| <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"> |
| <html xmlns="http://www.w3.org/1999/xhtml"> |
| <head> |
| <meta http-equiv="Content-Type" content="text/xhtml;charset=UTF-8"/> |
| <meta http-equiv="X-UA-Compatible" content="IE=9"/> |
| <meta name="generator" content="Doxygen 1.8.4"/> |
| <title>MADlib: hypothesis_tests.sql_in Source File</title> |
| <link href="tabs.css" rel="stylesheet" type="text/css"/> |
| <script type="text/javascript" src="jquery.js"></script> |
| <script type="text/javascript" src="dynsections.js"></script> |
| <link href="navtree.css" rel="stylesheet" type="text/css"/> |
| <script type="text/javascript" src="resize.js"></script> |
| <script type="text/javascript" src="navtree.js"></script> |
| <script type="text/javascript"> |
| $(document).ready(initResizable); |
| $(window).load(resizeHeight); |
| </script> |
| <link href="search/search.css" rel="stylesheet" type="text/css"/> |
| <script type="text/javascript" src="search/search.js"></script> |
| <script type="text/javascript"> |
| $(document).ready(function() { searchBox.OnSelectItem(0); }); |
| </script> |
| <script type="text/x-mathjax-config"> |
| MathJax.Hub.Config({ |
| extensions: ["tex2jax.js", "TeX/AMSmath.js", "TeX/AMSsymbols.js"], |
| jax: ["input/TeX","output/HTML-CSS"], |
| }); |
| </script><script src="../mathjax/MathJax.js"></script> |
| <link href="doxygen.css" rel="stylesheet" type="text/css" /> |
| </head> |
| <body> |
| <div id="top"><!-- do not remove this div, it is closed by doxygen! --> |
| <div id="titlearea"> |
| <table cellspacing="0" cellpadding="0"> |
| <tbody> |
| <tr style="height: 56px;"> |
| <td style="padding-left: 0.5em;"> |
| <div id="projectname">MADlib |
|  <span id="projectnumber">1.1</span> <span style="font-size:10pt; font-style:italic"><a href="../latest/./hypothesis__tests_8sql__in_source.html"> A newer version is available</a></span> |
| </div> |
| <div id="projectbrief">User Documentation</div> |
| </td> |
| </tr> |
| </tbody> |
| </table> |
| </div> |
| <!-- end header part --> |
| <!-- Generated by Doxygen 1.8.4 --> |
| <script type="text/javascript"> |
| var searchBox = new SearchBox("searchBox", "search",false,'Search'); |
| </script> |
| <div id="navrow1" class="tabs"> |
| <ul class="tablist"> |
| <li><a href="index.html"><span>Main Page</span></a></li> |
| <li><a href="modules.html"><span>Modules</span></a></li> |
| <li class="current"><a href="files.html"><span>Files</span></a></li> |
| <li> |
| <div id="MSearchBox" class="MSearchBoxInactive"> |
| <span class="left"> |
| <img id="MSearchSelect" src="search/mag_sel.png" |
| onmouseover="return searchBox.OnSearchSelectShow()" |
| onmouseout="return searchBox.OnSearchSelectHide()" |
| alt=""/> |
| <input type="text" id="MSearchField" value="Search" accesskey="S" |
| onfocus="searchBox.OnSearchFieldFocus(true)" |
| onblur="searchBox.OnSearchFieldFocus(false)" |
| onkeyup="searchBox.OnSearchFieldChange(event)"/> |
| </span><span class="right"> |
| <a id="MSearchClose" href="javascript:searchBox.CloseResultsWindow()"><img id="MSearchCloseImg" border="0" src="search/close.png" alt=""/></a> |
| </span> |
| </div> |
| </li> |
| </ul> |
| </div> |
| <div id="navrow2" class="tabs2"> |
| <ul class="tablist"> |
| <li><a href="files.html"><span>File List</span></a></li> |
| <li><a href="globals.html"><span>File Members</span></a></li> |
| </ul> |
| </div> |
| </div><!-- top --> |
| <div id="side-nav" class="ui-resizable side-nav-resizable"> |
| <div id="nav-tree"> |
| <div id="nav-tree-contents"> |
| <div id="nav-sync" class="sync"></div> |
| </div> |
| </div> |
| <div id="splitbar" style="-moz-user-select:none;" |
| class="ui-resizable-handle"> |
| </div> |
| </div> |
| <script type="text/javascript"> |
| $(document).ready(function(){initNavTree('hypothesis__tests_8sql__in_source.html','');}); |
| </script> |
| <div id="doc-content"> |
| <!-- window showing the filter options --> |
| <div id="MSearchSelectWindow" |
| onmouseover="return searchBox.OnSearchSelectShow()" |
| onmouseout="return searchBox.OnSearchSelectHide()" |
| onkeydown="return searchBox.OnSearchSelectKey(event)"> |
| <a class="SelectItem" href="javascript:void(0)" onclick="searchBox.OnSelectItem(0)"><span class="SelectionMark"> </span>All</a><a class="SelectItem" href="javascript:void(0)" onclick="searchBox.OnSelectItem(1)"><span class="SelectionMark"> </span>Files</a><a class="SelectItem" href="javascript:void(0)" onclick="searchBox.OnSelectItem(2)"><span class="SelectionMark"> </span>Functions</a><a class="SelectItem" href="javascript:void(0)" onclick="searchBox.OnSelectItem(3)"><span class="SelectionMark"> </span>Groups</a></div> |
| |
| <!-- iframe showing the search results (closed by default) --> |
| <div id="MSearchResultsWindow"> |
| <iframe src="javascript:void(0)" frameborder="0" |
| name="MSearchResults" id="MSearchResults"> |
| </iframe> |
| </div> |
| |
| <div class="header"> |
| <div class="headertitle"> |
| <div class="title">hypothesis_tests.sql_in</div> </div> |
| </div><!--header--> |
| <div class="contents"> |
| <a href="hypothesis__tests_8sql__in.html">Go to the documentation of this file.</a><div class="fragment"><div class="line"><a name="l00001"></a><span class="lineno"> 1</span> <span class="comment">/* ----------------------------------------------------------------------- */</span><span class="comment"></span></div> |
| <div class="line"><a name="l00002"></a><span class="lineno"> 2</span> <span class="comment">/**</span></div> |
| <div class="line"><a name="l00003"></a><span class="lineno"> 3</span> <span class="comment"> *</span></div> |
| <div class="line"><a name="l00004"></a><span class="lineno"> 4</span> <span class="comment"> * @file hypothesis_tests.sql_in</span></div> |
| <div class="line"><a name="l00005"></a><span class="lineno"> 5</span> <span class="comment"> *</span></div> |
| <div class="line"><a name="l00006"></a><span class="lineno"> 6</span> <span class="comment"> * @brief SQL functions for statistical hypothesis tests</span></div> |
| <div class="line"><a name="l00007"></a><span class="lineno"> 7</span> <span class="comment"> *</span></div> |
| <div class="line"><a name="l00008"></a><span class="lineno"> 8</span> <span class="comment"> * @sa For an overview of hypthesis-test functions, see the module</span></div> |
| <div class="line"><a name="l00009"></a><span class="lineno"> 9</span> <span class="comment"> * description \ref grp_stats_tests.</span></div> |
| <div class="line"><a name="l00010"></a><span class="lineno"> 10</span> <span class="comment"> *</span></div> |
| <div class="line"><a name="l00011"></a><span class="lineno"> 11</span> <span class="comment"> */</span></div> |
| <div class="line"><a name="l00012"></a><span class="lineno"> 12</span>  <span class="comment">/* ----------------------------------------------------------------------- */</span></div> |
| <div class="line"><a name="l00013"></a><span class="lineno"> 13</span> </div> |
| <div class="line"><a name="l00014"></a><span class="lineno"> 14</span> m4_include(`SQLCommon.m4<span class="stringliteral">')</span></div> |
| <div class="line"><a name="l00015"></a><span class="lineno"> 15</span> <span class="stringliteral">m4_changequote(<!,!>)</span></div> |
| <div class="line"><a name="l00016"></a><span class="lineno"> 16</span> <span class="stringliteral"></span><span class="comment"></span></div> |
| <div class="line"><a name="l00017"></a><span class="lineno"> 17</span> <span class="comment">/**</span></div> |
| <div class="line"><a name="l00018"></a><span class="lineno"> 18</span> <span class="comment">@addtogroup grp_stats_tests</span></div> |
| <div class="line"><a name="l00019"></a><span class="lineno"> 19</span> <span class="comment"></span></div> |
| <div class="line"><a name="l00020"></a><span class="lineno"> 20</span> <span class="comment">@about</span></div> |
| <div class="line"><a name="l00021"></a><span class="lineno"> 21</span> <span class="comment"></span></div> |
| <div class="line"><a name="l00022"></a><span class="lineno"> 22</span> <span class="comment">Hypothesis tests are used to confirm or reject a <em>“null” hypothesis</em></span></div> |
| <div class="line"><a name="l00023"></a><span class="lineno"> 23</span> <span class="comment">\f$ H_0 \f$ about the distribution of random variables, given realizations of</span></div> |
| <div class="line"><a name="l00024"></a><span class="lineno"> 24</span> <span class="comment">these random variables. Since in general it is not possible to make statements</span></div> |
| <div class="line"><a name="l00025"></a><span class="lineno"> 25</span> <span class="comment">with certainty, one is interested in the probability \f$ p \f$ of seeing random</span></div> |
| <div class="line"><a name="l00026"></a><span class="lineno"> 26</span> <span class="comment">variates at least as extreme as the ones observed, assuming that \f$ H_0 \f$ is</span></div> |
| <div class="line"><a name="l00027"></a><span class="lineno"> 27</span> <span class="comment">true. If this probability \f$ p \f$ is small, \f$ H_0 \f$ will be rejected by</span></div> |
| <div class="line"><a name="l00028"></a><span class="lineno"> 28</span> <span class="comment">the test with <em>significance level</em> \f$ p \f$. Falsifying \f$ H_0 \f$ is</span></div> |
| <div class="line"><a name="l00029"></a><span class="lineno"> 29</span> <span class="comment">the canonic goal when employing a hypothesis test. That is, hypothesis tests are</span></div> |
| <div class="line"><a name="l00030"></a><span class="lineno"> 30</span> <span class="comment">typically used in order to substantiate that instead the <em>alternative</span></div> |
| <div class="line"><a name="l00031"></a><span class="lineno"> 31</span> <span class="comment">hypothesis</em> \f$ H_1 \f$ is true.</span></div> |
| <div class="line"><a name="l00032"></a><span class="lineno"> 32</span> <span class="comment"></span></div> |
| <div class="line"><a name="l00033"></a><span class="lineno"> 33</span> <span class="comment">Hypothesis tests may be devided into parametric and non-parametric tests. A</span></div> |
| <div class="line"><a name="l00034"></a><span class="lineno"> 34</span> <span class="comment">parametric test assumes certain distributions and makes inferences about</span></div> |
| <div class="line"><a name="l00035"></a><span class="lineno"> 35</span> <span class="comment">parameters of the distributions (like, e.g., the mean of a normal distribution).</span></div> |
| <div class="line"><a name="l00036"></a><span class="lineno"> 36</span> <span class="comment">Formally, there is a given domain of possible parameters \f$ \Gamma \f$ and the</span></div> |
| <div class="line"><a name="l00037"></a><span class="lineno"> 37</span> <span class="comment">null hypothesis \f$ H_0 \f$ is the event that the true parameter</span></div> |
| <div class="line"><a name="l00038"></a><span class="lineno"> 38</span> <span class="comment">\f$ \gamma_0 \in \Gamma_0 \f$, where \f$ \Gamma_0 \subsetneq \Gamma \f$.</span></div> |
| <div class="line"><a name="l00039"></a><span class="lineno"> 39</span> <span class="comment">Non-parametric tests, on the other hand, do not assume any particular</span></div> |
| <div class="line"><a name="l00040"></a><span class="lineno"> 40</span> <span class="comment">distribution of the sample (e.g., a non-parametric test may simply test if two</span></div> |
| <div class="line"><a name="l00041"></a><span class="lineno"> 41</span> <span class="comment">distributions are similar).</span></div> |
| <div class="line"><a name="l00042"></a><span class="lineno"> 42</span> <span class="comment"></span></div> |
| <div class="line"><a name="l00043"></a><span class="lineno"> 43</span> <span class="comment">The first step of a hypothesis test is to compute a <em>test statistic</em>,</span></div> |
| <div class="line"><a name="l00044"></a><span class="lineno"> 44</span> <span class="comment">which is a function of the random variates, i.e., a random variate itself.</span></div> |
| <div class="line"><a name="l00045"></a><span class="lineno"> 45</span> <span class="comment">A hypothesis test relies on that the distribution of the test statistic is</span></div> |
| <div class="line"><a name="l00046"></a><span class="lineno"> 46</span> <span class="comment">(approximately) known. Now, the \f$ p \f$-value is the probability of seeing a</span></div> |
| <div class="line"><a name="l00047"></a><span class="lineno"> 47</span> <span class="comment">test statistic at least as extreme as the one observed, assuming that</span></div> |
| <div class="line"><a name="l00048"></a><span class="lineno"> 48</span> <span class="comment">\f$ H_0 \f$ is true. In a case where the null hypothesis corresponds to a family</span></div> |
| <div class="line"><a name="l00049"></a><span class="lineno"> 49</span> <span class="comment">of distributions (e.g., in a parametric test where \f$ \Gamma_0 \f$ is not a</span></div> |
| <div class="line"><a name="l00050"></a><span class="lineno"> 50</span> <span class="comment">singleton set), the \f$ p \f$-value is the supremum, over all possible</span></div> |
| <div class="line"><a name="l00051"></a><span class="lineno"> 51</span> <span class="comment">distributions according to the null hypothesis, of these probabilities.</span></div> |
| <div class="line"><a name="l00052"></a><span class="lineno"> 52</span> <span class="comment"></span></div> |
| <div class="line"><a name="l00053"></a><span class="lineno"> 53</span> <span class="comment">@input</span></div> |
| <div class="line"><a name="l00054"></a><span class="lineno"> 54</span> <span class="comment"></span></div> |
| <div class="line"><a name="l00055"></a><span class="lineno"> 55</span> <span class="comment">Input data is assumed to be normalized with all values stored row-wise. In</span></div> |
| <div class="line"><a name="l00056"></a><span class="lineno"> 56</span> <span class="comment">general, the following inputs are expected.</span></div> |
| <div class="line"><a name="l00057"></a><span class="lineno"> 57</span> <span class="comment"></span></div> |
| <div class="line"><a name="l00058"></a><span class="lineno"> 58</span> <span class="comment">One-sample tests expect the following form:</span></div> |
| <div class="line"><a name="l00059"></a><span class="lineno"> 59</span> <span class="comment"><pre>{TABLE|VIEW} <em>source</em> (</span></div> |
| <div class="line"><a name="l00060"></a><span class="lineno"> 60</span> <span class="comment"> ...</span></div> |
| <div class="line"><a name="l00061"></a><span class="lineno"> 61</span> <span class="comment"> <em>value</em> DOUBLE PRECISION</span></div> |
| <div class="line"><a name="l00062"></a><span class="lineno"> 62</span> <span class="comment"> ...</span></div> |
| <div class="line"><a name="l00063"></a><span class="lineno"> 63</span> <span class="comment">)</pre></span></div> |
| <div class="line"><a name="l00064"></a><span class="lineno"> 64</span> <span class="comment"></span></div> |
| <div class="line"><a name="l00065"></a><span class="lineno"> 65</span> <span class="comment">Two-sample tests expect the following form:</span></div> |
| <div class="line"><a name="l00066"></a><span class="lineno"> 66</span> <span class="comment"><pre>{TABLE|VIEW} <em>source</em> (</span></div> |
| <div class="line"><a name="l00067"></a><span class="lineno"> 67</span> <span class="comment"> ...</span></div> |
| <div class="line"><a name="l00068"></a><span class="lineno"> 68</span> <span class="comment"> <em>first</em> BOOLEAN,</span></div> |
| <div class="line"><a name="l00069"></a><span class="lineno"> 69</span> <span class="comment"> <em>value</em> DOUBLE PRECISION</span></div> |
| <div class="line"><a name="l00070"></a><span class="lineno"> 70</span> <span class="comment"> ...</span></div> |
| <div class="line"><a name="l00071"></a><span class="lineno"> 71</span> <span class="comment">)</pre></span></div> |
| <div class="line"><a name="l00072"></a><span class="lineno"> 72</span> <span class="comment">Here, \c first indicates whether a value is from the first (if \c TRUE) or the</span></div> |
| <div class="line"><a name="l00073"></a><span class="lineno"> 73</span> <span class="comment">second sample (if \c FALSE).</span></div> |
| <div class="line"><a name="l00074"></a><span class="lineno"> 74</span> <span class="comment"></span></div> |
| <div class="line"><a name="l00075"></a><span class="lineno"> 75</span> <span class="comment">Many-sample tests expect the following form:</span></div> |
| <div class="line"><a name="l00076"></a><span class="lineno"> 76</span> <span class="comment"><pre>{TABLE|VIEW} <em>source</em> (</span></div> |
| <div class="line"><a name="l00077"></a><span class="lineno"> 77</span> <span class="comment"> ...</span></div> |
| <div class="line"><a name="l00078"></a><span class="lineno"> 78</span> <span class="comment"> <em>group</em> INTEGER,</span></div> |
| <div class="line"><a name="l00079"></a><span class="lineno"> 79</span> <span class="comment"> <em>value</em> DOUBLE PRECISION</span></div> |
| <div class="line"><a name="l00080"></a><span class="lineno"> 80</span> <span class="comment"> ...</span></div> |
| <div class="line"><a name="l00081"></a><span class="lineno"> 81</span> <span class="comment">)</pre></span></div> |
| <div class="line"><a name="l00082"></a><span class="lineno"> 82</span> <span class="comment"></span></div> |
| <div class="line"><a name="l00083"></a><span class="lineno"> 83</span> <span class="comment">@usage</span></div> |
| <div class="line"><a name="l00084"></a><span class="lineno"> 84</span> <span class="comment"></span></div> |
| <div class="line"><a name="l00085"></a><span class="lineno"> 85</span> <span class="comment">All tests are implemented as aggregate functions. The non-parametric</span></div> |
| <div class="line"><a name="l00086"></a><span class="lineno"> 86</span> <span class="comment">(rank-based) tests are implemented as ordered aggregate functions and thus</span></div> |
| <div class="line"><a name="l00087"></a><span class="lineno"> 87</span> <span class="comment">necessitate an <tt>ORDER BY</tt> clause. In the following, the most simple</span></div> |
| <div class="line"><a name="l00088"></a><span class="lineno"> 88</span> <span class="comment">forms of usage are given. Specific function signatures, as described in</span></div> |
| <div class="line"><a name="l00089"></a><span class="lineno"> 89</span> <span class="comment">\ref hypothesis_tests.sql_in, may ask for more arguments or for a different</span></div> |
| <div class="line"><a name="l00090"></a><span class="lineno"> 90</span> <span class="comment"><tt>ORDER BY</tt> clause.</span></div> |
| <div class="line"><a name="l00091"></a><span class="lineno"> 91</span> <span class="comment"></span></div> |
| <div class="line"><a name="l00092"></a><span class="lineno"> 92</span> <span class="comment">- Run a parametric one-sample test:</span></div> |
| <div class="line"><a name="l00093"></a><span class="lineno"> 93</span> <span class="comment"> <pre>SELECT <em>test</em>(<em>value</em>) FROM <em>source</em></pre></span></div> |
| <div class="line"><a name="l00094"></a><span class="lineno"> 94</span> <span class="comment">- Run a parametric two-sample test:</span></div> |
| <div class="line"><a name="l00095"></a><span class="lineno"> 95</span> <span class="comment"> <pre>SELECT <em>test</em>(<em>first</em>, <em>value</em>) FROM <em>source</em></pre></span></div> |
| <div class="line"><a name="l00096"></a><span class="lineno"> 96</span> <span class="comment">- Run a non-parametric one-sample test:</span></div> |
| <div class="line"><a name="l00097"></a><span class="lineno"> 97</span> <span class="comment"> <pre>SELECT <em>test</em>(<em>value</em> ORDER BY <em>value</em>) FROM <em>source</em></pre></span></div> |
| <div class="line"><a name="l00098"></a><span class="lineno"> 98</span> <span class="comment">- Run a non-parametric two-sample test:</span></div> |
| <div class="line"><a name="l00099"></a><span class="lineno"> 99</span> <span class="comment"> <pre>SELECT <em>test</em>(<em>first</em>, <em>value</em> ORDER BY <em>value</em>) FROM <em>source</em></pre></span></div> |
| <div class="line"><a name="l00100"></a><span class="lineno"> 100</span> <span class="comment"></span></div> |
| <div class="line"><a name="l00101"></a><span class="lineno"> 101</span> <span class="comment">@examp</span></div> |
| <div class="line"><a name="l00102"></a><span class="lineno"> 102</span> <span class="comment"></span></div> |
| <div class="line"><a name="l00103"></a><span class="lineno"> 103</span> <span class="comment">See \ref hypothesis_tests.sql_in for examples for each of the aggregate</span></div> |
| <div class="line"><a name="l00104"></a><span class="lineno"> 104</span> <span class="comment">functions.</span></div> |
| <div class="line"><a name="l00105"></a><span class="lineno"> 105</span> <span class="comment"></span></div> |
| <div class="line"><a name="l00106"></a><span class="lineno"> 106</span> <span class="comment">@literature</span></div> |
| <div class="line"><a name="l00107"></a><span class="lineno"> 107</span> <span class="comment"></span></div> |
| <div class="line"><a name="l00108"></a><span class="lineno"> 108</span> <span class="comment">[1] M. Hollander, D. Wolfe: <em>Nonparametric Statistical Methods</em>,</span></div> |
| <div class="line"><a name="l00109"></a><span class="lineno"> 109</span> <span class="comment"> 2nd edition, Wiley, 1999</span></div> |
| <div class="line"><a name="l00110"></a><span class="lineno"> 110</span> <span class="comment"></span></div> |
| <div class="line"><a name="l00111"></a><span class="lineno"> 111</span> <span class="comment">[2] E. Lehmann, J. Romano: <em>Testing Statistical Hypotheses</em>, 3rd edition,</span></div> |
| <div class="line"><a name="l00112"></a><span class="lineno"> 112</span> <span class="comment"> Springer, 2005</span></div> |
| <div class="line"><a name="l00113"></a><span class="lineno"> 113</span> <span class="comment"></span></div> |
| <div class="line"><a name="l00114"></a><span class="lineno"> 114</span> <span class="comment">@sa File hypothesis_tests.sql_in documenting the SQL functions.</span></div> |
| <div class="line"><a name="l00115"></a><span class="lineno"> 115</span> <span class="comment">*/</span></div> |
| <div class="line"><a name="l00116"></a><span class="lineno"> 116</span> </div> |
| <div class="line"><a name="l00117"></a><span class="lineno"> 117</span> CREATE TYPE MADLIB_SCHEMA.t_test_result AS (</div> |
| <div class="line"><a name="l00118"></a><span class="lineno"> 118</span>  statistic DOUBLE PRECISION,</div> |
| <div class="line"><a name="l00119"></a><span class="lineno"> 119</span>  df DOUBLE PRECISION,</div> |
| <div class="line"><a name="l00120"></a><span class="lineno"> 120</span>  p_value_one_sided DOUBLE PRECISION,</div> |
| <div class="line"><a name="l00121"></a><span class="lineno"> 121</span>  p_value_two_sided DOUBLE PRECISION</div> |
| <div class="line"><a name="l00122"></a><span class="lineno"> 122</span> );</div> |
| <div class="line"><a name="l00123"></a><span class="lineno"> 123</span> </div> |
| <div class="line"><a name="l00124"></a><span class="lineno"> 124</span> CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.t_test_one_transition(</div> |
| <div class="line"><a name="l00125"></a><span class="lineno"> 125</span>  state DOUBLE PRECISION[],</div> |
| <div class="line"><a name="l00126"></a><span class="lineno"> 126</span>  value DOUBLE PRECISION</div> |
| <div class="line"><a name="l00127"></a><span class="lineno"> 127</span> ) RETURNS DOUBLE PRECISION[]</div> |
| <div class="line"><a name="l00128"></a><span class="lineno"> 128</span> AS 'MODULE_PATHNAME<span class="stringliteral">'</span></div> |
| <div class="line"><a name="l00129"></a><span class="lineno"> 129</span> <span class="stringliteral">LANGUAGE C</span></div> |
| <div class="line"><a name="l00130"></a><span class="lineno"> 130</span> <span class="stringliteral">IMMUTABLE</span></div> |
| <div class="line"><a name="l00131"></a><span class="lineno"> 131</span> <span class="stringliteral">STRICT;</span></div> |
| <div class="line"><a name="l00132"></a><span class="lineno"> 132</span> <span class="stringliteral"></span></div> |
| <div class="line"><a name="l00133"></a><span class="lineno"> 133</span> <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.t_test_merge_states(</span></div> |
| <div class="line"><a name="l00134"></a><span class="lineno"> 134</span> <span class="stringliteral"> state1 DOUBLE PRECISION[],</span></div> |
| <div class="line"><a name="l00135"></a><span class="lineno"> 135</span> <span class="stringliteral"> state2 DOUBLE PRECISION[])</span></div> |
| <div class="line"><a name="l00136"></a><span class="lineno"> 136</span> <span class="stringliteral">RETURNS DOUBLE PRECISION[]</span></div> |
| <div class="line"><a name="l00137"></a><span class="lineno"> 137</span> <span class="stringliteral">AS '</span>MODULE_PATHNAME<span class="stringliteral">'</span></div> |
| <div class="line"><a name="l00138"></a><span class="lineno"> 138</span> <span class="stringliteral">LANGUAGE C</span></div> |
| <div class="line"><a name="l00139"></a><span class="lineno"> 139</span> <span class="stringliteral">IMMUTABLE STRICT;</span></div> |
| <div class="line"><a name="l00140"></a><span class="lineno"> 140</span> <span class="stringliteral"></span></div> |
| <div class="line"><a name="l00141"></a><span class="lineno"> 141</span> <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.t_test_one_final(</span></div> |
| <div class="line"><a name="l00142"></a><span class="lineno"> 142</span> <span class="stringliteral"> state DOUBLE PRECISION[])</span></div> |
| <div class="line"><a name="l00143"></a><span class="lineno"> 143</span> <span class="stringliteral">RETURNS MADLIB_SCHEMA.t_test_result</span></div> |
| <div class="line"><a name="l00144"></a><span class="lineno"> 144</span> <span class="stringliteral">AS '</span>MODULE_PATHNAME<span class="stringliteral">'</span></div> |
| <div class="line"><a name="l00145"></a><span class="lineno"> 145</span> <span class="stringliteral">LANGUAGE C IMMUTABLE STRICT;</span></div> |
| <div class="line"><a name="l00146"></a><span class="lineno"> 146</span> <span class="stringliteral"></span></div> |
| <div class="line"><a name="l00147"></a><span class="lineno"> 147</span> <span class="stringliteral">CREATE TYPE MADLIB_SCHEMA.f_test_result AS (</span></div> |
| <div class="line"><a name="l00148"></a><span class="lineno"> 148</span> <span class="stringliteral"> statistic DOUBLE PRECISION,</span></div> |
| <div class="line"><a name="l00149"></a><span class="lineno"> 149</span> <span class="stringliteral"> df1 DOUBLE PRECISION,</span></div> |
| <div class="line"><a name="l00150"></a><span class="lineno"> 150</span> <span class="stringliteral"> df2 DOUBLE PRECISION,</span></div> |
| <div class="line"><a name="l00151"></a><span class="lineno"> 151</span> <span class="stringliteral"> p_value_one_sided DOUBLE PRECISION,</span></div> |
| <div class="line"><a name="l00152"></a><span class="lineno"> 152</span> <span class="stringliteral"> p_value_two_sided DOUBLE PRECISION</span></div> |
| <div class="line"><a name="l00153"></a><span class="lineno"> 153</span> <span class="stringliteral">);</span></div> |
| <div class="line"><a name="l00154"></a><span class="lineno"> 154</span> <span class="stringliteral"></span></div> |
| <div class="line"><a name="l00155"></a><span class="lineno"> 155</span> <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.f_test_final(</span></div> |
| <div class="line"><a name="l00156"></a><span class="lineno"> 156</span> <span class="stringliteral"> state DOUBLE PRECISION[])</span></div> |
| <div class="line"><a name="l00157"></a><span class="lineno"> 157</span> <span class="stringliteral">RETURNS MADLIB_SCHEMA.f_test_result</span></div> |
| <div class="line"><a name="l00158"></a><span class="lineno"> 158</span> <span class="stringliteral">AS '</span>MODULE_PATHNAME<span class="stringliteral">'</span></div> |
| <div class="line"><a name="l00159"></a><span class="lineno"> 159</span> <span class="stringliteral">LANGUAGE C IMMUTABLE STRICT;</span></div> |
| <div class="line"><a name="l00160"></a><span class="lineno"> 160</span> <span class="stringliteral"></span></div> |
| <div class="line"><a name="l00161"></a><span class="lineno"> 161</span> <span class="stringliteral"></span><span class="comment"></span></div> |
| <div class="line"><a name="l00162"></a><span class="lineno"> 162</span> <span class="comment">/**</span></div> |
| <div class="line"><a name="l00163"></a><span class="lineno"> 163</span> <span class="comment"> * @brief Perform one-sample or dependent paired Student t-test</span></div> |
| <div class="line"><a name="l00164"></a><span class="lineno"> 164</span> <span class="comment"> *</span></div> |
| <div class="line"><a name="l00165"></a><span class="lineno"> 165</span> <span class="comment"> * Given realizations \f$ x_1, \dots, x_n \f$ of i.i.d. random variables</span></div> |
| <div class="line"><a name="l00166"></a><span class="lineno"> 166</span> <span class="comment"> * \f$ X_1, \dots, X_n \sim N(\mu, \sigma^2) \f$ with unknown parameters \f$ \mu \f$ and</span></div> |
| <div class="line"><a name="l00167"></a><span class="lineno"> 167</span> <span class="comment"> * \f$ \sigma^2 \f$, test the null hypotheses \f$ H_0 : \mu \leq 0 \f$ and</span></div> |
| <div class="line"><a name="l00168"></a><span class="lineno"> 168</span> <span class="comment"> * \f$ H_0 : \mu = 0 \f$.</span></div> |
| <div class="line"><a name="l00169"></a><span class="lineno"> 169</span> <span class="comment"> *</span></div> |
| <div class="line"><a name="l00170"></a><span class="lineno"> 170</span> <span class="comment"> * @param value Value of random variate \f$ x_i \f$</span></div> |
| <div class="line"><a name="l00171"></a><span class="lineno"> 171</span> <span class="comment"> *</span></div> |
| <div class="line"><a name="l00172"></a><span class="lineno"> 172</span> <span class="comment"> * @return A composite value as follows. We denote by \f$ \bar x \f$ the</span></div> |
| <div class="line"><a name="l00173"></a><span class="lineno"> 173</span> <span class="comment"> * \ref sample_mean "sample mean" and by \f$ s^2 \f$ the</span></div> |
| <div class="line"><a name="l00174"></a><span class="lineno"> 174</span> <span class="comment"> * \ref sample_variance "sample variance".</span></div> |
| <div class="line"><a name="l00175"></a><span class="lineno"> 175</span> <span class="comment"> * - <tt>statistic FLOAT8</tt> - Statistic</span></div> |
| <div class="line"><a name="l00176"></a><span class="lineno"> 176</span> <span class="comment"> * \f[</span></div> |
| <div class="line"><a name="l00177"></a><span class="lineno"> 177</span> <span class="comment"> * t = \frac{\sqrt n \cdot \bar x}{s}</span></div> |
| <div class="line"><a name="l00178"></a><span class="lineno"> 178</span> <span class="comment"> * \f]</span></div> |
| <div class="line"><a name="l00179"></a><span class="lineno"> 179</span> <span class="comment"> * The corresponding random</span></div> |
| <div class="line"><a name="l00180"></a><span class="lineno"> 180</span> <span class="comment"> * variable is Student-t distributed with</span></div> |
| <div class="line"><a name="l00181"></a><span class="lineno"> 181</span> <span class="comment"> * \f$ (n - 1) \f$ degrees of freedom.</span></div> |
| <div class="line"><a name="l00182"></a><span class="lineno"> 182</span> <span class="comment"> * - <tt>df FLOAT8</tt> - Degrees of freedom \f$ (n - 1) \f$</span></div> |
| <div class="line"><a name="l00183"></a><span class="lineno"> 183</span> <span class="comment"> * - <tt>p_value_one_sided FLOAT8</tt> - Lower bound on one-sided p-value.</span></div> |
| <div class="line"><a name="l00184"></a><span class="lineno"> 184</span> <span class="comment"> * In detail, the result is \f$ \Pr[\bar X \geq \bar x \mid \mu = 0] \f$,</span></div> |
| <div class="line"><a name="l00185"></a><span class="lineno"> 185</span> <span class="comment"> * which is a lower bound on</span></div> |
| <div class="line"><a name="l00186"></a><span class="lineno"> 186</span> <span class="comment"> * \f$ \Pr[\bar X \geq \bar x \mid \mu \leq 0] \f$. Computed as</span></div> |
| <div class="line"><a name="l00187"></a><span class="lineno"> 187</span> <span class="comment"> * <tt>(1.0 - \ref students_t_cdf "students_t_cdf"(statistic))</tt>.</span></div> |
| <div class="line"><a name="l00188"></a><span class="lineno"> 188</span> <span class="comment"> * - <tt>p_value_two_sided FLOAT8</tt> - Two-sided p-value, i.e.,</span></div> |
| <div class="line"><a name="l00189"></a><span class="lineno"> 189</span> <span class="comment"> * \f$ \Pr[ |\bar X| \geq |\bar x| \mid \mu = 0] \f$. Computed as</span></div> |
| <div class="line"><a name="l00190"></a><span class="lineno"> 190</span> <span class="comment"> * <tt>(2 * \ref students_t_cdf "students_t_cdf"(-abs(statistic)))</tt>.</span></div> |
| <div class="line"><a name="l00191"></a><span class="lineno"> 191</span> <span class="comment"> *</span></div> |
| <div class="line"><a name="l00192"></a><span class="lineno"> 192</span> <span class="comment"> * @usage</span></div> |
| <div class="line"><a name="l00193"></a><span class="lineno"> 193</span> <span class="comment"> * - One-sample t-test: Test null hypothesis that the mean of a sample is at</span></div> |
| <div class="line"><a name="l00194"></a><span class="lineno"> 194</span> <span class="comment"> * most (or equal to, respectively) \f$ \mu_0 \f$:</span></div> |
| <div class="line"><a name="l00195"></a><span class="lineno"> 195</span> <span class="comment"> * <pre>SELECT (t_test_one(<em>value</em> - <em>mu_0</em>)).* FROM <em>source</em></pre></span></div> |
| <div class="line"><a name="l00196"></a><span class="lineno"> 196</span> <span class="comment"> * - Dependent paired t-test: Test null hypothesis that the mean difference</span></div> |
| <div class="line"><a name="l00197"></a><span class="lineno"> 197</span> <span class="comment"> * between the first and second value in each pair is at most (or equal to,</span></div> |
| <div class="line"><a name="l00198"></a><span class="lineno"> 198</span> <span class="comment"> * respectively) \f$ \mu_0 \f$:</span></div> |
| <div class="line"><a name="l00199"></a><span class="lineno"> 199</span> <span class="comment"> * <pre>SELECT (t_test_one(<em>first</em> - <em>second</em> - <em>mu_0</em>)).*</span></div> |
| <div class="line"><a name="l00200"></a><span class="lineno"> 200</span> <span class="comment"> * FROM <em>source</em></pre></span></div> |
| <div class="line"><a name="l00201"></a><span class="lineno"> 201</span> <span class="comment"> */</span></div> |
| <div class="line"><a name="l00202"></a><span class="lineno"> 202</span> CREATE AGGREGATE MADLIB_SCHEMA.t_test_one(</div> |
| <div class="line"><a name="l00203"></a><span class="lineno"> 203</span>  /*+ value */ DOUBLE PRECISION) (</div> |
| <div class="line"><a name="l00204"></a><span class="lineno"> 204</span> </div> |
| <div class="line"><a name="l00205"></a><span class="lineno"> 205</span>  SFUNC=MADLIB_SCHEMA.t_test_one_transition,</div> |
| <div class="line"><a name="l00206"></a><span class="lineno"> 206</span>  STYPE=DOUBLE PRECISION[],</div> |
| <div class="line"><a name="l00207"></a><span class="lineno"> 207</span>  FINALFUNC=MADLIB_SCHEMA.t_test_one_final,</div> |
| <div class="line"><a name="l00208"></a><span class="lineno"> 208</span>  m4_ifdef(<!__GREENPLUM__!>,<!PREFUNC=MADLIB_SCHEMA.t_test_merge_states,!>)</div> |
| <div class="line"><a name="l00209"></a><span class="lineno"> 209</span>  INITCOND='{0,0,0,0,0,0,0}<span class="stringliteral">'</span></div> |
| <div class="line"><a name="l00210"></a><span class="lineno"> 210</span> <span class="stringliteral">);</span></div> |
| <div class="line"><a name="l00211"></a><span class="lineno"> 211</span> <span class="stringliteral"></span></div> |
| <div class="line"><a name="l00212"></a><span class="lineno"> 212</span> <span class="stringliteral"></span></div> |
| <div class="line"><a name="l00213"></a><span class="lineno"> 213</span> <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.t_test_two_transition(</span></div> |
| <div class="line"><a name="l00214"></a><span class="lineno"> 214</span> <span class="stringliteral"> state DOUBLE PRECISION[],</span></div> |
| <div class="line"><a name="l00215"></a><span class="lineno"> 215</span> <span class="stringliteral"> "first" BOOLEAN,</span></div> |
| <div class="line"><a name="l00216"></a><span class="lineno"> 216</span> <span class="stringliteral"> "value" DOUBLE PRECISION)</span></div> |
| <div class="line"><a name="l00217"></a><span class="lineno"> 217</span> <span class="stringliteral">RETURNS DOUBLE PRECISION[]</span></div> |
| <div class="line"><a name="l00218"></a><span class="lineno"> 218</span> <span class="stringliteral">AS '</span>MODULE_PATHNAME<span class="stringliteral">'</span></div> |
| <div class="line"><a name="l00219"></a><span class="lineno"> 219</span> <span class="stringliteral">LANGUAGE C</span></div> |
| <div class="line"><a name="l00220"></a><span class="lineno"> 220</span> <span class="stringliteral">IMMUTABLE</span></div> |
| <div class="line"><a name="l00221"></a><span class="lineno"> 221</span> <span class="stringliteral">STRICT;</span></div> |
| <div class="line"><a name="l00222"></a><span class="lineno"> 222</span> <span class="stringliteral"></span></div> |
| <div class="line"><a name="l00223"></a><span class="lineno"> 223</span> <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.t_test_two_pooled_final(</span></div> |
| <div class="line"><a name="l00224"></a><span class="lineno"><a class="code" href="hypothesis__tests_8sql__in.html#ae7197f66a085f53d71167ac0a9029567"> 224</a></span> <span class="stringliteral"> state DOUBLE PRECISION[])</span></div> |
| <div class="line"><a name="l00225"></a><span class="lineno"> 225</span> <span class="stringliteral">RETURNS MADLIB_SCHEMA.t_test_result</span></div> |
| <div class="line"><a name="l00226"></a><span class="lineno"> 226</span> <span class="stringliteral">AS '</span>MODULE_PATHNAME<span class="stringliteral">'</span></div> |
| <div class="line"><a name="l00227"></a><span class="lineno"> 227</span> <span class="stringliteral">LANGUAGE C IMMUTABLE STRICT;</span></div> |
| <div class="line"><a name="l00228"></a><span class="lineno"> 228</span> <span class="stringliteral"></span><span class="comment"></span></div> |
| <div class="line"><a name="l00229"></a><span class="lineno"> 229</span> <span class="comment">/**</span></div> |
| <div class="line"><a name="l00230"></a><span class="lineno"> 230</span> <span class="comment"> * @brief Perform two-sample pooled (i.e., equal variances) Student t-test</span></div> |
| <div class="line"><a name="l00231"></a><span class="lineno"> 231</span> <span class="comment"> *</span></div> |
| <div class="line"><a name="l00232"></a><span class="lineno"> 232</span> <span class="comment"> * Given realizations \f$ x_1, \dots, x_n \f$ and \f$ y_1, \dots, y_m \f$ of</span></div> |
| <div class="line"><a name="l00233"></a><span class="lineno"> 233</span> <span class="comment"> * i.i.d. random variables \f$ X_1, \dots, X_n \sim N(\mu_X, \sigma^2) \f$ and</span></div> |
| <div class="line"><a name="l00234"></a><span class="lineno"> 234</span> <span class="comment"> * \f$ Y_1, \dots, Y_m \sim N(\mu_Y, \sigma^2) \f$ with unknown parameters</span></div> |
| <div class="line"><a name="l00235"></a><span class="lineno"> 235</span> <span class="comment"> * \f$ \mu_X, \mu_Y, \f$ and \f$ \sigma^2 \f$, test the null hypotheses</span></div> |
| <div class="line"><a name="l00236"></a><span class="lineno"> 236</span> <span class="comment"> * \f$ H_0 : \mu_X \leq \mu_Y \f$ and \f$ H_0 : \mu_X = \mu_Y \f$.</span></div> |
| <div class="line"><a name="l00237"></a><span class="lineno"> 237</span> <span class="comment"> *</span></div> |
| <div class="line"><a name="l00238"></a><span class="lineno"> 238</span> <span class="comment"> * @param first Indicator whether \c value is from first sample</span></div> |
| <div class="line"><a name="l00239"></a><span class="lineno"> 239</span> <span class="comment"> * \f$ x_1, \dots, x_n \f$ (if \c TRUE) or from second sample</span></div> |
| <div class="line"><a name="l00240"></a><span class="lineno"> 240</span> <span class="comment"> * \f$ y_1, \dots, y_m \f$ (if \c FALSE)</span></div> |
| <div class="line"><a name="l00241"></a><span class="lineno"> 241</span> <span class="comment"> * @param value Value of random variate \f$ x_i \f$ or \f$ y_i \f$</span></div> |
| <div class="line"><a name="l00242"></a><span class="lineno"> 242</span> <span class="comment"> *</span></div> |
| <div class="line"><a name="l00243"></a><span class="lineno"> 243</span> <span class="comment"> * @return A composite value as follows. We denote by \f$ \bar x, \bar y \f$</span></div> |
| <div class="line"><a name="l00244"></a><span class="lineno"> 244</span> <span class="comment"> * the \ref sample_mean "sample means" and by \f$ s_X^2, s_Y^2 \f$ the</span></div> |
| <div class="line"><a name="l00245"></a><span class="lineno"> 245</span> <span class="comment"> * \ref sample_variance "sample variances".</span></div> |
| <div class="line"><a name="l00246"></a><span class="lineno"> 246</span> <span class="comment"> * - <tt>statistic FLOAT8</tt> - Statistic</span></div> |
| <div class="line"><a name="l00247"></a><span class="lineno"> 247</span> <span class="comment"> * \f[</span></div> |
| <div class="line"><a name="l00248"></a><span class="lineno"> 248</span> <span class="comment"> * t = \frac{\bar x - \bar y}{s_p \sqrt{1/n + 1/m}}</span></div> |
| <div class="line"><a name="l00249"></a><span class="lineno"> 249</span> <span class="comment"> * \f]</span></div> |
| <div class="line"><a name="l00250"></a><span class="lineno"> 250</span> <span class="comment"> * where</span></div> |
| <div class="line"><a name="l00251"></a><span class="lineno"> 251</span> <span class="comment"> * \f[</span></div> |
| <div class="line"><a name="l00252"></a><span class="lineno"> 252</span> <span class="comment"> * s_p^2 = \frac{\sum_{i=1}^n (x_i - \bar x)^2</span></div> |
| <div class="line"><a name="l00253"></a><span class="lineno"> 253</span> <span class="comment"> * + \sum_{i=1}^m (y_i - \bar y)^2}</span></div> |
| <div class="line"><a name="l00254"></a><span class="lineno"> 254</span> <span class="comment"> * {n + m - 2}</span></div> |
| <div class="line"><a name="l00255"></a><span class="lineno"> 255</span> <span class="comment"> * \f]</span></div> |
| <div class="line"><a name="l00256"></a><span class="lineno"> 256</span> <span class="comment"> * is the <em>pooled variance</em>.</span></div> |
| <div class="line"><a name="l00257"></a><span class="lineno"> 257</span> <span class="comment"> * The corresponding random</span></div> |
| <div class="line"><a name="l00258"></a><span class="lineno"> 258</span> <span class="comment"> * variable is Student-t distributed with</span></div> |
| <div class="line"><a name="l00259"></a><span class="lineno"> 259</span> <span class="comment"> * \f$ (n + m - 2) \f$ degrees of freedom.</span></div> |
| <div class="line"><a name="l00260"></a><span class="lineno"> 260</span> <span class="comment"> * - <tt>df FLOAT8</tt> - Degrees of freedom \f$ (n + m - 2) \f$</span></div> |
| <div class="line"><a name="l00261"></a><span class="lineno"> 261</span> <span class="comment"> * - <tt>p_value_one_sided FLOAT8</tt> - Lower bound on one-sided p-value.</span></div> |
| <div class="line"><a name="l00262"></a><span class="lineno"> 262</span> <span class="comment"> * In detail, the result is \f$ \Pr[\bar X - \bar Y \geq \bar x - \bar y \mid \mu_X = \mu_Y] \f$,</span></div> |
| <div class="line"><a name="l00263"></a><span class="lineno"> 263</span> <span class="comment"> * which is a lower bound on</span></div> |
| <div class="line"><a name="l00264"></a><span class="lineno"> 264</span> <span class="comment"> * \f$ \Pr[\bar X - \bar Y \geq \bar x - \bar y \mid \mu_X \leq \mu_Y] \f$.</span></div> |
| <div class="line"><a name="l00265"></a><span class="lineno"> 265</span> <span class="comment"> * Computed as</span></div> |
| <div class="line"><a name="l00266"></a><span class="lineno"> 266</span> <span class="comment"> * <tt>(1.0 - \ref students_t_cdf "students_t_cdf"(statistic))</tt>.</span></div> |
| <div class="line"><a name="l00267"></a><span class="lineno"> 267</span> <span class="comment"> * - <tt>p_value_two_sided FLOAT8</tt> - Two-sided p-value, i.e.,</span></div> |
| <div class="line"><a name="l00268"></a><span class="lineno"> 268</span> <span class="comment"> * \f$ \Pr[ |\bar X - \bar Y| \geq |\bar x - \bar y| \mid \mu_X = \mu_Y] \f$.</span></div> |
| <div class="line"><a name="l00269"></a><span class="lineno"> 269</span> <span class="comment"> * Computed as</span></div> |
| <div class="line"><a name="l00270"></a><span class="lineno"> 270</span> <span class="comment"> * <tt>(2 * \ref students_t_cdf "students_t_cdf"(-abs(statistic)))</tt>.</span></div> |
| <div class="line"><a name="l00271"></a><span class="lineno"> 271</span> <span class="comment"> *</span></div> |
| <div class="line"><a name="l00272"></a><span class="lineno"> 272</span> <span class="comment"> * @usage</span></div> |
| <div class="line"><a name="l00273"></a><span class="lineno"> 273</span> <span class="comment"> * - Two-sample pooled t-test: Test null hypothesis that the mean of the first</span></div> |
| <div class="line"><a name="l00274"></a><span class="lineno"> 274</span> <span class="comment"> * sample is at most (or equal to, respectively) the mean of the second</span></div> |
| <div class="line"><a name="l00275"></a><span class="lineno"> 275</span> <span class="comment"> * sample:</span></div> |
| <div class="line"><a name="l00276"></a><span class="lineno"> 276</span> <span class="comment"> * <pre>SELECT (t_test_pooled(<em>first</em>, <em>value</em>)).* FROM <em>source</em></pre></span></div> |
| <div class="line"><a name="l00277"></a><span class="lineno"> 277</span> <span class="comment"> */</span></div> |
| <div class="line"><a name="l00278"></a><span class="lineno"> 278</span> CREATE AGGREGATE MADLIB_SCHEMA.t_test_two_pooled(</div> |
| <div class="line"><a name="l00279"></a><span class="lineno"> 279</span>  /*+ "first" */ BOOLEAN,</div> |
| <div class="line"><a name="l00280"></a><span class="lineno"> 280</span>  /*+ "value" */ DOUBLE PRECISION) (</div> |
| <div class="line"><a name="l00281"></a><span class="lineno"> 281</span> </div> |
| <div class="line"><a name="l00282"></a><span class="lineno"> 282</span>  SFUNC=MADLIB_SCHEMA.t_test_two_transition,</div> |
| <div class="line"><a name="l00283"></a><span class="lineno"> 283</span>  STYPE=DOUBLE PRECISION[],</div> |
| <div class="line"><a name="l00284"></a><span class="lineno"> 284</span>  FINALFUNC=MADLIB_SCHEMA.t_test_two_pooled_final,</div> |
| <div class="line"><a name="l00285"></a><span class="lineno"> 285</span>  m4_ifdef(<!__GREENPLUM__!>,<!PREFUNC=MADLIB_SCHEMA.t_test_merge_states,!>)</div> |
| <div class="line"><a name="l00286"></a><span class="lineno"> 286</span>  INITCOND='{0,0,0,0,0,0,0}<span class="stringliteral">'</span></div> |
| <div class="line"><a name="l00287"></a><span class="lineno"> 287</span> <span class="stringliteral">);</span></div> |
| <div class="line"><a name="l00288"></a><span class="lineno"> 288</span> <span class="stringliteral"></span></div> |
| <div class="line"><a name="l00289"></a><span class="lineno"> 289</span> <span class="stringliteral"></span></div> |
| <div class="line"><a name="l00290"></a><span class="lineno"> 290</span> <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.t_test_two_unpooled_final(</span></div> |
| <div class="line"><a name="l00291"></a><span class="lineno"> 291</span> <span class="stringliteral"> state DOUBLE PRECISION[])</span></div> |
| <div class="line"><a name="l00292"></a><span class="lineno"> 292</span> <span class="stringliteral">RETURNS MADLIB_SCHEMA.t_test_result</span></div> |
| <div class="line"><a name="l00293"></a><span class="lineno"> 293</span> <span class="stringliteral">AS '</span>MODULE_PATHNAME<span class="stringliteral">'</span></div> |
| <div class="line"><a name="l00294"></a><span class="lineno"> 294</span> <span class="stringliteral">LANGUAGE C IMMUTABLE STRICT;</span></div> |
| <div class="line"><a name="l00295"></a><span class="lineno"> 295</span> <span class="stringliteral"></span><span class="comment"></span></div> |
| <div class="line"><a name="l00296"></a><span class="lineno"> 296</span> <span class="comment">/**</span></div> |
| <div class="line"><a name="l00297"></a><span class="lineno"> 297</span> <span class="comment"> * @brief Perform unpooled (i.e., unequal variances) t-test (also known as</span></div> |
| <div class="line"><a name="l00298"></a><span class="lineno"> 298</span> <span class="comment"> * Welch's t-test)</span></div> |
| <div class="line"><a name="l00299"></a><span class="lineno"> 299</span> <span class="comment"> *</span></div> |
| <div class="line"><a name="l00300"></a><span class="lineno"><a class="code" href="hypothesis__tests_8sql__in.html#a74e6ef1522197957b5aa35bf67004364"> 300</a></span> <span class="comment"> * Given realizations \f$ x_1, \dots, x_n \f$ and \f$ y_1, \dots, y_m \f$ of</span></div> |
| <div class="line"><a name="l00301"></a><span class="lineno"> 301</span> <span class="comment"> * i.i.d. random variables \f$ X_1, \dots, X_n \sim N(\mu_X, \sigma_X^2) \f$ and</span></div> |
| <div class="line"><a name="l00302"></a><span class="lineno"> 302</span> <span class="comment"> * \f$ Y_1, \dots, Y_m \sim N(\mu_Y, \sigma_Y^2) \f$ with unknown parameters</span></div> |
| <div class="line"><a name="l00303"></a><span class="lineno"> 303</span> <span class="comment"> * \f$ \mu_X, \mu_Y, \sigma_X^2, \f$ and \f$ \sigma_Y^2 \f$, test the null</span></div> |
| <div class="line"><a name="l00304"></a><span class="lineno"> 304</span> <span class="comment"> * hypotheses \f$ H_0 : \mu_X \leq \mu_Y \f$ and \f$ H_0 : \mu_X = \mu_Y \f$.</span></div> |
| <div class="line"><a name="l00305"></a><span class="lineno"> 305</span> <span class="comment"> *</span></div> |
| <div class="line"><a name="l00306"></a><span class="lineno"> 306</span> <span class="comment"> * @param first Indicator whether \c value is from first sample</span></div> |
| <div class="line"><a name="l00307"></a><span class="lineno"> 307</span> <span class="comment"> * \f$ x_1, \dots, x_n \f$ (if \c TRUE) or from second sample</span></div> |
| <div class="line"><a name="l00308"></a><span class="lineno"> 308</span> <span class="comment"> * \f$ y_1, \dots, y_m \f$ (if \c FALSE)</span></div> |
| <div class="line"><a name="l00309"></a><span class="lineno"> 309</span> <span class="comment"> * @param value Value of random variate \f$ x_i \f$ or \f$ y_i \f$</span></div> |
| <div class="line"><a name="l00310"></a><span class="lineno"> 310</span> <span class="comment"> *</span></div> |
| <div class="line"><a name="l00311"></a><span class="lineno"> 311</span> <span class="comment"> * @return A composite value as follows. We denote by \f$ \bar x, \bar y \f$</span></div> |
| <div class="line"><a name="l00312"></a><span class="lineno"> 312</span> <span class="comment"> * the \ref sample_mean "sample means" and by \f$ s_X^2, s_Y^2 \f$ the</span></div> |
| <div class="line"><a name="l00313"></a><span class="lineno"> 313</span> <span class="comment"> * \ref sample_variance "sample variances".</span></div> |
| <div class="line"><a name="l00314"></a><span class="lineno"> 314</span> <span class="comment"> * - <tt>statistic FLOAT8</tt> - Statistic</span></div> |
| <div class="line"><a name="l00315"></a><span class="lineno"> 315</span> <span class="comment"> * \f[</span></div> |
| <div class="line"><a name="l00316"></a><span class="lineno"> 316</span> <span class="comment"> * t = \frac{\bar x - \bar y}{\sqrt{s_X^2/n + s_Y^2/m}}</span></div> |
| <div class="line"><a name="l00317"></a><span class="lineno"> 317</span> <span class="comment"> * \f]</span></div> |
| <div class="line"><a name="l00318"></a><span class="lineno"> 318</span> <span class="comment"> * The corresponding random variable is approximately Student-t distributed</span></div> |
| <div class="line"><a name="l00319"></a><span class="lineno"> 319</span> <span class="comment"> * with</span></div> |
| <div class="line"><a name="l00320"></a><span class="lineno"> 320</span> <span class="comment"> * \f[</span></div> |
| <div class="line"><a name="l00321"></a><span class="lineno"> 321</span> <span class="comment"> * \frac{(s_X^2 / n + s_Y^2 / m)^2}{(s_X^2 / n)^2/(n-1) + (s_Y^2 / m)^2/(m-1)}</span></div> |
| <div class="line"><a name="l00322"></a><span class="lineno"> 322</span> <span class="comment"> * \f]</span></div> |
| <div class="line"><a name="l00323"></a><span class="lineno"> 323</span> <span class="comment"> * degrees of freedom (Welch–Satterthwaite formula).</span></div> |
| <div class="line"><a name="l00324"></a><span class="lineno"> 324</span> <span class="comment"> * - <tt>df FLOAT8</tt> - Degrees of freedom (as above)</span></div> |
| <div class="line"><a name="l00325"></a><span class="lineno"> 325</span> <span class="comment"> * - <tt>p_value_one_sided FLOAT8</tt> - Lower bound on one-sided p-value.</span></div> |
| <div class="line"><a name="l00326"></a><span class="lineno"> 326</span> <span class="comment"> * In detail, the result is \f$ \Pr[\bar X - \bar Y \geq \bar x - \bar y \mid \mu_X = \mu_Y] \f$,</span></div> |
| <div class="line"><a name="l00327"></a><span class="lineno"> 327</span> <span class="comment"> * which is a lower bound on</span></div> |
| <div class="line"><a name="l00328"></a><span class="lineno"> 328</span> <span class="comment"> * \f$ \Pr[\bar X - \bar Y \geq \bar x - \bar y \mid \mu_X \leq \mu_Y] \f$.</span></div> |
| <div class="line"><a name="l00329"></a><span class="lineno"> 329</span> <span class="comment"> * Computed as</span></div> |
| <div class="line"><a name="l00330"></a><span class="lineno"> 330</span> <span class="comment"> * <tt>(1.0 - \ref students_t_cdf "students_t_cdf"(statistic))</tt>.</span></div> |
| <div class="line"><a name="l00331"></a><span class="lineno"> 331</span> <span class="comment"> * - <tt>p_value_two_sided FLOAT8</tt> - Two-sided p-value, i.e.,</span></div> |
| <div class="line"><a name="l00332"></a><span class="lineno"> 332</span> <span class="comment"> * \f$ \Pr[ |\bar X - \bar Y| \geq |\bar x - \bar y| \mid \mu_X = \mu_Y] \f$.</span></div> |
| <div class="line"><a name="l00333"></a><span class="lineno"> 333</span> <span class="comment"> * Computed as</span></div> |
| <div class="line"><a name="l00334"></a><span class="lineno"> 334</span> <span class="comment"> * <tt>(2 * \ref students_t_cdf "students_t_cdf"(-abs(statistic)))</tt>.</span></div> |
| <div class="line"><a name="l00335"></a><span class="lineno"> 335</span> <span class="comment"> *</span></div> |
| <div class="line"><a name="l00336"></a><span class="lineno"> 336</span> <span class="comment"> * @usage</span></div> |
| <div class="line"><a name="l00337"></a><span class="lineno"> 337</span> <span class="comment"> * - Two-sample unpooled t-test: Test null hypothesis that the mean of the</span></div> |
| <div class="line"><a name="l00338"></a><span class="lineno"> 338</span> <span class="comment"> * first sample is at most (or equal to, respectively) the mean of the second</span></div> |
| <div class="line"><a name="l00339"></a><span class="lineno"> 339</span> <span class="comment"> * sample:</span></div> |
| <div class="line"><a name="l00340"></a><span class="lineno"> 340</span> <span class="comment"> * <pre>SELECT (t_test_unpooled(<em>first</em>, <em>value</em>)).* FROM <em>source</em></pre></span></div> |
| <div class="line"><a name="l00341"></a><span class="lineno"> 341</span> <span class="comment"> */</span></div> |
| <div class="line"><a name="l00342"></a><span class="lineno"> 342</span> CREATE AGGREGATE MADLIB_SCHEMA.t_test_two_unpooled(</div> |
| <div class="line"><a name="l00343"></a><span class="lineno"> 343</span>  /*+ "first" */ BOOLEAN,</div> |
| <div class="line"><a name="l00344"></a><span class="lineno"> 344</span>  /*+ "value" */ DOUBLE PRECISION) (</div> |
| <div class="line"><a name="l00345"></a><span class="lineno"> 345</span> </div> |
| <div class="line"><a name="l00346"></a><span class="lineno"> 346</span>  SFUNC=MADLIB_SCHEMA.t_test_two_transition,</div> |
| <div class="line"><a name="l00347"></a><span class="lineno"> 347</span>  STYPE=DOUBLE PRECISION[],</div> |
| <div class="line"><a name="l00348"></a><span class="lineno"> 348</span>  FINALFUNC=MADLIB_SCHEMA.t_test_two_unpooled_final,</div> |
| <div class="line"><a name="l00349"></a><span class="lineno"> 349</span>  m4_ifdef(<!__GREENPLUM__!>,<!PREFUNC=MADLIB_SCHEMA.t_test_merge_states,!>)</div> |
| <div class="line"><a name="l00350"></a><span class="lineno"> 350</span>  INITCOND='{0,0,0,0,0,0,0}<span class="stringliteral">'</span></div> |
| <div class="line"><a name="l00351"></a><span class="lineno"> 351</span> <span class="stringliteral">);</span></div> |
| <div class="line"><a name="l00352"></a><span class="lineno"> 352</span> <span class="stringliteral"></span><span class="comment"></span></div> |
| <div class="line"><a name="l00353"></a><span class="lineno"> 353</span> <span class="comment">/**</span></div> |
| <div class="line"><a name="l00354"></a><span class="lineno"> 354</span> <span class="comment"> * @brief Perform Fisher F-test</span></div> |
| <div class="line"><a name="l00355"></a><span class="lineno"> 355</span> <span class="comment"> *</span></div> |
| <div class="line"><a name="l00356"></a><span class="lineno"> 356</span> <span class="comment"> * Given realizations \f$ x_1, \dots, x_m \f$ and \f$ y_1, \dots, y_n \f$ of</span></div> |
| <div class="line"><a name="l00357"></a><span class="lineno"> 357</span> <span class="comment"> * i.i.d. random variables \f$ X_1, \dots, X_m \sim N(\mu_X, \sigma^2) \f$ and</span></div> |
| <div class="line"><a name="l00358"></a><span class="lineno"> 358</span> <span class="comment"> * \f$ Y_1, \dots, Y_n \sim N(\mu_Y, \sigma^2) \f$ with unknown parameters</span></div> |
| <div class="line"><a name="l00359"></a><span class="lineno"> 359</span> <span class="comment"> * \f$ \mu_X, \mu_Y, \f$ and \f$ \sigma^2 \f$, test the null hypotheses</span></div> |
| <div class="line"><a name="l00360"></a><span class="lineno"> 360</span> <span class="comment"> * \f$ H_0 : \sigma_X < \sigma_Y \f$ and \f$ H_0 : \sigma_X = \sigma_Y \f$.</span></div> |
| <div class="line"><a name="l00361"></a><span class="lineno"> 361</span> <span class="comment"> *</span></div> |
| <div class="line"><a name="l00362"></a><span class="lineno"> 362</span> <span class="comment"> * @param first Indicator whether \c value is from first sample</span></div> |
| <div class="line"><a name="l00363"></a><span class="lineno"> 363</span> <span class="comment"> * \f$ x_1, \dots, x_m \f$ (if \c TRUE) or from second sample</span></div> |
| <div class="line"><a name="l00364"></a><span class="lineno"><a class="code" href="hypothesis__tests_8sql__in.html#aa95e5a0c8b4841c113c84a393b8b4868"> 364</a></span> <span class="comment"> * \f$ y_1, \dots, y_n \f$ (if \c FALSE)</span></div> |
| <div class="line"><a name="l00365"></a><span class="lineno"> 365</span> <span class="comment"> * @param value Value of random variate \f$ x_i \f$ or \f$ y_i \f$</span></div> |
| <div class="line"><a name="l00366"></a><span class="lineno"> 366</span> <span class="comment"> *</span></div> |
| <div class="line"><a name="l00367"></a><span class="lineno"> 367</span> <span class="comment"> * @return A composite value as follows. We denote by \f$ \bar x, \bar y \f$</span></div> |
| <div class="line"><a name="l00368"></a><span class="lineno"> 368</span> <span class="comment"> * the \ref sample_mean "sample means" and by \f$ s_X^2, s_Y^2 \f$ the</span></div> |
| <div class="line"><a name="l00369"></a><span class="lineno"> 369</span> <span class="comment"> * \ref sample_variance "sample variances".</span></div> |
| <div class="line"><a name="l00370"></a><span class="lineno"> 370</span> <span class="comment"> * - <tt>statistic FLOAT8</tt> - Statistic</span></div> |
| <div class="line"><a name="l00371"></a><span class="lineno"> 371</span> <span class="comment"> * \f[</span></div> |
| <div class="line"><a name="l00372"></a><span class="lineno"> 372</span> <span class="comment"> * f = \frac{s_Y^2}{s_X^2}</span></div> |
| <div class="line"><a name="l00373"></a><span class="lineno"> 373</span> <span class="comment"> * \f]</span></div> |
| <div class="line"><a name="l00374"></a><span class="lineno"> 374</span> <span class="comment"> * The corresponding random</span></div> |
| <div class="line"><a name="l00375"></a><span class="lineno"> 375</span> <span class="comment"> * variable is F-distributed with</span></div> |
| <div class="line"><a name="l00376"></a><span class="lineno"> 376</span> <span class="comment"> * \f$ (n - 1) \f$ degrees of freedom in the numerator and</span></div> |
| <div class="line"><a name="l00377"></a><span class="lineno"> 377</span> <span class="comment"> * \f$ (m - 1) \f$ degrees of freedom in the denominator.</span></div> |
| <div class="line"><a name="l00378"></a><span class="lineno"> 378</span> <span class="comment"> * - <tt>df1 BIGINT</tt> - Degrees of freedom in the numerator \f$ (n - 1) \f$</span></div> |
| <div class="line"><a name="l00379"></a><span class="lineno"> 379</span> <span class="comment"> * - <tt>df2 BIGINT</tt> - Degrees of freedom in the denominator \f$ (m - 1) \f$</span></div> |
| <div class="line"><a name="l00380"></a><span class="lineno"> 380</span> <span class="comment"> * - <tt>p_value_one_sided FLOAT8</tt> - Lower bound on one-sided p-value.</span></div> |
| <div class="line"><a name="l00381"></a><span class="lineno"> 381</span> <span class="comment"> * In detail, the result is \f$ \Pr[F \geq f \mid \sigma_X = \sigma_Y] \f$,</span></div> |
| <div class="line"><a name="l00382"></a><span class="lineno"> 382</span> <span class="comment"> * which is a lower bound on</span></div> |
| <div class="line"><a name="l00383"></a><span class="lineno"> 383</span> <span class="comment"> * \f$ \Pr[F \geq f \mid \sigma_X \leq \sigma_Y] \f$. Computed as</span></div> |
| <div class="line"><a name="l00384"></a><span class="lineno"> 384</span> <span class="comment"> * <tt>(1.0 - \ref fisher_f_cdf "fisher_f_cdf"(statistic))</tt>.</span></div> |
| <div class="line"><a name="l00385"></a><span class="lineno"> 385</span> <span class="comment"> * - <tt>p_value_two_sided FLOAT8</tt> - Two-sided p-value, i.e.,</span></div> |
| <div class="line"><a name="l00386"></a><span class="lineno"> 386</span> <span class="comment"> * \f$ 2 \cdot \min \{ p, 1 - p \} \f$ where</span></div> |
| <div class="line"><a name="l00387"></a><span class="lineno"> 387</span> <span class="comment"> * \f$ p = \Pr[ F \geq f \mid \sigma_X = \sigma_Y] \f$. Computed as</span></div> |
| <div class="line"><a name="l00388"></a><span class="lineno"> 388</span> <span class="comment"> * <tt>(min(p_value_one_sided, 1. - p_value_one_sided))</tt>.</span></div> |
| <div class="line"><a name="l00389"></a><span class="lineno"> 389</span> <span class="comment"> *</span></div> |
| <div class="line"><a name="l00390"></a><span class="lineno"> 390</span> <span class="comment"> * @usage</span></div> |
| <div class="line"><a name="l00391"></a><span class="lineno"> 391</span> <span class="comment"> * - Test null hypothesis that the variance of the first sample is at most (or</span></div> |
| <div class="line"><a name="l00392"></a><span class="lineno"> 392</span> <span class="comment"> * equal to, respectively) the variance of the second sample:</span></div> |
| <div class="line"><a name="l00393"></a><span class="lineno"> 393</span> <span class="comment"> * <pre>SELECT (f_test(<em>first</em>, <em>value</em>)).* FROM <em>source</em></pre></span></div> |
| <div class="line"><a name="l00394"></a><span class="lineno"> 394</span> <span class="comment"> *</span></div> |
| <div class="line"><a name="l00395"></a><span class="lineno"> 395</span> <span class="comment"> * @internal We reuse the two-sample t-test transition and merge functions.</span></div> |
| <div class="line"><a name="l00396"></a><span class="lineno"> 396</span> <span class="comment"> */</span></div> |
| <div class="line"><a name="l00397"></a><span class="lineno"> 397</span> CREATE AGGREGATE MADLIB_SCHEMA.f_test(</div> |
| <div class="line"><a name="l00398"></a><span class="lineno"> 398</span>  /*+ "first" */ BOOLEAN,</div> |
| <div class="line"><a name="l00399"></a><span class="lineno"> 399</span>  /*+ "value" */ DOUBLE PRECISION) (</div> |
| <div class="line"><a name="l00400"></a><span class="lineno"> 400</span> </div> |
| <div class="line"><a name="l00401"></a><span class="lineno"> 401</span>  SFUNC=MADLIB_SCHEMA.t_test_two_transition,</div> |
| <div class="line"><a name="l00402"></a><span class="lineno"> 402</span>  STYPE=DOUBLE PRECISION[],</div> |
| <div class="line"><a name="l00403"></a><span class="lineno"> 403</span>  FINALFUNC=MADLIB_SCHEMA.f_test_final,</div> |
| <div class="line"><a name="l00404"></a><span class="lineno"> 404</span>  m4_ifdef(<!__GREENPLUM__!>,<!PREFUNC=MADLIB_SCHEMA.t_test_merge_states,!>)</div> |
| <div class="line"><a name="l00405"></a><span class="lineno"> 405</span>  INITCOND='{0,0,0,0,0,0,0}<span class="stringliteral">'</span></div> |
| <div class="line"><a name="l00406"></a><span class="lineno"> 406</span> <span class="stringliteral">);</span></div> |
| <div class="line"><a name="l00407"></a><span class="lineno"> 407</span> <span class="stringliteral"></span></div> |
| <div class="line"><a name="l00408"></a><span class="lineno"> 408</span> <span class="stringliteral"></span></div> |
| <div class="line"><a name="l00409"></a><span class="lineno"> 409</span> <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.chi2_gof_test_transition(</span></div> |
| <div class="line"><a name="l00410"></a><span class="lineno"> 410</span> <span class="stringliteral"> state DOUBLE PRECISION[],</span></div> |
| <div class="line"><a name="l00411"></a><span class="lineno"> 411</span> <span class="stringliteral"> observed BIGINT,</span></div> |
| <div class="line"><a name="l00412"></a><span class="lineno"> 412</span> <span class="stringliteral"> expected DOUBLE PRECISION,</span></div> |
| <div class="line"><a name="l00413"></a><span class="lineno"> 413</span> <span class="stringliteral"> df BIGINT</span></div> |
| <div class="line"><a name="l00414"></a><span class="lineno"> 414</span> <span class="stringliteral">) RETURNS DOUBLE PRECISION[]</span></div> |
| <div class="line"><a name="l00415"></a><span class="lineno"> 415</span> <span class="stringliteral">AS '</span>MODULE_PATHNAME<span class="stringliteral">'</span></div> |
| <div class="line"><a name="l00416"></a><span class="lineno"> 416</span> <span class="stringliteral">LANGUAGE C</span></div> |
| <div class="line"><a name="l00417"></a><span class="lineno"> 417</span> <span class="stringliteral">IMMUTABLE</span></div> |
| <div class="line"><a name="l00418"></a><span class="lineno"> 418</span> <span class="stringliteral">STRICT;</span></div> |
| <div class="line"><a name="l00419"></a><span class="lineno"><a class="code" href="hypothesis__tests_8sql__in.html#a8f90d2f805a6ab3034f80a5967dffa1d"> 419</a></span> <span class="stringliteral"></span></div> |
| <div class="line"><a name="l00420"></a><span class="lineno"> 420</span> <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.chi2_gof_test_transition(</span></div> |
| <div class="line"><a name="l00421"></a><span class="lineno"> 421</span> <span class="stringliteral"> state DOUBLE PRECISION[],</span></div> |
| <div class="line"><a name="l00422"></a><span class="lineno"> 422</span> <span class="stringliteral"> observed BIGINT,</span></div> |
| <div class="line"><a name="l00423"></a><span class="lineno"> 423</span> <span class="stringliteral"> expected DOUBLE PRECISION</span></div> |
| <div class="line"><a name="l00424"></a><span class="lineno"> 424</span> <span class="stringliteral">) RETURNS DOUBLE PRECISION[]</span></div> |
| <div class="line"><a name="l00425"></a><span class="lineno"> 425</span> <span class="stringliteral">AS '</span>MODULE_PATHNAME<span class="stringliteral">'</span></div> |
| <div class="line"><a name="l00426"></a><span class="lineno"> 426</span> <span class="stringliteral">LANGUAGE C</span></div> |
| <div class="line"><a name="l00427"></a><span class="lineno"> 427</span> <span class="stringliteral">IMMUTABLE</span></div> |
| <div class="line"><a name="l00428"></a><span class="lineno"> 428</span> <span class="stringliteral">STRICT;</span></div> |
| <div class="line"><a name="l00429"></a><span class="lineno"> 429</span> <span class="stringliteral"></span></div> |
| <div class="line"><a name="l00430"></a><span class="lineno"> 430</span> <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.chi2_gof_test_transition(</span></div> |
| <div class="line"><a name="l00431"></a><span class="lineno"> 431</span> <span class="stringliteral"> state DOUBLE PRECISION[],</span></div> |
| <div class="line"><a name="l00432"></a><span class="lineno"> 432</span> <span class="stringliteral"> observed BIGINT</span></div> |
| <div class="line"><a name="l00433"></a><span class="lineno"> 433</span> <span class="stringliteral">) RETURNS DOUBLE PRECISION[]</span></div> |
| <div class="line"><a name="l00434"></a><span class="lineno"> 434</span> <span class="stringliteral">AS '</span>MODULE_PATHNAME<span class="stringliteral">'</span></div> |
| <div class="line"><a name="l00435"></a><span class="lineno"> 435</span> <span class="stringliteral">LANGUAGE C</span></div> |
| <div class="line"><a name="l00436"></a><span class="lineno"> 436</span> <span class="stringliteral">IMMUTABLE</span></div> |
| <div class="line"><a name="l00437"></a><span class="lineno"> 437</span> <span class="stringliteral">STRICT;</span></div> |
| <div class="line"><a name="l00438"></a><span class="lineno"> 438</span> <span class="stringliteral"></span></div> |
| <div class="line"><a name="l00439"></a><span class="lineno"> 439</span> <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.chi2_gof_test_merge_states(</span></div> |
| <div class="line"><a name="l00440"></a><span class="lineno"> 440</span> <span class="stringliteral"> state1 DOUBLE PRECISION[],</span></div> |
| <div class="line"><a name="l00441"></a><span class="lineno"> 441</span> <span class="stringliteral"> state2 DOUBLE PRECISION[])</span></div> |
| <div class="line"><a name="l00442"></a><span class="lineno"> 442</span> <span class="stringliteral">RETURNS DOUBLE PRECISION[]</span></div> |
| <div class="line"><a name="l00443"></a><span class="lineno"> 443</span> <span class="stringliteral">AS '</span>MODULE_PATHNAME<span class="stringliteral">'</span></div> |
| <div class="line"><a name="l00444"></a><span class="lineno"> 444</span> <span class="stringliteral">LANGUAGE C</span></div> |
| <div class="line"><a name="l00445"></a><span class="lineno"> 445</span> <span class="stringliteral">IMMUTABLE</span></div> |
| <div class="line"><a name="l00446"></a><span class="lineno"> 446</span> <span class="stringliteral">STRICT;</span></div> |
| <div class="line"><a name="l00447"></a><span class="lineno"> 447</span> <span class="stringliteral"></span></div> |
| <div class="line"><a name="l00448"></a><span class="lineno"> 448</span> <span class="stringliteral">CREATE TYPE MADLIB_SCHEMA.chi2_test_result AS (</span></div> |
| <div class="line"><a name="l00449"></a><span class="lineno"> 449</span> <span class="stringliteral"> statistic DOUBLE PRECISION,</span></div> |
| <div class="line"><a name="l00450"></a><span class="lineno"> 450</span> <span class="stringliteral"> p_value DOUBLE PRECISION,</span></div> |
| <div class="line"><a name="l00451"></a><span class="lineno"> 451</span> <span class="stringliteral"> df BIGINT,</span></div> |
| <div class="line"><a name="l00452"></a><span class="lineno"> 452</span> <span class="stringliteral"> phi DOUBLE PRECISION,</span></div> |
| <div class="line"><a name="l00453"></a><span class="lineno"> 453</span> <span class="stringliteral"> contingency_coef DOUBLE PRECISION</span></div> |
| <div class="line"><a name="l00454"></a><span class="lineno"> 454</span> <span class="stringliteral">);</span></div> |
| <div class="line"><a name="l00455"></a><span class="lineno"> 455</span> <span class="stringliteral"></span></div> |
| <div class="line"><a name="l00456"></a><span class="lineno"> 456</span> <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.chi2_gof_test_final(</span></div> |
| <div class="line"><a name="l00457"></a><span class="lineno"> 457</span> <span class="stringliteral"> state DOUBLE PRECISION[]</span></div> |
| <div class="line"><a name="l00458"></a><span class="lineno"> 458</span> <span class="stringliteral">) RETURNS MADLIB_SCHEMA.chi2_test_result</span></div> |
| <div class="line"><a name="l00459"></a><span class="lineno"> 459</span> <span class="stringliteral">AS '</span>MODULE_PATHNAME<span class="stringliteral">'</span></div> |
| <div class="line"><a name="l00460"></a><span class="lineno"> 460</span> <span class="stringliteral">LANGUAGE C</span></div> |
| <div class="line"><a name="l00461"></a><span class="lineno"> 461</span> <span class="stringliteral">IMMUTABLE</span></div> |
| <div class="line"><a name="l00462"></a><span class="lineno"> 462</span> <span class="stringliteral">STRICT;</span></div> |
| <div class="line"><a name="l00463"></a><span class="lineno"> 463</span> <span class="stringliteral"></span><span class="comment"></span></div> |
| <div class="line"><a name="l00464"></a><span class="lineno"> 464</span> <span class="comment">/**</span></div> |
| <div class="line"><a name="l00465"></a><span class="lineno"> 465</span> <span class="comment"> * @brief Perform Pearson's chi-squared goodness-of-fit test</span></div> |
| <div class="line"><a name="l00466"></a><span class="lineno"> 466</span> <span class="comment"> *</span></div> |
| <div class="line"><a name="l00467"></a><span class="lineno"> 467</span> <span class="comment"> * Let \f$ n_1, \dots, n_k \f$ be a realization of a (vector) random variable</span></div> |
| <div class="line"><a name="l00468"></a><span class="lineno"> 468</span> <span class="comment"> * \f$ N = (N_1, \dots, N_k) \f$ that follows the multinomial distribution with</span></div> |
| <div class="line"><a name="l00469"></a><span class="lineno"> 469</span> <span class="comment"> * parameters \f$ k \f$ and \f$ p = (p_1, \dots, p_k) \f$. Test the null</span></div> |
| <div class="line"><a name="l00470"></a><span class="lineno"> 470</span> <span class="comment"> * hypothesis \f$ H_0 : p = p^0 \f$.</span></div> |
| <div class="line"><a name="l00471"></a><span class="lineno"> 471</span> <span class="comment"> *</span></div> |
| <div class="line"><a name="l00472"></a><span class="lineno"> 472</span> <span class="comment"> * @param observed Number \f$ n_i \f$ of observations of the current event/row</span></div> |
| <div class="line"><a name="l00473"></a><span class="lineno"> 473</span> <span class="comment"> * @param expected Expected number of observations of current event/row. This</span></div> |
| <div class="line"><a name="l00474"></a><span class="lineno"> 474</span> <span class="comment"> * number is not required to be normalized. That is, \f$ p^0_i \f$ will be</span></div> |
| <div class="line"><a name="l00475"></a><span class="lineno"> 475</span> <span class="comment"> * taken as \c expected divided by <tt>sum(expected)</tt>. Hence, if this</span></div> |
| <div class="line"><a name="l00476"></a><span class="lineno"> 476</span> <span class="comment"> * parameter is not specified, chi2_test() will by default use</span></div> |
| <div class="line"><a name="l00477"></a><span class="lineno"> 477</span> <span class="comment"> * \f$ p^0 = (\frac 1k, \dots, \frac 1k) \f$, i.e., test that \f$ p \f$ is a</span></div> |
| <div class="line"><a name="l00478"></a><span class="lineno"> 478</span> <span class="comment"> * discrete uniform distribution.</span></div> |
| <div class="line"><a name="l00479"></a><span class="lineno"> 479</span> <span class="comment"> * @param df Degrees of freedom. This is the number of events reduced by the</span></div> |
| <div class="line"><a name="l00480"></a><span class="lineno"> 480</span> <span class="comment"> * degree of freedom lost by using the observed numbers for defining the</span></div> |
| <div class="line"><a name="l00481"></a><span class="lineno"> 481</span> <span class="comment"> * expected number of observations. If this parameter is 0, the degree</span></div> |
| <div class="line"><a name="l00482"></a><span class="lineno"> 482</span> <span class="comment"> * of freedom is taken as \f$ (k - 1) \f$.</span></div> |
| <div class="line"><a name="l00483"></a><span class="lineno"> 483</span> <span class="comment"> *</span></div> |
| <div class="line"><a name="l00484"></a><span class="lineno"> 484</span> <span class="comment"> * @return A composite value as follows. Let \f$ n = \sum_{i=1}^n n_i \f$.</span></div> |
| <div class="line"><a name="l00485"></a><span class="lineno"> 485</span> <span class="comment"> * - <tt>statistic FLOAT8</tt> - Statistic</span></div> |
| <div class="line"><a name="l00486"></a><span class="lineno"> 486</span> <span class="comment"> * \f[</span></div> |
| <div class="line"><a name="l00487"></a><span class="lineno"> 487</span> <span class="comment"> * \chi^2 = \sum_{i=1}^k \frac{(n_i - np_i)^2}{np_i}</span></div> |
| <div class="line"><a name="l00488"></a><span class="lineno"> 488</span> <span class="comment"> * \f]</span></div> |
| <div class="line"><a name="l00489"></a><span class="lineno"> 489</span> <span class="comment"> * The corresponding random</span></div> |
| <div class="line"><a name="l00490"></a><span class="lineno"> 490</span> <span class="comment"> * variable is approximately chi-squared distributed with</span></div> |
| <div class="line"><a name="l00491"></a><span class="lineno"> 491</span> <span class="comment"> * \c df degrees of freedom.</span></div> |
| <div class="line"><a name="l00492"></a><span class="lineno"> 492</span> <span class="comment"> * - <tt>df BIGINT</tt> - Degrees of freedom</span></div> |
| <div class="line"><a name="l00493"></a><span class="lineno"> 493</span> <span class="comment"> * - <tt>p_value FLOAT8</tt> - Approximate p-value, i.e.,</span></div> |
| <div class="line"><a name="l00494"></a><span class="lineno"> 494</span> <span class="comment"> * \f$ \Pr[X^2 \geq \chi^2 \mid p = p^0] \f$. Computed as</span></div> |
| <div class="line"><a name="l00495"></a><span class="lineno"> 495</span> <span class="comment"> * <tt>(1.0 - \ref chi_squared_cdf "chi_squared_cdf"(statistic))</tt>.</span></div> |
| <div class="line"><a name="l00496"></a><span class="lineno"> 496</span> <span class="comment"> * - <tt>phi FLOAT8</tt> - Phi coefficient, i.e.,</span></div> |
| <div class="line"><a name="l00497"></a><span class="lineno"> 497</span> <span class="comment"> * \f$ \phi = \sqrt{\frac{\chi^2}{n}} \f$</span></div> |
| <div class="line"><a name="l00498"></a><span class="lineno"> 498</span> <span class="comment"> * - <tt>contingency_coef FLOAT8</tt> - Contingency coefficient, i.e.,</span></div> |
| <div class="line"><a name="l00499"></a><span class="lineno"> 499</span> <span class="comment"> * \f$ \sqrt{\frac{\chi^2}{n + \chi^2}} \f$</span></div> |
| <div class="line"><a name="l00500"></a><span class="lineno"> 500</span> <span class="comment"> *</span></div> |
| <div class="line"><a name="l00501"></a><span class="lineno"> 501</span> <span class="comment"> * @usage</span></div> |
| <div class="line"><a name="l00502"></a><span class="lineno"> 502</span> <span class="comment"> * - Test null hypothesis that all possible outcomes of a categorical variable</span></div> |
| <div class="line"><a name="l00503"></a><span class="lineno"> 503</span> <span class="comment"> * are equally likely:</span></div> |
| <div class="line"><a name="l00504"></a><span class="lineno"> 504</span> <span class="comment"> * <pre>SELECT (chi2_gof_test(<em>observed</em>, 1, NULL)).* FROM <em>source</em></pre></span></div> |
| <div class="line"><a name="l00505"></a><span class="lineno"> 505</span> <span class="comment"> * - Test null hypothesis that two categorical variables are independent.</span></div> |
| <div class="line"><a name="l00506"></a><span class="lineno"> 506</span> <span class="comment"> * Such data is often shown in a <em>contingency table</em> (also known as</span></div> |
| <div class="line"><a name="l00507"></a><span class="lineno"> 507</span> <span class="comment"> * \em crosstab). A crosstab is a matrix where possible values for the first</span></div> |
| <div class="line"><a name="l00508"></a><span class="lineno"> 508</span> <span class="comment"> * variable correspond to rows and values for the second variable to</span></div> |
| <div class="line"><a name="l00509"></a><span class="lineno"> 509</span> <span class="comment"> * columns. The matrix elements are the observation frequencies of the</span></div> |
| <div class="line"><a name="l00510"></a><span class="lineno"> 510</span> <span class="comment"> * joint occurrence of the respective values.</span></div> |
| <div class="line"><a name="l00511"></a><span class="lineno"> 511</span> <span class="comment"> * chi2_gof_test() assumes that the crosstab is stored in normalized form,</span></div> |
| <div class="line"><a name="l00512"></a><span class="lineno"> 512</span> <span class="comment"> * i.e., there are three columns <tt><em>var1</em></tt>,</span></div> |
| <div class="line"><a name="l00513"></a><span class="lineno"> 513</span> <span class="comment"> * <tt><em>var2</em></tt>, <tt><em>observed</em></tt>.</span></div> |
| <div class="line"><a name="l00514"></a><span class="lineno"> 514</span> <span class="comment"> * <pre>SELECT (chi2_gof_test(<em>observed</em>, expected, deg_freedom)).*</span></div> |
| <div class="line"><a name="l00515"></a><span class="lineno"> 515</span> <span class="comment"> *FROM (</span></div> |
| <div class="line"><a name="l00516"></a><span class="lineno"> 516</span> <span class="comment"> * SELECT</span></div> |
| <div class="line"><a name="l00517"></a><span class="lineno"> 517</span> <span class="comment"> * <em>observed</em>,</span></div> |
| <div class="line"><a name="l00518"></a><span class="lineno"> 518</span> <span class="comment"> * sum(<em>observed</em>) OVER (PARTITION BY var1)::DOUBLE PRECISION</span></div> |
| <div class="line"><a name="l00519"></a><span class="lineno"> 519</span> <span class="comment"> * * sum(<em>observed</em>) OVER (PARTITION BY var2) AS expected</span></div> |
| <div class="line"><a name="l00520"></a><span class="lineno"> 520</span> <span class="comment"> * FROM <em>source</em></span></div> |
| <div class="line"><a name="l00521"></a><span class="lineno"> 521</span> <span class="comment"> *) p, (</span></div> |
| <div class="line"><a name="l00522"></a><span class="lineno"> 522</span> <span class="comment"> * SELECT</span></div> |
| <div class="line"><a name="l00523"></a><span class="lineno"> 523</span> <span class="comment"> * (count(DISTINCT <em>var1</em>) - 1) * (count(DISTINCT <em>var2</em>) - 1) AS deg_freedom</span></div> |
| <div class="line"><a name="l00524"></a><span class="lineno"> 524</span> <span class="comment"> * FROM <em>source</em></span></div> |
| <div class="line"><a name="l00525"></a><span class="lineno"> 525</span> <span class="comment"> *) q;</pre></span></div> |
| <div class="line"><a name="l00526"></a><span class="lineno"> 526</span> <span class="comment"> */</span></div> |
| <div class="line"><a name="l00527"></a><span class="lineno"> 527</span> CREATE AGGREGATE MADLIB_SCHEMA.chi2_gof_test(</div> |
| <div class="line"><a name="l00528"></a><span class="lineno"> 528</span>  /*+ observed */ BIGINT,</div> |
| <div class="line"><a name="l00529"></a><span class="lineno"> 529</span>  /*+ expected */ DOUBLE PRECISION /*+ DEFAULT 1 */,</div> |
| <div class="line"><a name="l00530"></a><span class="lineno"> 530</span>  /*+ df */ BIGINT /*+ DEFAULT 0 */</div> |
| <div class="line"><a name="l00531"></a><span class="lineno"> 531</span> ) (</div> |
| <div class="line"><a name="l00532"></a><span class="lineno"> 532</span>  SFUNC=MADLIB_SCHEMA.chi2_gof_test_transition,</div> |
| <div class="line"><a name="l00533"></a><span class="lineno"> 533</span>  STYPE=DOUBLE PRECISION[],</div> |
| <div class="line"><a name="l00534"></a><span class="lineno"> 534</span>  FINALFUNC=MADLIB_SCHEMA.chi2_gof_test_final,</div> |
| <div class="line"><a name="l00535"></a><span class="lineno"> 535</span>  m4_ifdef(<!__GREENPLUM__!>,<!PREFUNC=MADLIB_SCHEMA.chi2_gof_test_merge_states,!>)</div> |
| <div class="line"><a name="l00536"></a><span class="lineno"> 536</span>  INITCOND='{0,0,0,0,0,0}<span class="stringliteral">'</span></div> |
| <div class="line"><a name="l00537"></a><span class="lineno"> 537</span> <span class="stringliteral">);</span></div> |
| <div class="line"><a name="l00538"></a><span class="lineno"> 538</span> <span class="stringliteral"></span></div> |
| <div class="line"><a name="l00539"></a><span class="lineno"> 539</span> <span class="stringliteral">CREATE AGGREGATE MADLIB_SCHEMA.chi2_gof_test(</span></div> |
| <div class="line"><a name="l00540"></a><span class="lineno"> 540</span> <span class="stringliteral"> /*+ observed */ BIGINT,</span></div> |
| <div class="line"><a name="l00541"></a><span class="lineno"> 541</span> <span class="stringliteral"> /*+ expected */ DOUBLE PRECISION</span></div> |
| <div class="line"><a name="l00542"></a><span class="lineno"> 542</span> <span class="stringliteral">) (</span></div> |
| <div class="line"><a name="l00543"></a><span class="lineno"> 543</span> <span class="stringliteral"> SFUNC=MADLIB_SCHEMA.chi2_gof_test_transition,</span></div> |
| <div class="line"><a name="l00544"></a><span class="lineno"> 544</span> <span class="stringliteral"> STYPE=DOUBLE PRECISION[],</span></div> |
| <div class="line"><a name="l00545"></a><span class="lineno"> 545</span> <span class="stringliteral"> FINALFUNC=MADLIB_SCHEMA.chi2_gof_test_final,</span></div> |
| <div class="line"><a name="l00546"></a><span class="lineno"> 546</span> <span class="stringliteral"> m4_ifdef(<!__GREENPLUM__!>,<!PREFUNC=MADLIB_SCHEMA.chi2_gof_test_merge_states,!>)</span></div> |
| <div class="line"><a name="l00547"></a><span class="lineno"> 547</span> <span class="stringliteral"> INITCOND='</span>{0,0,0,0,0,0,0}<span class="stringliteral">'</span></div> |
| <div class="line"><a name="l00548"></a><span class="lineno"> 548</span> <span class="stringliteral">);</span></div> |
| <div class="line"><a name="l00549"></a><span class="lineno"><a class="code" href="hypothesis__tests_8sql__in.html#afc6a7ac3eada83df681bc6efeddfd9eb"> 549</a></span> <span class="stringliteral"></span></div> |
| <div class="line"><a name="l00550"></a><span class="lineno"> 550</span> <span class="stringliteral">CREATE AGGREGATE MADLIB_SCHEMA.chi2_gof_test(</span></div> |
| <div class="line"><a name="l00551"></a><span class="lineno"> 551</span> <span class="stringliteral"> /*+ observed */ BIGINT</span></div> |
| <div class="line"><a name="l00552"></a><span class="lineno"> 552</span> <span class="stringliteral">) (</span></div> |
| <div class="line"><a name="l00553"></a><span class="lineno"> 553</span> <span class="stringliteral"> SFUNC=MADLIB_SCHEMA.chi2_gof_test_transition,</span></div> |
| <div class="line"><a name="l00554"></a><span class="lineno"> 554</span> <span class="stringliteral"> STYPE=DOUBLE PRECISION[],</span></div> |
| <div class="line"><a name="l00555"></a><span class="lineno"> 555</span> <span class="stringliteral"> FINALFUNC=MADLIB_SCHEMA.chi2_gof_test_final,</span></div> |
| <div class="line"><a name="l00556"></a><span class="lineno"> 556</span> <span class="stringliteral"> m4_ifdef(<!__GREENPLUM__!>,<!PREFUNC=MADLIB_SCHEMA.chi2_gof_test_merge_states,!>)</span></div> |
| <div class="line"><a name="l00557"></a><span class="lineno"> 557</span> <span class="stringliteral"> INITCOND='</span>{0,0,0,0,0,0,0}<span class="stringliteral">'</span></div> |
| <div class="line"><a name="l00558"></a><span class="lineno"> 558</span> <span class="stringliteral">);</span></div> |
| <div class="line"><a name="l00559"></a><span class="lineno"> 559</span> <span class="stringliteral"></span></div> |
| <div class="line"><a name="l00560"></a><span class="lineno"> 560</span> <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.ks_test_transition(</span></div> |
| <div class="line"><a name="l00561"></a><span class="lineno"> 561</span> <span class="stringliteral"> state DOUBLE PRECISION[],</span></div> |
| <div class="line"><a name="l00562"></a><span class="lineno"> 562</span> <span class="stringliteral"> "first" BOOLEAN,</span></div> |
| <div class="line"><a name="l00563"></a><span class="lineno"> 563</span> <span class="stringliteral"> "value" DOUBLE PRECISION,</span></div> |
| <div class="line"><a name="l00564"></a><span class="lineno"> 564</span> <span class="stringliteral"> "numFirst" BIGINT,</span></div> |
| <div class="line"><a name="l00565"></a><span class="lineno"> 565</span> <span class="stringliteral"> "numSecond" BIGINT</span></div> |
| <div class="line"><a name="l00566"></a><span class="lineno"> 566</span> <span class="stringliteral">) RETURNS DOUBLE PRECISION[]</span></div> |
| <div class="line"><a name="l00567"></a><span class="lineno"> 567</span> <span class="stringliteral">AS '</span>MODULE_PATHNAME<span class="stringliteral">'</span></div> |
| <div class="line"><a name="l00568"></a><span class="lineno"> 568</span> <span class="stringliteral">LANGUAGE C</span></div> |
| <div class="line"><a name="l00569"></a><span class="lineno"> 569</span> <span class="stringliteral">IMMUTABLE</span></div> |
| <div class="line"><a name="l00570"></a><span class="lineno"> 570</span> <span class="stringliteral">STRICT;</span></div> |
| <div class="line"><a name="l00571"></a><span class="lineno"> 571</span> <span class="stringliteral"></span></div> |
| <div class="line"><a name="l00572"></a><span class="lineno"> 572</span> <span class="stringliteral">CREATE TYPE MADLIB_SCHEMA.ks_test_result AS (</span></div> |
| <div class="line"><a name="l00573"></a><span class="lineno"> 573</span> <span class="stringliteral"> statistic DOUBLE PRECISION,</span></div> |
| <div class="line"><a name="l00574"></a><span class="lineno"> 574</span> <span class="stringliteral"> k_statistic DOUBLE PRECISION,</span></div> |
| <div class="line"><a name="l00575"></a><span class="lineno"> 575</span> <span class="stringliteral"> p_value DOUBLE PRECISION</span></div> |
| <div class="line"><a name="l00576"></a><span class="lineno"> 576</span> <span class="stringliteral">);</span></div> |
| <div class="line"><a name="l00577"></a><span class="lineno"> 577</span> <span class="stringliteral"></span></div> |
| <div class="line"><a name="l00578"></a><span class="lineno"> 578</span> <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.ks_test_final(</span></div> |
| <div class="line"><a name="l00579"></a><span class="lineno"> 579</span> <span class="stringliteral"> state DOUBLE PRECISION[])</span></div> |
| <div class="line"><a name="l00580"></a><span class="lineno"> 580</span> <span class="stringliteral">RETURNS MADLIB_SCHEMA.ks_test_result</span></div> |
| <div class="line"><a name="l00581"></a><span class="lineno"> 581</span> <span class="stringliteral">AS '</span>MODULE_PATHNAME<span class="stringliteral">'</span></div> |
| <div class="line"><a name="l00582"></a><span class="lineno"> 582</span> <span class="stringliteral">LANGUAGE C IMMUTABLE STRICT;</span></div> |
| <div class="line"><a name="l00583"></a><span class="lineno"> 583</span> <span class="stringliteral"></span><span class="comment"></span></div> |
| <div class="line"><a name="l00584"></a><span class="lineno"> 584</span> <span class="comment">/**</span></div> |
| <div class="line"><a name="l00585"></a><span class="lineno"> 585</span> <span class="comment"> * @brief Perform Kolmogorov-Smirnov test</span></div> |
| <div class="line"><a name="l00586"></a><span class="lineno"> 586</span> <span class="comment"> *</span></div> |
| <div class="line"><a name="l00587"></a><span class="lineno"> 587</span> <span class="comment"> * Given realizations \f$ x_1, \dots, x_m \f$ and \f$ y_1, \dots, y_m \f$ of</span></div> |
| <div class="line"><a name="l00588"></a><span class="lineno"> 588</span> <span class="comment"> * i.i.d. random variables \f$ X_1, \dots, X_m \f$ and i.i.d.</span></div> |
| <div class="line"><a name="l00589"></a><span class="lineno"> 589</span> <span class="comment"> * \f$ Y_1, \dots, Y_n \f$, respectively, test the null hypothesis that the</span></div> |
| <div class="line"><a name="l00590"></a><span class="lineno"> 590</span> <span class="comment"> * underlying distributions function \f$ F_X, F_Y \f$ are identical, i.e.,</span></div> |
| <div class="line"><a name="l00591"></a><span class="lineno"> 591</span> <span class="comment"> * \f$ H_0 : F_X = F_Y \f$.</span></div> |
| <div class="line"><a name="l00592"></a><span class="lineno"> 592</span> <span class="comment"> *</span></div> |
| <div class="line"><a name="l00593"></a><span class="lineno"> 593</span> <span class="comment"> * @param first Determines whether the value belongs to the first</span></div> |
| <div class="line"><a name="l00594"></a><span class="lineno"> 594</span> <span class="comment"> * (if \c TRUE) or the second sample (if \c FALSE)</span></div> |
| <div class="line"><a name="l00595"></a><span class="lineno"> 595</span> <span class="comment"> * @param value Value of random variate \f$ x_i \f$ or \f$ y_i \f$</span></div> |
| <div class="line"><a name="l00596"></a><span class="lineno"> 596</span> <span class="comment"> * @param m Size \f$ m \f$ of the first sample. See usage instructions below.</span></div> |
| <div class="line"><a name="l00597"></a><span class="lineno"> 597</span> <span class="comment"> * @param n Size of the second sample. See usage instructions below.</span></div> |
| <div class="line"><a name="l00598"></a><span class="lineno"> 598</span> <span class="comment"> *</span></div> |
| <div class="line"><a name="l00599"></a><span class="lineno"> 599</span> <span class="comment"> * @return A composite value.</span></div> |
| <div class="line"><a name="l00600"></a><span class="lineno"> 600</span> <span class="comment"> * - <tt>statistic FLOAT8</tt> - Kolmogorov–Smirnov statistic</span></div> |
| <div class="line"><a name="l00601"></a><span class="lineno"> 601</span> <span class="comment"> * \f[</span></div> |
| <div class="line"><a name="l00602"></a><span class="lineno"> 602</span> <span class="comment"> * d = \max_{t \in \mathbb R} |F_x(t) - F_y(t)|</span></div> |
| <div class="line"><a name="l00603"></a><span class="lineno"> 603</span> <span class="comment"> * \f]</span></div> |
| <div class="line"><a name="l00604"></a><span class="lineno"> 604</span> <span class="comment"> * where \f$ F_x(t) := \frac 1m |\{ i \mid x_i \leq t \}| \f$ and</span></div> |
| <div class="line"><a name="l00605"></a><span class="lineno"> 605</span> <span class="comment"> * \f$ F_y \f$ (defined likewise) are the empirical distribution functions.</span></div> |
| <div class="line"><a name="l00606"></a><span class="lineno"> 606</span> <span class="comment"> * - <tt>k_statistic FLOAT8</tt> - Kolmogorov statistic</span></div> |
| <div class="line"><a name="l00607"></a><span class="lineno"> 607</span> <span class="comment"> * \f$</span></div> |
| <div class="line"><a name="l00608"></a><span class="lineno"> 608</span> <span class="comment"> * k = r + 0.12 + \frac{0.11}{r}</span></div> |
| <div class="line"><a name="l00609"></a><span class="lineno"> 609</span> <span class="comment"> * \f$</span></div> |
| <div class="line"><a name="l00610"></a><span class="lineno"> 610</span> <span class="comment"> * where</span></div> |
| <div class="line"><a name="l00611"></a><span class="lineno"> 611</span> <span class="comment"> * \f$</span></div> |
| <div class="line"><a name="l00612"></a><span class="lineno"> 612</span> <span class="comment"> * r = \sqrt{\frac{m n}{m+n}}.</span></div> |
| <div class="line"><a name="l00613"></a><span class="lineno"> 613</span> <span class="comment"> * \f$</span></div> |
| <div class="line"><a name="l00614"></a><span class="lineno"> 614</span> <span class="comment"> * Then \f$ k \f$ is approximately Kolmogorov distributed.</span></div> |
| <div class="line"><a name="l00615"></a><span class="lineno"> 615</span> <span class="comment"> * - <tt>p_value FLOAT8</tt> - Approximate p-value, i.e., an approximate value</span></div> |
| <div class="line"><a name="l00616"></a><span class="lineno"> 616</span> <span class="comment"> * for \f$ \Pr[D \geq d \mid F_X = F_Y] \f$. Computed as</span></div> |
| <div class="line"><a name="l00617"></a><span class="lineno"> 617</span> <span class="comment"> * <tt>(1.0 - \ref kolmogorov_cdf "kolmogorov_cdf"(k_statistic))</tt>.</span></div> |
| <div class="line"><a name="l00618"></a><span class="lineno"> 618</span> <span class="comment"> *</span></div> |
| <div class="line"><a name="l00619"></a><span class="lineno"> 619</span> <span class="comment"> * @usage</span></div> |
| <div class="line"><a name="l00620"></a><span class="lineno"> 620</span> <span class="comment"> * - Test null hypothesis that two samples stem from the same distribution:</span></div> |
| <div class="line"><a name="l00621"></a><span class="lineno"> 621</span> <span class="comment"> * <pre>SELECT (ks_test(<em>first</em>, <em>value</em>,</span></div> |
| <div class="line"><a name="l00622"></a><span class="lineno"> 622</span> <span class="comment"> * (SELECT count(<em>value</em>) FROM <em>source</em> WHERE <em>first</em>),</span></div> |
| <div class="line"><a name="l00623"></a><span class="lineno"> 623</span> <span class="comment"> * (SELECT count(<em>value</em>) FROM <em>source</em> WHERE NOT <em>first</em>)</span></div> |
| <div class="line"><a name="l00624"></a><span class="lineno"> 624</span> <span class="comment"> * ORDER BY <em>value</em></span></div> |
| <div class="line"><a name="l00625"></a><span class="lineno"> 625</span> <span class="comment"> *)).* FROM <em>source</em></pre></span></div> |
| <div class="line"><a name="l00626"></a><span class="lineno"> 626</span> <span class="comment"> *</span></div> |
| <div class="line"><a name="l00627"></a><span class="lineno"> 627</span> <span class="comment"> * @note</span></div> |
| <div class="line"><a name="l00628"></a><span class="lineno"> 628</span> <span class="comment"> * This aggregate must be used as an ordered aggregate</span></div> |
| <div class="line"><a name="l00629"></a><span class="lineno"> 629</span> <span class="comment"> * (<tt>ORDER BY \em value</tt>) and will raise an exception if values are</span></div> |
| <div class="line"><a name="l00630"></a><span class="lineno"> 630</span> <span class="comment"> * not ordered.</span></div> |
| <div class="line"><a name="l00631"></a><span class="lineno"> 631</span> <span class="comment"> */</span></div> |
| <div class="line"><a name="l00632"></a><span class="lineno"> 632</span> m4_ifdef(<!__HAS_ORDERED_AGGREGATES__!>,<!</div> |
| <div class="line"><a name="l00633"></a><span class="lineno"> 633</span> CREATE</div> |
| <div class="line"><a name="l00634"></a><span class="lineno"> 634</span> m4_ifdef(<!__GREENPLUM__!>,<!ORDERED!>)</div> |
| <div class="line"><a name="l00635"></a><span class="lineno"> 635</span> AGGREGATE MADLIB_SCHEMA.ks_test(</div> |
| <div class="line"><a name="l00636"></a><span class="lineno"> 636</span>  /*+ "first" */ BOOLEAN,</div> |
| <div class="line"><a name="l00637"></a><span class="lineno"> 637</span>  /*+ "value" */ DOUBLE PRECISION,</div> |
| <div class="line"><a name="l00638"></a><span class="lineno"> 638</span>  /*+ m */ BIGINT,</div> |
| <div class="line"><a name="l00639"></a><span class="lineno"> 639</span>  /*+ n */ BIGINT</div> |
| <div class="line"><a name="l00640"></a><span class="lineno"> 640</span> ) (</div> |
| <div class="line"><a name="l00641"></a><span class="lineno"> 641</span>  SFUNC=MADLIB_SCHEMA.ks_test_transition,</div> |
| <div class="line"><a name="l00642"></a><span class="lineno"> 642</span>  STYPE=DOUBLE PRECISION[],</div> |
| <div class="line"><a name="l00643"></a><span class="lineno"> 643</span>  FINALFUNC=MADLIB_SCHEMA.ks_test_final,</div> |
| <div class="line"><a name="l00644"></a><span class="lineno"> 644</span>  INITCOND='{0,0,0,0,0,0,0}<span class="stringliteral">'</span></div> |
| <div class="line"><a name="l00645"></a><span class="lineno"> 645</span> <span class="stringliteral">);</span></div> |
| <div class="line"><a name="l00646"></a><span class="lineno"> 646</span> <span class="stringliteral">!>)</span></div> |
| <div class="line"><a name="l00647"></a><span class="lineno"> 647</span> <span class="stringliteral"></span></div> |
| <div class="line"><a name="l00648"></a><span class="lineno"> 648</span> <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.mw_test_transition(</span></div> |
| <div class="line"><a name="l00649"></a><span class="lineno"> 649</span> <span class="stringliteral"> state DOUBLE PRECISION[],</span></div> |
| <div class="line"><a name="l00650"></a><span class="lineno"> 650</span> <span class="stringliteral"> "first" BOOLEAN,</span></div> |
| <div class="line"><a name="l00651"></a><span class="lineno"> 651</span> <span class="stringliteral"> "value" DOUBLE PRECISION</span></div> |
| <div class="line"><a name="l00652"></a><span class="lineno"> 652</span> <span class="stringliteral">) RETURNS DOUBLE PRECISION[]</span></div> |
| <div class="line"><a name="l00653"></a><span class="lineno"> 653</span> <span class="stringliteral">AS '</span>MODULE_PATHNAME<span class="stringliteral">'</span></div> |
| <div class="line"><a name="l00654"></a><span class="lineno"> 654</span> <span class="stringliteral">LANGUAGE C</span></div> |
| <div class="line"><a name="l00655"></a><span class="lineno"><a class="code" href="hypothesis__tests_8sql__in.html#af45ae9d1275d385bbacd18bff688ba7f"> 655</a></span> <span class="stringliteral">IMMUTABLE</span></div> |
| <div class="line"><a name="l00656"></a><span class="lineno"> 656</span> <span class="stringliteral">STRICT;</span></div> |
| <div class="line"><a name="l00657"></a><span class="lineno"> 657</span> <span class="stringliteral"></span></div> |
| <div class="line"><a name="l00658"></a><span class="lineno"> 658</span> <span class="stringliteral">CREATE TYPE MADLIB_SCHEMA.mw_test_result AS (</span></div> |
| <div class="line"><a name="l00659"></a><span class="lineno"> 659</span> <span class="stringliteral"> statistic DOUBLE PRECISION,</span></div> |
| <div class="line"><a name="l00660"></a><span class="lineno"> 660</span> <span class="stringliteral"> u_statistic DOUBLE PRECISION,</span></div> |
| <div class="line"><a name="l00661"></a><span class="lineno"> 661</span> <span class="stringliteral"> p_value_one_sided DOUBLE PRECISION,</span></div> |
| <div class="line"><a name="l00662"></a><span class="lineno"> 662</span> <span class="stringliteral"> p_value_two_sided DOUBLE PRECISION</span></div> |
| <div class="line"><a name="l00663"></a><span class="lineno"> 663</span> <span class="stringliteral">);</span></div> |
| <div class="line"><a name="l00664"></a><span class="lineno"> 664</span> <span class="stringliteral"></span></div> |
| <div class="line"><a name="l00665"></a><span class="lineno"> 665</span> <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.mw_test_final(</span></div> |
| <div class="line"><a name="l00666"></a><span class="lineno"> 666</span> <span class="stringliteral"> state DOUBLE PRECISION[])</span></div> |
| <div class="line"><a name="l00667"></a><span class="lineno"> 667</span> <span class="stringliteral">RETURNS MADLIB_SCHEMA.mw_test_result</span></div> |
| <div class="line"><a name="l00668"></a><span class="lineno"> 668</span> <span class="stringliteral">AS '</span>MODULE_PATHNAME<span class="stringliteral">'</span></div> |
| <div class="line"><a name="l00669"></a><span class="lineno"> 669</span> <span class="stringliteral">LANGUAGE C IMMUTABLE STRICT;</span></div> |
| <div class="line"><a name="l00670"></a><span class="lineno"> 670</span> <span class="stringliteral"></span><span class="comment"></span></div> |
| <div class="line"><a name="l00671"></a><span class="lineno"> 671</span> <span class="comment">/**</span></div> |
| <div class="line"><a name="l00672"></a><span class="lineno"> 672</span> <span class="comment"> * @brief Perform Mann-Whitney test</span></div> |
| <div class="line"><a name="l00673"></a><span class="lineno"> 673</span> <span class="comment"> *</span></div> |
| <div class="line"><a name="l00674"></a><span class="lineno"> 674</span> <span class="comment"> * Given realizations \f$ x_1, \dots, x_m \f$ and \f$ y_1, \dots, y_m \f$ of</span></div> |
| <div class="line"><a name="l00675"></a><span class="lineno"> 675</span> <span class="comment"> * i.i.d. random variables \f$ X_1, \dots, X_m \f$ and i.i.d.</span></div> |
| <div class="line"><a name="l00676"></a><span class="lineno"> 676</span> <span class="comment"> * \f$ Y_1, \dots, Y_n \f$, respectively, test the null hypothesis that the</span></div> |
| <div class="line"><a name="l00677"></a><span class="lineno"> 677</span> <span class="comment"> * underlying distributions are equal, i.e.,</span></div> |
| <div class="line"><a name="l00678"></a><span class="lineno"> 678</span> <span class="comment"> * \f$ H_0 : \forall i,j: \Pr[X_i > Y_j] + \frac{\Pr[X_i = Y_j]}{2} = \frac 12 \f$.</span></div> |
| <div class="line"><a name="l00679"></a><span class="lineno"> 679</span> <span class="comment"> *</span></div> |
| <div class="line"><a name="l00680"></a><span class="lineno"> 680</span> <span class="comment"> * @param first Determines whether the value belongs to the first</span></div> |
| <div class="line"><a name="l00681"></a><span class="lineno"> 681</span> <span class="comment"> * (if \c TRUE) or the second sample (if \c FALSE)</span></div> |
| <div class="line"><a name="l00682"></a><span class="lineno"> 682</span> <span class="comment"> * @param value Value of random variate \f$ x_i \f$ or \f$ y_i \f$</span></div> |
| <div class="line"><a name="l00683"></a><span class="lineno"> 683</span> <span class="comment"> *</span></div> |
| <div class="line"><a name="l00684"></a><span class="lineno"> 684</span> <span class="comment"> * @return A composite value.</span></div> |
| <div class="line"><a name="l00685"></a><span class="lineno"> 685</span> <span class="comment"> * - <tt>statistic FLOAT8</tt> - Statistic</span></div> |
| <div class="line"><a name="l00686"></a><span class="lineno"> 686</span> <span class="comment"> * \f[</span></div> |
| <div class="line"><a name="l00687"></a><span class="lineno"> 687</span> <span class="comment"> * z = \frac{u - \bar x}{\sqrt{\frac{mn(m+n+1)}{12}}}</span></div> |
| <div class="line"><a name="l00688"></a><span class="lineno"> 688</span> <span class="comment"> * \f]</span></div> |
| <div class="line"><a name="l00689"></a><span class="lineno"> 689</span> <span class="comment"> * where \f$ u \f$ is the u-statistic computed as follows. The z-statistic</span></div> |
| <div class="line"><a name="l00690"></a><span class="lineno"> 690</span> <span class="comment"> * is approximately standard normally distributed.</span></div> |
| <div class="line"><a name="l00691"></a><span class="lineno"> 691</span> <span class="comment"> * - <tt>u_statistic FLOAT8</tt> - Statistic</span></div> |
| <div class="line"><a name="l00692"></a><span class="lineno"> 692</span> <span class="comment"> * \f$ u = \min \{ u_x, u_y \} \f$ where</span></div> |
| <div class="line"><a name="l00693"></a><span class="lineno"> 693</span> <span class="comment"> * \f[</span></div> |
| <div class="line"><a name="l00694"></a><span class="lineno"> 694</span> <span class="comment"> * u_x = mn + \binom{m+1}{2} - \sum_{i=1}^m r_{x,i}</span></div> |
| <div class="line"><a name="l00695"></a><span class="lineno"> 695</span> <span class="comment"> * \f]</span></div> |
| <div class="line"><a name="l00696"></a><span class="lineno"> 696</span> <span class="comment"> * where</span></div> |
| <div class="line"><a name="l00697"></a><span class="lineno"> 697</span> <span class="comment"> * \f[</span></div> |
| <div class="line"><a name="l00698"></a><span class="lineno"> 698</span> <span class="comment"> * r_{x,i}</span></div> |
| <div class="line"><a name="l00699"></a><span class="lineno"> 699</span> <span class="comment"> * = \{ j \mid x_j < x_i \} + \{ j \mid y_j < x_i \} +</span></div> |
| <div class="line"><a name="l00700"></a><span class="lineno"> 700</span> <span class="comment"> * \frac{\{ j \mid x_j = x_i \} + \{ j \mid y_j = x_i \} + 1}{2}</span></div> |
| <div class="line"><a name="l00701"></a><span class="lineno"> 701</span> <span class="comment"> * \f]</span></div> |
| <div class="line"><a name="l00702"></a><span class="lineno"> 702</span> <span class="comment"> * is defined as the rank of \f$ x_i \f$ in the combined list of all</span></div> |
| <div class="line"><a name="l00703"></a><span class="lineno"> 703</span> <span class="comment"> * \f$ m+n \f$ observations. For ties, the average rank of all equal values</span></div> |
| <div class="line"><a name="l00704"></a><span class="lineno"> 704</span> <span class="comment"> * is used.</span></div> |
| <div class="line"><a name="l00705"></a><span class="lineno"> 705</span> <span class="comment"> * - <tt>p_value_one_sided FLOAT8</tt> - Approximate one-sided p-value, i.e.,</span></div> |
| <div class="line"><a name="l00706"></a><span class="lineno"> 706</span> <span class="comment"> * an approximate value for \f$ \Pr[Z \geq z \mid H_0] \f$. Computed as</span></div> |
| <div class="line"><a name="l00707"></a><span class="lineno"> 707</span> <span class="comment"> * <tt>(1.0 - \ref normal_cdf "normal_cdf"(z_statistic))</tt>.</span></div> |
| <div class="line"><a name="l00708"></a><span class="lineno"> 708</span> <span class="comment"> * - <tt>p_value_two_sided FLOAT8</tt> - Approximate two-sided p-value, i.e.,</span></div> |
| <div class="line"><a name="l00709"></a><span class="lineno"> 709</span> <span class="comment"> * an approximate value for \f$ \Pr[|Z| \geq |z| \mid H_0] \f$. Computed as</span></div> |
| <div class="line"><a name="l00710"></a><span class="lineno"> 710</span> <span class="comment"> * <tt>(2 * \ref normal_cdf "normal_cdf"(-abs(z_statistic)))</tt>.</span></div> |
| <div class="line"><a name="l00711"></a><span class="lineno"> 711</span> <span class="comment"> *</span></div> |
| <div class="line"><a name="l00712"></a><span class="lineno"> 712</span> <span class="comment"> * @usage</span></div> |
| <div class="line"><a name="l00713"></a><span class="lineno"> 713</span> <span class="comment"> * - Test null hypothesis that two samples stem from the same distribution:</span></div> |
| <div class="line"><a name="l00714"></a><span class="lineno"> 714</span> <span class="comment"> * <pre>SELECT (mw_test(<em>first</em>, <em>value</em> ORDER BY <em>value</em>)).* FROM <em>source</em></pre></span></div> |
| <div class="line"><a name="l00715"></a><span class="lineno"> 715</span> <span class="comment"> *</span></div> |
| <div class="line"><a name="l00716"></a><span class="lineno"> 716</span> <span class="comment"> * @note</span></div> |
| <div class="line"><a name="l00717"></a><span class="lineno"> 717</span> <span class="comment"> * This aggregate must be used as an ordered aggregate</span></div> |
| <div class="line"><a name="l00718"></a><span class="lineno"> 718</span> <span class="comment"> * (<tt>ORDER BY \em value</tt>) and will raise an exception if values are</span></div> |
| <div class="line"><a name="l00719"></a><span class="lineno"> 719</span> <span class="comment"> * not ordered.</span></div> |
| <div class="line"><a name="l00720"></a><span class="lineno"> 720</span> <span class="comment"> */</span></div> |
| <div class="line"><a name="l00721"></a><span class="lineno"> 721</span> m4_ifdef(<!__HAS_ORDERED_AGGREGATES__!>,<!</div> |
| <div class="line"><a name="l00722"></a><span class="lineno"> 722</span> CREATE</div> |
| <div class="line"><a name="l00723"></a><span class="lineno"> 723</span> m4_ifdef(<!__GREENPLUM__!>,<!ORDERED!>)</div> |
| <div class="line"><a name="l00724"></a><span class="lineno"> 724</span> AGGREGATE MADLIB_SCHEMA.mw_test(</div> |
| <div class="line"><a name="l00725"></a><span class="lineno"> 725</span>  /*+ "first" */ BOOLEAN,</div> |
| <div class="line"><a name="l00726"></a><span class="lineno"> 726</span>  /*+ "value" */ DOUBLE PRECISION</div> |
| <div class="line"><a name="l00727"></a><span class="lineno"> 727</span> ) (</div> |
| <div class="line"><a name="l00728"></a><span class="lineno"> 728</span>  SFUNC=MADLIB_SCHEMA.mw_test_transition,</div> |
| <div class="line"><a name="l00729"></a><span class="lineno"> 729</span>  STYPE=DOUBLE PRECISION[],</div> |
| <div class="line"><a name="l00730"></a><span class="lineno"> 730</span>  FINALFUNC=MADLIB_SCHEMA.mw_test_final,</div> |
| <div class="line"><a name="l00731"></a><span class="lineno"> 731</span>  INITCOND='{0,0,0,0,0,0,0}<span class="stringliteral">'</span></div> |
| <div class="line"><a name="l00732"></a><span class="lineno"> 732</span> <span class="stringliteral">);</span></div> |
| <div class="line"><a name="l00733"></a><span class="lineno"> 733</span> <span class="stringliteral">!>)</span></div> |
| <div class="line"><a name="l00734"></a><span class="lineno"> 734</span> <span class="stringliteral"></span></div> |
| <div class="line"><a name="l00735"></a><span class="lineno"> 735</span> <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.wsr_test_transition(</span></div> |
| <div class="line"><a name="l00736"></a><span class="lineno"> 736</span> <span class="stringliteral"> state DOUBLE PRECISION[],</span></div> |
| <div class="line"><a name="l00737"></a><span class="lineno"> 737</span> <span class="stringliteral"> value DOUBLE PRECISION,</span></div> |
| <div class="line"><a name="l00738"></a><span class="lineno"> 738</span> <span class="stringliteral"> "precision" DOUBLE PRECISION</span></div> |
| <div class="line"><a name="l00739"></a><span class="lineno"> 739</span> <span class="stringliteral">) RETURNS DOUBLE PRECISION[]</span></div> |
| <div class="line"><a name="l00740"></a><span class="lineno"> 740</span> <span class="stringliteral">AS '</span>MODULE_PATHNAME<span class="stringliteral">'</span></div> |
| <div class="line"><a name="l00741"></a><span class="lineno"> 741</span> <span class="stringliteral">LANGUAGE C</span></div> |
| <div class="line"><a name="l00742"></a><span class="lineno"> 742</span> <span class="stringliteral">IMMUTABLE</span></div> |
| <div class="line"><a name="l00743"></a><span class="lineno"> 743</span> <span class="stringliteral">STRICT;</span></div> |
| <div class="line"><a name="l00744"></a><span class="lineno"><a class="code" href="hypothesis__tests_8sql__in.html#a32cdc58e8a5d149dd90304805de07fbd"> 744</a></span> <span class="stringliteral"></span></div> |
| <div class="line"><a name="l00745"></a><span class="lineno"> 745</span> <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.wsr_test_transition(</span></div> |
| <div class="line"><a name="l00746"></a><span class="lineno"> 746</span> <span class="stringliteral"> state DOUBLE PRECISION[],</span></div> |
| <div class="line"><a name="l00747"></a><span class="lineno"> 747</span> <span class="stringliteral"> value DOUBLE PRECISION</span></div> |
| <div class="line"><a name="l00748"></a><span class="lineno"> 748</span> <span class="stringliteral">) RETURNS DOUBLE PRECISION[]</span></div> |
| <div class="line"><a name="l00749"></a><span class="lineno"> 749</span> <span class="stringliteral">AS '</span>MODULE_PATHNAME<span class="stringliteral">'</span></div> |
| <div class="line"><a name="l00750"></a><span class="lineno"> 750</span> <span class="stringliteral">LANGUAGE C</span></div> |
| <div class="line"><a name="l00751"></a><span class="lineno"> 751</span> <span class="stringliteral">IMMUTABLE</span></div> |
| <div class="line"><a name="l00752"></a><span class="lineno"> 752</span> <span class="stringliteral">STRICT;</span></div> |
| <div class="line"><a name="l00753"></a><span class="lineno"> 753</span> <span class="stringliteral"></span></div> |
| <div class="line"><a name="l00754"></a><span class="lineno"> 754</span> <span class="stringliteral"></span></div> |
| <div class="line"><a name="l00755"></a><span class="lineno"> 755</span> <span class="stringliteral">CREATE TYPE MADLIB_SCHEMA.wsr_test_result AS (</span></div> |
| <div class="line"><a name="l00756"></a><span class="lineno"> 756</span> <span class="stringliteral"> statistic DOUBLE PRECISION,</span></div> |
| <div class="line"><a name="l00757"></a><span class="lineno"> 757</span> <span class="stringliteral"> rank_sum_pos FLOAT8,</span></div> |
| <div class="line"><a name="l00758"></a><span class="lineno"> 758</span> <span class="stringliteral"> rank_sum_neg FLOAT8,</span></div> |
| <div class="line"><a name="l00759"></a><span class="lineno"> 759</span> <span class="stringliteral"> num BIGINT,</span></div> |
| <div class="line"><a name="l00760"></a><span class="lineno"> 760</span> <span class="stringliteral"> z_statistic DOUBLE PRECISION,</span></div> |
| <div class="line"><a name="l00761"></a><span class="lineno"> 761</span> <span class="stringliteral"> p_value_one_sided DOUBLE PRECISION,</span></div> |
| <div class="line"><a name="l00762"></a><span class="lineno"> 762</span> <span class="stringliteral"> p_value_two_sided DOUBLE PRECISION</span></div> |
| <div class="line"><a name="l00763"></a><span class="lineno"> 763</span> <span class="stringliteral">);</span></div> |
| <div class="line"><a name="l00764"></a><span class="lineno"> 764</span> <span class="stringliteral"></span></div> |
| <div class="line"><a name="l00765"></a><span class="lineno"> 765</span> <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.wsr_test_final(</span></div> |
| <div class="line"><a name="l00766"></a><span class="lineno"> 766</span> <span class="stringliteral"> state DOUBLE PRECISION[])</span></div> |
| <div class="line"><a name="l00767"></a><span class="lineno"> 767</span> <span class="stringliteral">RETURNS MADLIB_SCHEMA.wsr_test_result</span></div> |
| <div class="line"><a name="l00768"></a><span class="lineno"> 768</span> <span class="stringliteral">AS '</span>MODULE_PATHNAME<span class="stringliteral">'</span></div> |
| <div class="line"><a name="l00769"></a><span class="lineno"> 769</span> <span class="stringliteral">LANGUAGE C IMMUTABLE STRICT;</span></div> |
| <div class="line"><a name="l00770"></a><span class="lineno"> 770</span> <span class="stringliteral"></span><span class="comment"></span></div> |
| <div class="line"><a name="l00771"></a><span class="lineno"> 771</span> <span class="comment">/**</span></div> |
| <div class="line"><a name="l00772"></a><span class="lineno"> 772</span> <span class="comment"> * @brief Perform Wilcoxon-Signed-Rank test</span></div> |
| <div class="line"><a name="l00773"></a><span class="lineno"> 773</span> <span class="comment"> *</span></div> |
| <div class="line"><a name="l00774"></a><span class="lineno"> 774</span> <span class="comment"> * Given realizations \f$ x_1, \dots, x_n \f$ of i.i.d. random variables</span></div> |
| <div class="line"><a name="l00775"></a><span class="lineno"> 775</span> <span class="comment"> * \f$ X_1, \dots, X_n \f$ with unknown mean \f$ \mu \f$, test the null</span></div> |
| <div class="line"><a name="l00776"></a><span class="lineno"> 776</span> <span class="comment"> * hypotheses \f$ H_0 : \mu \leq 0 \f$ and \f$ H_0 : \mu = 0 \f$.</span></div> |
| <div class="line"><a name="l00777"></a><span class="lineno"> 777</span> <span class="comment"> *</span></div> |
| <div class="line"><a name="l00778"></a><span class="lineno"> 778</span> <span class="comment"> * @param value Value of random variate \f$ x_i \f$ or \f$ y_i \f$. Values of 0</span></div> |
| <div class="line"><a name="l00779"></a><span class="lineno"> 779</span> <span class="comment"> * are ignored (i.e., they do not count towards \f$ n \f$).</span></div> |
| <div class="line"><a name="l00780"></a><span class="lineno"> 780</span> <span class="comment"> * @param precision The precision \f$ \epsilon_i \f$ with which value is known.</span></div> |
| <div class="line"><a name="l00781"></a><span class="lineno"> 781</span> <span class="comment"> * The precision determines the handling of ties. The current value</span></div> |
| <div class="line"><a name="l00782"></a><span class="lineno"> 782</span> <span class="comment"> * \f$ v_i \f$ is regarded a tie with the previous value \f$ v_{i-1} \f$ if</span></div> |
| <div class="line"><a name="l00783"></a><span class="lineno"> 783</span> <span class="comment"> * \f$ v_i - \epsilon_i \leq \max_{j=1, \dots, i-1} v_j + \epsilon_j \f$.</span></div> |
| <div class="line"><a name="l00784"></a><span class="lineno"> 784</span> <span class="comment"> * If \c precision is negative, then it will be treated as</span></div> |
| <div class="line"><a name="l00785"></a><span class="lineno"> 785</span> <span class="comment"> * <tt>value * 2^(-52)</tt>. (Note that \f$ 2^{-52} \f$ is the machine</span></div> |
| <div class="line"><a name="l00786"></a><span class="lineno"> 786</span> <span class="comment"> * epsilon for type <tt>DOUBLE PRECISION</tt>.)</span></div> |
| <div class="line"><a name="l00787"></a><span class="lineno"> 787</span> <span class="comment"> *</span></div> |
| <div class="line"><a name="l00788"></a><span class="lineno"> 788</span> <span class="comment"> * @return A composite value:</span></div> |
| <div class="line"><a name="l00789"></a><span class="lineno"> 789</span> <span class="comment"> * - <tt>statistic FLOAT8</tt> - statistic computed as follows. Let</span></div> |
| <div class="line"><a name="l00790"></a><span class="lineno"> 790</span> <span class="comment"> * \f$</span></div> |
| <div class="line"><a name="l00791"></a><span class="lineno"> 791</span> <span class="comment"> * w^+ = \sum_{i \mid x_i > 0} r_i</span></div> |
| <div class="line"><a name="l00792"></a><span class="lineno"> 792</span> <span class="comment"> * \f$</span></div> |
| <div class="line"><a name="l00793"></a><span class="lineno"> 793</span> <span class="comment"> * and</span></div> |
| <div class="line"><a name="l00794"></a><span class="lineno"> 794</span> <span class="comment"> * \f$</span></div> |
| <div class="line"><a name="l00795"></a><span class="lineno"> 795</span> <span class="comment"> * w^- = \sum_{i \mid x_i < 0} r_i</span></div> |
| <div class="line"><a name="l00796"></a><span class="lineno"> 796</span> <span class="comment"> * \f$</span></div> |
| <div class="line"><a name="l00797"></a><span class="lineno"> 797</span> <span class="comment"> * be the <em>signed rank sums</em> where</span></div> |
| <div class="line"><a name="l00798"></a><span class="lineno"> 798</span> <span class="comment"> * \f[</span></div> |
| <div class="line"><a name="l00799"></a><span class="lineno"> 799</span> <span class="comment"> * r_i</span></div> |
| <div class="line"><a name="l00800"></a><span class="lineno"> 800</span> <span class="comment"> * = \{ j \mid |x_j| < |x_i| \}</span></div> |
| <div class="line"><a name="l00801"></a><span class="lineno"> 801</span> <span class="comment"> * + \frac{\{ j \mid |x_j| = |x_i| \} + 1}{2}.</span></div> |
| <div class="line"><a name="l00802"></a><span class="lineno"> 802</span> <span class="comment"> * \f]</span></div> |
| <div class="line"><a name="l00803"></a><span class="lineno"> 803</span> <span class="comment"> * The Wilcoxon signed-rank statistic is \f$ w = \min \{ w^+, w^- \} \f$.</span></div> |
| <div class="line"><a name="l00804"></a><span class="lineno"> 804</span> <span class="comment"> * - <tt>rank_sum_pos FLOAT8</tt> - rank sum of all positive values, i.e., \f$ w^+ \f$</span></div> |
| <div class="line"><a name="l00805"></a><span class="lineno"> 805</span> <span class="comment"> * - <tt>rank_sum_neg FLOAT8</tt> - rank sum of all negative values, i.e., \f$ w^- \f$</span></div> |
| <div class="line"><a name="l00806"></a><span class="lineno"> 806</span> <span class="comment"> * - <tt>num BIGINT</tt> - number \f$ n \f$ of non-zero values</span></div> |
| <div class="line"><a name="l00807"></a><span class="lineno"> 807</span> <span class="comment"> * - <tt>z_statistic FLOAT8</tt> - z-statistic</span></div> |
| <div class="line"><a name="l00808"></a><span class="lineno"> 808</span> <span class="comment"> * \f[</span></div> |
| <div class="line"><a name="l00809"></a><span class="lineno"> 809</span> <span class="comment"> * z = \frac{w^+ - \frac{n(n+1)}{4}}</span></div> |
| <div class="line"><a name="l00810"></a><span class="lineno"> 810</span> <span class="comment"> * {\sqrt{\frac{n(n+1)(2n+1)}{24}</span></div> |
| <div class="line"><a name="l00811"></a><span class="lineno"> 811</span> <span class="comment"> * - \sum_{i=1}^n \frac{t_i^2 - 1}{48}}}</span></div> |
| <div class="line"><a name="l00812"></a><span class="lineno"> 812</span> <span class="comment"> * \f]</span></div> |
| <div class="line"><a name="l00813"></a><span class="lineno"> 813</span> <span class="comment"> * where \f$ t_i \f$ is the number of</span></div> |
| <div class="line"><a name="l00814"></a><span class="lineno"> 814</span> <span class="comment"> * values with absolute value equal to \f$ |x_i| \f$. The corresponding</span></div> |
| <div class="line"><a name="l00815"></a><span class="lineno"> 815</span> <span class="comment"> * random variable is approximately standard normally distributed.</span></div> |
| <div class="line"><a name="l00816"></a><span class="lineno"> 816</span> <span class="comment"> * - <tt>p_value_one_sided FLOAT8</tt> - One-sided p-value i.e.,</span></div> |
| <div class="line"><a name="l00817"></a><span class="lineno"> 817</span> <span class="comment"> * \f$ \Pr[Z \geq z \mid \mu \leq 0] \f$. Computed as</span></div> |
| <div class="line"><a name="l00818"></a><span class="lineno"> 818</span> <span class="comment"> * <tt>(1.0 - \ref normal_cdf "normal_cdf"(z_statistic))</tt>.</span></div> |
| <div class="line"><a name="l00819"></a><span class="lineno"> 819</span> <span class="comment"> * - <tt>p_value_two_sided FLOAT8</tt> - Two-sided p-value, i.e.,</span></div> |
| <div class="line"><a name="l00820"></a><span class="lineno"> 820</span> <span class="comment"> * \f$ \Pr[ |Z| \geq |z| \mid \mu = 0] \f$. Computed as</span></div> |
| <div class="line"><a name="l00821"></a><span class="lineno"> 821</span> <span class="comment"> * <tt>(2 * \ref normal_cdf "normal_cdf"(-abs(z_statistic)))</tt>.</span></div> |
| <div class="line"><a name="l00822"></a><span class="lineno"> 822</span> <span class="comment"> *</span></div> |
| <div class="line"><a name="l00823"></a><span class="lineno"> 823</span> <span class="comment"> * @usage</span></div> |
| <div class="line"><a name="l00824"></a><span class="lineno"> 824</span> <span class="comment"> * - One-sample test: Test null hypothesis that the mean of a sample is at</span></div> |
| <div class="line"><a name="l00825"></a><span class="lineno"> 825</span> <span class="comment"> * most (or equal to, respectively) \f$ \mu_0 \f$:</span></div> |
| <div class="line"><a name="l00826"></a><span class="lineno"> 826</span> <span class="comment"> * <pre>SELECT (wsr_test(<em>value</em> - <em>mu_0</em> ORDER BY abs(<em>value</em>))).* FROM <em>source</em></pre></span></div> |
| <div class="line"><a name="l00827"></a><span class="lineno"> 827</span> <span class="comment"> * - Dependent paired test: Test null hypothesis that the mean difference</span></div> |
| <div class="line"><a name="l00828"></a><span class="lineno"> 828</span> <span class="comment"> * between the first and second value in a pair is at most (or equal to,</span></div> |
| <div class="line"><a name="l00829"></a><span class="lineno"> 829</span> <span class="comment"> * respectively) \f$ \mu_0 \f$:</span></div> |
| <div class="line"><a name="l00830"></a><span class="lineno"> 830</span> <span class="comment"> * <pre>SELECT (wsr_test(<em>first</em> - <em>second</em> - <em>mu_0</em> ORDER BY abs(<em>first</em> - <em>second</em>))).* FROM <em>source</em></pre></span></div> |
| <div class="line"><a name="l00831"></a><span class="lineno"> 831</span> <span class="comment"> * If correctly determining ties is important (e.g., you may want to do so</span></div> |
| <div class="line"><a name="l00832"></a><span class="lineno"> 832</span> <span class="comment"> * when comparing to software products that take \c first, \c second,</span></div> |
| <div class="line"><a name="l00833"></a><span class="lineno"> 833</span> <span class="comment"> * and \c mu_0 as individual parameters), supply the precision parameter.</span></div> |
| <div class="line"><a name="l00834"></a><span class="lineno"> 834</span> <span class="comment"> * This can be done as follows:</span></div> |
| <div class="line"><a name="l00835"></a><span class="lineno"> 835</span> <span class="comment"> * <pre>SELECT (wsr_test(</span></div> |
| <div class="line"><a name="l00836"></a><span class="lineno"> 836</span> <span class="comment"> <em>first</em> - <em>second</em> - <em>mu_0</em>,</span></div> |
| <div class="line"><a name="l00837"></a><span class="lineno"> 837</span> <span class="comment"> 3 * 2^(-52) * greatest(first, second, mu_0)</span></div> |
| <div class="line"><a name="l00838"></a><span class="lineno"> 838</span> <span class="comment"> ORDER BY abs(<em>first</em> - <em>second</em>)</span></div> |
| <div class="line"><a name="l00839"></a><span class="lineno"> 839</span> <span class="comment">)).* FROM <em>source</em></pre></span></div> |
| <div class="line"><a name="l00840"></a><span class="lineno"> 840</span> <span class="comment"> * Here \f$ 2^{-52} \f$ is the machine epsilon, which we scale to the</span></div> |
| <div class="line"><a name="l00841"></a><span class="lineno"> 841</span> <span class="comment"> * magnitude of the input data and multiply with 3 because we have a sum with</span></div> |
| <div class="line"><a name="l00842"></a><span class="lineno"> 842</span> <span class="comment"> * three terms.</span></div> |
| <div class="line"><a name="l00843"></a><span class="lineno"> 843</span> <span class="comment"> *</span></div> |
| <div class="line"><a name="l00844"></a><span class="lineno"> 844</span> <span class="comment"> * @note</span></div> |
| <div class="line"><a name="l00845"></a><span class="lineno"> 845</span> <span class="comment"> * This aggregate must be used as an ordered aggregate</span></div> |
| <div class="line"><a name="l00846"></a><span class="lineno"> 846</span> <span class="comment"> * (<tt>ORDER BY abs(\em value</tt>)) and will raise an exception if the</span></div> |
| <div class="line"><a name="l00847"></a><span class="lineno"> 847</span> <span class="comment"> * absolute values are not ordered.</span></div> |
| <div class="line"><a name="l00848"></a><span class="lineno"> 848</span> <span class="comment"> */</span></div> |
| <div class="line"><a name="l00849"></a><span class="lineno"> 849</span> m4_ifdef(<!__HAS_ORDERED_AGGREGATES__!>,<!</div> |
| <div class="line"><a name="l00850"></a><span class="lineno"> 850</span> CREATE</div> |
| <div class="line"><a name="l00851"></a><span class="lineno"> 851</span> m4_ifdef(<!__GREENPLUM__!>,<!ORDERED!>)</div> |
| <div class="line"><a name="l00852"></a><span class="lineno"> 852</span> AGGREGATE MADLIB_SCHEMA.wsr_test(</div> |
| <div class="line"><a name="l00853"></a><span class="lineno"> 853</span>  /*+ "value" */ DOUBLE PRECISION,</div> |
| <div class="line"><a name="l00854"></a><span class="lineno"> 854</span>  /*+ "precision" */ DOUBLE PRECISION /*+ DEFAULT -1 */</div> |
| <div class="line"><a name="l00855"></a><span class="lineno"> 855</span> ) (</div> |
| <div class="line"><a name="l00856"></a><span class="lineno"> 856</span>  SFUNC=MADLIB_SCHEMA.wsr_test_transition,</div> |
| <div class="line"><a name="l00857"></a><span class="lineno"> 857</span>  STYPE=DOUBLE PRECISION[],</div> |
| <div class="line"><a name="l00858"></a><span class="lineno"> 858</span>  FINALFUNC=MADLIB_SCHEMA.wsr_test_final,</div> |
| <div class="line"><a name="l00859"></a><span class="lineno"> 859</span>  INITCOND='{0,0,0,0,0,0,0,0,0}<span class="stringliteral">'</span></div> |
| <div class="line"><a name="l00860"></a><span class="lineno"> 860</span> <span class="stringliteral">);</span></div> |
| <div class="line"><a name="l00861"></a><span class="lineno"> 861</span> <span class="stringliteral">!>)</span></div> |
| <div class="line"><a name="l00862"></a><span class="lineno"> 862</span> <span class="stringliteral"></span></div> |
| <div class="line"><a name="l00863"></a><span class="lineno"> 863</span> <span class="stringliteral">m4_ifdef(<!__HAS_ORDERED_AGGREGATES__!>,<!</span></div> |
| <div class="line"><a name="l00864"></a><span class="lineno"> 864</span> <span class="stringliteral">CREATE</span></div> |
| <div class="line"><a name="l00865"></a><span class="lineno"> 865</span> <span class="stringliteral">m4_ifdef(<!__GREENPLUM__!>,<!ORDERED!>)</span></div> |
| <div class="line"><a name="l00866"></a><span class="lineno"> 866</span> <span class="stringliteral">AGGREGATE MADLIB_SCHEMA.wsr_test(</span></div> |
| <div class="line"><a name="l00867"></a><span class="lineno"> 867</span> <span class="stringliteral"> /*+ value */ DOUBLE PRECISION</span></div> |
| <div class="line"><a name="l00868"></a><span class="lineno"> 868</span> <span class="stringliteral">) (</span></div> |
| <div class="line"><a name="l00869"></a><span class="lineno"> 869</span> <span class="stringliteral"> SFUNC=MADLIB_SCHEMA.wsr_test_transition,</span></div> |
| <div class="line"><a name="l00870"></a><span class="lineno"> 870</span> <span class="stringliteral"> STYPE=DOUBLE PRECISION[],</span></div> |
| <div class="line"><a name="l00871"></a><span class="lineno"> 871</span> <span class="stringliteral"> FINALFUNC=MADLIB_SCHEMA.wsr_test_final,</span></div> |
| <div class="line"><a name="l00872"></a><span class="lineno"><a class="code" href="hypothesis__tests_8sql__in.html#afea2309e99477df6ebbfbcea11272507"> 872</a></span> <span class="stringliteral"> INITCOND='</span>{0,0,0,0,0,0,0,0,0}<span class="stringliteral">'</span></div> |
| <div class="line"><a name="l00873"></a><span class="lineno"> 873</span> <span class="stringliteral">);</span></div> |
| <div class="line"><a name="l00874"></a><span class="lineno"> 874</span> <span class="stringliteral">!>)</span></div> |
| <div class="line"><a name="l00875"></a><span class="lineno"> 875</span> <span class="stringliteral"></span></div> |
| <div class="line"><a name="l00876"></a><span class="lineno"> 876</span> <span class="stringliteral">CREATE TYPE MADLIB_SCHEMA.one_way_anova_result AS (</span></div> |
| <div class="line"><a name="l00877"></a><span class="lineno"> 877</span> <span class="stringliteral"> sum_squares_between DOUBLE PRECISION,</span></div> |
| <div class="line"><a name="l00878"></a><span class="lineno"> 878</span> <span class="stringliteral"> sum_squares_within DOUBLE PRECISION,</span></div> |
| <div class="line"><a name="l00879"></a><span class="lineno"> 879</span> <span class="stringliteral"> df_between BIGINT,</span></div> |
| <div class="line"><a name="l00880"></a><span class="lineno"> 880</span> <span class="stringliteral"> df_within BIGINT,</span></div> |
| <div class="line"><a name="l00881"></a><span class="lineno"> 881</span> <span class="stringliteral"> mean_squares_between DOUBLE PRECISION,</span></div> |
| <div class="line"><a name="l00882"></a><span class="lineno"> 882</span> <span class="stringliteral"> mean_squares_within DOUBLE PRECISION,</span></div> |
| <div class="line"><a name="l00883"></a><span class="lineno"> 883</span> <span class="stringliteral"> statistic DOUBLE PRECISION,</span></div> |
| <div class="line"><a name="l00884"></a><span class="lineno"> 884</span> <span class="stringliteral"> p_value DOUBLE PRECISION</span></div> |
| <div class="line"><a name="l00885"></a><span class="lineno"> 885</span> <span class="stringliteral">);</span></div> |
| <div class="line"><a name="l00886"></a><span class="lineno"> 886</span> <span class="stringliteral"></span></div> |
| <div class="line"><a name="l00887"></a><span class="lineno"> 887</span> <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.one_way_anova_transition(</span></div> |
| <div class="line"><a name="l00888"></a><span class="lineno"> 888</span> <span class="stringliteral"> state DOUBLE PRECISION[],</span></div> |
| <div class="line"><a name="l00889"></a><span class="lineno"> 889</span> <span class="stringliteral"> "group" INTEGER,</span></div> |
| <div class="line"><a name="l00890"></a><span class="lineno"> 890</span> <span class="stringliteral"> value DOUBLE PRECISION)</span></div> |
| <div class="line"><a name="l00891"></a><span class="lineno"> 891</span> <span class="stringliteral">RETURNS DOUBLE PRECISION[]</span></div> |
| <div class="line"><a name="l00892"></a><span class="lineno"> 892</span> <span class="stringliteral">AS '</span>MODULE_PATHNAME<span class="stringliteral">'</span></div> |
| <div class="line"><a name="l00893"></a><span class="lineno"> 893</span> <span class="stringliteral">LANGUAGE C</span></div> |
| <div class="line"><a name="l00894"></a><span class="lineno"> 894</span> <span class="stringliteral">IMMUTABLE</span></div> |
| <div class="line"><a name="l00895"></a><span class="lineno"> 895</span> <span class="stringliteral">STRICT;</span></div> |
| <div class="line"><a name="l00896"></a><span class="lineno"> 896</span> <span class="stringliteral"></span></div> |
| <div class="line"><a name="l00897"></a><span class="lineno"> 897</span> <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.one_way_anova_merge_states(</span></div> |
| <div class="line"><a name="l00898"></a><span class="lineno"> 898</span> <span class="stringliteral"> state1 DOUBLE PRECISION[],</span></div> |
| <div class="line"><a name="l00899"></a><span class="lineno"> 899</span> <span class="stringliteral"> state2 DOUBLE PRECISION[])</span></div> |
| <div class="line"><a name="l00900"></a><span class="lineno"> 900</span> <span class="stringliteral">RETURNS DOUBLE PRECISION[]</span></div> |
| <div class="line"><a name="l00901"></a><span class="lineno"> 901</span> <span class="stringliteral">AS '</span>MODULE_PATHNAME<span class="stringliteral">'</span></div> |
| <div class="line"><a name="l00902"></a><span class="lineno"> 902</span> <span class="stringliteral">LANGUAGE C</span></div> |
| <div class="line"><a name="l00903"></a><span class="lineno"> 903</span> <span class="stringliteral">IMMUTABLE STRICT;</span></div> |
| <div class="line"><a name="l00904"></a><span class="lineno"> 904</span> <span class="stringliteral"></span></div> |
| <div class="line"><a name="l00905"></a><span class="lineno"> 905</span> <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.one_way_anova_final(</span></div> |
| <div class="line"><a name="l00906"></a><span class="lineno"> 906</span> <span class="stringliteral"> state DOUBLE PRECISION[])</span></div> |
| <div class="line"><a name="l00907"></a><span class="lineno"> 907</span> <span class="stringliteral">RETURNS MADLIB_SCHEMA.one_way_anova_result</span></div> |
| <div class="line"><a name="l00908"></a><span class="lineno"> 908</span> <span class="stringliteral">AS '</span>MODULE_PATHNAME<span class="stringliteral">'</span></div> |
| <div class="line"><a name="l00909"></a><span class="lineno"> 909</span> <span class="stringliteral">LANGUAGE C IMMUTABLE STRICT;</span></div> |
| <div class="line"><a name="l00910"></a><span class="lineno"> 910</span> <span class="stringliteral"></span><span class="comment"></span></div> |
| <div class="line"><a name="l00911"></a><span class="lineno"> 911</span> <span class="comment">/**</span></div> |
| <div class="line"><a name="l00912"></a><span class="lineno"> 912</span> <span class="comment"> * @brief Perform one-way analysis of variance</span></div> |
| <div class="line"><a name="l00913"></a><span class="lineno"> 913</span> <span class="comment"> *</span></div> |
| <div class="line"><a name="l00914"></a><span class="lineno"> 914</span> <span class="comment"> * Given realizations</span></div> |
| <div class="line"><a name="l00915"></a><span class="lineno"> 915</span> <span class="comment"> * \f$ x_{1,1}, \dots, x_{1, n_1}, x_{2,1}, \dots, x_{2,n_2}, \dots, x_{k,n_k} \f$</span></div> |
| <div class="line"><a name="l00916"></a><span class="lineno"> 916</span> <span class="comment"> * of i.i.d. random variables \f$ X_{i,j} \sim N(\mu_i, \sigma^2) \f$ with</span></div> |
| <div class="line"><a name="l00917"></a><span class="lineno"> 917</span> <span class="comment"> * unknown parameters \f$ \mu_1, \dots, \mu_k \f$ and \f$ \sigma^2 \f$, test the</span></div> |
| <div class="line"><a name="l00918"></a><span class="lineno"> 918</span> <span class="comment"> * null hypotheses \f$ H_0 : \mu_1 = \dots = \mu_k \f$.</span></div> |
| <div class="line"><a name="l00919"></a><span class="lineno"> 919</span> <span class="comment"> *</span></div> |
| <div class="line"><a name="l00920"></a><span class="lineno"> 920</span> <span class="comment"> * @param group Group which \c value is from. Note that \c group can assume</span></div> |
| <div class="line"><a name="l00921"></a><span class="lineno"> 921</span> <span class="comment"> * arbitary value not limited to a continguous range of integers.</span></div> |
| <div class="line"><a name="l00922"></a><span class="lineno"> 922</span> <span class="comment"> * @param value Value of random variate \f$ x_{i,j} \f$</span></div> |
| <div class="line"><a name="l00923"></a><span class="lineno"> 923</span> <span class="comment"> *</span></div> |
| <div class="line"><a name="l00924"></a><span class="lineno"> 924</span> <span class="comment"> * @return A composite value as follows. Let \f$ n := \sum_{i=1}^k n_i \f$ be</span></div> |
| <div class="line"><a name="l00925"></a><span class="lineno"> 925</span> <span class="comment"> * the total size of all samples. Denote by \f$ \bar x \f$ the grand</span></div> |
| <div class="line"><a name="l00926"></a><span class="lineno"> 926</span> <span class="comment"> * \ref sample_mean "mean", by \f$ \overline{x_i} \f$ the group</span></div> |
| <div class="line"><a name="l00927"></a><span class="lineno"> 927</span> <span class="comment"> * \ref sample_mean "sample means", and by \f$ s_i^2 \f$ the group</span></div> |
| <div class="line"><a name="l00928"></a><span class="lineno"> 928</span> <span class="comment"> * \ref sample_variance "sample variances".</span></div> |
| <div class="line"><a name="l00929"></a><span class="lineno"> 929</span> <span class="comment"> * - <tt>sum_squares_between DOUBLE PRECISION</tt> - sum of squares between the</span></div> |
| <div class="line"><a name="l00930"></a><span class="lineno"> 930</span> <span class="comment"> * group means, i.e.,</span></div> |
| <div class="line"><a name="l00931"></a><span class="lineno"> 931</span> <span class="comment"> * \f$</span></div> |
| <div class="line"><a name="l00932"></a><span class="lineno"> 932</span> <span class="comment"> * \mathit{SS}_b = \sum_{i=1}^k n_i (\overline{x_i} - \bar x)^2.</span></div> |
| <div class="line"><a name="l00933"></a><span class="lineno"> 933</span> <span class="comment"> * \f$</span></div> |
| <div class="line"><a name="l00934"></a><span class="lineno"> 934</span> <span class="comment"> * - <tt>sum_squares_within DOUBLE PRECISION</tt> - sum of squares within the</span></div> |
| <div class="line"><a name="l00935"></a><span class="lineno"> 935</span> <span class="comment"> * groups, i.e.,</span></div> |
| <div class="line"><a name="l00936"></a><span class="lineno"> 936</span> <span class="comment"> * \f$</span></div> |
| <div class="line"><a name="l00937"></a><span class="lineno"> 937</span> <span class="comment"> * \mathit{SS}_w = \sum_{i=1}^k (n_i - 1) s_i^2.</span></div> |
| <div class="line"><a name="l00938"></a><span class="lineno"> 938</span> <span class="comment"> * \f$</span></div> |
| <div class="line"><a name="l00939"></a><span class="lineno"> 939</span> <span class="comment"> * - <tt>df_between BIGINT</tt> - degree of freedom for between-group variation \f$ (k-1) \f$</span></div> |
| <div class="line"><a name="l00940"></a><span class="lineno"> 940</span> <span class="comment"> * - <tt>df_within BIGINT</tt> - degree of freedom for within-group variation \f$ (n-k) \f$</span></div> |
| <div class="line"><a name="l00941"></a><span class="lineno"> 941</span> <span class="comment"> * - <tt>mean_squares_between DOUBLE PRECISION</tt> - mean square between</span></div> |
| <div class="line"><a name="l00942"></a><span class="lineno"> 942</span> <span class="comment"> * groups, i.e.,</span></div> |
| <div class="line"><a name="l00943"></a><span class="lineno"> 943</span> <span class="comment"> * \f$</span></div> |
| <div class="line"><a name="l00944"></a><span class="lineno"> 944</span> <span class="comment"> * s_b^2 := \frac{\mathit{SS}_b}{k-1}</span></div> |
| <div class="line"><a name="l00945"></a><span class="lineno"> 945</span> <span class="comment"> * \f$</span></div> |
| <div class="line"><a name="l00946"></a><span class="lineno"> 946</span> <span class="comment"> * - <tt>mean_squares_within DOUBLE PRECISION</tt> - mean square within</span></div> |
| <div class="line"><a name="l00947"></a><span class="lineno"> 947</span> <span class="comment"> * groups, i.e.,</span></div> |
| <div class="line"><a name="l00948"></a><span class="lineno"> 948</span> <span class="comment"> * \f$</span></div> |
| <div class="line"><a name="l00949"></a><span class="lineno"> 949</span> <span class="comment"> * s_w^2 := \frac{\mathit{SS}_w}{n-k}</span></div> |
| <div class="line"><a name="l00950"></a><span class="lineno"> 950</span> <span class="comment"> * \f$</span></div> |
| <div class="line"><a name="l00951"></a><span class="lineno"> 951</span> <span class="comment"> * - <tt>statistic DOUBLE PRECISION</tt> - Statistic computed as</span></div> |
| <div class="line"><a name="l00952"></a><span class="lineno"> 952</span> <span class="comment"> * \f[</span></div> |
| <div class="line"><a name="l00953"></a><span class="lineno"> 953</span> <span class="comment"> * f = \frac{s_b^2}{s_w^2}.</span></div> |
| <div class="line"><a name="l00954"></a><span class="lineno"> 954</span> <span class="comment"> * \f]</span></div> |
| <div class="line"><a name="l00955"></a><span class="lineno"> 955</span> <span class="comment"> * This statistic is Fisher F-distributed with \f$ (k-1) \f$ degrees of</span></div> |
| <div class="line"><a name="l00956"></a><span class="lineno"> 956</span> <span class="comment"> * freedom in the numerator and \f$ (n-k) \f$ degrees of freedom in the</span></div> |
| <div class="line"><a name="l00957"></a><span class="lineno"> 957</span> <span class="comment"> * denominator.</span></div> |
| <div class="line"><a name="l00958"></a><span class="lineno"> 958</span> <span class="comment"> * - <tt>p_value DOUBLE PRECISION</tt> - p-value, i.e.,</span></div> |
| <div class="line"><a name="l00959"></a><span class="lineno"> 959</span> <span class="comment"> * \f$ \Pr[ F \geq f \mid H_0] \f$.</span></div> |
| <div class="line"><a name="l00960"></a><span class="lineno"> 960</span> <span class="comment"> *</span></div> |
| <div class="line"><a name="l00961"></a><span class="lineno"> 961</span> <span class="comment"> * @usage</span></div> |
| <div class="line"><a name="l00962"></a><span class="lineno"> 962</span> <span class="comment"> * - Test null hypothesis that the mean of the all samples is equal:</span></div> |
| <div class="line"><a name="l00963"></a><span class="lineno"> 963</span> <span class="comment"> * <pre>SELECT (one_way_anova(<em>group</em>, <em>value</em>)).* FROM <em>source</em></pre></span></div> |
| <div class="line"><a name="l00964"></a><span class="lineno"> 964</span> <span class="comment"> */</span></div> |
| <div class="line"><a name="l00965"></a><span class="lineno"> 965</span> CREATE AGGREGATE MADLIB_SCHEMA.one_way_anova(</div> |
| <div class="line"><a name="l00966"></a><span class="lineno"> 966</span>  /*+ group */ INTEGER,</div> |
| <div class="line"><a name="l00967"></a><span class="lineno"> 967</span>  /*+ value */ DOUBLE PRECISION) (</div> |
| <div class="line"><a name="l00968"></a><span class="lineno"> 968</span> </div> |
| <div class="line"><a name="l00969"></a><span class="lineno"> 969</span>  SFUNC=MADLIB_SCHEMA.one_way_anova_transition,</div> |
| <div class="line"><a name="l00970"></a><span class="lineno"> 970</span>  STYPE=DOUBLE PRECISION[],</div> |
| <div class="line"><a name="l00971"></a><span class="lineno"> 971</span>  FINALFUNC=MADLIB_SCHEMA.one_way_anova_final,</div> |
| <div class="line"><a name="l00972"></a><span class="lineno"> 972</span>  m4_ifdef(<!__GREENPLUM__!>,<!PREFUNC=MADLIB_SCHEMA.one_way_anova_merge_states,!>)</div> |
| <div class="line"><a name="l00973"></a><span class="lineno"> 973</span>  INITCOND='{0,0}<span class="stringliteral">'</span></div> |
| <div class="line"><a name="l00974"></a><span class="lineno"> 974</span> <span class="stringliteral">);</span></div> |
| <div class="line"><a name="l00975"></a><span class="lineno"> 975</span> <span class="stringliteral"></span></div> |
| <div class="line"><a name="l00976"></a><span class="lineno"> 976</span> <span class="stringliteral">m4_changequote(<!`!>,<!'</span>!>)</div> |
| </div><!-- fragment --></div><!-- contents --> |
| </div><!-- doc-content --> |
| <!-- start footer part --> |
| <div id="nav-path" class="navpath"><!-- id is needed for treeview function! --> |
| <ul> |
| <li class="navelem"><a class="el" href="dir_68267d1309a1af8e8297ef4c3efbcdba.html">src</a></li><li class="navelem"><a class="el" href="dir_efbcf68973d247bbf15f9eecae7f24e3.html">ports</a></li><li class="navelem"><a class="el" href="dir_a4a48839224ef8488facbffa8a397967.html">postgres</a></li><li class="navelem"><a class="el" href="dir_dc596537ad427a4d866006d1a3e1fe29.html">modules</a></li><li class="navelem"><a class="el" href="dir_505cd743a8a717435eca324f49291a46.html">stats</a></li><li class="navelem"><a class="el" href="hypothesis__tests_8sql__in.html">hypothesis_tests.sql_in</a></li> |
| <li class="footer">Generated on Wed Aug 21 2013 16:09:52 for MADlib by |
| <a href="http://www.doxygen.org/index.html"> |
| <img class="footer" src="doxygen.png" alt="doxygen"/></a> 1.8.4 </li> |
| </ul> |
| </div> |
| </body> |
| </html> |
