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2 results for “Least Squares Approximation”
FEX3-ECG/Charts01: Least Squares Approximation of ECG Signals with Rational Functions
<p> </p> <p> 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 is one result of the approximations and its charts.</p> <p><br> The original ECG signals:<br> DOI: <a href="https://doi.org/10.13026/C28C71">https://doi.org/10.13026/C28C71</a><br> License: Open Data Commons Attribution License v1.0</p> <p>We approached the following signal from the database above:<br> Patient009, the file: S0035_RE.XYZ, vy(Frank lead system)<br> The location of the QRS: 8669th point (8.669 sec)</p> <p>That is:<br> The approximated signal (P QRS T) section: 1:1139, the location of the QRS 371<br> The first signal point = 8299th data point.<br> Detailes:<br> locations of data points : 8299 8300 8301 .. 8669 .. 9437<br> indexes of signal points : 1 2 3 .. 371 .. 1139<br> values of signal points : 291 282 307 .. 352</p> <p>The signal described above is approximated by the following parameters of rational function in the example_1a.m short program.</p> <p><em>The </em><em>grids of charts:</em><br> X axis: 40ms (The sampling rate is 1000Hz)<br> Y axis: 0.1mV (200 A/D units)<br> This is on the original medical ECG paper: 1mm x 1mm.</p> <p> </p> <p><strong>FILES of Charts01.zip:<br> example_1a.m</strong> This file contains a short script of charts.(GNU Octave or MATLAB?)<br> Input: no. <strong><em>The parameters are in this</em></strong><strong><em> short</em></strong><strong><em> program.</em></strong><br> Output: Figure_b_1 and Figure_b_2<br> <strong>PQRST_sgnl_a.m</strong> Subroutine (of example_1a.m)</p> <p><strong>Figure_b_1.jpg</strong> First output chart of the example_1a.m<br> blue An approximation of the P wave<br> red An approximation of the QRS wave<br> yellow An approximation of the T wave<br> magenta An approximation of the Ta wave</p> <p><strong>Figure_b_2.jpg</strong> Second output chart of the example_1a.m<br> blue An approximation of all the waves (the sum of the above)</p> <p><strong>data_a_c.csv</strong> Result spreadsheet of the approximation program (decimal comma)<br> <strong>data_a_</strong><strong>p</strong><strong>.csv</strong> Result spreadsheet of the approximation program (decimal point)<br> Columns:<br> R relative QRS relative indexing<br> Original The original ECG signal<br> Approx An approximation of the ECG signal<br> Err220128 Error and noise (+date: yymmdd)<br> BL174355 Baseline (+time: hhmmss)<br> P(+Ta) An approximation of the P wave (+Ta wave)<br> QRS An approximation of the QRS wave<br> T An approximation of the T wave<br> V__idy 4 The length of the ECG vector (normalized, max. = 1mV)</p> <p><strong>image_a_</strong><strong>1</strong><strong>.gif</strong> First chart of the columns in the data_a_?.csv (MS Excel)<br> Original<br> Approx<br> Err220128<br> BL174355</p> <p><strong>image_a_</strong><strong>2</strong><strong>.gif</strong> Second chart of the columns in the data_a_?.csv (MS Excel)<br> P(+Ta)<br> QRS<br> T</p> <p> Kobzos, Laszlo<br> Location: HU (Budapest)<br> email: zehu.kola.ci@gmail.com</p> <p> </p>
FEX3-ECG/Charts02: Least Squares Approximation of ECG Signals with Rational Functions
<p><br> 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.<br> <br> The original ECG signals:<br> DOI: <a href="https://doi.org/10.13026/C28C71">https://doi.org/10.13026/C28C71</a><br> License: Open Data Commons Attribution License v1.0</p> <p><strong><em>Notes:</em></strong></p> <p>- We refer to the database records as follows: s.... = serial number, p... = patient. For example: <strong>s0508_p269</strong>. 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: <strong>s0508_p269, vy, 6689</strong> (Where "vy" is one of the Frank leads.).</p> <p>- 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.</p> <p>- 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 <strong>example_1a.m </strong>in <a href="https://doi.org/10.5281/zenodo.6479410">https://doi.org/10.5281/zenodo.6479410</a> is from <strong>From_series\s0035_p009\vy\ppaarraamm.txt</strong> also found below.<br> <em>Mini <strong>Hu==>En dictionary</strong>: </em>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).</p> <p>- 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 <strong>d</strong>), while 5 is the number of pole <strong>pairs</strong> (factors) of the denominator. So the number of approximation parameters is 3 + 2×5=13.<br> The meaning of Ta here: the approximation includes the Ta wave.</p> <p>- Below is the meaning of the 2×6 digit number, for example in filename: <br> <em>yymmdd hhmmss</em>: year month day hour minute second (time stamp).<br> <br> <strong><em>Notes on pole-zero representation:</em></strong></p> <p>- In the figures, the poles are marked with <strong>x</strong> and the zeros with <strong>o</strong>.</p> <p>- 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.</p> <p>- The pole of the Ta wave was not marked.</p> <p>- In the<strong> older figures</strong>, 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 <strong>t</strong>) is good, but the imaginary part was taken into account in a different way at that time.</p> <p>- Drawing the imaginary values of the roots on the same diagram as the signs on the <strong>newer figures</strong>, we multiplied them by 50, 100 and 10 for the P, QRS and T waves, respectively.<br> <br> <strong><em>The grids of charts </em></strong>(on the original medical ECG paper: "small squares" = 1mm x 1mm):<br> X axis: 40ms (The sampling rate is 1000Hz)<br> Y axis: 0.1mV (200 A/D units)</p> <p>=====================================================================================</p> <p><strong>Unid-Alte\</strong></p> <p>Diagram of two multipliers.<br> Here, the imaginary part of the poles and zeros is multiplied by 0.3.<br> It can be seen that the pole and zero locations do not coincide with the extremum locations on the alternating direction diagram.<br> <br> <strong>Histogram\</strong></p> <p>All 125 periods of the <strong>s0177_p050</strong> signal were approximated (<strong>vx, vy, vz</strong>). The figure is a histogram of one of the parameters of the T wave of <strong>vz</strong> lead (excluding the 2 extreme values).<br> <br> <strong>s0021_p005-vy\</strong></p> <p>These figures are the steps for approximating a QRS (<strong>s0021_p005, vy, 6810</strong>).<br> <em>Mini <strong>Hu==>En dictionary</strong>:</em> kozelito==>approximate; err......==>discrepancy; kend volt==>this had to be approached</p> <p>rajz-r-y-002---1.gif result of QRS11<br> rajz-r-y-002---2.gif result of QRS12<br> rajz-r-y-002---3.gif result of QRS13<br> 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.<br> rajz-r-y-002---4.gif result of QRS34<br> rajz-r-y-002---5.gif result of QRS35 Acceptable.<br> </p> <p>rajz-r-y-002---6.gif after a later approximation, we compared the results of the new QRS77 and QRS35.</p> <p>rajz-y-002-221022_201157_1.gif The final results of this new approximation (indicating the poles and zeros).<br> rajz-y-002-221022_201157_2.gif</p> <p><em>Mini <strong>Hu==>En dictionary</strong></em>: R.hely==>the relative <strong>t</strong> coordinate calculated from the origin; Eredeti==>raw; Approx==>approximate; Err......==>discrepancy, BL......==>baseline<br> <br> <strong>Shapes\</strong></p> <p>Approximation of some important shape types.<br> <em>Mini <strong>Hu==>En dictionary</strong></em>: R.hely==>the relative <strong>t</strong> coordinate calculated from the origin; Eredet==>raw; Approx==>approximate; Err......==>discrepancy, BL......==>baseline; pólusok==>poles</p> <p>s0025_p005 V3 7009 Dome Ta P32 T34 QRS34<br> s0047_p015 II 2758 Digitalis P22 T33 QRS34 One pair of poles of QRS and T coincide in the drawing (<strong>t</strong>=32).<br> s0146_p044 V4 3725 Coronaria T Ta P22 T22 QRS23<br> s0508_p269 aVR 5019 Block Ta P12 T12 QRS24<br> s0542_p283 vz 2581 QRS W-shaped P__ T22 QRS34 There is no P wave in any of the leads. (Atrial fibrillation)<br> <br> <strong>From_series\</strong></p> <p> <em>Mini <strong>Hu==>En dictionary</strong></em>: R.hely==>the relative <strong>t</strong> coordinate calculated from the origin; Eredet==>raw; Approx==>approximate; Err......==>discrepancy, BL......==>baseline; pólusok==>poles</p> <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><strong>vx</strong></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><strong>vy</strong></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><strong>vz</strong></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> <p>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 <strong>t</strong> axis (vx, vy, vz) is determined from the curve of the vector length.</p> <p>The figures for <strong>s0001_p119, vx, 10291</strong> are in the s0001_p119_U\vx directory. This is from the same <strong>vx</strong> 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 (Data<strong>1</strong> 6938 and Data<strong>5</strong> 10291) of the vx signal.</p> <p>The figures of <strong>s0002_p196,</strong> <strong>vx, vy, vz, 9067</strong> show that the approximation on the left side of the QRS should be improved, probably by increasing the degree of the denominator.</p> <p>The figures for <strong>s0034_p163, vx, 7580</strong> 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) :<br> rajz-x-003-221031_174534_2-zoom.gif.</p> <p>The figures for <strong>s0035_p009, vy, 8669</strong> 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.<br> 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.</p> <p> Kobzos, Laszlo<br> Location: HU (Budapest)<br> email: zehu.kola.ci@gmail.com</p>
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