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| <a href="hypothesis__tests_8sql__in.html">Go to the documentation of this file.</a><div class="fragment"><pre class="fragment"><a name="l00001"></a>00001 <span class="comment">/* ----------------------------------------------------------------------- */</span><span class="comment">/**</span> |
| <a name="l00002"></a>00002 <span class="comment"> *</span> |
| <a name="l00003"></a>00003 <span class="comment"> * @file hypothesis_tests.sql_in</span> |
| <a name="l00004"></a>00004 <span class="comment"> *</span> |
| <a name="l00005"></a>00005 <span class="comment"> * @brief SQL functions for statistical hypothesis tests</span> |
| <a name="l00006"></a>00006 <span class="comment"> *</span> |
| <a name="l00007"></a>00007 <span class="comment"> * @sa For an overview of hypthesis-test functions, see the module</span> |
| <a name="l00008"></a>00008 <span class="comment"> * description \ref grp_stats_tests.</span> |
| <a name="l00009"></a>00009 <span class="comment"> *</span> |
| <a name="l00010"></a>00010 <span class="comment"> */</span><span class="comment">/* ----------------------------------------------------------------------- */</span> |
| <a name="l00011"></a>00011 |
| <a name="l00012"></a>00012 m4_include(`SQLCommon.m4<span class="stringliteral">')</span> |
| <a name="l00013"></a>00013 <span class="stringliteral">m4_changequote(<!,!>)</span> |
| <a name="l00014"></a>00014 <span class="stringliteral"></span><span class="comment"></span> |
| <a name="l00015"></a>00015 <span class="comment">/**</span> |
| <a name="l00016"></a>00016 <span class="comment">@addtogroup grp_stats_tests</span> |
| <a name="l00017"></a>00017 <span class="comment"></span> |
| <a name="l00018"></a>00018 <span class="comment">@about</span> |
| <a name="l00019"></a>00019 <span class="comment"></span> |
| <a name="l00020"></a>00020 <span class="comment">Hypothesis tests are used to confirm or reject a <em>“null” hypothesis</em></span> |
| <a name="l00021"></a>00021 <span class="comment">\f$ H_0 \f$ about the distribution of random variables, given realizations of</span> |
| <a name="l00022"></a>00022 <span class="comment">these random variables. Since in general it is not possible to make statements</span> |
| <a name="l00023"></a>00023 <span class="comment">with certainty, one is interested in the probability \f$ p \f$ of seeing random</span> |
| <a name="l00024"></a>00024 <span class="comment">variates at least as extreme as the ones observed, assuming that \f$ H_0 \f$ is</span> |
| <a name="l00025"></a>00025 <span class="comment">true. If this probability \f$ p \f$ is small, \f$ H_0 \f$ will be rejected by</span> |
| <a name="l00026"></a>00026 <span class="comment">the test with <em>significance level</em> \f$ p \f$. Falsifying \f$ H_0 \f$ is</span> |
| <a name="l00027"></a>00027 <span class="comment">the canonic goal when employing a hypothesis test. That is, hypothesis tests are</span> |
| <a name="l00028"></a>00028 <span class="comment">typically used in order to substantiate that instead the <em>alternative</span> |
| <a name="l00029"></a>00029 <span class="comment">hypothesis</em> \f$ H_1 \f$ is true.</span> |
| <a name="l00030"></a>00030 <span class="comment"></span> |
| <a name="l00031"></a>00031 <span class="comment">Hypothesis tests may be devided into parametric and non-parametric tests. A</span> |
| <a name="l00032"></a>00032 <span class="comment">parametric test assumes certain distributions and makes inferences about</span> |
| <a name="l00033"></a>00033 <span class="comment">parameters of the distributions (like, e.g., the mean of a normal distribution).</span> |
| <a name="l00034"></a>00034 <span class="comment">Formally, there is a given domain of possible parameters \f$ \Gamma \f$ and the</span> |
| <a name="l00035"></a>00035 <span class="comment">null hypothesis \f$ H_0 \f$ is the event that the true parameter</span> |
| <a name="l00036"></a>00036 <span class="comment">\f$ \gamma_0 \in \Gamma_0 \f$, where \f$ \Gamma_0 \subsetneq \Gamma \f$.</span> |
| <a name="l00037"></a>00037 <span class="comment">Non-parametric tests, on the other hand, do not assume any particular</span> |
| <a name="l00038"></a>00038 <span class="comment">distribution of the sample (e.g., a non-parametric test may simply test if two</span> |
| <a name="l00039"></a>00039 <span class="comment">distributions are similar).</span> |
| <a name="l00040"></a>00040 <span class="comment"></span> |
| <a name="l00041"></a>00041 <span class="comment">The first step of a hypothesis test is to compute a <em>test statistic</em>,</span> |
| <a name="l00042"></a>00042 <span class="comment">which is a function of the random variates, i.e., a random variate itself.</span> |
| <a name="l00043"></a>00043 <span class="comment">A hypothesis test relies on that the distribution of the test statistic is</span> |
| <a name="l00044"></a>00044 <span class="comment">(approximately) known. Now, the \f$ p \f$-value is the probability of seeing a</span> |
| <a name="l00045"></a>00045 <span class="comment">test statistic at least as extreme as the one observed, assuming that</span> |
| <a name="l00046"></a>00046 <span class="comment">\f$ H_0 \f$ is true. In a case where the null hypothesis corresponds to a family</span> |
| <a name="l00047"></a>00047 <span class="comment">of distributions (e.g., in a parametric test where \f$ \Gamma_0 \f$ is not a</span> |
| <a name="l00048"></a>00048 <span class="comment">singleton set), the \f$ p \f$-value is the supremum, over all possible</span> |
| <a name="l00049"></a>00049 <span class="comment">distributions according to the null hypothesis, of these probabilities.</span> |
| <a name="l00050"></a>00050 <span class="comment"></span> |
| <a name="l00051"></a>00051 <span class="comment">@input</span> |
| <a name="l00052"></a>00052 <span class="comment"></span> |
| <a name="l00053"></a>00053 <span class="comment">Input data is assumed to be normalized with all values stored row-wise. In</span> |
| <a name="l00054"></a>00054 <span class="comment">general, the following inputs are expected.</span> |
| <a name="l00055"></a>00055 <span class="comment"></span> |
| <a name="l00056"></a>00056 <span class="comment">One-sample tests expect the following form:</span> |
| <a name="l00057"></a>00057 <span class="comment"><pre>{TABLE|VIEW} <em>source</em> (</span> |
| <a name="l00058"></a>00058 <span class="comment"> ...</span> |
| <a name="l00059"></a>00059 <span class="comment"> <em>value</em> DOUBLE PRECISION</span> |
| <a name="l00060"></a>00060 <span class="comment"> ...</span> |
| <a name="l00061"></a>00061 <span class="comment">)</pre></span> |
| <a name="l00062"></a>00062 <span class="comment"></span> |
| <a name="l00063"></a>00063 <span class="comment">Two-sample tests expect the following form:</span> |
| <a name="l00064"></a>00064 <span class="comment"><pre>{TABLE|VIEW} <em>source</em> (</span> |
| <a name="l00065"></a>00065 <span class="comment"> ...</span> |
| <a name="l00066"></a>00066 <span class="comment"> <em>first</em> BOOLEAN,</span> |
| <a name="l00067"></a>00067 <span class="comment"> <em>value</em> DOUBLE PRECISION</span> |
| <a name="l00068"></a>00068 <span class="comment"> ...</span> |
| <a name="l00069"></a>00069 <span class="comment">)</pre></span> |
| <a name="l00070"></a>00070 <span class="comment">Here, \c first indicates whether a value is from the first (if \c TRUE) or the</span> |
| <a name="l00071"></a>00071 <span class="comment">second sample (if \c FALSE).</span> |
| <a name="l00072"></a>00072 <span class="comment"></span> |
| <a name="l00073"></a>00073 <span class="comment">Many-sample tests expect the following form:</span> |
| <a name="l00074"></a>00074 <span class="comment"><pre>{TABLE|VIEW} <em>source</em> (</span> |
| <a name="l00075"></a>00075 <span class="comment"> ...</span> |
| <a name="l00076"></a>00076 <span class="comment"> <em>group</em> INTEGER,</span> |
| <a name="l00077"></a>00077 <span class="comment"> <em>value</em> DOUBLE PRECISION</span> |
| <a name="l00078"></a>00078 <span class="comment"> ...</span> |
| <a name="l00079"></a>00079 <span class="comment">)</pre></span> |
| <a name="l00080"></a>00080 <span class="comment"></span> |
| <a name="l00081"></a>00081 <span class="comment">@usage</span> |
| <a name="l00082"></a>00082 <span class="comment"></span> |
| <a name="l00083"></a>00083 <span class="comment">All tests are implemented as aggregate functions. The non-parametric</span> |
| <a name="l00084"></a>00084 <span class="comment">(rank-based) tests are implemented as ordered aggregate functions and thus</span> |
| <a name="l00085"></a>00085 <span class="comment">necessitate an <tt>ORDER BY</tt> clause. In the following, the most simple</span> |
| <a name="l00086"></a>00086 <span class="comment">forms of usage are given. Specific function signatures, as described in</span> |
| <a name="l00087"></a>00087 <span class="comment">\ref hypothesis_tests.sql_in, may ask for more arguments or for a different</span> |
| <a name="l00088"></a>00088 <span class="comment"><tt>ORDER BY</tt> clause.</span> |
| <a name="l00089"></a>00089 <span class="comment"></span> |
| <a name="l00090"></a>00090 <span class="comment">- Run a parametric one-sample test:</span> |
| <a name="l00091"></a>00091 <span class="comment"> <pre>SELECT <em>test</em>(<em>value</em>) FROM <em>source</em></pre></span> |
| <a name="l00092"></a>00092 <span class="comment">- Run a parametric two-sample test:</span> |
| <a name="l00093"></a>00093 <span class="comment"> <pre>SELECT <em>test</em>(<em>first</em>, <em>value</em>) FROM <em>source</em></pre></span> |
| <a name="l00094"></a>00094 <span class="comment">- Run a non-parametric one-sample test:</span> |
| <a name="l00095"></a>00095 <span class="comment"> <pre>SELECT <em>test</em>(<em>value</em> ORDER BY <em>value</em>) FROM <em>source</em></pre></span> |
| <a name="l00096"></a>00096 <span class="comment">- Run a non-parametric two-sample test:</span> |
| <a name="l00097"></a>00097 <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> |
| <a name="l00098"></a>00098 <span class="comment"></span> |
| <a name="l00099"></a>00099 <span class="comment">@examp</span> |
| <a name="l00100"></a>00100 <span class="comment"></span> |
| <a name="l00101"></a>00101 <span class="comment">See \ref hypothesis_tests.sql_in for examples for each of the aggregate</span> |
| <a name="l00102"></a>00102 <span class="comment">functions.</span> |
| <a name="l00103"></a>00103 <span class="comment"></span> |
| <a name="l00104"></a>00104 <span class="comment">@literature</span> |
| <a name="l00105"></a>00105 <span class="comment"></span> |
| <a name="l00106"></a>00106 <span class="comment">[1] M. Hollander, D. Wolfe: <em>Nonparametric Statistical Methods</em>,</span> |
| <a name="l00107"></a>00107 <span class="comment"> 2nd edition, Wiley, 1999</span> |
| <a name="l00108"></a>00108 <span class="comment"></span> |
| <a name="l00109"></a>00109 <span class="comment">[2] E. Lehmann, J. Romano: <em>Testing Statistical Hypotheses</em>, 3rd edition,</span> |
| <a name="l00110"></a>00110 <span class="comment"> Springer, 2005</span> |
| <a name="l00111"></a>00111 <span class="comment"></span> |
| <a name="l00112"></a>00112 <span class="comment">@sa File hypothesis_tests.sql_in documenting the SQL functions.</span> |
| <a name="l00113"></a>00113 <span class="comment">*/</span> |
| <a name="l00114"></a>00114 |
| <a name="l00115"></a>00115 CREATE TYPE MADLIB_SCHEMA.t_test_result AS ( |
| <a name="l00116"></a>00116 statistic DOUBLE PRECISION, |
| <a name="l00117"></a>00117 df DOUBLE PRECISION, |
| <a name="l00118"></a>00118 p_value_one_sided DOUBLE PRECISION, |
| <a name="l00119"></a>00119 p_value_two_sided DOUBLE PRECISION |
| <a name="l00120"></a>00120 ); |
| <a name="l00121"></a>00121 |
| <a name="l00122"></a>00122 CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.t_test_one_transition( |
| <a name="l00123"></a>00123 state DOUBLE PRECISION[], |
| <a name="l00124"></a>00124 value DOUBLE PRECISION |
| <a name="l00125"></a>00125 ) RETURNS DOUBLE PRECISION[] |
| <a name="l00126"></a>00126 AS 'MODULE_PATHNAME<span class="stringliteral">'</span> |
| <a name="l00127"></a>00127 <span class="stringliteral">LANGUAGE C</span> |
| <a name="l00128"></a>00128 <span class="stringliteral">IMMUTABLE</span> |
| <a name="l00129"></a>00129 <span class="stringliteral">STRICT;</span> |
| <a name="l00130"></a>00130 <span class="stringliteral"></span> |
| <a name="l00131"></a>00131 <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.t_test_merge_states(</span> |
| <a name="l00132"></a>00132 <span class="stringliteral"> state1 DOUBLE PRECISION[],</span> |
| <a name="l00133"></a>00133 <span class="stringliteral"> state2 DOUBLE PRECISION[])</span> |
| <a name="l00134"></a>00134 <span class="stringliteral">RETURNS DOUBLE PRECISION[]</span> |
| <a name="l00135"></a>00135 <span class="stringliteral">AS '</span>MODULE_PATHNAME<span class="stringliteral">'</span> |
| <a name="l00136"></a>00136 <span class="stringliteral">LANGUAGE C</span> |
| <a name="l00137"></a>00137 <span class="stringliteral">IMMUTABLE STRICT;</span> |
| <a name="l00138"></a>00138 <span class="stringliteral"></span> |
| <a name="l00139"></a>00139 <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.t_test_one_final(</span> |
| <a name="l00140"></a>00140 <span class="stringliteral"> state DOUBLE PRECISION[])</span> |
| <a name="l00141"></a>00141 <span class="stringliteral">RETURNS MADLIB_SCHEMA.t_test_result</span> |
| <a name="l00142"></a>00142 <span class="stringliteral">AS '</span>MODULE_PATHNAME<span class="stringliteral">'</span> |
| <a name="l00143"></a>00143 <span class="stringliteral">LANGUAGE C IMMUTABLE STRICT;</span> |
| <a name="l00144"></a>00144 <span class="stringliteral"></span> |
| <a name="l00145"></a>00145 <span class="stringliteral">CREATE TYPE MADLIB_SCHEMA.f_test_result AS (</span> |
| <a name="l00146"></a>00146 <span class="stringliteral"> statistic DOUBLE PRECISION,</span> |
| <a name="l00147"></a>00147 <span class="stringliteral"> df1 DOUBLE PRECISION,</span> |
| <a name="l00148"></a>00148 <span class="stringliteral"> df2 DOUBLE PRECISION,</span> |
| <a name="l00149"></a>00149 <span class="stringliteral"> p_value_one_sided DOUBLE PRECISION,</span> |
| <a name="l00150"></a>00150 <span class="stringliteral"> p_value_two_sided DOUBLE PRECISION</span> |
| <a name="l00151"></a>00151 <span class="stringliteral">);</span> |
| <a name="l00152"></a>00152 <span class="stringliteral"></span> |
| <a name="l00153"></a>00153 <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.f_test_final(</span> |
| <a name="l00154"></a>00154 <span class="stringliteral"> state DOUBLE PRECISION[])</span> |
| <a name="l00155"></a>00155 <span class="stringliteral">RETURNS MADLIB_SCHEMA.f_test_result</span> |
| <a name="l00156"></a>00156 <span class="stringliteral">AS '</span>MODULE_PATHNAME<span class="stringliteral">'</span> |
| <a name="l00157"></a>00157 <span class="stringliteral">LANGUAGE C IMMUTABLE STRICT;</span> |
| <a name="l00158"></a>00158 <span class="stringliteral"></span> |
| <a name="l00159"></a>00159 <span class="stringliteral"></span><span class="comment"></span> |
| <a name="l00160"></a>00160 <span class="comment">/**</span> |
| <a name="l00161"></a>00161 <span class="comment"> * @brief Perform one-sample or dependent paired Student t-test</span> |
| <a name="l00162"></a>00162 <span class="comment"> *</span> |
| <a name="l00163"></a>00163 <span class="comment"> * Given realizations \f$ x_1, \dots, x_n \f$ of i.i.d. random variables</span> |
| <a name="l00164"></a>00164 <span class="comment"> * \f$ X_1, \dots, X_n \sim N(\mu, \sigma^2) \f$ with unknown parameters \f$ \mu \f$ and</span> |
| <a name="l00165"></a>00165 <span class="comment"> * \f$ \sigma^2 \f$, test the null hypotheses \f$ H_0 : \mu \leq 0 \f$ and</span> |
| <a name="l00166"></a>00166 <span class="comment"> * \f$ H_0 : \mu = 0 \f$.</span> |
| <a name="l00167"></a>00167 <span class="comment"> *</span> |
| <a name="l00168"></a>00168 <span class="comment"> * @param value Value of random variate \f$ x_i \f$</span> |
| <a name="l00169"></a>00169 <span class="comment"> *</span> |
| <a name="l00170"></a>00170 <span class="comment"> * @return A composite value as follows. We denote by \f$ \bar x \f$ the</span> |
| <a name="l00171"></a>00171 <span class="comment"> * \ref sample_mean "sample mean" and by \f$ s^2 \f$ the</span> |
| <a name="l00172"></a>00172 <span class="comment"> * \ref sample_variance "sample variance".</span> |
| <a name="l00173"></a>00173 <span class="comment"> * - <tt>statistic FLOAT8</tt> - Statistic</span> |
| <a name="l00174"></a>00174 <span class="comment"> * \f[</span> |
| <a name="l00175"></a>00175 <span class="comment"> * t = \frac{\sqrt n \cdot \bar x}{s}</span> |
| <a name="l00176"></a>00176 <span class="comment"> * \f]</span> |
| <a name="l00177"></a>00177 <span class="comment"> * The corresponding random</span> |
| <a name="l00178"></a>00178 <span class="comment"> * variable is Student-t distributed with</span> |
| <a name="l00179"></a>00179 <span class="comment"> * \f$ (n - 1) \f$ degrees of freedom.</span> |
| <a name="l00180"></a>00180 <span class="comment"> * - <tt>df FLOAT8</tt> - Degrees of freedom \f$ (n - 1) \f$</span> |
| <a name="l00181"></a>00181 <span class="comment"> * - <tt>p_value_one_sided FLOAT8</tt> - Lower bound on one-sided p-value.</span> |
| <a name="l00182"></a>00182 <span class="comment"> * In detail, the result is \f$ \Pr[\bar X \geq \bar x \mid \mu = 0] \f$,</span> |
| <a name="l00183"></a>00183 <span class="comment"> * which is a lower bound on</span> |
| <a name="l00184"></a>00184 <span class="comment"> * \f$ \Pr[\bar X \geq \bar x \mid \mu \leq 0] \f$. Computed as</span> |
| <a name="l00185"></a>00185 <span class="comment"> * <tt>(1.0 - \ref students_t_cdf "students_t_cdf"(statistic))</tt>.</span> |
| <a name="l00186"></a>00186 <span class="comment"> * - <tt>p_value_two_sided FLOAT8</tt> - Two-sided p-value, i.e.,</span> |
| <a name="l00187"></a>00187 <span class="comment"> * \f$ \Pr[ |\bar X| \geq |\bar x| \mid \mu = 0] \f$. Computed as</span> |
| <a name="l00188"></a>00188 <span class="comment"> * <tt>(2 * \ref students_t_cdf "students_t_cdf"(-abs(statistic)))</tt>.</span> |
| <a name="l00189"></a>00189 <span class="comment"> *</span> |
| <a name="l00190"></a>00190 <span class="comment"> * @usage</span> |
| <a name="l00191"></a>00191 <span class="comment"> * - One-sample t-test: Test null hypothesis that the mean of a sample is at</span> |
| <a name="l00192"></a>00192 <span class="comment"> * most (or equal to, respectively) \f$ \mu_0 \f$:</span> |
| <a name="l00193"></a>00193 <span class="comment"> * <pre>SELECT (t_test_one(<em>value</em> - <em>mu_0</em>)).* FROM <em>source</em></pre></span> |
| <a name="l00194"></a>00194 <span class="comment"> * - Dependent paired t-test: Test null hypothesis that the mean difference</span> |
| <a name="l00195"></a>00195 <span class="comment"> * between the first and second value in each pair is at most (or equal to,</span> |
| <a name="l00196"></a>00196 <span class="comment"> * respectively) \f$ \mu_0 \f$:</span> |
| <a name="l00197"></a>00197 <span class="comment"> * <pre>SELECT (t_test_one(<em>first</em> - <em>second</em> - <em>mu_0</em>)).* FROM <em>source</em></pre></span> |
| <a name="l00198"></a>00198 <span class="comment"> */</span> |
| <a name="l00199"></a>00199 CREATE AGGREGATE MADLIB_SCHEMA.t_test_one( |
| <a name="l00200"></a>00200 /*+ value */ DOUBLE PRECISION) ( |
| <a name="l00201"></a>00201 |
| <a name="l00202"></a>00202 SFUNC=MADLIB_SCHEMA.t_test_one_transition, |
| <a name="l00203"></a>00203 STYPE=DOUBLE PRECISION[], |
| <a name="l00204"></a>00204 FINALFUNC=MADLIB_SCHEMA.t_test_one_final, |
| <a name="l00205"></a>00205 m4_ifdef(<!__GREENPLUM__!>,<!PREFUNC=MADLIB_SCHEMA.t_test_merge_states,!>) |
| <a name="l00206"></a>00206 INITCOND='{0,0,0,0,0,0,0}<span class="stringliteral">'</span> |
| <a name="l00207"></a>00207 <span class="stringliteral">);</span> |
| <a name="l00208"></a>00208 <span class="stringliteral"></span> |
| <a name="l00209"></a>00209 <span class="stringliteral"></span> |
| <a name="l00210"></a>00210 <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.t_test_two_transition(</span> |
| <a name="l00211"></a>00211 <span class="stringliteral"> state DOUBLE PRECISION[],</span> |
| <a name="l00212"></a>00212 <span class="stringliteral"> "first" BOOLEAN,</span> |
| <a name="l00213"></a>00213 <span class="stringliteral"> "value" DOUBLE PRECISION)</span> |
| <a name="l00214"></a>00214 <span class="stringliteral">RETURNS DOUBLE PRECISION[]</span> |
| <a name="l00215"></a>00215 <span class="stringliteral">AS '</span>MODULE_PATHNAME<span class="stringliteral">'</span> |
| <a name="l00216"></a>00216 <span class="stringliteral">LANGUAGE C</span> |
| <a name="l00217"></a>00217 <span class="stringliteral">IMMUTABLE</span> |
| <a name="l00218"></a>00218 <span class="stringliteral">STRICT;</span> |
| <a name="l00219"></a>00219 <span class="stringliteral"></span> |
| <a name="l00220"></a>00220 <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.t_test_two_pooled_final(</span> |
| <a name="l00221"></a><a class="code" href="hypothesis__tests_8sql__in.html#ae7197f66a085f53d71167ac0a9029567">00221</a> <span class="stringliteral"> state DOUBLE PRECISION[])</span> |
| <a name="l00222"></a>00222 <span class="stringliteral">RETURNS MADLIB_SCHEMA.t_test_result</span> |
| <a name="l00223"></a>00223 <span class="stringliteral">AS '</span>MODULE_PATHNAME<span class="stringliteral">'</span> |
| <a name="l00224"></a>00224 <span class="stringliteral">LANGUAGE C IMMUTABLE STRICT;</span> |
| <a name="l00225"></a>00225 <span class="stringliteral"></span><span class="comment"></span> |
| <a name="l00226"></a>00226 <span class="comment">/**</span> |
| <a name="l00227"></a>00227 <span class="comment"> * @brief Perform two-sample pooled (i.e., equal variances) Student t-test</span> |
| <a name="l00228"></a>00228 <span class="comment"> *</span> |
| <a name="l00229"></a>00229 <span class="comment"> * Given realizations \f$ x_1, \dots, x_n \f$ and \f$ y_1, \dots, y_m \f$ of</span> |
| <a name="l00230"></a>00230 <span class="comment"> * i.i.d. random variables \f$ X_1, \dots, X_n \sim N(\mu_X, \sigma^2) \f$ and</span> |
| <a name="l00231"></a>00231 <span class="comment"> * \f$ Y_1, \dots, Y_m \sim N(\mu_Y, \sigma^2) \f$ with unknown parameters</span> |
| <a name="l00232"></a>00232 <span class="comment"> * \f$ \mu_X, \mu_Y, \f$ and \f$ \sigma^2 \f$, test the null hypotheses</span> |
| <a name="l00233"></a>00233 <span class="comment"> * \f$ H_0 : \mu_X \leq \mu_Y \f$ and \f$ H_0 : \mu_X = \mu_Y \f$.</span> |
| <a name="l00234"></a>00234 <span class="comment"> *</span> |
| <a name="l00235"></a>00235 <span class="comment"> * @param first Indicator whether \c value is from first sample</span> |
| <a name="l00236"></a>00236 <span class="comment"> * \f$ x_1, \dots, x_n \f$ (if \c TRUE) or from second sample</span> |
| <a name="l00237"></a>00237 <span class="comment"> * \f$ y_1, \dots, y_m \f$ (if \c FALSE)</span> |
| <a name="l00238"></a>00238 <span class="comment"> * @param value Value of random variate \f$ x_i \f$ or \f$ y_i \f$</span> |
| <a name="l00239"></a>00239 <span class="comment"> *</span> |
| <a name="l00240"></a>00240 <span class="comment"> * @return A composite value as follows. We denote by \f$ \bar x, \bar y \f$</span> |
| <a name="l00241"></a>00241 <span class="comment"> * the \ref sample_mean "sample means" and by \f$ s_X^2, s_Y^2 \f$ the</span> |
| <a name="l00242"></a>00242 <span class="comment"> * \ref sample_variance "sample variances".</span> |
| <a name="l00243"></a>00243 <span class="comment"> * - <tt>statistic FLOAT8</tt> - Statistic</span> |
| <a name="l00244"></a>00244 <span class="comment"> * \f[</span> |
| <a name="l00245"></a>00245 <span class="comment"> * t = \frac{\bar x - \bar y}{s_p \sqrt{1/n + 1/m}}</span> |
| <a name="l00246"></a>00246 <span class="comment"> * \f]</span> |
| <a name="l00247"></a>00247 <span class="comment"> * where</span> |
| <a name="l00248"></a>00248 <span class="comment"> * \f[</span> |
| <a name="l00249"></a>00249 <span class="comment"> * s_p^2 = \frac{\sum_{i=1}^n (x_i - \bar x)^2</span> |
| <a name="l00250"></a>00250 <span class="comment"> * + \sum_{i=1}^m (y_i - \bar y)^2}</span> |
| <a name="l00251"></a>00251 <span class="comment"> * {n + m - 2}</span> |
| <a name="l00252"></a>00252 <span class="comment"> * \f]</span> |
| <a name="l00253"></a>00253 <span class="comment"> * is the <em>pooled variance</em>.</span> |
| <a name="l00254"></a>00254 <span class="comment"> * The corresponding random</span> |
| <a name="l00255"></a>00255 <span class="comment"> * variable is Student-t distributed with</span> |
| <a name="l00256"></a>00256 <span class="comment"> * \f$ (n + m - 2) \f$ degrees of freedom.</span> |
| <a name="l00257"></a>00257 <span class="comment"> * - <tt>df FLOAT8</tt> - Degrees of freedom \f$ (n + m - 2) \f$</span> |
| <a name="l00258"></a>00258 <span class="comment"> * - <tt>p_value_one_sided FLOAT8</tt> - Lower bound on one-sided p-value.</span> |
| <a name="l00259"></a>00259 <span class="comment"> * In detail, the result is \f$ \Pr[\bar X - \bar Y \geq \bar x - \bar y \mid \mu_X = \mu_Y] \f$,</span> |
| <a name="l00260"></a>00260 <span class="comment"> * which is a lower bound on</span> |
| <a name="l00261"></a>00261 <span class="comment"> * \f$ \Pr[\bar X - \bar Y \geq \bar x - \bar y \mid \mu_X \leq \mu_Y] \f$.</span> |
| <a name="l00262"></a>00262 <span class="comment"> * Computed as</span> |
| <a name="l00263"></a>00263 <span class="comment"> * <tt>(1.0 - \ref students_t_cdf "students_t_cdf"(statistic))</tt>.</span> |
| <a name="l00264"></a>00264 <span class="comment"> * - <tt>p_value_two_sided FLOAT8</tt> - Two-sided p-value, i.e.,</span> |
| <a name="l00265"></a>00265 <span class="comment"> * \f$ \Pr[ |\bar X - \bar Y| \geq |\bar x - \bar y| \mid \mu_X = \mu_Y] \f$.</span> |
| <a name="l00266"></a>00266 <span class="comment"> * Computed as</span> |
| <a name="l00267"></a>00267 <span class="comment"> * <tt>(2 * \ref students_t_cdf "students_t_cdf"(-abs(statistic)))</tt>.</span> |
| <a name="l00268"></a>00268 <span class="comment"> *</span> |
| <a name="l00269"></a>00269 <span class="comment"> * @usage</span> |
| <a name="l00270"></a>00270 <span class="comment"> * - Two-sample pooled t-test: Test null hypothesis that the mean of the first</span> |
| <a name="l00271"></a>00271 <span class="comment"> * sample is at most (or equal to, respectively) the mean of the second</span> |
| <a name="l00272"></a>00272 <span class="comment"> * sample:</span> |
| <a name="l00273"></a>00273 <span class="comment"> * <pre>SELECT (t_test_pooled(<em>first</em>, <em>value</em>)).* FROM <em>source</em></pre></span> |
| <a name="l00274"></a>00274 <span class="comment"> */</span> |
| <a name="l00275"></a>00275 CREATE AGGREGATE MADLIB_SCHEMA.t_test_two_pooled( |
| <a name="l00276"></a>00276 /*+ "first" */ BOOLEAN, |
| <a name="l00277"></a>00277 /*+ "value" */ DOUBLE PRECISION) ( |
| <a name="l00278"></a>00278 |
| <a name="l00279"></a>00279 SFUNC=MADLIB_SCHEMA.t_test_two_transition, |
| <a name="l00280"></a>00280 STYPE=DOUBLE PRECISION[], |
| <a name="l00281"></a>00281 FINALFUNC=MADLIB_SCHEMA.t_test_two_pooled_final, |
| <a name="l00282"></a>00282 m4_ifdef(<!__GREENPLUM__!>,<!PREFUNC=MADLIB_SCHEMA.t_test_merge_states,!>) |
| <a name="l00283"></a>00283 INITCOND='{0,0,0,0,0,0,0}<span class="stringliteral">'</span> |
| <a name="l00284"></a>00284 <span class="stringliteral">);</span> |
| <a name="l00285"></a>00285 <span class="stringliteral"></span> |
| <a name="l00286"></a>00286 <span class="stringliteral"></span> |
| <a name="l00287"></a>00287 <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.t_test_two_unpooled_final(</span> |
| <a name="l00288"></a>00288 <span class="stringliteral"> state DOUBLE PRECISION[])</span> |
| <a name="l00289"></a>00289 <span class="stringliteral">RETURNS MADLIB_SCHEMA.t_test_result</span> |
| <a name="l00290"></a>00290 <span class="stringliteral">AS '</span>MODULE_PATHNAME<span class="stringliteral">'</span> |
| <a name="l00291"></a>00291 <span class="stringliteral">LANGUAGE C IMMUTABLE STRICT;</span> |
| <a name="l00292"></a>00292 <span class="stringliteral"></span><span class="comment"></span> |
| <a name="l00293"></a>00293 <span class="comment">/**</span> |
| <a name="l00294"></a>00294 <span class="comment"> * @brief Perform unpooled (i.e., unequal variances) t-test (also known as</span> |
| <a name="l00295"></a>00295 <span class="comment"> * Welch's t-test)</span> |
| <a name="l00296"></a>00296 <span class="comment"> *</span> |
| <a name="l00297"></a><a class="code" href="hypothesis__tests_8sql__in.html#a74e6ef1522197957b5aa35bf67004364">00297</a> <span class="comment"> * Given realizations \f$ x_1, \dots, x_n \f$ and \f$ y_1, \dots, y_m \f$ of</span> |
| <a name="l00298"></a>00298 <span class="comment"> * i.i.d. random variables \f$ X_1, \dots, X_n \sim N(\mu_X, \sigma_X^2) \f$ and</span> |
| <a name="l00299"></a>00299 <span class="comment"> * \f$ Y_1, \dots, Y_m \sim N(\mu_Y, \sigma_Y^2) \f$ with unknown parameters</span> |
| <a name="l00300"></a>00300 <span class="comment"> * \f$ \mu_X, \mu_Y, \sigma_X^2, \f$ and \f$ \sigma_Y^2 \f$, test the null</span> |
| <a name="l00301"></a>00301 <span class="comment"> * hypotheses \f$ H_0 : \mu_X \leq \mu_Y \f$ and \f$ H_0 : \mu_X = \mu_Y \f$.</span> |
| <a name="l00302"></a>00302 <span class="comment"> *</span> |
| <a name="l00303"></a>00303 <span class="comment"> * @param first Indicator whether \c value is from first sample</span> |
| <a name="l00304"></a>00304 <span class="comment"> * \f$ x_1, \dots, x_n \f$ (if \c TRUE) or from second sample</span> |
| <a name="l00305"></a>00305 <span class="comment"> * \f$ y_1, \dots, y_m \f$ (if \c FALSE)</span> |
| <a name="l00306"></a>00306 <span class="comment"> * @param value Value of random variate \f$ x_i \f$ or \f$ y_i \f$</span> |
| <a name="l00307"></a>00307 <span class="comment"> *</span> |
| <a name="l00308"></a>00308 <span class="comment"> * @return A composite value as follows. We denote by \f$ \bar x, \bar y \f$</span> |
| <a name="l00309"></a>00309 <span class="comment"> * the \ref sample_mean "sample means" and by \f$ s_X^2, s_Y^2 \f$ the</span> |
| <a name="l00310"></a>00310 <span class="comment"> * \ref sample_variance "sample variances".</span> |
| <a name="l00311"></a>00311 <span class="comment"> * - <tt>statistic FLOAT8</tt> - Statistic</span> |
| <a name="l00312"></a>00312 <span class="comment"> * \f[</span> |
| <a name="l00313"></a>00313 <span class="comment"> * t = \frac{\bar x - \bar y}{\sqrt{s_X^2/n + s_Y^2/m}}</span> |
| <a name="l00314"></a>00314 <span class="comment"> * \f]</span> |
| <a name="l00315"></a>00315 <span class="comment"> * The corresponding random variable is approximately Student-t distributed</span> |
| <a name="l00316"></a>00316 <span class="comment"> * with</span> |
| <a name="l00317"></a>00317 <span class="comment"> * \f[</span> |
| <a name="l00318"></a>00318 <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> |
| <a name="l00319"></a>00319 <span class="comment"> * \f]</span> |
| <a name="l00320"></a>00320 <span class="comment"> * degrees of freedom (Welch–Satterthwaite formula).</span> |
| <a name="l00321"></a>00321 <span class="comment"> * - <tt>df FLOAT8</tt> - Degrees of freedom (as above)</span> |
| <a name="l00322"></a>00322 <span class="comment"> * - <tt>p_value_one_sided FLOAT8</tt> - Lower bound on one-sided p-value.</span> |
| <a name="l00323"></a>00323 <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$,</span> |
| <a name="l00324"></a>00324 <span class="comment"> * which is a lower bound on</span> |
| <a name="l00325"></a>00325 <span class="comment"> * \f$ \Pr[\bar X - \bar Y \geq \bar x - \bar y \mid \mu_X \leq \mu_Y] \f$.</span> |
| <a name="l00326"></a>00326 <span class="comment"> * Computed as</span> |
| <a name="l00327"></a>00327 <span class="comment"> * <tt>(1.0 - \ref students_t_cdf "students_t_cdf"(statistic))</tt>.</span> |
| <a name="l00328"></a>00328 <span class="comment"> * - <tt>p_value_two_sided FLOAT8</tt> - Two-sided p-value, i.e.,</span> |
| <a name="l00329"></a>00329 <span class="comment"> * \f$ \Pr[ |\bar X - \bar Y| \geq |\bar x - \bar y| \mid \mu_X = \mu_Y] \f$.</span> |
| <a name="l00330"></a>00330 <span class="comment"> * Computed as</span> |
| <a name="l00331"></a>00331 <span class="comment"> * <tt>(2 * \ref students_t_cdf "students_t_cdf"(-abs(statistic)))</tt>.</span> |
| <a name="l00332"></a>00332 <span class="comment"> *</span> |
| <a name="l00333"></a>00333 <span class="comment"> * @usage</span> |
| <a name="l00334"></a>00334 <span class="comment"> * - Two-sample unpooled t-test: Test null hypothesis that the mean of the</span> |
| <a name="l00335"></a>00335 <span class="comment"> * first sample is at most (or equal to, respectively) the mean of the second</span> |
| <a name="l00336"></a>00336 <span class="comment"> * sample:</span> |
| <a name="l00337"></a>00337 <span class="comment"> * <pre>SELECT (t_test_unpooled(<em>first</em>, <em>value</em>)).* FROM <em>source</em></pre></span> |
| <a name="l00338"></a>00338 <span class="comment"> */</span> |
| <a name="l00339"></a>00339 CREATE AGGREGATE MADLIB_SCHEMA.t_test_two_unpooled( |
| <a name="l00340"></a>00340 /*+ "first" */ BOOLEAN, |
| <a name="l00341"></a>00341 /*+ "value" */ DOUBLE PRECISION) ( |
| <a name="l00342"></a>00342 |
| <a name="l00343"></a>00343 SFUNC=MADLIB_SCHEMA.t_test_two_transition, |
| <a name="l00344"></a>00344 STYPE=DOUBLE PRECISION[], |
| <a name="l00345"></a>00345 FINALFUNC=MADLIB_SCHEMA.t_test_two_unpooled_final, |
| <a name="l00346"></a>00346 m4_ifdef(<!__GREENPLUM__!>,<!PREFUNC=MADLIB_SCHEMA.t_test_merge_states,!>) |
| <a name="l00347"></a>00347 INITCOND='{0,0,0,0,0,0,0}<span class="stringliteral">'</span> |
| <a name="l00348"></a>00348 <span class="stringliteral">);</span> |
| <a name="l00349"></a>00349 <span class="stringliteral"></span><span class="comment"></span> |
| <a name="l00350"></a>00350 <span class="comment">/**</span> |
| <a name="l00351"></a>00351 <span class="comment"> * @brief Perform Fisher F-test</span> |
| <a name="l00352"></a>00352 <span class="comment"> *</span> |
| <a name="l00353"></a>00353 <span class="comment"> * Given realizations \f$ x_1, \dots, x_m \f$ and \f$ y_1, \dots, y_n \f$ of</span> |
| <a name="l00354"></a>00354 <span class="comment"> * i.i.d. random variables \f$ X_1, \dots, X_m \sim N(\mu_X, \sigma^2) \f$ and</span> |
| <a name="l00355"></a>00355 <span class="comment"> * \f$ Y_1, \dots, Y_n \sim N(\mu_Y, \sigma^2) \f$ with unknown parameters</span> |
| <a name="l00356"></a>00356 <span class="comment"> * \f$ \mu_X, \mu_Y, \f$ and \f$ \sigma^2 \f$, test the null hypotheses</span> |
| <a name="l00357"></a>00357 <span class="comment"> * \f$ H_0 : \sigma_X < \sigma_Y \f$ and \f$ H_0 : \sigma_X = \sigma_Y \f$.</span> |
| <a name="l00358"></a>00358 <span class="comment"> *</span> |
| <a name="l00359"></a>00359 <span class="comment"> * @param first Indicator whether \c value is from first sample</span> |
| <a name="l00360"></a>00360 <span class="comment"> * \f$ x_1, \dots, x_m \f$ (if \c TRUE) or from second sample</span> |
| <a name="l00361"></a><a class="code" href="hypothesis__tests_8sql__in.html#aa95e5a0c8b4841c113c84a393b8b4868">00361</a> <span class="comment"> * \f$ y_1, \dots, y_n \f$ (if \c FALSE)</span> |
| <a name="l00362"></a>00362 <span class="comment"> * @param value Value of random variate \f$ x_i \f$ or \f$ y_i \f$</span> |
| <a name="l00363"></a>00363 <span class="comment"> *</span> |
| <a name="l00364"></a>00364 <span class="comment"> * @return A composite value as follows. We denote by \f$ \bar x, \bar y \f$</span> |
| <a name="l00365"></a>00365 <span class="comment"> * the \ref sample_mean "sample means" and by \f$ s_X^2, s_Y^2 \f$ the</span> |
| <a name="l00366"></a>00366 <span class="comment"> * \ref sample_variance "sample variances".</span> |
| <a name="l00367"></a>00367 <span class="comment"> * - <tt>statistic FLOAT8</tt> - Statistic</span> |
| <a name="l00368"></a>00368 <span class="comment"> * \f[</span> |
| <a name="l00369"></a>00369 <span class="comment"> * f = \frac{s_Y^2}{s_X^2}</span> |
| <a name="l00370"></a>00370 <span class="comment"> * \f]</span> |
| <a name="l00371"></a>00371 <span class="comment"> * The corresponding random</span> |
| <a name="l00372"></a>00372 <span class="comment"> * variable is F-distributed with</span> |
| <a name="l00373"></a>00373 <span class="comment"> * \f$ (n - 1) \f$ degrees of freedom in the numerator and</span> |
| <a name="l00374"></a>00374 <span class="comment"> * \f$ (m - 1) \f$ degrees of freedom in the denominator.</span> |
| <a name="l00375"></a>00375 <span class="comment"> * - <tt>df1 BIGINT</tt> - Degrees of freedom in the numerator \f$ (n - 1) \f$</span> |
| <a name="l00376"></a>00376 <span class="comment"> * - <tt>df2 BIGINT</tt> - Degrees of freedom in the denominator \f$ (m - 1) \f$</span> |
| <a name="l00377"></a>00377 <span class="comment"> * - <tt>p_value_one_sided FLOAT8</tt> - Lower bound on one-sided p-value.</span> |
| <a name="l00378"></a>00378 <span class="comment"> * In detail, the result is \f$ \Pr[F \geq f \mid \sigma_X = \sigma_Y] \f$,</span> |
| <a name="l00379"></a>00379 <span class="comment"> * which is a lower bound on</span> |
| <a name="l00380"></a>00380 <span class="comment"> * \f$ \Pr[F \geq f \mid \sigma_X \leq \sigma_Y] \f$. Computed as</span> |
| <a name="l00381"></a>00381 <span class="comment"> * <tt>(1.0 - \ref fisher_f_cdf "fisher_f_cdf"(statistic))</tt>.</span> |
| <a name="l00382"></a>00382 <span class="comment"> * - <tt>p_value_two_sided FLOAT8</tt> - Two-sided p-value, i.e.,</span> |
| <a name="l00383"></a>00383 <span class="comment"> * \f$ 2 \cdot \min \{ p, 1 - p \} \f$ where</span> |
| <a name="l00384"></a>00384 <span class="comment"> * \f$ p = \Pr[ F \geq f \mid \sigma_X = \sigma_Y] \f$. Computed as</span> |
| <a name="l00385"></a>00385 <span class="comment"> * <tt>(min(p_value_one_sided, 1. - p_value_one_sided))</tt>.</span> |
| <a name="l00386"></a>00386 <span class="comment"> *</span> |
| <a name="l00387"></a>00387 <span class="comment"> * @usage</span> |
| <a name="l00388"></a>00388 <span class="comment"> * - Test null hypothesis that the variance of the first sample is at most (or</span> |
| <a name="l00389"></a>00389 <span class="comment"> * equal to, respectively) the variance of the second sample:</span> |
| <a name="l00390"></a>00390 <span class="comment"> * <pre>SELECT (f_test(<em>first</em>, <em>value</em>)).* FROM <em>source</em></pre></span> |
| <a name="l00391"></a>00391 <span class="comment"> *</span> |
| <a name="l00392"></a>00392 <span class="comment"> * @internal We reuse the two-sample t-test transition and merge functions.</span> |
| <a name="l00393"></a>00393 <span class="comment"> */</span> |
| <a name="l00394"></a>00394 CREATE AGGREGATE MADLIB_SCHEMA.f_test( |
| <a name="l00395"></a>00395 /*+ "first" */ BOOLEAN, |
| <a name="l00396"></a>00396 /*+ "value" */ DOUBLE PRECISION) ( |
| <a name="l00397"></a>00397 |
| <a name="l00398"></a>00398 SFUNC=MADLIB_SCHEMA.t_test_two_transition, |
| <a name="l00399"></a>00399 STYPE=DOUBLE PRECISION[], |
| <a name="l00400"></a>00400 FINALFUNC=MADLIB_SCHEMA.f_test_final, |
| <a name="l00401"></a>00401 m4_ifdef(<!__GREENPLUM__!>,<!PREFUNC=MADLIB_SCHEMA.t_test_merge_states,!>) |
| <a name="l00402"></a>00402 INITCOND='{0,0,0,0,0,0,0}<span class="stringliteral">'</span> |
| <a name="l00403"></a>00403 <span class="stringliteral">);</span> |
| <a name="l00404"></a>00404 <span class="stringliteral"></span> |
| <a name="l00405"></a>00405 <span class="stringliteral"></span> |
| <a name="l00406"></a>00406 <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.chi2_gof_test_transition(</span> |
| <a name="l00407"></a>00407 <span class="stringliteral"> state DOUBLE PRECISION[],</span> |
| <a name="l00408"></a>00408 <span class="stringliteral"> observed BIGINT,</span> |
| <a name="l00409"></a>00409 <span class="stringliteral"> expected DOUBLE PRECISION,</span> |
| <a name="l00410"></a>00410 <span class="stringliteral"> df BIGINT</span> |
| <a name="l00411"></a>00411 <span class="stringliteral">) RETURNS DOUBLE PRECISION[]</span> |
| <a name="l00412"></a>00412 <span class="stringliteral">AS '</span>MODULE_PATHNAME<span class="stringliteral">'</span> |
| <a name="l00413"></a>00413 <span class="stringliteral">LANGUAGE C</span> |
| <a name="l00414"></a>00414 <span class="stringliteral">IMMUTABLE</span> |
| <a name="l00415"></a>00415 <span class="stringliteral">STRICT;</span> |
| <a name="l00416"></a><a class="code" href="hypothesis__tests_8sql__in.html#a8f90d2f805a6ab3034f80a5967dffa1d">00416</a> <span class="stringliteral"></span> |
| <a name="l00417"></a>00417 <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.chi2_gof_test_transition(</span> |
| <a name="l00418"></a>00418 <span class="stringliteral"> state DOUBLE PRECISION[],</span> |
| <a name="l00419"></a>00419 <span class="stringliteral"> observed BIGINT,</span> |
| <a name="l00420"></a>00420 <span class="stringliteral"> expected DOUBLE PRECISION</span> |
| <a name="l00421"></a>00421 <span class="stringliteral">) RETURNS DOUBLE PRECISION[]</span> |
| <a name="l00422"></a>00422 <span class="stringliteral">AS '</span>MODULE_PATHNAME<span class="stringliteral">'</span> |
| <a name="l00423"></a>00423 <span class="stringliteral">LANGUAGE C</span> |
| <a name="l00424"></a>00424 <span class="stringliteral">IMMUTABLE</span> |
| <a name="l00425"></a>00425 <span class="stringliteral">STRICT;</span> |
| <a name="l00426"></a>00426 <span class="stringliteral"></span> |
| <a name="l00427"></a>00427 <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.chi2_gof_test_transition(</span> |
| <a name="l00428"></a>00428 <span class="stringliteral"> state DOUBLE PRECISION[],</span> |
| <a name="l00429"></a>00429 <span class="stringliteral"> observed BIGINT</span> |
| <a name="l00430"></a>00430 <span class="stringliteral">) RETURNS DOUBLE PRECISION[]</span> |
| <a name="l00431"></a>00431 <span class="stringliteral">AS '</span>MODULE_PATHNAME<span class="stringliteral">'</span> |
| <a name="l00432"></a>00432 <span class="stringliteral">LANGUAGE C</span> |
| <a name="l00433"></a>00433 <span class="stringliteral">IMMUTABLE</span> |
| <a name="l00434"></a>00434 <span class="stringliteral">STRICT;</span> |
| <a name="l00435"></a>00435 <span class="stringliteral"></span> |
| <a name="l00436"></a>00436 <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.chi2_gof_test_merge_states(</span> |
| <a name="l00437"></a>00437 <span class="stringliteral"> state1 DOUBLE PRECISION[],</span> |
| <a name="l00438"></a>00438 <span class="stringliteral"> state2 DOUBLE PRECISION[])</span> |
| <a name="l00439"></a>00439 <span class="stringliteral">RETURNS DOUBLE PRECISION[]</span> |
| <a name="l00440"></a>00440 <span class="stringliteral">AS '</span>MODULE_PATHNAME<span class="stringliteral">'</span> |
| <a name="l00441"></a>00441 <span class="stringliteral">LANGUAGE C</span> |
| <a name="l00442"></a>00442 <span class="stringliteral">IMMUTABLE</span> |
| <a name="l00443"></a>00443 <span class="stringliteral">STRICT;</span> |
| <a name="l00444"></a>00444 <span class="stringliteral"></span> |
| <a name="l00445"></a>00445 <span class="stringliteral">CREATE TYPE MADLIB_SCHEMA.chi2_test_result AS (</span> |
| <a name="l00446"></a>00446 <span class="stringliteral"> statistic DOUBLE PRECISION,</span> |
| <a name="l00447"></a>00447 <span class="stringliteral"> p_value DOUBLE PRECISION,</span> |
| <a name="l00448"></a>00448 <span class="stringliteral"> df BIGINT,</span> |
| <a name="l00449"></a>00449 <span class="stringliteral"> phi DOUBLE PRECISION,</span> |
| <a name="l00450"></a>00450 <span class="stringliteral"> contingency_coef DOUBLE PRECISION</span> |
| <a name="l00451"></a>00451 <span class="stringliteral">);</span> |
| <a name="l00452"></a>00452 <span class="stringliteral"></span> |
| <a name="l00453"></a>00453 <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.chi2_gof_test_final(</span> |
| <a name="l00454"></a>00454 <span class="stringliteral"> state DOUBLE PRECISION[]</span> |
| <a name="l00455"></a>00455 <span class="stringliteral">) RETURNS MADLIB_SCHEMA.chi2_test_result</span> |
| <a name="l00456"></a>00456 <span class="stringliteral">AS '</span>MODULE_PATHNAME<span class="stringliteral">'</span> |
| <a name="l00457"></a>00457 <span class="stringliteral">LANGUAGE C</span> |
| <a name="l00458"></a>00458 <span class="stringliteral">IMMUTABLE</span> |
| <a name="l00459"></a>00459 <span class="stringliteral">STRICT;</span> |
| <a name="l00460"></a>00460 <span class="stringliteral"></span><span class="comment"></span> |
| <a name="l00461"></a>00461 <span class="comment">/**</span> |
| <a name="l00462"></a>00462 <span class="comment"> * @brief Perform Pearson's chi-squared goodness-of-fit test</span> |
| <a name="l00463"></a>00463 <span class="comment"> *</span> |
| <a name="l00464"></a>00464 <span class="comment"> * Let \f$ n_1, \dots, n_k \f$ be a realization of a (vector) random variable</span> |
| <a name="l00465"></a>00465 <span class="comment"> * \f$ N = (N_1, \dots, N_k) \f$ that follows the multinomial distribution with</span> |
| <a name="l00466"></a>00466 <span class="comment"> * parameters \f$ k \f$ and \f$ p = (p_1, \dots, p_k) \f$. Test the null</span> |
| <a name="l00467"></a>00467 <span class="comment"> * hypothesis \f$ H_0 : p = p^0 \f$.</span> |
| <a name="l00468"></a>00468 <span class="comment"> *</span> |
| <a name="l00469"></a>00469 <span class="comment"> * @param observed Number \f$ n_i \f$ of observations of the current event/row</span> |
| <a name="l00470"></a>00470 <span class="comment"> * @param expected Expected number of observations of current event/row. This</span> |
| <a name="l00471"></a>00471 <span class="comment"> * number is not required to be normalized. That is, \f$ p^0_i \f$ will be</span> |
| <a name="l00472"></a>00472 <span class="comment"> * taken as \c expected divided by <tt>sum(expected)</tt>. Hence, if this</span> |
| <a name="l00473"></a>00473 <span class="comment"> * parameter is not specified, chi2_test() will by default use</span> |
| <a name="l00474"></a>00474 <span class="comment"> * \f$ p^0 = (\frac 1k, \dots, \frac 1k) \f$, i.e., test that \f$ p \f$ is a</span> |
| <a name="l00475"></a>00475 <span class="comment"> * discrete uniform distribution.</span> |
| <a name="l00476"></a>00476 <span class="comment"> * @param df Degrees of freedom. This is the number of events reduced by the</span> |
| <a name="l00477"></a>00477 <span class="comment"> * degree of freedom lost by using the observed numbers for defining the</span> |
| <a name="l00478"></a>00478 <span class="comment"> * expected number of observations. If this parameter is 0, the degree</span> |
| <a name="l00479"></a>00479 <span class="comment"> * of freedom is taken as \f$ (k - 1) \f$.</span> |
| <a name="l00480"></a>00480 <span class="comment"> *</span> |
| <a name="l00481"></a>00481 <span class="comment"> * @return A composite value as follows. Let \f$ n = \sum_{i=1}^n n_i \f$.</span> |
| <a name="l00482"></a>00482 <span class="comment"> * - <tt>statistic FLOAT8</tt> - Statistic</span> |
| <a name="l00483"></a>00483 <span class="comment"> * \f[</span> |
| <a name="l00484"></a>00484 <span class="comment"> * \chi^2 = \sum_{i=1}^k \frac{(n_i - np_i)^2}{np_i}</span> |
| <a name="l00485"></a>00485 <span class="comment"> * \f]</span> |
| <a name="l00486"></a>00486 <span class="comment"> * The corresponding random</span> |
| <a name="l00487"></a>00487 <span class="comment"> * variable is approximately chi-squared distributed with</span> |
| <a name="l00488"></a>00488 <span class="comment"> * \c df degrees of freedom.</span> |
| <a name="l00489"></a>00489 <span class="comment"> * - <tt>df BIGINT</tt> - Degrees of freedom</span> |
| <a name="l00490"></a>00490 <span class="comment"> * - <tt>p_value FLOAT8</tt> - Approximate p-value, i.e.,</span> |
| <a name="l00491"></a>00491 <span class="comment"> * \f$ \Pr[X^2 \geq \chi^2 \mid p = p^0] \f$. Computed as</span> |
| <a name="l00492"></a>00492 <span class="comment"> * <tt>(1.0 - \ref chi_squared_cdf "chi_squared_cdf"(statistic))</tt>.</span> |
| <a name="l00493"></a>00493 <span class="comment"> * - <tt>phi FLOAT8</tt> - Phi coefficient, i.e.,</span> |
| <a name="l00494"></a>00494 <span class="comment"> * \f$ \phi = \sqrt{\frac{\chi^2}{n}} \f$</span> |
| <a name="l00495"></a>00495 <span class="comment"> * - <tt>contingency_coef FLOAT8</tt> - Contingency coefficient, i.e.,</span> |
| <a name="l00496"></a>00496 <span class="comment"> * \f$ \sqrt{\frac{\chi^2}{n + \chi^2}} \f$</span> |
| <a name="l00497"></a>00497 <span class="comment"> *</span> |
| <a name="l00498"></a>00498 <span class="comment"> * @usage</span> |
| <a name="l00499"></a>00499 <span class="comment"> * - Test null hypothesis that all possible outcomes of a categorical variable</span> |
| <a name="l00500"></a>00500 <span class="comment"> * are equally likely:</span> |
| <a name="l00501"></a>00501 <span class="comment"> * <pre>SELECT (chi2_gof_test(<em>observed</em>, 1, NULL)).* FROM <em>source</em></pre></span> |
| <a name="l00502"></a>00502 <span class="comment"> * - Test null hypothesis that two categorical variables are independent.</span> |
| <a name="l00503"></a>00503 <span class="comment"> * Such data is often shown in a <em>contingency table</em> (also known as</span> |
| <a name="l00504"></a>00504 <span class="comment"> * \em crosstab). A crosstab is a matrix where possible values for the first</span> |
| <a name="l00505"></a>00505 <span class="comment"> * variable correspond to rows and values for the second variable to</span> |
| <a name="l00506"></a>00506 <span class="comment"> * columns. The matrix elements are the observation frequencies of the</span> |
| <a name="l00507"></a>00507 <span class="comment"> * joint occurrence of the respective values.</span> |
| <a name="l00508"></a>00508 <span class="comment"> * chi2_gof_test() assumes that the crosstab is stored in normalized form,</span> |
| <a name="l00509"></a>00509 <span class="comment"> * i.e., there are three columns <tt><em>var1</em></tt>,</span> |
| <a name="l00510"></a>00510 <span class="comment"> * <tt><em>var2</em></tt>, <tt><em>observed</em></tt>.</span> |
| <a name="l00511"></a>00511 <span class="comment"> * <pre>SELECT (chi2_gof_test(<em>observed</em>, expected, deg_freedom)).*</span> |
| <a name="l00512"></a>00512 <span class="comment"> *FROM (</span> |
| <a name="l00513"></a>00513 <span class="comment"> * SELECT</span> |
| <a name="l00514"></a>00514 <span class="comment"> * <em>observed</em>,</span> |
| <a name="l00515"></a>00515 <span class="comment"> * sum(<em>observed</em>) OVER (PARTITION BY var1)::DOUBLE PRECISION</span> |
| <a name="l00516"></a>00516 <span class="comment"> * * sum(<em>observed</em>) OVER (PARTITION BY var2) AS expected</span> |
| <a name="l00517"></a>00517 <span class="comment"> * FROM <em>source</em></span> |
| <a name="l00518"></a>00518 <span class="comment"> *) p, (</span> |
| <a name="l00519"></a>00519 <span class="comment"> * SELECT</span> |
| <a name="l00520"></a>00520 <span class="comment"> * (count(DISTINCT <em>var1</em>) - 1) * (count(DISTINCT <em>var2</em>) - 1) AS deg_freedom</span> |
| <a name="l00521"></a>00521 <span class="comment"> * FROM <em>source</em></span> |
| <a name="l00522"></a>00522 <span class="comment"> *) q;</pre></span> |
| <a name="l00523"></a>00523 <span class="comment"> */</span> |
| <a name="l00524"></a>00524 CREATE AGGREGATE MADLIB_SCHEMA.chi2_gof_test( |
| <a name="l00525"></a>00525 /*+ observed */ BIGINT, |
| <a name="l00526"></a>00526 /*+ expected */ DOUBLE PRECISION /*+ DEFAULT 1 */, |
| <a name="l00527"></a>00527 /*+ df */ BIGINT /*+ DEFAULT 0 */ |
| <a name="l00528"></a>00528 ) ( |
| <a name="l00529"></a>00529 SFUNC=MADLIB_SCHEMA.chi2_gof_test_transition, |
| <a name="l00530"></a>00530 STYPE=DOUBLE PRECISION[], |
| <a name="l00531"></a>00531 FINALFUNC=MADLIB_SCHEMA.chi2_gof_test_final, |
| <a name="l00532"></a>00532 m4_ifdef(<!__GREENPLUM__!>,<!PREFUNC=MADLIB_SCHEMA.chi2_gof_test_merge_states,!>) |
| <a name="l00533"></a>00533 INITCOND='{0,0,0,0,0,0}<span class="stringliteral">'</span> |
| <a name="l00534"></a>00534 <span class="stringliteral">);</span> |
| <a name="l00535"></a>00535 <span class="stringliteral"></span> |
| <a name="l00536"></a>00536 <span class="stringliteral">CREATE AGGREGATE MADLIB_SCHEMA.chi2_gof_test(</span> |
| <a name="l00537"></a>00537 <span class="stringliteral"> /*+ observed */ BIGINT,</span> |
| <a name="l00538"></a>00538 <span class="stringliteral"> /*+ expected */ DOUBLE PRECISION</span> |
| <a name="l00539"></a>00539 <span class="stringliteral">) (</span> |
| <a name="l00540"></a>00540 <span class="stringliteral"> SFUNC=MADLIB_SCHEMA.chi2_gof_test_transition,</span> |
| <a name="l00541"></a>00541 <span class="stringliteral"> STYPE=DOUBLE PRECISION[],</span> |
| <a name="l00542"></a>00542 <span class="stringliteral"> FINALFUNC=MADLIB_SCHEMA.chi2_gof_test_final,</span> |
| <a name="l00543"></a>00543 <span class="stringliteral"> m4_ifdef(<!__GREENPLUM__!>,<!PREFUNC=MADLIB_SCHEMA.chi2_gof_test_merge_states,!>)</span> |
| <a name="l00544"></a>00544 <span class="stringliteral"> INITCOND='</span>{0,0,0,0,0,0,0}<span class="stringliteral">'</span> |
| <a name="l00545"></a>00545 <span class="stringliteral">);</span> |
| <a name="l00546"></a><a class="code" href="hypothesis__tests_8sql__in.html#afc6a7ac3eada83df681bc6efeddfd9eb">00546</a> <span class="stringliteral"></span> |
| <a name="l00547"></a>00547 <span class="stringliteral">CREATE AGGREGATE MADLIB_SCHEMA.chi2_gof_test(</span> |
| <a name="l00548"></a>00548 <span class="stringliteral"> /*+ observed */ BIGINT</span> |
| <a name="l00549"></a>00549 <span class="stringliteral">) (</span> |
| <a name="l00550"></a>00550 <span class="stringliteral"> SFUNC=MADLIB_SCHEMA.chi2_gof_test_transition,</span> |
| <a name="l00551"></a>00551 <span class="stringliteral"> STYPE=DOUBLE PRECISION[],</span> |
| <a name="l00552"></a>00552 <span class="stringliteral"> FINALFUNC=MADLIB_SCHEMA.chi2_gof_test_final,</span> |
| <a name="l00553"></a>00553 <span class="stringliteral"> m4_ifdef(<!__GREENPLUM__!>,<!PREFUNC=MADLIB_SCHEMA.chi2_gof_test_merge_states,!>)</span> |
| <a name="l00554"></a>00554 <span class="stringliteral"> INITCOND='</span>{0,0,0,0,0,0,0}<span class="stringliteral">'</span> |
| <a name="l00555"></a>00555 <span class="stringliteral">);</span> |
| <a name="l00556"></a>00556 <span class="stringliteral"></span> |
| <a name="l00557"></a>00557 <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.ks_test_transition(</span> |
| <a name="l00558"></a>00558 <span class="stringliteral"> state DOUBLE PRECISION[],</span> |
| <a name="l00559"></a>00559 <span class="stringliteral"> "first" BOOLEAN,</span> |
| <a name="l00560"></a>00560 <span class="stringliteral"> "value" DOUBLE PRECISION,</span> |
| <a name="l00561"></a>00561 <span class="stringliteral"> "numFirst" BIGINT,</span> |
| <a name="l00562"></a>00562 <span class="stringliteral"> "numSecond" BIGINT</span> |
| <a name="l00563"></a>00563 <span class="stringliteral">) RETURNS DOUBLE PRECISION[]</span> |
| <a name="l00564"></a>00564 <span class="stringliteral">AS '</span>MODULE_PATHNAME<span class="stringliteral">'</span> |
| <a name="l00565"></a>00565 <span class="stringliteral">LANGUAGE C</span> |
| <a name="l00566"></a>00566 <span class="stringliteral">IMMUTABLE</span> |
| <a name="l00567"></a>00567 <span class="stringliteral">STRICT;</span> |
| <a name="l00568"></a>00568 <span class="stringliteral"></span> |
| <a name="l00569"></a>00569 <span class="stringliteral">CREATE TYPE MADLIB_SCHEMA.ks_test_result AS (</span> |
| <a name="l00570"></a>00570 <span class="stringliteral"> statistic DOUBLE PRECISION,</span> |
| <a name="l00571"></a>00571 <span class="stringliteral"> k_statistic DOUBLE PRECISION,</span> |
| <a name="l00572"></a>00572 <span class="stringliteral"> p_value DOUBLE PRECISION</span> |
| <a name="l00573"></a>00573 <span class="stringliteral">);</span> |
| <a name="l00574"></a>00574 <span class="stringliteral"></span> |
| <a name="l00575"></a>00575 <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.ks_test_final(</span> |
| <a name="l00576"></a>00576 <span class="stringliteral"> state DOUBLE PRECISION[])</span> |
| <a name="l00577"></a>00577 <span class="stringliteral">RETURNS MADLIB_SCHEMA.ks_test_result</span> |
| <a name="l00578"></a>00578 <span class="stringliteral">AS '</span>MODULE_PATHNAME<span class="stringliteral">'</span> |
| <a name="l00579"></a>00579 <span class="stringliteral">LANGUAGE C IMMUTABLE STRICT;</span> |
| <a name="l00580"></a>00580 <span class="stringliteral"></span><span class="comment"></span> |
| <a name="l00581"></a>00581 <span class="comment">/**</span> |
| <a name="l00582"></a>00582 <span class="comment"> * @brief Perform Kolmogorov-Smirnov test</span> |
| <a name="l00583"></a>00583 <span class="comment"> *</span> |
| <a name="l00584"></a>00584 <span class="comment"> * Given realizations \f$ x_1, \dots, x_m \f$ and \f$ y_1, \dots, y_m \f$ of</span> |
| <a name="l00585"></a>00585 <span class="comment"> * i.i.d. random variables \f$ X_1, \dots, X_m \f$ and i.i.d.</span> |
| <a name="l00586"></a>00586 <span class="comment"> * \f$ Y_1, \dots, Y_n \f$, respectively, test the null hypothesis that the</span> |
| <a name="l00587"></a>00587 <span class="comment"> * underlying distributions function \f$ F_X, F_Y \f$ are identical, i.e.,</span> |
| <a name="l00588"></a>00588 <span class="comment"> * \f$ H_0 : F_X = F_Y \f$.</span> |
| <a name="l00589"></a>00589 <span class="comment"> *</span> |
| <a name="l00590"></a>00590 <span class="comment"> * @param first Determines whether the value belongs to the first</span> |
| <a name="l00591"></a>00591 <span class="comment"> * (if \c TRUE) or the second sample (if \c FALSE)</span> |
| <a name="l00592"></a>00592 <span class="comment"> * @param value Value of random variate \f$ x_i \f$ or \f$ y_i \f$</span> |
| <a name="l00593"></a>00593 <span class="comment"> * @param m Size \f$ m \f$ of the first sample. See usage instructions below.</span> |
| <a name="l00594"></a>00594 <span class="comment"> * @param n Size of the second sample. See usage instructions below.</span> |
| <a name="l00595"></a>00595 <span class="comment"> *</span> |
| <a name="l00596"></a>00596 <span class="comment"> * @return A composite value.</span> |
| <a name="l00597"></a>00597 <span class="comment"> * - <tt>statistic FLOAT8</tt> - Kolmogorov–Smirnov statistic</span> |
| <a name="l00598"></a>00598 <span class="comment"> * \f[</span> |
| <a name="l00599"></a>00599 <span class="comment"> * d = \max_{t \in \mathbb R} |F_x(t) - F_y(t)|</span> |
| <a name="l00600"></a>00600 <span class="comment"> * \f]</span> |
| <a name="l00601"></a>00601 <span class="comment"> * where \f$ F_x(t) := \frac 1m |\{ i \mid x_i \leq t \}| \f$ and</span> |
| <a name="l00602"></a>00602 <span class="comment"> * \f$ F_y \f$ (defined likewise) are the empirical distribution functions.</span> |
| <a name="l00603"></a>00603 <span class="comment"> * - <tt>k_statistic FLOAT8</tt> - Kolmogorov statistic</span> |
| <a name="l00604"></a>00604 <span class="comment"> * \f$</span> |
| <a name="l00605"></a>00605 <span class="comment"> * k = r + 0.12 + \frac{0.11}{r}</span> |
| <a name="l00606"></a>00606 <span class="comment"> * \f$</span> |
| <a name="l00607"></a>00607 <span class="comment"> * where</span> |
| <a name="l00608"></a>00608 <span class="comment"> * \f$</span> |
| <a name="l00609"></a>00609 <span class="comment"> * r = \sqrt{\frac{m n}{m+n}}.</span> |
| <a name="l00610"></a>00610 <span class="comment"> * \f$</span> |
| <a name="l00611"></a>00611 <span class="comment"> * Then \f$ k \f$ is approximately Kolmogorov distributed.</span> |
| <a name="l00612"></a>00612 <span class="comment"> * - <tt>p_value FLOAT8</tt> - Approximate p-value, i.e., an approximate value</span> |
| <a name="l00613"></a>00613 <span class="comment"> * for \f$ \Pr[D \geq d \mid F_X = F_Y] \f$. Computed as</span> |
| <a name="l00614"></a>00614 <span class="comment"> * <tt>(1.0 - \ref kolmogorov_cdf "kolmogorov_cdf"(k_statistic))</tt>.</span> |
| <a name="l00615"></a>00615 <span class="comment"> *</span> |
| <a name="l00616"></a>00616 <span class="comment"> * @usage</span> |
| <a name="l00617"></a>00617 <span class="comment"> * - Test null hypothesis that two samples stem from the same distribution:</span> |
| <a name="l00618"></a>00618 <span class="comment"> * <pre>SELECT (ks_test(<em>first</em>, <em>value</em>,</span> |
| <a name="l00619"></a>00619 <span class="comment"> * (SELECT count(<em>value</em>) FROM <em>source</em> WHERE <em>first</em>),</span> |
| <a name="l00620"></a>00620 <span class="comment"> * (SELECT count(<em>value</em>) FROM <em>source</em> WHERE NOT <em>first</em>)</span> |
| <a name="l00621"></a>00621 <span class="comment"> * ORDER BY <em>value</em></span> |
| <a name="l00622"></a>00622 <span class="comment"> *)).* FROM <em>source</em></pre></span> |
| <a name="l00623"></a>00623 <span class="comment"> *</span> |
| <a name="l00624"></a>00624 <span class="comment"> * @note</span> |
| <a name="l00625"></a>00625 <span class="comment"> * This aggregate must be used as an ordered aggregate</span> |
| <a name="l00626"></a>00626 <span class="comment"> * (<tt>ORDER BY \em value</tt>) and will raise an exception if values are</span> |
| <a name="l00627"></a>00627 <span class="comment"> * not ordered.</span> |
| <a name="l00628"></a>00628 <span class="comment"> */</span> |
| <a name="l00629"></a>00629 m4_ifdef(<!__HAS_ORDERED_AGGREGATES__!>,<! |
| <a name="l00630"></a>00630 CREATE |
| <a name="l00631"></a>00631 m4_ifdef(<!__GREENPLUM__!>,<!ORDERED!>) |
| <a name="l00632"></a>00632 AGGREGATE MADLIB_SCHEMA.ks_test( |
| <a name="l00633"></a>00633 /*+ "first" */ BOOLEAN, |
| <a name="l00634"></a>00634 /*+ "value" */ DOUBLE PRECISION, |
| <a name="l00635"></a>00635 /*+ m */ BIGINT, |
| <a name="l00636"></a>00636 /*+ n */ BIGINT |
| <a name="l00637"></a>00637 ) ( |
| <a name="l00638"></a>00638 SFUNC=MADLIB_SCHEMA.ks_test_transition, |
| <a name="l00639"></a>00639 STYPE=DOUBLE PRECISION[], |
| <a name="l00640"></a>00640 FINALFUNC=MADLIB_SCHEMA.ks_test_final, |
| <a name="l00641"></a>00641 INITCOND='{0,0,0,0,0,0,0}<span class="stringliteral">'</span> |
| <a name="l00642"></a>00642 <span class="stringliteral">);</span> |
| <a name="l00643"></a>00643 <span class="stringliteral">!>)</span> |
| <a name="l00644"></a>00644 <span class="stringliteral"></span> |
| <a name="l00645"></a>00645 <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.mw_test_transition(</span> |
| <a name="l00646"></a>00646 <span class="stringliteral"> state DOUBLE PRECISION[],</span> |
| <a name="l00647"></a>00647 <span class="stringliteral"> "first" BOOLEAN,</span> |
| <a name="l00648"></a>00648 <span class="stringliteral"> "value" DOUBLE PRECISION</span> |
| <a name="l00649"></a>00649 <span class="stringliteral">) RETURNS DOUBLE PRECISION[]</span> |
| <a name="l00650"></a>00650 <span class="stringliteral">AS '</span>MODULE_PATHNAME<span class="stringliteral">'</span> |
| <a name="l00651"></a>00651 <span class="stringliteral">LANGUAGE C</span> |
| <a name="l00652"></a><a class="code" href="hypothesis__tests_8sql__in.html#af45ae9d1275d385bbacd18bff688ba7f">00652</a> <span class="stringliteral">IMMUTABLE</span> |
| <a name="l00653"></a>00653 <span class="stringliteral">STRICT;</span> |
| <a name="l00654"></a>00654 <span class="stringliteral"></span> |
| <a name="l00655"></a>00655 <span class="stringliteral">CREATE TYPE MADLIB_SCHEMA.mw_test_result AS (</span> |
| <a name="l00656"></a>00656 <span class="stringliteral"> statistic DOUBLE PRECISION,</span> |
| <a name="l00657"></a>00657 <span class="stringliteral"> u_statistic DOUBLE PRECISION,</span> |
| <a name="l00658"></a>00658 <span class="stringliteral"> p_value_one_sided DOUBLE PRECISION,</span> |
| <a name="l00659"></a>00659 <span class="stringliteral"> p_value_two_sided DOUBLE PRECISION</span> |
| <a name="l00660"></a>00660 <span class="stringliteral">);</span> |
| <a name="l00661"></a>00661 <span class="stringliteral"></span> |
| <a name="l00662"></a>00662 <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.mw_test_final(</span> |
| <a name="l00663"></a>00663 <span class="stringliteral"> state DOUBLE PRECISION[])</span> |
| <a name="l00664"></a>00664 <span class="stringliteral">RETURNS MADLIB_SCHEMA.mw_test_result</span> |
| <a name="l00665"></a>00665 <span class="stringliteral">AS '</span>MODULE_PATHNAME<span class="stringliteral">'</span> |
| <a name="l00666"></a>00666 <span class="stringliteral">LANGUAGE C IMMUTABLE STRICT;</span> |
| <a name="l00667"></a>00667 <span class="stringliteral"></span><span class="comment"></span> |
| <a name="l00668"></a>00668 <span class="comment">/**</span> |
| <a name="l00669"></a>00669 <span class="comment"> * @brief Perform Mann-Whitney test</span> |
| <a name="l00670"></a>00670 <span class="comment"> *</span> |
| <a name="l00671"></a>00671 <span class="comment"> * Given realizations \f$ x_1, \dots, x_m \f$ and \f$ y_1, \dots, y_m \f$ of</span> |
| <a name="l00672"></a>00672 <span class="comment"> * i.i.d. random variables \f$ X_1, \dots, X_m \f$ and i.i.d.</span> |
| <a name="l00673"></a>00673 <span class="comment"> * \f$ Y_1, \dots, Y_n \f$, respectively, test the null hypothesis that the</span> |
| <a name="l00674"></a>00674 <span class="comment"> * underlying distributions are equal, i.e.,</span> |
| <a name="l00675"></a>00675 <span class="comment"> * \f$ H_0 : \forall i,j: \Pr[X_i > Y_j] + \frac{\Pr[X_i = Y_j]}{2} = \frac 12 \f$.</span> |
| <a name="l00676"></a>00676 <span class="comment"> *</span> |
| <a name="l00677"></a>00677 <span class="comment"> * @param first Determines whether the value belongs to the first</span> |
| <a name="l00678"></a>00678 <span class="comment"> * (if \c TRUE) or the second sample (if \c FALSE)</span> |
| <a name="l00679"></a>00679 <span class="comment"> * @param value Value of random variate \f$ x_i \f$ or \f$ y_i \f$</span> |
| <a name="l00680"></a>00680 <span class="comment"> *</span> |
| <a name="l00681"></a>00681 <span class="comment"> * @return A composite value.</span> |
| <a name="l00682"></a>00682 <span class="comment"> * - <tt>statistic FLOAT8</tt> - Statistic</span> |
| <a name="l00683"></a>00683 <span class="comment"> * \f[</span> |
| <a name="l00684"></a>00684 <span class="comment"> * z = \frac{u - \bar x}{\sqrt{\frac{mn(m+n+1)}{12}}}</span> |
| <a name="l00685"></a>00685 <span class="comment"> * \f]</span> |
| <a name="l00686"></a>00686 <span class="comment"> * where \f$ u \f$ is the u-statistic computed as follows. The z-statistic</span> |
| <a name="l00687"></a>00687 <span class="comment"> * is approximately standard normally distributed.</span> |
| <a name="l00688"></a>00688 <span class="comment"> * - <tt>u_statistic FLOAT8</tt> - Statistic</span> |
| <a name="l00689"></a>00689 <span class="comment"> * \f$ u = \min \{ u_x, u_y \} \f$ where</span> |
| <a name="l00690"></a>00690 <span class="comment"> * \f[</span> |
| <a name="l00691"></a>00691 <span class="comment"> * u_x = mn + \binom{m+1}{2} - \sum_{i=1}^m r_{x,i}</span> |
| <a name="l00692"></a>00692 <span class="comment"> * \f]</span> |
| <a name="l00693"></a>00693 <span class="comment"> * where</span> |
| <a name="l00694"></a>00694 <span class="comment"> * \f[</span> |
| <a name="l00695"></a>00695 <span class="comment"> * r_{x,i}</span> |
| <a name="l00696"></a>00696 <span class="comment"> * = \{ j \mid x_j < x_i \} + \{ j \mid y_j < x_i \} +</span> |
| <a name="l00697"></a>00697 <span class="comment"> * \frac{\{ j \mid x_j = x_i \} + \{ j \mid y_j = x_i \} + 1}{2}</span> |
| <a name="l00698"></a>00698 <span class="comment"> * \f]</span> |
| <a name="l00699"></a>00699 <span class="comment"> * is defined as the rank of \f$ x_i \f$ in the combined list of all</span> |
| <a name="l00700"></a>00700 <span class="comment"> * \f$ m+n \f$ observations. For ties, the average rank of all equal values</span> |
| <a name="l00701"></a>00701 <span class="comment"> * is used.</span> |
| <a name="l00702"></a>00702 <span class="comment"> * - <tt>p_value_one_sided FLOAT8</tt> - Approximate one-sided p-value, i.e.,</span> |
| <a name="l00703"></a>00703 <span class="comment"> * an approximate value for \f$ \Pr[Z \geq z \mid H_0] \f$. Computed as</span> |
| <a name="l00704"></a>00704 <span class="comment"> * <tt>(1.0 - \ref normal_cdf "normal_cdf"(z_statistic))</tt>.</span> |
| <a name="l00705"></a>00705 <span class="comment"> * - <tt>p_value_two_sided FLOAT8</tt> - Approximate two-sided p-value, i.e.,</span> |
| <a name="l00706"></a>00706 <span class="comment"> * an approximate value for \f$ \Pr[|Z| \geq |z| \mid H_0] \f$. Computed as</span> |
| <a name="l00707"></a>00707 <span class="comment"> * <tt>(2 * \ref normal_cdf "normal_cdf"(-abs(z_statistic)))</tt>.</span> |
| <a name="l00708"></a>00708 <span class="comment"> *</span> |
| <a name="l00709"></a>00709 <span class="comment"> * @usage</span> |
| <a name="l00710"></a>00710 <span class="comment"> * - Test null hypothesis that two samples stem from the same distribution:</span> |
| <a name="l00711"></a>00711 <span class="comment"> * <pre>SELECT (mw_test(<em>first</em>, <em>value</em> ORDER BY <em>value</em>)).* FROM <em>source</em></pre></span> |
| <a name="l00712"></a>00712 <span class="comment"> *</span> |
| <a name="l00713"></a>00713 <span class="comment"> * @note</span> |
| <a name="l00714"></a>00714 <span class="comment"> * This aggregate must be used as an ordered aggregate</span> |
| <a name="l00715"></a>00715 <span class="comment"> * (<tt>ORDER BY \em value</tt>) and will raise an exception if values are</span> |
| <a name="l00716"></a>00716 <span class="comment"> * not ordered.</span> |
| <a name="l00717"></a>00717 <span class="comment"> */</span> |
| <a name="l00718"></a>00718 m4_ifdef(<!__HAS_ORDERED_AGGREGATES__!>,<! |
| <a name="l00719"></a>00719 CREATE |
| <a name="l00720"></a>00720 m4_ifdef(<!__GREENPLUM__!>,<!ORDERED!>) |
| <a name="l00721"></a>00721 AGGREGATE MADLIB_SCHEMA.mw_test( |
| <a name="l00722"></a>00722 /*+ "first" */ BOOLEAN, |
| <a name="l00723"></a>00723 /*+ "value" */ DOUBLE PRECISION |
| <a name="l00724"></a>00724 ) ( |
| <a name="l00725"></a>00725 SFUNC=MADLIB_SCHEMA.mw_test_transition, |
| <a name="l00726"></a>00726 STYPE=DOUBLE PRECISION[], |
| <a name="l00727"></a>00727 FINALFUNC=MADLIB_SCHEMA.mw_test_final, |
| <a name="l00728"></a>00728 INITCOND='{0,0,0,0,0,0,0}<span class="stringliteral">'</span> |
| <a name="l00729"></a>00729 <span class="stringliteral">);</span> |
| <a name="l00730"></a>00730 <span class="stringliteral">!>)</span> |
| <a name="l00731"></a>00731 <span class="stringliteral"></span> |
| <a name="l00732"></a>00732 <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.wsr_test_transition(</span> |
| <a name="l00733"></a>00733 <span class="stringliteral"> state DOUBLE PRECISION[],</span> |
| <a name="l00734"></a>00734 <span class="stringliteral"> value DOUBLE PRECISION,</span> |
| <a name="l00735"></a>00735 <span class="stringliteral"> "precision" DOUBLE PRECISION</span> |
| <a name="l00736"></a>00736 <span class="stringliteral">) RETURNS DOUBLE PRECISION[]</span> |
| <a name="l00737"></a>00737 <span class="stringliteral">AS '</span>MODULE_PATHNAME<span class="stringliteral">'</span> |
| <a name="l00738"></a>00738 <span class="stringliteral">LANGUAGE C</span> |
| <a name="l00739"></a>00739 <span class="stringliteral">IMMUTABLE</span> |
| <a name="l00740"></a>00740 <span class="stringliteral">STRICT;</span> |
| <a name="l00741"></a><a class="code" href="hypothesis__tests_8sql__in.html#a32cdc58e8a5d149dd90304805de07fbd">00741</a> <span class="stringliteral"></span> |
| <a name="l00742"></a>00742 <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.wsr_test_transition(</span> |
| <a name="l00743"></a>00743 <span class="stringliteral"> state DOUBLE PRECISION[],</span> |
| <a name="l00744"></a>00744 <span class="stringliteral"> value DOUBLE PRECISION</span> |
| <a name="l00745"></a>00745 <span class="stringliteral">) RETURNS DOUBLE PRECISION[]</span> |
| <a name="l00746"></a>00746 <span class="stringliteral">AS '</span>MODULE_PATHNAME<span class="stringliteral">'</span> |
| <a name="l00747"></a>00747 <span class="stringliteral">LANGUAGE C</span> |
| <a name="l00748"></a>00748 <span class="stringliteral">IMMUTABLE</span> |
| <a name="l00749"></a>00749 <span class="stringliteral">STRICT;</span> |
| <a name="l00750"></a>00750 <span class="stringliteral"></span> |
| <a name="l00751"></a>00751 <span class="stringliteral"></span> |
| <a name="l00752"></a>00752 <span class="stringliteral">CREATE TYPE MADLIB_SCHEMA.wsr_test_result AS (</span> |
| <a name="l00753"></a>00753 <span class="stringliteral"> statistic DOUBLE PRECISION,</span> |
| <a name="l00754"></a>00754 <span class="stringliteral"> rank_sum_pos FLOAT8,</span> |
| <a name="l00755"></a>00755 <span class="stringliteral"> rank_sum_neg FLOAT8,</span> |
| <a name="l00756"></a>00756 <span class="stringliteral"> num BIGINT,</span> |
| <a name="l00757"></a>00757 <span class="stringliteral"> z_statistic DOUBLE PRECISION,</span> |
| <a name="l00758"></a>00758 <span class="stringliteral"> p_value_one_sided DOUBLE PRECISION,</span> |
| <a name="l00759"></a>00759 <span class="stringliteral"> p_value_two_sided DOUBLE PRECISION</span> |
| <a name="l00760"></a>00760 <span class="stringliteral">);</span> |
| <a name="l00761"></a>00761 <span class="stringliteral"></span> |
| <a name="l00762"></a>00762 <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.wsr_test_final(</span> |
| <a name="l00763"></a>00763 <span class="stringliteral"> state DOUBLE PRECISION[])</span> |
| <a name="l00764"></a>00764 <span class="stringliteral">RETURNS MADLIB_SCHEMA.wsr_test_result</span> |
| <a name="l00765"></a>00765 <span class="stringliteral">AS '</span>MODULE_PATHNAME<span class="stringliteral">'</span> |
| <a name="l00766"></a>00766 <span class="stringliteral">LANGUAGE C IMMUTABLE STRICT;</span> |
| <a name="l00767"></a>00767 <span class="stringliteral"></span><span class="comment"></span> |
| <a name="l00768"></a>00768 <span class="comment">/**</span> |
| <a name="l00769"></a>00769 <span class="comment"> * @brief Perform Wilcoxon-Signed-Rank test</span> |
| <a name="l00770"></a>00770 <span class="comment"> *</span> |
| <a name="l00771"></a>00771 <span class="comment"> * Given realizations \f$ x_1, \dots, x_n \f$ of i.i.d. random variables</span> |
| <a name="l00772"></a>00772 <span class="comment"> * \f$ X_1, \dots, X_n \f$ with unknown mean \f$ \mu \f$, test the null</span> |
| <a name="l00773"></a>00773 <span class="comment"> * hypotheses \f$ H_0 : \mu \leq 0 \f$ and \f$ H_0 : \mu = 0 \f$.</span> |
| <a name="l00774"></a>00774 <span class="comment"> *</span> |
| <a name="l00775"></a>00775 <span class="comment"> * @param value Value of random variate \f$ x_i \f$ or \f$ y_i \f$. Values of 0</span> |
| <a name="l00776"></a>00776 <span class="comment"> * are ignored (i.e., they do not count towards \f$ n \f$).</span> |
| <a name="l00777"></a>00777 <span class="comment"> * @param precision The precision \f$ \epsilon_i \f$ with which value is known.</span> |
| <a name="l00778"></a>00778 <span class="comment"> * The precision determines the handling of ties. The current value</span> |
| <a name="l00779"></a>00779 <span class="comment"> * \f$ v_i \f$ is regarded a tie with the previous value \f$ v_{i-1} \f$ if</span> |
| <a name="l00780"></a>00780 <span class="comment"> * \f$ v_i - \epsilon_i \leq \max_{j=1, \dots, i-1} v_j + \epsilon_j \f$.</span> |
| <a name="l00781"></a>00781 <span class="comment"> * If \c precision is negative, then it will be treated as</span> |
| <a name="l00782"></a>00782 <span class="comment"> * <tt>value * 2^(-52)</tt>. (Note that \f$ 2^{-52} \f$ is the machine</span> |
| <a name="l00783"></a>00783 <span class="comment"> * epsilon for type <tt>DOUBLE PRECISION</tt>.)</span> |
| <a name="l00784"></a>00784 <span class="comment"> *</span> |
| <a name="l00785"></a>00785 <span class="comment"> * @return A composite value:</span> |
| <a name="l00786"></a>00786 <span class="comment"> * - <tt>statistic FLOAT8</tt> - statistic computed as follows. Let</span> |
| <a name="l00787"></a>00787 <span class="comment"> * \f$</span> |
| <a name="l00788"></a>00788 <span class="comment"> * w^+ = \sum_{i \mid x_i > 0} r_i</span> |
| <a name="l00789"></a>00789 <span class="comment"> * \f$</span> |
| <a name="l00790"></a>00790 <span class="comment"> * and</span> |
| <a name="l00791"></a>00791 <span class="comment"> * \f$</span> |
| <a name="l00792"></a>00792 <span class="comment"> * w^- = \sum_{i \mid x_i < 0} r_i</span> |
| <a name="l00793"></a>00793 <span class="comment"> * \f$</span> |
| <a name="l00794"></a>00794 <span class="comment"> * be the <em>signed rank sums</em> where</span> |
| <a name="l00795"></a>00795 <span class="comment"> * \f[</span> |
| <a name="l00796"></a>00796 <span class="comment"> * r_i</span> |
| <a name="l00797"></a>00797 <span class="comment"> * = \{ j \mid |x_j| < |x_i| \}</span> |
| <a name="l00798"></a>00798 <span class="comment"> * + \frac{\{ j \mid |x_j| = |x_i| \} + 1}{2}.</span> |
| <a name="l00799"></a>00799 <span class="comment"> * \f]</span> |
| <a name="l00800"></a>00800 <span class="comment"> * The Wilcoxon signed-rank statistic is \f$ w = \min \{ w^+, w^- \} \f$.</span> |
| <a name="l00801"></a>00801 <span class="comment"> * - <tt>rank_sum_pos FLOAT8</tt> - rank sum of all positive values, i.e., \f$ w^+ \f$</span> |
| <a name="l00802"></a>00802 <span class="comment"> * - <tt>rank_sum_neg FLOAT8</tt> - rank sum of all negative values, i.e., \f$ w^- \f$</span> |
| <a name="l00803"></a>00803 <span class="comment"> * - <tt>num BIGINT</tt> - number \f$ n \f$ of non-zero values</span> |
| <a name="l00804"></a>00804 <span class="comment"> * - <tt>z_statistic FLOAT8</tt> - z-statistic</span> |
| <a name="l00805"></a>00805 <span class="comment"> * \f[</span> |
| <a name="l00806"></a>00806 <span class="comment"> * z = \frac{w^+ - \frac{n(n+1)}{4}}</span> |
| <a name="l00807"></a>00807 <span class="comment"> * {\sqrt{\frac{n(n+1)(2n+1)}{24}</span> |
| <a name="l00808"></a>00808 <span class="comment"> * - \sum_{i=1}^n \frac{t_i^2 - 1}{48}}}</span> |
| <a name="l00809"></a>00809 <span class="comment"> * \f]</span> |
| <a name="l00810"></a>00810 <span class="comment"> * where \f$ t_i \f$ is the number of</span> |
| <a name="l00811"></a>00811 <span class="comment"> * values with absolute value equal to \f$ |x_i| \f$. The corresponding</span> |
| <a name="l00812"></a>00812 <span class="comment"> * random variable is approximately standard normally distributed.</span> |
| <a name="l00813"></a>00813 <span class="comment"> * - <tt>p_value_one_sided FLOAT8</tt> - One-sided p-value i.e.,</span> |
| <a name="l00814"></a>00814 <span class="comment"> * \f$ \Pr[Z \geq z \mid \mu \leq 0] \f$. Computed as</span> |
| <a name="l00815"></a>00815 <span class="comment"> * <tt>(1.0 - \ref normal_cdf "normal_cdf"(z_statistic))</tt>.</span> |
| <a name="l00816"></a>00816 <span class="comment"> * - <tt>p_value_two_sided FLOAT8</tt> - Two-sided p-value, i.e.,</span> |
| <a name="l00817"></a>00817 <span class="comment"> * \f$ \Pr[ |Z| \geq |z| \mid \mu = 0] \f$. Computed as</span> |
| <a name="l00818"></a>00818 <span class="comment"> * <tt>(2 * \ref normal_cdf "normal_cdf"(-abs(z_statistic)))</tt>.</span> |
| <a name="l00819"></a>00819 <span class="comment"> *</span> |
| <a name="l00820"></a>00820 <span class="comment"> * @usage</span> |
| <a name="l00821"></a>00821 <span class="comment"> * - One-sample test: Test null hypothesis that the mean of a sample is at</span> |
| <a name="l00822"></a>00822 <span class="comment"> * most (or equal to, respectively) \f$ \mu_0 \f$:</span> |
| <a name="l00823"></a>00823 <span class="comment"> * <pre>SELECT (wsr_test(<em>value</em> - <em>mu_0</em> ORDER BY abs(<em>value</em>))).* FROM <em>source</em></pre></span> |
| <a name="l00824"></a>00824 <span class="comment"> * - Dependent paired test: Test null hypothesis that the mean difference</span> |
| <a name="l00825"></a>00825 <span class="comment"> * between the first and second value in a pair is at most (or equal to,</span> |
| <a name="l00826"></a>00826 <span class="comment"> * respectively) \f$ \mu_0 \f$:</span> |
| <a name="l00827"></a>00827 <span class="comment"> * <pre>SELECT (wsr_test(<em>first</em> - <em>second</em> - <em>mu_0</em> ORDER BY abs(<em>first</em> - <em>second</em>))).* FROM <em>source</em></pre></span> |
| <a name="l00828"></a>00828 <span class="comment"> * If correctly determining ties is important (e.g., you may want to do so</span> |
| <a name="l00829"></a>00829 <span class="comment"> * when comparing to software products that take \c first, \c second,</span> |
| <a name="l00830"></a>00830 <span class="comment"> * and \c mu_0 as individual parameters), supply the precision parameter.</span> |
| <a name="l00831"></a>00831 <span class="comment"> * This can be done as follows:</span> |
| <a name="l00832"></a>00832 <span class="comment"> * <pre>SELECT (wsr_test(</span> |
| <a name="l00833"></a>00833 <span class="comment"> <em>first</em> - <em>second</em> - <em>mu_0</em>,</span> |
| <a name="l00834"></a>00834 <span class="comment"> 3 * 2^(-52) * greatest(first, second, mu_0)</span> |
| <a name="l00835"></a>00835 <span class="comment"> ORDER BY abs(<em>first</em> - <em>second</em>)</span> |
| <a name="l00836"></a>00836 <span class="comment">)).* FROM <em>source</em></pre></span> |
| <a name="l00837"></a>00837 <span class="comment"> * Here \f$ 2^{-52} \f$ is the machine epsilon, which we scale to the</span> |
| <a name="l00838"></a>00838 <span class="comment"> * magnitude of the input data and multiply with 3 because we have a sum with</span> |
| <a name="l00839"></a>00839 <span class="comment"> * three terms.</span> |
| <a name="l00840"></a>00840 <span class="comment"> *</span> |
| <a name="l00841"></a>00841 <span class="comment"> * @note</span> |
| <a name="l00842"></a>00842 <span class="comment"> * This aggregate must be used as an ordered aggregate</span> |
| <a name="l00843"></a>00843 <span class="comment"> * (<tt>ORDER BY abs(\em value</tt>)) and will raise an exception if the</span> |
| <a name="l00844"></a>00844 <span class="comment"> * absolute values are not ordered.</span> |
| <a name="l00845"></a>00845 <span class="comment"> */</span> |
| <a name="l00846"></a>00846 m4_ifdef(<!__HAS_ORDERED_AGGREGATES__!>,<! |
| <a name="l00847"></a>00847 CREATE |
| <a name="l00848"></a>00848 m4_ifdef(<!__GREENPLUM__!>,<!ORDERED!>) |
| <a name="l00849"></a>00849 AGGREGATE MADLIB_SCHEMA.wsr_test( |
| <a name="l00850"></a>00850 /*+ "value" */ DOUBLE PRECISION, |
| <a name="l00851"></a>00851 /*+ "precision" */ DOUBLE PRECISION /*+ DEFAULT -1 */ |
| <a name="l00852"></a>00852 ) ( |
| <a name="l00853"></a>00853 SFUNC=MADLIB_SCHEMA.wsr_test_transition, |
| <a name="l00854"></a>00854 STYPE=DOUBLE PRECISION[], |
| <a name="l00855"></a>00855 FINALFUNC=MADLIB_SCHEMA.wsr_test_final, |
| <a name="l00856"></a>00856 INITCOND='{0,0,0,0,0,0,0,0,0}<span class="stringliteral">'</span> |
| <a name="l00857"></a>00857 <span class="stringliteral">);</span> |
| <a name="l00858"></a>00858 <span class="stringliteral">!>)</span> |
| <a name="l00859"></a>00859 <span class="stringliteral"></span> |
| <a name="l00860"></a>00860 <span class="stringliteral">m4_ifdef(<!__HAS_ORDERED_AGGREGATES__!>,<!</span> |
| <a name="l00861"></a>00861 <span class="stringliteral">CREATE</span> |
| <a name="l00862"></a>00862 <span class="stringliteral">m4_ifdef(<!__GREENPLUM__!>,<!ORDERED!>)</span> |
| <a name="l00863"></a>00863 <span class="stringliteral">AGGREGATE MADLIB_SCHEMA.wsr_test(</span> |
| <a name="l00864"></a>00864 <span class="stringliteral"> /*+ value */ DOUBLE PRECISION</span> |
| <a name="l00865"></a>00865 <span class="stringliteral">) (</span> |
| <a name="l00866"></a>00866 <span class="stringliteral"> SFUNC=MADLIB_SCHEMA.wsr_test_transition,</span> |
| <a name="l00867"></a>00867 <span class="stringliteral"> STYPE=DOUBLE PRECISION[],</span> |
| <a name="l00868"></a>00868 <span class="stringliteral"> FINALFUNC=MADLIB_SCHEMA.wsr_test_final,</span> |
| <a name="l00869"></a><a class="code" href="hypothesis__tests_8sql__in.html#afea2309e99477df6ebbfbcea11272507">00869</a> <span class="stringliteral"> INITCOND='</span>{0,0,0,0,0,0,0,0,0}<span class="stringliteral">'</span> |
| <a name="l00870"></a>00870 <span class="stringliteral">);</span> |
| <a name="l00871"></a>00871 <span class="stringliteral">!>)</span> |
| <a name="l00872"></a>00872 <span class="stringliteral"></span> |
| <a name="l00873"></a>00873 <span class="stringliteral">CREATE TYPE MADLIB_SCHEMA.one_way_anova_result AS (</span> |
| <a name="l00874"></a>00874 <span class="stringliteral"> sum_squares_between DOUBLE PRECISION,</span> |
| <a name="l00875"></a>00875 <span class="stringliteral"> sum_squares_within DOUBLE PRECISION,</span> |
| <a name="l00876"></a>00876 <span class="stringliteral"> df_between BIGINT,</span> |
| <a name="l00877"></a>00877 <span class="stringliteral"> df_within BIGINT,</span> |
| <a name="l00878"></a>00878 <span class="stringliteral"> mean_squares_between DOUBLE PRECISION,</span> |
| <a name="l00879"></a>00879 <span class="stringliteral"> mean_squares_within DOUBLE PRECISION,</span> |
| <a name="l00880"></a>00880 <span class="stringliteral"> statistic DOUBLE PRECISION,</span> |
| <a name="l00881"></a>00881 <span class="stringliteral"> p_value DOUBLE PRECISION</span> |
| <a name="l00882"></a>00882 <span class="stringliteral">);</span> |
| <a name="l00883"></a>00883 <span class="stringliteral"></span> |
| <a name="l00884"></a>00884 <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.one_way_anova_transition(</span> |
| <a name="l00885"></a>00885 <span class="stringliteral"> state DOUBLE PRECISION[],</span> |
| <a name="l00886"></a>00886 <span class="stringliteral"> "group" INTEGER,</span> |
| <a name="l00887"></a>00887 <span class="stringliteral"> value DOUBLE PRECISION)</span> |
| <a name="l00888"></a>00888 <span class="stringliteral">RETURNS DOUBLE PRECISION[]</span> |
| <a name="l00889"></a>00889 <span class="stringliteral">AS '</span>MODULE_PATHNAME<span class="stringliteral">'</span> |
| <a name="l00890"></a>00890 <span class="stringliteral">LANGUAGE C</span> |
| <a name="l00891"></a>00891 <span class="stringliteral">IMMUTABLE</span> |
| <a name="l00892"></a>00892 <span class="stringliteral">STRICT;</span> |
| <a name="l00893"></a>00893 <span class="stringliteral"></span> |
| <a name="l00894"></a>00894 <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.one_way_anova_merge_states(</span> |
| <a name="l00895"></a>00895 <span class="stringliteral"> state1 DOUBLE PRECISION[],</span> |
| <a name="l00896"></a>00896 <span class="stringliteral"> state2 DOUBLE PRECISION[])</span> |
| <a name="l00897"></a>00897 <span class="stringliteral">RETURNS DOUBLE PRECISION[]</span> |
| <a name="l00898"></a>00898 <span class="stringliteral">AS '</span>MODULE_PATHNAME<span class="stringliteral">'</span> |
| <a name="l00899"></a>00899 <span class="stringliteral">LANGUAGE C</span> |
| <a name="l00900"></a>00900 <span class="stringliteral">IMMUTABLE STRICT;</span> |
| <a name="l00901"></a>00901 <span class="stringliteral"></span> |
| <a name="l00902"></a>00902 <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.one_way_anova_final(</span> |
| <a name="l00903"></a>00903 <span class="stringliteral"> state DOUBLE PRECISION[])</span> |
| <a name="l00904"></a>00904 <span class="stringliteral">RETURNS MADLIB_SCHEMA.one_way_anova_result</span> |
| <a name="l00905"></a>00905 <span class="stringliteral">AS '</span>MODULE_PATHNAME<span class="stringliteral">'</span> |
| <a name="l00906"></a>00906 <span class="stringliteral">LANGUAGE C IMMUTABLE STRICT;</span> |
| <a name="l00907"></a>00907 <span class="stringliteral"></span><span class="comment"></span> |
| <a name="l00908"></a>00908 <span class="comment">/**</span> |
| <a name="l00909"></a>00909 <span class="comment"> * @brief Perform one-way analysis of variance</span> |
| <a name="l00910"></a>00910 <span class="comment"> *</span> |
| <a name="l00911"></a>00911 <span class="comment"> * Given realizations</span> |
| <a name="l00912"></a>00912 <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$</span> |
| <a name="l00913"></a>00913 <span class="comment"> * of i.i.d. random variables \f$ X_{i,j} \sim N(\mu_i, \sigma^2) \f$ with</span> |
| <a name="l00914"></a>00914 <span class="comment"> * unknown parameters \f$ \mu_1, \dots, \mu_k \f$ and \f$ \sigma^2 \f$, test the</span> |
| <a name="l00915"></a>00915 <span class="comment"> * null hypotheses \f$ H_0 : \mu_1 = \dots = \mu_k \f$.</span> |
| <a name="l00916"></a>00916 <span class="comment"> *</span> |
| <a name="l00917"></a>00917 <span class="comment"> * @param group Group which \c value is from. Note that \c group can assume</span> |
| <a name="l00918"></a>00918 <span class="comment"> * arbitary value not limited to a continguous range of integers.</span> |
| <a name="l00919"></a>00919 <span class="comment"> * @param value Value of random variate \f$ x_{i,j} \f$</span> |
| <a name="l00920"></a>00920 <span class="comment"> *</span> |
| <a name="l00921"></a>00921 <span class="comment"> * @return A composite value as follows. Let \f$ n := \sum_{i=1}^k n_i \f$ be</span> |
| <a name="l00922"></a>00922 <span class="comment"> * the total size of all samples. Denote by \f$ \bar x \f$ the grand</span> |
| <a name="l00923"></a>00923 <span class="comment"> * \ref sample_mean "mean", by \f$ \overline{x_i} \f$ the group</span> |
| <a name="l00924"></a>00924 <span class="comment"> * \ref sample_mean "sample means", and by \f$ s_i^2 \f$ the group</span> |
| <a name="l00925"></a>00925 <span class="comment"> * \ref sample_variance "sample variances".</span> |
| <a name="l00926"></a>00926 <span class="comment"> * - <tt>sum_squares_between DOUBLE PRECISION</tt> - sum of squares between the</span> |
| <a name="l00927"></a>00927 <span class="comment"> * group means, i.e.,</span> |
| <a name="l00928"></a>00928 <span class="comment"> * \f$</span> |
| <a name="l00929"></a>00929 <span class="comment"> * \mathit{SS}_b = \sum_{i=1}^k n_i (\overline{x_i} - \bar x)^2.</span> |
| <a name="l00930"></a>00930 <span class="comment"> * \f$</span> |
| <a name="l00931"></a>00931 <span class="comment"> * - <tt>sum_squares_within DOUBLE PRECISION</tt> - sum of squares within the</span> |
| <a name="l00932"></a>00932 <span class="comment"> * groups, i.e.,</span> |
| <a name="l00933"></a>00933 <span class="comment"> * \f$</span> |
| <a name="l00934"></a>00934 <span class="comment"> * \mathit{SS}_w = \sum_{i=1}^k (n_i - 1) s_i^2.</span> |
| <a name="l00935"></a>00935 <span class="comment"> * \f$</span> |
| <a name="l00936"></a>00936 <span class="comment"> * - <tt>df_between BIGINT</tt> - degree of freedom for between-group variation \f$ (k-1) \f$</span> |
| <a name="l00937"></a>00937 <span class="comment"> * - <tt>df_within BIGINT</tt> - degree of freedom for within-group variation \f$ (n-k) \f$</span> |
| <a name="l00938"></a>00938 <span class="comment"> * - <tt>mean_squares_between DOUBLE PRECISION</tt> - mean square between</span> |
| <a name="l00939"></a>00939 <span class="comment"> * groups, i.e.,</span> |
| <a name="l00940"></a>00940 <span class="comment"> * \f$</span> |
| <a name="l00941"></a>00941 <span class="comment"> * s_b^2 := \frac{\mathit{SS}_b}{k-1}</span> |
| <a name="l00942"></a>00942 <span class="comment"> * \f$</span> |
| <a name="l00943"></a>00943 <span class="comment"> * - <tt>mean_squares_within DOUBLE PRECISION</tt> - mean square within</span> |
| <a name="l00944"></a>00944 <span class="comment"> * groups, i.e.,</span> |
| <a name="l00945"></a>00945 <span class="comment"> * \f$</span> |
| <a name="l00946"></a>00946 <span class="comment"> * s_w^2 := \frac{\mathit{SS}_w}{n-k}</span> |
| <a name="l00947"></a>00947 <span class="comment"> * \f$</span> |
| <a name="l00948"></a>00948 <span class="comment"> * - <tt>statistic DOUBLE PRECISION</tt> - Statistic computed as</span> |
| <a name="l00949"></a>00949 <span class="comment"> * \f[</span> |
| <a name="l00950"></a>00950 <span class="comment"> * f = \frac{s_b^2}{s_w^2}.</span> |
| <a name="l00951"></a>00951 <span class="comment"> * \f]</span> |
| <a name="l00952"></a>00952 <span class="comment"> * This statistic is Fisher F-distributed with \f$ (k-1) \f$ degrees of</span> |
| <a name="l00953"></a>00953 <span class="comment"> * freedom in the numerator and \f$ (n-k) \f$ degrees of freedom in the</span> |
| <a name="l00954"></a>00954 <span class="comment"> * denominator.</span> |
| <a name="l00955"></a>00955 <span class="comment"> * - <tt>p_value DOUBLE PRECISION</tt> - p-value, i.e.,</span> |
| <a name="l00956"></a>00956 <span class="comment"> * \f$ \Pr[ F \geq f \mid H_0] \f$.</span> |
| <a name="l00957"></a>00957 <span class="comment"> *</span> |
| <a name="l00958"></a>00958 <span class="comment"> * @usage</span> |
| <a name="l00959"></a>00959 <span class="comment"> * - Test null hypothesis that the mean of the all samples is equal:</span> |
| <a name="l00960"></a>00960 <span class="comment"> * <pre>SELECT (one_way_anova(<em>group</em>, <em>value</em>)).* FROM <em>source</em></pre></span> |
| <a name="l00961"></a>00961 <span class="comment"> */</span> |
| <a name="l00962"></a>00962 CREATE AGGREGATE MADLIB_SCHEMA.one_way_anova( |
| <a name="l00963"></a>00963 /*+ group */ INTEGER, |
| <a name="l00964"></a>00964 /*+ value */ DOUBLE PRECISION) ( |
| <a name="l00965"></a>00965 |
| <a name="l00966"></a>00966 SFUNC=MADLIB_SCHEMA.one_way_anova_transition, |
| <a name="l00967"></a>00967 STYPE=DOUBLE PRECISION[], |
| <a name="l00968"></a>00968 FINALFUNC=MADLIB_SCHEMA.one_way_anova_final, |
| <a name="l00969"></a>00969 m4_ifdef(<!__GREENPLUM__!>,<!PREFUNC=MADLIB_SCHEMA.one_way_anova_merge_states,!>) |
| <a name="l00970"></a>00970 INITCOND='{0,0}<span class="stringliteral">'</span> |
| <a name="l00971"></a>00971 <span class="stringliteral">);</span> |
| <a name="l00972"></a>00972 <span class="stringliteral"></span> |
| <a name="l00973"></a>00973 <span class="stringliteral">m4_changequote(<!`!>,<!'</span>!>) |
| </pre></div></div> |
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