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  1. <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
  2. "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
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  4. <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
  5. <meta name="Author" lang="en" content="Miguel Casquilho" />
  7. <title>M&middot;C, IST: Monte Carlo sigma from sums</title>
  9. <link rel="stylesheet" type="text/css"
  10. href="http://sasws.tecnico.ulisboa.pt/common/x_commonstyle.css" />
  11. <style type="text/css"> .hil {background-color: #a0c0ff}</style>
  12. </head> <!-- 20Jan2018 -->
  14. <body class="sm">
  15. <form method="post" action=
  16. "../common/interface_png.php?silent">
  17. <input name="executable" type="hidden" value="silent.exe" />
  19. <input type="hidden" name="servers[]" value="http://sasws.ist.utl.pt/cisti" /><input type="hidden" name="servers[]" value="http://sasws.ist.utl.pt/cisti" />
  20. <table width="90%" bgcolor="#eeeeee" align="center" border="1" class="sm">
  21. <tr><td colspan="3">
  22. <table width="100%" class="sm">
  23. <tr><td width="50px"><a href="https://fenix.ist.utl.pt/departamentos/deq"><img
  24. src="../common/emblem.png" alt="emblem" border="0px" width="40" height="41" /></a>
  25. </td><!--title-->
  26. <td><div style="font-size: 13.5pt; text-align: center"><!-- title --><b
  27. onmouseout="this.style.color='blue', this.style.fontStyle='normal'"
  28. onmouseover="this.style.color='red',
  29. this.style.fontStyle='bold'">
  30. <i>&sigma;</i>
  31. from unequal sample sizes</b></div><center>Finds <i>&sigma;</i>
  32. from bag batches' sums of weights: a conjecture.<!--b>
  33. UN DERCONSTRUCTION</b--></center></td>
  34. <td width="112px"><table width='100%'>
  35. <tr><td align="center" style="background-color: white">
  36. <input type="submit"
  37. value="&nbsp; Execute &nbsp;" style="background-color: #00bfff"
  38. onmouseout="this.style.fontStyle='normal', this.style.fontWeight='normal'"
  39. onmouseover="this.style.fontStyle='normal', this.style.fontWeight='bold'"/></td>
  40. </tr><tr><td align='center'><span style='font-size: xx-small'>2020.Mar.17 01:17:05</span></td></tr></table>
  41. </td></tr></table>
  42. </td></tr> <!-- ................................................... -->
  44. <tr><td width="15%" align="center"><b>n</b></td>
  45. <td width="45%"><textarea name="lots" rows="1"
  46. cols="45"> 5 11 26 58</textarea><input
  47. name="eodn" type="hidden" value="EOD" /></td>
  48. <td width="40%">Lot <i>sizes</i>, <i>n<sub>i</sub></i>,
  49. <i>i</i> = 1..<i>m</i>. &bull;</td></tr>
  51. <tr><td align="center"><i>&mu;</i>, <i>&sigma;</i></td>
  52. <td><input name="mu" type="text" value="25" size="6"
  53. maxlength="8" /><input name="sig" type="text" value="0.2" size="6"
  54. maxlength="8" /> (kg) &emsp; (theoretical)</td>
  55. <td>Gaussian parameters for the bag weight. &bull;</td></tr>
  57. <tr><td align="center">Criterion</td>
  58. <td><i>c</i> =
  59. <label for="crit0" class="hil"><input name="crit" id="crit0"
  60. type="radio" value="0" />&nbsp;0&nbsp;</label> &nbsp;
  61. <label for="crit1" class="hil"><input name="crit" id="crit1"
  62. type="radio" value="1" checked="checked" />&nbsp;1&nbsp;</label> &nbsp;
  63. <label for="crit2" class="hil"><input name="crit" id="crit2"
  64. type="radio" value="2" />&nbsp;2&nbsp;</label> &nbsp;
  65. <input name="altcrit" type="text" value="?" size="4"
  66. maxlength="10" style="background-color: #a0c0ff" /> (iff given)</td>
  67. <td><i>Criterion</i> for weighting. &bull;</td></tr>
  69. <tr><td align="center">Lg(trials), .seed</td>
  70. <td><input name="trialslg" type="text" value="5." size="6"
  71. maxlength="10" /><input name="iseed" type="text" value="0" size="6"
  72. maxlength="10" /></td>
  73. <td>N. of <i>trials</i> and random n. generator <i>seed</i>. &bull;</td></tr>
  75. <tr><td align="center">Show values</td>
  76. <td><select name="ishow"><option value="0" selected="selected">No</option>
  77. <option value="1">Yes</option></select></td>
  78. <td>Show the graph coordinates.</td></tr>
  80. <!-- plot 'gnudata.dat' \ (or anything: this line is a MANDATORY dummy) -->
  82. <tr><td colspan="3"><input name="gnuplot" type="hidden"
  83. value="wid = 600
  84. set terminal png enhanced font 'Times, 16' size wid, wid/1.618
  85. set timestamp offset 0, +0.75 font 'Helvetica-Bold, 9'
  86. set key top left font 'Times-Italic, 13'
  87. set title 'From loads of bags to {/Symbol s}' offset 0, -1.
  88. set xlabel '{/Times-Italic a}' offset 14, +0.95
  89. set ylabel ' {/Times-Italic f}' rotate by 0 offset +1.5, +4.25
  90. set yrange [0:*]; set ytics nomirror
  91. set y2label '{/Times-Italic F}' rotate by 0 offset -2.9, +4.25
  92. set y2range [0:1]; set y2tics 0.2 nomirror
  93. set grid; set tics font 'Times, 13'
  94. set style line 11 lc rgb 'red' lw 2
  95. set style line 12 lc rgb '#09ad00' lw 4 # green
  96. set style line 13 lc rgb '#0000ff' lw 2 # blue
  98. plot anyname.dat \
  99. usi 1:2 axis x1y1 title '{/Times-Italic f}' with lines ls 11, \
  100. '' usi 1:3 axis x1y2 title '{/Times-Italic F}' with lines ls 12" /><p class="ij">Tries,
  101. from simulated weights of batches of possibly
  102. <i>unequal</i> sizes of bags (or other collective items),
  103. to determine the <i>mean</i> and <i>standard deviation</i>,
  104. <i>&mu;</i> and <i>&sigma;</i>, of the Gaussian distribution
  105. assumed for the weight of the <i>individual</i> bags.
  106. The sizes of the <i>m</i> batches (indeed, commercial <i>lots</i>)
  107. are <i>n<sub>i</sub></i>, <i>i</i> = 1..<i>m</i>,
  108. given in the vector&nbsp;<b>n</b>.</p>
  110. <p class="ij">The determination of <i>&sigma;</i>
  111. is done by a weighting criterion applied to the various lot sizes,
  112. a conjecture. The weights are, for <i>i</i> = 1..<i>m</i>,
  114. <span class="nobreak"><i>w<sub>i</sub></i> = <i>n<sub>i</sub></i><sup><i>c</i></sup> &frasl; &Sigma;<i>n<sub>i</sub></i><sup>c</sup></span>.
  115. (A criterion of <i>c</i> = 0 leads, of course, to the arithmetic mean,
  116. which will also be the case for any <i>c</i> if all the
  117. <i>n<sub>i</sub></i> are equal.)</p>
  119. <p class="ij">The best criterion appears to be
  120. <i>c</i>&nbsp;=&nbsp;1.</p>
  122. <p class="ij">This is an attempt to estimate <i>&sigma;</i>
  123. (estimating <i>&mu;</i> is trivial) from a set of experimental data
  124. that are here simulated but which routinely exist in many industries.
  125. (The base data are inspired in real bags of fertilizers.)</p>
  127. <p class="ij">As this computation is rather lengthy,
  128. if the number of trials is excessive (for the total lot sizes),
  129. it is reduced (trials &times; lot total &le; 10<sup><b>8</b></sup>).</p>
  131. <p class="ij">Draws a plot of <i>f</i>, the 'pdf', and <i>F</i>,
  132. the 'cdf', of the averages of bags,
  133. <i>a</i> (certainly, around <i>&mu;</i>), only to verify
  134. the sufficiency of the Monte Carlo simulation.</p></td></tr>
  136. <tr><td align="center"><input type="reset" style="font-size: smaller"
  137. value=" Reset " /></td><td><i>References:</i></td>
  138. <td align="center"><i>Plate:</i> SigmaFromSums</td></tr>
  140. <tr><td colspan="3"><!--p class="ref">&bull;&nbsp;&bull;&nbsp;#Wikipedia: <a href=
  141. "https://en.wikipedia.org/wiki/Variance">variance</a></p-->
  143. <p class="ref">&bull;&nbsp;Google search: <a href=
  144. "https://www.google.com/search?q=%22different+sample+size%22+-unequal&amp;oq=
  145. %22different+sample+size%22+-unequal"
  146. >"different sample size" -unequal</a>; <a href=
  147. "https://www.google.com/search?q=%22unequal+sample+size%22+-different&amp;oq=
  148. %22unequal+sample+size%22+-different"
  149. >"unequal sample size" -different</a></p>
  151. <p class="ref">&bull;&nbsp;<a href=
  152. "http://www.adp-fertilizantes.pt/templates/SP.aspx?M=112&amp;F=7"
  153. >Catálogo de fertilizantes</a> (<a href=
  154. "http://www.adp-fertilizantes.pt/">ADP fertilizantes, SA</a>,
  155. "fertilizers"). (<a href=
  156. "http://www.sapecagro.pt/internet/asp/home.asp"
  157. >SAPEC</a>; <a href=
  158. "http://www.tradecorp.es/productos/nutricomplex-rojo"
  159. >Tradecorp</a>)</p>
  162. <p class="ref">&bull;&nbsp;1842-11-12: <a href=
  163. "http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Rayleigh.html"
  164. ><span class="sc">Strutt</span></a>, John William (Lord Rayleigh)
  165. (&dagger; 1919-06-30).</p></td></tr>
  166. </table>
  167. </form>
  169. <div style="line-height: 12pt">&nbsp;</div>
  170. <div style="line-height: 1.5pt; color: black; background-color: red">&nbsp;</div>
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  181. <td style="width: 70%" align=
  182. "right"><i>http://sasws.ist.utl.pt/cisti/Fx-sigmaFromSums-sas.php</i><br />
  183. <i>Created:</i> 2017-11-12 &mdash; <i>Last modified:</i>
  184. 2018-01-20</td></tr></table>
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