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  2. <!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
    3. 

  3. <html><head><title>Python: module telemetry.util.statistics</title>
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  4. <meta charset="utf-8">
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  5. </head><body bgcolor="#f0f0f8">
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  6. 

  7. <table width="100%" cellspacing=0 cellpadding=2 border=0 summary="heading">
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  8. <tr bgcolor="#7799ee">
    9. 

  9. <td valign=bottom>&nbsp;<br>
    10. 

  10. <font color="#ffffff" face="helvetica, arial">&nbsp;<br><big><big><strong><a href="telemetry.html"><font color="#ffffff">telemetry</font></a>.<a href="telemetry.util.html"><font color="#ffffff">util</font></a>.statistics</strong></big></big></font></td
    11. 

  11. ><td align=right valign=bottom
    12. 

  12. ><font color="#ffffff" face="helvetica, arial"><a href=".">index</a><br><a href="../telemetry/util/statistics.py">telemetry/util/statistics.py</a></font></td></tr></table>
    13. 

  13.     <p><tt>A&nbsp;collection&nbsp;of&nbsp;statistical&nbsp;utility&nbsp;functions&nbsp;to&nbsp;be&nbsp;used&nbsp;by&nbsp;metrics.</tt></p>
    14. 

  14. <p>
    15. 

  15. <table width="100%" cellspacing=0 cellpadding=2 border=0 summary="section">
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  16. <tr bgcolor="#aa55cc">
    17. 

  17. <td colspan=3 valign=bottom>&nbsp;<br>
    18. 

  18. <font color="#ffffff" face="helvetica, arial"><big><strong>Modules</strong></big></font></td></tr>
    19. 

  19.     
    20. 

  20. <tr><td bgcolor="#aa55cc"><tt>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</tt></td><td>&nbsp;</td>
    21. 

  21. <td width="100%"><table width="100%" summary="list"><tr><td width="25%" valign=top><a href="math.html">math</a><br>
    22. 

  22. </td><td width="25%" valign=top></td><td width="25%" valign=top></td><td width="25%" valign=top></td></tr></table></td></tr></table><p>
    23. 

  23. <table width="100%" cellspacing=0 cellpadding=2 border=0 summary="section">
    24. 

  24. <tr bgcolor="#eeaa77">
    25. 

  25. <td colspan=3 valign=bottom>&nbsp;<br>
    26. 

  26. <font color="#ffffff" face="helvetica, arial"><big><strong>Functions</strong></big></font></td></tr>
    27. 

  27.     
    28. 

  28. <tr><td bgcolor="#eeaa77"><tt>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</tt></td><td>&nbsp;</td>
    29. 

  29. <td width="100%"><dl><dt><a name="-ArithmeticMean"><strong>ArithmeticMean</strong></a>(data)</dt><dd><tt>Calculates&nbsp;arithmetic&nbsp;mean.<br>
    30. 

  30. &nbsp;<br>
    31. 

  31. Args:<br>
    32. 

  32. &nbsp;&nbsp;data:&nbsp;A&nbsp;list&nbsp;of&nbsp;samples.<br>
    33. 

  33. &nbsp;<br>
    34. 

  34. Returns:<br>
    35. 

  35. &nbsp;&nbsp;The&nbsp;arithmetic&nbsp;mean&nbsp;value,&nbsp;or&nbsp;0&nbsp;if&nbsp;the&nbsp;list&nbsp;is&nbsp;empty.</tt></dd></dl>
    36. 

  36.  <dl><dt><a name="-Clamp"><strong>Clamp</strong></a>(value, low<font color="#909090">=0.0</font>, high<font color="#909090">=1.0</font>)</dt><dd><tt>Clamp&nbsp;a&nbsp;value&nbsp;between&nbsp;some&nbsp;low&nbsp;and&nbsp;high&nbsp;value.</tt></dd></dl>
    37. 

  37.  <dl><dt><a name="-Discrepancy"><strong>Discrepancy</strong></a>(samples, location_count<font color="#909090">=None</font>)</dt><dd><tt>Computes&nbsp;the&nbsp;discrepancy&nbsp;of&nbsp;a&nbsp;set&nbsp;of&nbsp;1D&nbsp;samples&nbsp;from&nbsp;the&nbsp;interval&nbsp;[0,1].<br>
    38. 

  38. &nbsp;<br>
    39. 

  39. The&nbsp;samples&nbsp;must&nbsp;be&nbsp;sorted.&nbsp;We&nbsp;define&nbsp;the&nbsp;discrepancy&nbsp;of&nbsp;an&nbsp;empty&nbsp;set<br>
    40. 

  40. of&nbsp;samples&nbsp;to&nbsp;be&nbsp;zero.<br>
    41. 

  41. &nbsp;<br>
    42. 

  42. <a href="http://en.wikipedia.org/wiki/Low-discrepancy_sequence">http://en.wikipedia.org/wiki/Low-discrepancy_sequence</a><br>
    43. 

  43. <a href="http://mathworld.wolfram.com/Discrepancy.html">http://mathworld.wolfram.com/Discrepancy.html</a></tt></dd></dl>
    44. 

  44.  <dl><dt><a name="-DivideIfPossibleOrZero"><strong>DivideIfPossibleOrZero</strong></a>(numerator, denominator)</dt><dd><tt>Returns&nbsp;the&nbsp;quotient,&nbsp;or&nbsp;zero&nbsp;if&nbsp;the&nbsp;denominator&nbsp;is&nbsp;zero.</tt></dd></dl>
    45. 

  45.  <dl><dt><a name="-DurationsDiscrepancy"><strong>DurationsDiscrepancy</strong></a>(durations, absolute<font color="#909090">=True</font>, location_count<font color="#909090">=None</font>)</dt><dd><tt>A&nbsp;discrepancy&nbsp;based&nbsp;metric&nbsp;for&nbsp;measuring&nbsp;duration&nbsp;jank.<br>
    46. 

  46. &nbsp;<br>
    47. 

  47. DurationsDiscrepancy&nbsp;computes&nbsp;a&nbsp;jank&nbsp;metric&nbsp;which&nbsp;measures&nbsp;how&nbsp;irregular&nbsp;a<br>
    48. 

  48. given&nbsp;sequence&nbsp;of&nbsp;intervals&nbsp;is.&nbsp;In&nbsp;order&nbsp;to&nbsp;minimize&nbsp;jank,&nbsp;each&nbsp;duration<br>
    49. 

  49. should&nbsp;be&nbsp;equally&nbsp;long.&nbsp;This&nbsp;is&nbsp;similar&nbsp;to&nbsp;how&nbsp;timestamp&nbsp;jank&nbsp;works,<br>
    50. 

  50. and&nbsp;we&nbsp;therefore&nbsp;reuse&nbsp;the&nbsp;timestamp&nbsp;discrepancy&nbsp;function&nbsp;above&nbsp;to&nbsp;compute&nbsp;a<br>
    51. 

  51. similar&nbsp;duration&nbsp;discrepancy&nbsp;number.<br>
    52. 

  52. &nbsp;<br>
    53. 

  53. Because&nbsp;timestamp&nbsp;discrepancy&nbsp;is&nbsp;defined&nbsp;in&nbsp;terms&nbsp;of&nbsp;timestamps,&nbsp;we&nbsp;first<br>
    54. 

  54. convert&nbsp;the&nbsp;list&nbsp;of&nbsp;durations&nbsp;to&nbsp;monotonically&nbsp;increasing&nbsp;timestamps.<br>
    55. 

  55. &nbsp;<br>
    56. 

  56. Args:<br>
    57. 

  57. &nbsp;&nbsp;durations:&nbsp;List&nbsp;of&nbsp;interval&nbsp;lengths&nbsp;in&nbsp;milliseconds.<br>
    58. 

  58. &nbsp;&nbsp;absolute:&nbsp;See&nbsp;TimestampsDiscrepancy.<br>
    59. 

  59. &nbsp;&nbsp;interval_multiplier:&nbsp;See&nbsp;TimestampsDiscrepancy.</tt></dd></dl>
    60. 

  60.  <dl><dt><a name="-GeneralizedMean"><strong>GeneralizedMean</strong></a>(values, exponent)</dt><dd><tt>See&nbsp;<a href="http://en.wikipedia.org/wiki/Generalized_mean">http://en.wikipedia.org/wiki/Generalized_mean</a></tt></dd></dl>
    61. 

  61.  <dl><dt><a name="-GeometricMean"><strong>GeometricMean</strong></a>(values)</dt><dd><tt>Compute&nbsp;a&nbsp;rounded&nbsp;geometric&nbsp;mean&nbsp;from&nbsp;an&nbsp;array&nbsp;of&nbsp;values.</tt></dd></dl>
    62. 

  62.  <dl><dt><a name="-Median"><strong>Median</strong></a>(values)</dt><dd><tt>Gets&nbsp;the&nbsp;median&nbsp;of&nbsp;a&nbsp;list&nbsp;of&nbsp;values.</tt></dd></dl>
    63. 

  63.  <dl><dt><a name="-NormalizeSamples"><strong>NormalizeSamples</strong></a>(samples)</dt><dd><tt>Sorts&nbsp;the&nbsp;samples,&nbsp;and&nbsp;map&nbsp;them&nbsp;linearly&nbsp;to&nbsp;the&nbsp;range&nbsp;[0,1].<br>
    64. 

  64. &nbsp;<br>
    65. 

  65. They're&nbsp;mapped&nbsp;such&nbsp;that&nbsp;for&nbsp;the&nbsp;N&nbsp;samples,&nbsp;the&nbsp;first&nbsp;sample&nbsp;is&nbsp;0.5/N&nbsp;and&nbsp;the<br>
    66. 

  66. last&nbsp;sample&nbsp;is&nbsp;(N-0.5)/N.<br>
    67. 

  67. &nbsp;<br>
    68. 

  68. Background:&nbsp;The&nbsp;discrepancy&nbsp;of&nbsp;the&nbsp;sample&nbsp;set&nbsp;i/(N-1);&nbsp;i=0,&nbsp;...,&nbsp;N-1&nbsp;is&nbsp;2/N,<br>
    69. 

  69. twice&nbsp;the&nbsp;discrepancy&nbsp;of&nbsp;the&nbsp;sample&nbsp;set&nbsp;(i+1/2)/N;&nbsp;i=0,&nbsp;...,&nbsp;N-1.&nbsp;In&nbsp;our&nbsp;case<br>
    70. 

  70. we&nbsp;don't&nbsp;want&nbsp;to&nbsp;distinguish&nbsp;between&nbsp;these&nbsp;two&nbsp;cases,&nbsp;as&nbsp;our&nbsp;original&nbsp;domain<br>
    71. 

  71. is&nbsp;not&nbsp;bounded&nbsp;(it&nbsp;is&nbsp;for&nbsp;Monte&nbsp;Carlo&nbsp;integration,&nbsp;where&nbsp;discrepancy&nbsp;was<br>
    72. 

  72. first&nbsp;used).</tt></dd></dl>
    73. 

  73.  <dl><dt><a name="-Percentile"><strong>Percentile</strong></a>(values, percentile)</dt><dd><tt>Calculates&nbsp;the&nbsp;value&nbsp;below&nbsp;which&nbsp;a&nbsp;given&nbsp;percentage&nbsp;of&nbsp;values&nbsp;fall.<br>
    74. 

  74. &nbsp;<br>
    75. 

  75. For&nbsp;example,&nbsp;if&nbsp;17%&nbsp;of&nbsp;the&nbsp;values&nbsp;are&nbsp;less&nbsp;than&nbsp;5.0,&nbsp;then&nbsp;5.0&nbsp;is&nbsp;the&nbsp;17th<br>
    76. 

  76. percentile&nbsp;for&nbsp;this&nbsp;set&nbsp;of&nbsp;values.&nbsp;When&nbsp;the&nbsp;percentage&nbsp;doesn't&nbsp;exactly<br>
    77. 

  77. match&nbsp;a&nbsp;rank&nbsp;in&nbsp;the&nbsp;list&nbsp;of&nbsp;values,&nbsp;the&nbsp;percentile&nbsp;is&nbsp;computed&nbsp;using&nbsp;linear<br>
    78. 

  78. interpolation&nbsp;between&nbsp;closest&nbsp;ranks.<br>
    79. 

  79. &nbsp;<br>
    80. 

  80. Args:<br>
    81. 

  81. &nbsp;&nbsp;values:&nbsp;A&nbsp;list&nbsp;of&nbsp;numerical&nbsp;values.<br>
    82. 

  82. &nbsp;&nbsp;percentile:&nbsp;A&nbsp;number&nbsp;between&nbsp;0&nbsp;and&nbsp;100.<br>
    83. 

  83. &nbsp;<br>
    84. 

  84. Returns:<br>
    85. 

  85. &nbsp;&nbsp;The&nbsp;Nth&nbsp;percentile&nbsp;for&nbsp;the&nbsp;list&nbsp;of&nbsp;values,&nbsp;where&nbsp;N&nbsp;is&nbsp;the&nbsp;given&nbsp;percentage.</tt></dd></dl>
    86. 

  86.  <dl><dt><a name="-StandardDeviation"><strong>StandardDeviation</strong></a>(data)</dt><dd><tt>Calculates&nbsp;the&nbsp;standard&nbsp;deviation.<br>
    87. 

  87. &nbsp;<br>
    88. 

  88. Args:<br>
    89. 

  89. &nbsp;&nbsp;data:&nbsp;A&nbsp;list&nbsp;of&nbsp;samples.<br>
    90. 

  90. &nbsp;<br>
    91. 

  91. Returns:<br>
    92. 

  92. &nbsp;&nbsp;The&nbsp;standard&nbsp;deviation&nbsp;of&nbsp;the&nbsp;samples&nbsp;provided.</tt></dd></dl>
    93. 

  93.  <dl><dt><a name="-TimestampsDiscrepancy"><strong>TimestampsDiscrepancy</strong></a>(timestamps, absolute<font color="#909090">=True</font>, location_count<font color="#909090">=None</font>)</dt><dd><tt>A&nbsp;discrepancy&nbsp;based&nbsp;metric&nbsp;for&nbsp;measuring&nbsp;timestamp&nbsp;jank.<br>
    94. 

  94. &nbsp;<br>
    95. 

  95. TimestampsDiscrepancy&nbsp;quantifies&nbsp;the&nbsp;largest&nbsp;area&nbsp;of&nbsp;jank&nbsp;observed&nbsp;in&nbsp;a&nbsp;series<br>
    96. 

  96. of&nbsp;timestamps.&nbsp;&nbsp;Note&nbsp;that&nbsp;this&nbsp;is&nbsp;different&nbsp;from&nbsp;metrics&nbsp;based&nbsp;on&nbsp;the<br>
    97. 

  97. max_time_interval.&nbsp;For&nbsp;example,&nbsp;the&nbsp;time&nbsp;stamp&nbsp;series&nbsp;A&nbsp;=&nbsp;[0,1,2,3,5,6]&nbsp;and<br>
    98. 

  98. B&nbsp;=&nbsp;[0,1,2,3,5,7]&nbsp;have&nbsp;the&nbsp;same&nbsp;max_time_interval&nbsp;=&nbsp;2,&nbsp;but<br>
    99. 

  99. <a href="#-Discrepancy">Discrepancy</a>(B)&nbsp;&gt;&nbsp;<a href="#-Discrepancy">Discrepancy</a>(A).<br>
    100. 

  100. &nbsp;<br>
    101. 

  101. Two&nbsp;variants&nbsp;of&nbsp;discrepancy&nbsp;can&nbsp;be&nbsp;computed:<br>
    102. 

  102. &nbsp;<br>
    103. 

  103. Relative&nbsp;discrepancy&nbsp;is&nbsp;following&nbsp;the&nbsp;original&nbsp;definition&nbsp;of<br>
    104. 

  104. discrepancy.&nbsp;It&nbsp;characterized&nbsp;the&nbsp;largest&nbsp;area&nbsp;of&nbsp;jank,&nbsp;relative&nbsp;to&nbsp;the<br>
    105. 

  105. duration&nbsp;of&nbsp;the&nbsp;entire&nbsp;time&nbsp;stamp&nbsp;series.&nbsp;&nbsp;We&nbsp;normalize&nbsp;the&nbsp;raw&nbsp;results,<br>
    106. 

  106. because&nbsp;the&nbsp;best&nbsp;case&nbsp;discrepancy&nbsp;for&nbsp;a&nbsp;set&nbsp;of&nbsp;N&nbsp;samples&nbsp;is&nbsp;1/N&nbsp;(for<br>
    107. 

  107. equally&nbsp;spaced&nbsp;samples),&nbsp;and&nbsp;we&nbsp;want&nbsp;our&nbsp;metric&nbsp;to&nbsp;report&nbsp;0.0&nbsp;in&nbsp;that<br>
    108. 

  108. case.<br>
    109. 

  109. &nbsp;<br>
    110. 

  110. Absolute&nbsp;discrepancy&nbsp;also&nbsp;characterizes&nbsp;the&nbsp;largest&nbsp;area&nbsp;of&nbsp;jank,&nbsp;but&nbsp;its<br>
    111. 

  111. value&nbsp;wouldn't&nbsp;change&nbsp;(except&nbsp;for&nbsp;imprecisions&nbsp;due&nbsp;to&nbsp;a&nbsp;low<br>
    112. 

  112. |interval_multiplier|)&nbsp;if&nbsp;additional&nbsp;'good'&nbsp;intervals&nbsp;were&nbsp;added&nbsp;to&nbsp;an<br>
    113. 

  113. exisiting&nbsp;list&nbsp;of&nbsp;time&nbsp;stamps.&nbsp;&nbsp;Its&nbsp;range&nbsp;is&nbsp;[0,inf]&nbsp;and&nbsp;the&nbsp;unit&nbsp;is<br>
    114. 

  114. milliseconds.<br>
    115. 

  115. &nbsp;<br>
    116. 

  116. The&nbsp;time&nbsp;stamp&nbsp;series&nbsp;C&nbsp;=&nbsp;[0,2,3,4]&nbsp;and&nbsp;D&nbsp;=&nbsp;[0,2,3,4,5]&nbsp;have&nbsp;the&nbsp;same<br>
    117. 

  117. absolute&nbsp;discrepancy,&nbsp;but&nbsp;D&nbsp;has&nbsp;lower&nbsp;relative&nbsp;discrepancy&nbsp;than&nbsp;C.<br>
    118. 

  118. &nbsp;<br>
    119. 

  119. |timestamps|&nbsp;may&nbsp;be&nbsp;a&nbsp;list&nbsp;of&nbsp;lists&nbsp;S&nbsp;=&nbsp;[S_1,&nbsp;S_2,&nbsp;...,&nbsp;S_N],&nbsp;where&nbsp;each<br>
    120. 

  120. S_i&nbsp;is&nbsp;a&nbsp;time&nbsp;stamp&nbsp;series.&nbsp;In&nbsp;that&nbsp;case,&nbsp;the&nbsp;discrepancy&nbsp;D(S)&nbsp;is:<br>
    121. 

  121. D(S)&nbsp;=&nbsp;max(D(S_1),&nbsp;D(S_2),&nbsp;...,&nbsp;D(S_N))</tt></dd></dl>
    122. 

  122.  <dl><dt><a name="-Total"><strong>Total</strong></a>(data)</dt><dd><tt>Returns&nbsp;the&nbsp;float&nbsp;value&nbsp;of&nbsp;a&nbsp;number&nbsp;or&nbsp;the&nbsp;sum&nbsp;of&nbsp;a&nbsp;list.</tt></dd></dl>
    123. 

  123.  <dl><dt><a name="-TrapezoidalRule"><strong>TrapezoidalRule</strong></a>(data, dx)</dt><dd><tt>Calculate&nbsp;the&nbsp;integral&nbsp;according&nbsp;to&nbsp;the&nbsp;trapezoidal&nbsp;rule<br>
    124. 

  124. &nbsp;<br>
    125. 

  125. TrapezoidalRule&nbsp;approximates&nbsp;the&nbsp;definite&nbsp;integral&nbsp;of&nbsp;f&nbsp;from&nbsp;a&nbsp;to&nbsp;b&nbsp;by<br>
    126. 

  126. the&nbsp;composite&nbsp;trapezoidal&nbsp;rule,&nbsp;using&nbsp;n&nbsp;subintervals.<br>
    127. 

  127. <a href="http://en.wikipedia.org/wiki/Trapezoidal_rule#Uniform_grid">http://en.wikipedia.org/wiki/Trapezoidal_rule#Uniform_grid</a><br>
    128. 

  128. &nbsp;<br>
    129. 

  129. Args:<br>
    130. 

  130. &nbsp;&nbsp;data:&nbsp;A&nbsp;list&nbsp;of&nbsp;samples<br>
    131. 

  131. &nbsp;&nbsp;dx:&nbsp;The&nbsp;uniform&nbsp;distance&nbsp;along&nbsp;the&nbsp;x&nbsp;axis&nbsp;between&nbsp;any&nbsp;two&nbsp;samples<br>
    132. 

  132. &nbsp;<br>
    133. 

  133. Returns:<br>
    134. 

  134. &nbsp;&nbsp;The&nbsp;area&nbsp;under&nbsp;the&nbsp;curve&nbsp;defined&nbsp;by&nbsp;the&nbsp;samples&nbsp;and&nbsp;the&nbsp;uniform&nbsp;distance<br>
    135. 

  135. &nbsp;&nbsp;according&nbsp;to&nbsp;the&nbsp;trapezoidal&nbsp;rule.</tt></dd></dl>
    136. 

  136. </td></tr></table>
    137. 

  137. </body></html>
    138.