different way at that time. - Drawing the imaginary values of the roots on the same diagram as the signs on the newer figures, we multiplied them by 50, 100 and 10 for the P, QRS and T waves, respectively. The grids of charts (on the original medical ECG paper: "small squares" = 1mm x 1mm):     X axis: 40ms  (The sampling rate is 1000Hz)     Y axis: 0.1mV (200 A/D units) ===================================================================================== Unid-Alte\ Diagram of two multipliers. Here, the imaginary part of the poles and zeros is multiplied by 0.3. It can be seen that the pole and zero locations do not coincide with the extremum locations on the alternating direction diagram. Histogram\ All 125 periods of the s0177_p050 signal were approximated (vx, vy, vz). The figure is a histogram of one of the parameters of the T wave of vz lead (excluding the 2 extreme values). s0021_p005-vy\ These figures are the steps for approximating a QRS (s0021_p005, vy, 6810). Mini Hu==>En dictionary: kozelito==>approximate; err......==>discrepancy; kend volt==>this had to be approached rajz-r-y-002---1.gif    result of QRS11 rajz-r-y-002---2.gif    result of QRS12 rajz-r-y-002---3.gif    result of QRS13     The approximation of the small wave on the right side of the QRS is still missing. The approximation below the left side of the QRS is not yet good enough. We entered 2 zeros and multiplied the denominator by a complex conjugate root pairs. rajz-r-y-002---4.gif    result of QRS34 rajz-r-y-002---5.gif    result of QRS35  Acceptable.   rajz-r-y-002---6.gif    after a later approximation, we compared the results of the new QRS77 and QRS35. rajz-y-002-221022_201157_1.gif     The final results of this new approximation (indicating the poles and zeros). rajz-y-002-221022_201157_2.gif Mini Hu==>En dictionary: R.hely==>the relative t coordinate calculated from the origin; Eredeti==>raw; Approx==>approximate; Err......==>discrepancy, BL......==>baseline Shapes\ Approximation of some important shape types. Mini Hu==>En dictionary: R.hely==>the relative t coordinate calculated from the origin; Eredet==>raw; Approx==>approximate; Err......==>discrepancy, BL......==>baseline; pólusok==>poles s0025_p005   V3 7009   Dome                 Ta  P32  T34  QRS34 s0047_p015     II  2758   Digitalis                    P22  T33  QRS34    One pair of poles of QRS and T coincide in the drawing (t=32). s0146_p044   V4 3725   Coronaria T        Ta  P22  T22  QRS23 s0508_p269 aVR 5019   Block                  Ta  P12  T12  QRS24 s0542_p283    vz  2581  QRS W-shaped         P__   T22  QRS34    There is no P wave in any of the leads. (Atrial fibrillation) From_series\   Mini Hu==>En dictionary: R.hely==>the relative t coordinate calculated from the origin; Eredet==>raw; Approx==>approximate; Err......==>discrepancy, BL......==>baseline; pólusok==>poles <table> <tbody> <tr> <td>s.... = S/N, p... = patient:</td> <td>       s0001_p119</td> <td>s0002_p196</td> <td>s0034_p163</td> <td>s0035_p009</td> </tr> <tr> <td>               QRS_location:</td> <td>6938              10291</td> <td>9067</td> <td>7580</td> <td>8669</td> </tr> <tr> <td>vx</td> <td>Ta                   Ta</td> <td>Ta</td> <td> </td> <td>Ta</td> </tr> <tr> <td> </td> <td>P12                 P12</td> <td>P12</td> <td>P65</td> <td>P13</td> </tr> <tr> <td> </td> <td>T23                 T34</td> <td>T25</td> <td>T35</td> <td>T34</td> </tr> <tr> <td> </td> <td>QRS24            QRS24</td> <td>QRS15</td> <td>QRS89</td> <td>QRS57</td> </tr> <tr> <td>vy</td> <td>Ta</td> <td> </td> <td>Ta</td> <td>Ta</td> </tr> <tr> <td> </td> <td>P24</td> <td>P55</td> <td>P34</td> <td>P13</td> </tr> <tr> <td> </td> <td>T54</td> <td>T24</td> <td>T26</td> <td>T13</td> </tr> <tr> <td> </td> <td>QRS55</td> <td>QRS25</td> <td>QRS47</td> <td>QRS48</td> </tr> <tr> <td>vz</td> <td> </td> <td> </td> <td>Ta</td> <td>Ta</td> </tr> <tr> <td> </td> <td>P65</td> <td>P43</td> <td>P23</td> <td>P11</td> </tr> <tr> <td> </td> <td>T14</td> <td>T33</td> <td>T24</td> <td>T44</td> </tr> <tr> <td> </td> <td>QRS25</td> <td>QRS25</td> <td>QRS27</td> <td>QRS26</td> </tr> </tbody> </table> The s0001_p119\vy\rajz-y-001-190918_210304_vlc.gif image has been expanded with the length of the vector ("V__idy 1"). The origin of the t axis (vx, vy, vz) is determined from the curve of the vector length. The figures for s0001_p119, vx, 10291 are in the s0001_p119_U\vx directory. This is from the same vx signal as the previous one (6938), just 4 periods later. There is a U wave in it, T34 was needed instead of T23 to approximate it. The image Rajz-001-005a.gif is in the same directory. On this, we enlarged the part between the T and the next P wave at two periods (Data1 6938 and Data5 10291) of the vx signal. The figures of s0002_p196, vx, vy, vz, 9067 show that the approximation on the left side of the QRS should be improved, probably by increasing the degree of the denominator. The figures for s0034_p163, vx, 7580 are in the s0034_p163\vx\ directory. In the figure "rajz-x-003-221031_174534_2.gif" at t=-128, a pole and a zero coincide in the drawing, so we have enlarged this part (around P and QRS) : rajz-x-003-221031_174534_2-zoom.gif. The figures for s0035_p009, vy, 8669 are in the s0035_p009\vy\ directory. In the figure "rajz-y-004-221031_114914_2.gif" at t=-21, a pole and a zero almost coincide in the drawing. As already written above, the data of example_1a.m (Octave, Matlab) on <a href="https://doi.org/10.5281/zenodo.6479410">https://doi.org/10.5281/zenodo.6479410</a> comes from the file ppaarraamm.txt in the same directory.                                                                                              Kobzos, Laszlo                                                                                              Location: HU (Budapest)                                                                                              email:   zehu.kola.ci@gmail.com

ShareScore

40/100

Overall dataset sharing score

Score breakdown

These five areas show where the dataset supports — or may limit — practical reuse.

Stewardship: 8
Harmonization: 8
Access: 16
Reuse readiness: 8
Engagement: 0

Topics

ECG Feature Extraction Least Squares Approximation Rational Functions