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Posted to commits@madlib.apache.org by xt...@apache.org on 2016/04/07 23:47:25 UTC
[22/51] [abbrv] [partial] incubator-madlib-site git commit: Update
doc for 1.9 release
http://git-wip-us.apache.org/repos/asf/incubator-madlib-site/blob/c506dd05/docs/latest/hypothesis__tests_8sql__in.html
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diff --git a/docs/latest/hypothesis__tests_8sql__in.html b/docs/latest/hypothesis__tests_8sql__in.html
index 356639a..d147634 100644
--- a/docs/latest/hypothesis__tests_8sql__in.html
+++ b/docs/latest/hypothesis__tests_8sql__in.html
@@ -24,14 +24,8 @@
<script type="text/javascript">
$(document).ready(function() { init_search(); });
</script>
-<script type="text/x-mathjax-config">
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-</script><script src="../mathjax/MathJax.js"></script>
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+<script type="text/javascript" src="eigen_navtree_hacks.js"></script>
<link href="doxygen.css" rel="stylesheet" type="text/css" />
<link href="madlib_extra.css" rel="stylesheet" type="text/css"/>
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+ ga('create', 'UA-45382226-1', 'madlib.net');
ga('send', 'pageview');
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</head>
@@ -50,10 +44,10 @@
<table cellspacing="0" cellpadding="0">
<tbody>
<tr style="height: 56px;">
- <td id="projectlogo"><a href="http://madlib.incubator.apache.org"><img alt="Logo" src="madlib.png" height="50" style="padding-left:0.5em;" border="0"/ ></a></td>
+ <td id="projectlogo"><a href="http://madlib.net"><img alt="Logo" src="madlib.png" height="50" style="padding-left:0.5em;" border="0"/ ></a></td>
<td style="padding-left: 0.5em;">
<div id="projectname">
- <span id="projectnumber">1.8</span>
+ <span id="projectnumber">1.9</span>
</div>
<div id="projectbrief">User Documentation for MADlib</div>
</td>
@@ -225,24 +219,24 @@ Functions</h2></td></tr>
</tr>
</table>
</div><div class="memdoc">
-<p>Let \( n_1, \dots, n_k \) be a realization of a (vector) random variable \( N = (N_1, \dots, N_k) \) that follows the multinomial distribution with parameters \( k \) and \( p = (p_1, \dots, p_k) \). Test the null hypothesis \( H_0 : p = p^0 \).</p>
+<p>Let <img class="formulaInl" alt="$ n_1, \dots, n_k $" src="form_445.png"/> be a realization of a (vector) random variable <img class="formulaInl" alt="$ N = (N_1, \dots, N_k) $" src="form_446.png"/> that follows the multinomial distribution with parameters <img class="formulaInl" alt="$ k $" src="form_97.png"/> and <img class="formulaInl" alt="$ p = (p_1, \dots, p_k) $" src="form_447.png"/>. Test the null hypothesis <img class="formulaInl" alt="$ H_0 : p = p^0 $" src="form_448.png"/>.</p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
- <tr><td class="paramname">observed</td><td>Number \( n_i \) of observations of the current event/row </td></tr>
- <tr><td class="paramname">expected</td><td>Expected number of observations of current event/row. This number is not required to be normalized. That is, \( p^0_i \) will be taken as <code>expected</code> divided by <code>sum(expected)</code>. Hence, if this parameter is not specified, chi2_test() will by default use \( p^0 = (\frac 1k, \dots, \frac 1k) \), i.e., test that \( p \) is a discrete uniform distribution. </td></tr>
- <tr><td class="paramname">df</td><td>Degrees of freedom. This is the number of events reduced by the degree of freedom lost by using the observed numbers for defining the expected number of observations. If this parameter is 0, the degree of freedom is taken as \( (k - 1) \).</td></tr>
+ <tr><td class="paramname">observed</td><td>Number <img class="formulaInl" alt="$ n_i $" src="form_449.png"/> of observations of the current event/row </td></tr>
+ <tr><td class="paramname">expected</td><td>Expected number of observations of current event/row. This number is not required to be normalized. That is, <img class="formulaInl" alt="$ p^0_i $" src="form_450.png"/> will be taken as <code>expected</code> divided by <code>sum(expected)</code>. Hence, if this parameter is not specified, chi2_test() will by default use <img class="formulaInl" alt="$ p^0 = (\frac 1k, \dots, \frac 1k) $" src="form_451.png"/>, i.e., test that <img class="formulaInl" alt="$ p $" src="form_110.png"/> is a discrete uniform distribution. </td></tr>
+ <tr><td class="paramname">df</td><td>Degrees of freedom. This is the number of events reduced by the degree of freedom lost by using the observed numbers for defining the expected number of observations. If this parameter is 0, the degree of freedom is taken as <img class="formulaInl" alt="$ (k - 1) $" src="form_452.png"/>.</td></tr>
</table>
</dd>
</dl>
-<dl class="section return"><dt>Returns</dt><dd>A composite value as follows. Let \( n = \sum_{i=1}^n n_i \).<ul>
+<dl class="section return"><dt>Returns</dt><dd>A composite value as follows. Let <img class="formulaInl" alt="$ n = \sum_{i=1}^n n_i $" src="form_453.png"/>.<ul>
<li><code>statistic FLOAT8</code> - Statistic <p class="formulaDsp">
-\[ \chi^2 = \sum_{i=1}^k \frac{(n_i - np_i)^2}{np_i} \]
+<img class="formulaDsp" alt="\[ \chi^2 = \sum_{i=1}^k \frac{(n_i - np_i)^2}{np_i} \]" src="form_454.png"/>
</p>
The corresponding random variable is approximately chi-squared distributed with <code>df</code> degrees of freedom.</li>
<li><code>df BIGINT</code> - Degrees of freedom</li>
-<li><code>p_value FLOAT8</code> - Approximate p-value, i.e., \( \Pr[X^2 \geq \chi^2 \mid p = p^0] \). Computed as <code>(1.0 - <a class="el" href="prob_8sql__in.html#a230513b6b549d5b445cbacbdbab42c15">chi_squared_cdf</a>(statistic))</code>.</li>
-<li><code>phi FLOAT8</code> - Phi coefficient, i.e., \( \phi = \sqrt{\frac{\chi^2}{n}} \)</li>
-<li><code>contingency_coef FLOAT8</code> - Contingency coefficient, i.e., \( \sqrt{\frac{\chi^2}{n + \chi^2}} \)</li>
+<li><code>p_value FLOAT8</code> - Approximate p-value, i.e., <img class="formulaInl" alt="$ \Pr[X^2 \geq \chi^2 \mid p = p^0] $" src="form_455.png"/>. Computed as <code>(1.0 - <a class="el" href="prob_8sql__in.html#a230513b6b549d5b445cbacbdbab42c15">chi_squared_cdf</a>(statistic))</code>.</li>
+<li><code>phi FLOAT8</code> - Phi coefficient, i.e., <img class="formulaInl" alt="$ \phi = \sqrt{\frac{\chi^2}{n}} $" src="form_456.png"/></li>
+<li><code>contingency_coef FLOAT8</code> - Contingency coefficient, i.e., <img class="formulaInl" alt="$ \sqrt{\frac{\chi^2}{n + \chi^2}} $" src="form_457.png"/></li>
</ul>
</dd></dl>
<dl class="section user"><dt>Usage</dt><dd><ul>
@@ -467,23 +461,23 @@ FROM (
</tr>
</table>
</div><div class="memdoc">
-<p>Given realizations \( x_1, \dots, x_m \) and \( y_1, \dots, y_n \) of i.i.d. random variables \( X_1, \dots, X_m \sim N(\mu_X, \sigma^2) \) and \( Y_1, \dots, Y_n \sim N(\mu_Y, \sigma^2) \) with unknown parameters \( \mu_X, \mu_Y, \) and \( \sigma^2 \), test the null hypotheses \( H_0 : \sigma_X < \sigma_Y \) and \( H_0 : \sigma_X = \sigma_Y \).</p>
+<p>Given realizations <img class="formulaInl" alt="$ x_1, \dots, x_m $" src="form_433.png"/> and <img class="formulaInl" alt="$ y_1, \dots, y_n $" src="form_434.png"/> of i.i.d. random variables <img class="formulaInl" alt="$ X_1, \dots, X_m \sim N(\mu_X, \sigma^2) $" src="form_435.png"/> and <img class="formulaInl" alt="$ Y_1, \dots, Y_n \sim N(\mu_Y, \sigma^2) $" src="form_436.png"/> with unknown parameters <img class="formulaInl" alt="$ \mu_X, \mu_Y, $" src="form_416.png"/> and <img class="formulaInl" alt="$ \sigma^2 $" src="form_304.png"/>, test the null hypotheses <img class="formulaInl" alt="$ H_0 : \sigma_X < \sigma_Y $" src="form_437.png"/> and <img class="formulaInl" alt="$ H_0 : \sigma_X = \sigma_Y $" src="form_438.png"/>.</p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
- <tr><td class="paramname">first</td><td>Indicator whether <code>value</code> is from first sample \( x_1, \dots, x_m \) (if <code>TRUE</code>) or from second sample \( y_1, \dots, y_n \) (if <code>FALSE</code>) </td></tr>
- <tr><td class="paramname">value</td><td>Value of random variate \( x_i \) or \( y_i \)</td></tr>
+ <tr><td class="paramname">first</td><td>Indicator whether <code>value</code> is from first sample <img class="formulaInl" alt="$ x_1, \dots, x_m $" src="form_433.png"/> (if <code>TRUE</code>) or from second sample <img class="formulaInl" alt="$ y_1, \dots, y_n $" src="form_434.png"/> (if <code>FALSE</code>) </td></tr>
+ <tr><td class="paramname">value</td><td>Value of random variate <img class="formulaInl" alt="$ x_i $" src="form_62.png"/> or <img class="formulaInl" alt="$ y_i $" src="form_60.png"/></td></tr>
</table>
</dd>
</dl>
-<dl class="section return"><dt>Returns</dt><dd>A composite value as follows. We denote by \( \bar x, \bar y \) the sample means and by \( s_X^2, s_Y^2 \) the sample variances.<ul>
+<dl class="section return"><dt>Returns</dt><dd>A composite value as follows. We denote by <img class="formulaInl" alt="$ \bar x, \bar y $" src="form_419.png"/> the sample means and by <img class="formulaInl" alt="$ s_X^2, s_Y^2 $" src="form_420.png"/> the sample variances.<ul>
<li><code>statistic FLOAT8</code> - Statistic <p class="formulaDsp">
-\[ f = \frac{s_Y^2}{s_X^2} \]
+<img class="formulaDsp" alt="\[ f = \frac{s_Y^2}{s_X^2} \]" src="form_439.png"/>
</p>
- The corresponding random variable is F-distributed with \( (n - 1) \) degrees of freedom in the numerator and \( (m - 1) \) degrees of freedom in the denominator.</li>
-<li><code>df1 BIGINT</code> - Degrees of freedom in the numerator \( (n - 1) \)</li>
-<li><code>df2 BIGINT</code> - Degrees of freedom in the denominator \( (m - 1) \)</li>
-<li><code>p_value_one_sided FLOAT8</code> - Lower bound on one-sided p-value. In detail, the result is \( \Pr[F \geq f \mid \sigma_X = \sigma_Y] \), which is a lower bound on \( \Pr[F \geq f \mid \sigma_X \leq \sigma_Y] \). Computed as <code>(1.0 - <a class="el" href="prob_8sql__in.html#a6c5b3e35531e44098f9d0cbef14cb8a6">fisher_f_cdf</a>(statistic))</code>.</li>
-<li><code>p_value_two_sided FLOAT8</code> - Two-sided p-value, i.e., \( 2 \cdot \min \{ p, 1 - p \} \) where \( p = \Pr[ F \geq f \mid \sigma_X = \sigma_Y] \). Computed as <code>(min(p_value_one_sided, 1. - p_value_one_sided))</code>.</li>
+ The corresponding random variable is F-distributed with <img class="formulaInl" alt="$ (n - 1) $" src="form_408.png"/> degrees of freedom in the numerator and <img class="formulaInl" alt="$ (m - 1) $" src="form_440.png"/> degrees of freedom in the denominator.</li>
+<li><code>df1 BIGINT</code> - Degrees of freedom in the numerator <img class="formulaInl" alt="$ (n - 1) $" src="form_408.png"/></li>
+<li><code>df2 BIGINT</code> - Degrees of freedom in the denominator <img class="formulaInl" alt="$ (m - 1) $" src="form_440.png"/></li>
+<li><code>p_value_one_sided FLOAT8</code> - Lower bound on one-sided p-value. In detail, the result is <img class="formulaInl" alt="$ \Pr[F \geq f \mid \sigma_X = \sigma_Y] $" src="form_441.png"/>, which is a lower bound on <img class="formulaInl" alt="$ \Pr[F \geq f \mid \sigma_X \leq \sigma_Y] $" src="form_442.png"/>. Computed as <code>(1.0 - <a class="el" href="prob_8sql__in.html#a6c5b3e35531e44098f9d0cbef14cb8a6">fisher_f_cdf</a>(statistic))</code>.</li>
+<li><code>p_value_two_sided FLOAT8</code> - Two-sided p-value, i.e., <img class="formulaInl" alt="$ 2 \cdot \min \{ p, 1 - p \} $" src="form_443.png"/> where <img class="formulaInl" alt="$ p = \Pr[ F \geq f \mid \sigma_X = \sigma_Y] $" src="form_444.png"/>. Computed as <code>(min(p_value_one_sided, 1. - p_value_one_sided))</code>.</li>
</ul>
</dd></dl>
<dl class="section user"><dt>Usage</dt><dd><ul>
@@ -614,23 +608,23 @@ FROM (
</tr>
</table>
</div><div class="memdoc">
-<p>Given realizations \( x_1, \dots, x_m \) and \( y_1, \dots, y_m \) of i.i.d. random variables \( X_1, \dots, X_m \) and i.i.d. \( Y_1, \dots, Y_n \), respectively, test the null hypothesis that the underlying distributions function \( F_X, F_Y \) are identical, i.e., \( H_0 : F_X = F_Y \).</p>
+<p>Given realizations <img class="formulaInl" alt="$ x_1, \dots, x_m $" src="form_433.png"/> and <img class="formulaInl" alt="$ y_1, \dots, y_m $" src="form_413.png"/> of i.i.d. random variables <img class="formulaInl" alt="$ X_1, \dots, X_m $" src="form_458.png"/> and i.i.d. <img class="formulaInl" alt="$ Y_1, \dots, Y_n $" src="form_459.png"/>, respectively, test the null hypothesis that the underlying distributions function <img class="formulaInl" alt="$ F_X, F_Y $" src="form_460.png"/> are identical, i.e., <img class="formulaInl" alt="$ H_0 : F_X = F_Y $" src="form_461.png"/>.</p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">first</td><td>Determines whether the value belongs to the first (if <code>TRUE</code>) or the second sample (if <code>FALSE</code>) </td></tr>
- <tr><td class="paramname">value</td><td>Value of random variate \( x_i \) or \( y_i \) </td></tr>
- <tr><td class="paramname">m</td><td>Size \( m \) of the first sample. See usage instructions below. </td></tr>
+ <tr><td class="paramname">value</td><td>Value of random variate <img class="formulaInl" alt="$ x_i $" src="form_62.png"/> or <img class="formulaInl" alt="$ y_i $" src="form_60.png"/> </td></tr>
+ <tr><td class="paramname">m</td><td>Size <img class="formulaInl" alt="$ m $" src="form_291.png"/> of the first sample. See usage instructions below. </td></tr>
<tr><td class="paramname">n</td><td>Size of the second sample. See usage instructions below.</td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>A composite value.<ul>
<li><code>statistic FLOAT8</code> - Kolmogorov–Smirnov statistic <p class="formulaDsp">
-\[ d = \max_{t \in \mathbb R} |F_x(t) - F_y(t)| \]
+<img class="formulaDsp" alt="\[ d = \max_{t \in \mathbb R} |F_x(t) - F_y(t)| \]" src="form_462.png"/>
</p>
- where \( F_x(t) := \frac 1m |\{ i \mid x_i \leq t \}| \) and \( F_y \) (defined likewise) are the empirical distribution functions.</li>
-<li><code>k_statistic FLOAT8</code> - Kolmogorov statistic \( k = (r + 0.12 + \frac{0.11}{r}) \cdot d \) where \( r = \sqrt{\frac{m n}{m+n}}. \) and \( d \) is the statistic. Then \( k \) is approximately Kolmogorov distributed.</li>
-<li><code>p_value FLOAT8</code> - Approximate p-value, i.e., an approximate value for \( \Pr[D \geq d \mid F_X = F_Y] \). Computed as <code>(1.0 - <a class="el" href="prob_8sql__in.html#aeef43f74f583bdff17bd074d9c0d9607">kolmogorov_cdf</a>(k_statistic))</code>.</li>
+ where <img class="formulaInl" alt="$ F_x(t) := \frac 1m |\{ i \mid x_i \leq t \}| $" src="form_463.png"/> and <img class="formulaInl" alt="$ F_y $" src="form_464.png"/> (defined likewise) are the empirical distribution functions.</li>
+<li><code>k_statistic FLOAT8</code> - Kolmogorov statistic <img class="formulaInl" alt="$ k = (r + 0.12 + \frac{0.11}{r}) \cdot d $" src="form_465.png"/> where <img class="formulaInl" alt="$ r = \sqrt{\frac{m n}{m+n}}. $" src="form_466.png"/> and <img class="formulaInl" alt="$ d $" src="form_467.png"/> is the statistic. Then <img class="formulaInl" alt="$ k $" src="form_97.png"/> is approximately Kolmogorov distributed.</li>
+<li><code>p_value FLOAT8</code> - Approximate p-value, i.e., an approximate value for <img class="formulaInl" alt="$ \Pr[D \geq d \mid F_X = F_Y] $" src="form_468.png"/>. Computed as <code>(1.0 - <a class="el" href="prob_8sql__in.html#aeef43f74f583bdff17bd074d9c0d9607">kolmogorov_cdf</a>(k_statistic))</code>.</li>
</ul>
</dd></dl>
<dl class="section user"><dt>Usage</dt><dd><ul>
@@ -668,26 +662,26 @@ FROM (
</tr>
</table>
</div><div class="memdoc">
-<p>Given realizations \( x_{1,1}, \dots, x_{1, n_1}, x_{2,1}, \dots, x_{2,n_2}, \dots, x_{k,n_k} \) of i.i.d. random variables \( X_{i,j} \sim N(\mu_i, \sigma^2) \) with unknown parameters \( \mu_1, \dots, \mu_k \) and \( \sigma^2 \), test the null hypotheses \( H_0 : \mu_1 = \dots = \mu_k \).</p>
+<p>Given realizations <img class="formulaInl" alt="$ x_{1,1}, \dots, x_{1, n_1}, x_{2,1}, \dots, x_{2,n_2}, \dots, x_{k,n_k} $" src="form_493.png"/> of i.i.d. random variables <img class="formulaInl" alt="$ X_{i,j} \sim N(\mu_i, \sigma^2) $" src="form_494.png"/> with unknown parameters <img class="formulaInl" alt="$ \mu_1, \dots, \mu_k $" src="form_495.png"/> and <img class="formulaInl" alt="$ \sigma^2 $" src="form_304.png"/>, test the null hypotheses <img class="formulaInl" alt="$ H_0 : \mu_1 = \dots = \mu_k $" src="form_496.png"/>.</p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">group</td><td>Group which <code>value</code> is from. Note that <code>group</code> can assume arbitary value not limited to a continguous range of integers. </td></tr>
- <tr><td class="paramname">value</td><td>Value of random variate \( x_{i,j} \)</td></tr>
+ <tr><td class="paramname">value</td><td>Value of random variate <img class="formulaInl" alt="$ x_{i,j} $" src="form_497.png"/></td></tr>
</table>
</dd>
</dl>
-<dl class="section return"><dt>Returns</dt><dd>A composite value as follows. Let \( n := \sum_{i=1}^k n_i \) be the total size of all samples. Denote by \( \bar x \) the grand mean, by \( \overline{x_i} \) the group sample means, and by \( s_i^2 \) the group sample variances.<ul>
-<li><code>sum_squares_between DOUBLE PRECISION</code> - sum of squares between the group means, i.e., \( \mathit{SS}_b = \sum_{i=1}^k n_i (\overline{x_i} - \bar x)^2. \)</li>
-<li><code>sum_squares_within DOUBLE PRECISION</code> - sum of squares within the groups, i.e., \( \mathit{SS}_w = \sum_{i=1}^k (n_i - 1) s_i^2. \)</li>
-<li><code>df_between BIGINT</code> - degree of freedom for between-group variation \( (k-1) \)</li>
-<li><code>df_within BIGINT</code> - degree of freedom for within-group variation \( (n-k) \)</li>
-<li><code>mean_squares_between DOUBLE PRECISION</code> - mean square between groups, i.e., \( s_b^2 := \frac{\mathit{SS}_b}{k-1} \)</li>
-<li><code>mean_squares_within DOUBLE PRECISION</code> - mean square within groups, i.e., \( s_w^2 := \frac{\mathit{SS}_w}{n-k} \)</li>
+<dl class="section return"><dt>Returns</dt><dd>A composite value as follows. Let <img class="formulaInl" alt="$ n := \sum_{i=1}^k n_i $" src="form_498.png"/> be the total size of all samples. Denote by <img class="formulaInl" alt="$ \bar x $" src="form_405.png"/> the grand mean, by <img class="formulaInl" alt="$ \overline{x_i} $" src="form_499.png"/> the group sample means, and by <img class="formulaInl" alt="$ s_i^2 $" src="form_500.png"/> the group sample variances.<ul>
+<li><code>sum_squares_between DOUBLE PRECISION</code> - sum of squares between the group means, i.e., <img class="formulaInl" alt="$ \mathit{SS}_b = \sum_{i=1}^k n_i (\overline{x_i} - \bar x)^2. $" src="form_501.png"/></li>
+<li><code>sum_squares_within DOUBLE PRECISION</code> - sum of squares within the groups, i.e., <img class="formulaInl" alt="$ \mathit{SS}_w = \sum_{i=1}^k (n_i - 1) s_i^2. $" src="form_502.png"/></li>
+<li><code>df_between BIGINT</code> - degree of freedom for between-group variation <img class="formulaInl" alt="$ (k-1) $" src="form_503.png"/></li>
+<li><code>df_within BIGINT</code> - degree of freedom for within-group variation <img class="formulaInl" alt="$ (n-k) $" src="form_504.png"/></li>
+<li><code>mean_squares_between DOUBLE PRECISION</code> - mean square between groups, i.e., <img class="formulaInl" alt="$ s_b^2 := \frac{\mathit{SS}_b}{k-1} $" src="form_505.png"/></li>
+<li><code>mean_squares_within DOUBLE PRECISION</code> - mean square within groups, i.e., <img class="formulaInl" alt="$ s_w^2 := \frac{\mathit{SS}_w}{n-k} $" src="form_506.png"/></li>
<li><code>statistic DOUBLE PRECISION</code> - Statistic computed as <p class="formulaDsp">
-\[ f = \frac{s_b^2}{s_w^2}. \]
+<img class="formulaDsp" alt="\[ f = \frac{s_b^2}{s_w^2}. \]" src="form_507.png"/>
</p>
- This statistic is Fisher F-distributed with \( (k-1) \) degrees of freedom in the numerator and \( (n-k) \) degrees of freedom in the denominator.</li>
-<li><code>p_value DOUBLE PRECISION</code> - p-value, i.e., \( \Pr[ F \geq f \mid H_0] \).</li>
+ This statistic is Fisher F-distributed with <img class="formulaInl" alt="$ (k-1) $" src="form_503.png"/> degrees of freedom in the numerator and <img class="formulaInl" alt="$ (n-k) $" src="form_504.png"/> degrees of freedom in the denominator.</li>
+<li><code>p_value DOUBLE PRECISION</code> - p-value, i.e., <img class="formulaInl" alt="$ \Pr[ F \geq f \mid H_0] $" src="form_508.png"/>.</li>
</ul>
</dd></dl>
<dl class="section user"><dt>Usage</dt><dd><ul>
@@ -768,37 +762,37 @@ FROM (
</tr>
</table>
</div><div class="memdoc">
-<p>Given realizations \( x_1, \dots, x_n \) of i.i.d. random variables \( X_1, \dots, X_n \) with unknown mean \( \mu \), test the null hypotheses \( H_0 : \mu \leq 0 \) and \( H_0 : \mu = 0 \).</p>
+<p>Given realizations <img class="formulaInl" alt="$ x_1, \dots, x_n $" src="form_183.png"/> of i.i.d. random variables <img class="formulaInl" alt="$ X_1, \dots, X_n $" src="form_478.png"/> with unknown mean <img class="formulaInl" alt="$ \mu $" src="form_286.png"/>, test the null hypotheses <img class="formulaInl" alt="$ H_0 : \mu \leq 0 $" src="form_403.png"/> and <img class="formulaInl" alt="$ H_0 : \mu = 0 $" src="form_404.png"/>.</p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
- <tr><td class="paramname">value</td><td>Value of random variate \( x_i \) or \( y_i \). Values of 0 are ignored (i.e., they do not count towards \( n \)). </td></tr>
- <tr><td class="paramname">precision</td><td>The precision \( \epsilon_i \) with which value is known. The precision determines the handling of ties. The current value \( v_i \) is regarded a tie with the previous value \( v_{i-1} \) if \( v_i - \epsilon_i \leq \max_{j=1, \dots, i-1} v_j + \epsilon_j \). If <code>precision</code> is negative, then it will be treated as <code>value * 2^(-52)</code>. (Note that \( 2^{-52} \) is the machine epsilon for type <code>DOUBLE PRECISION</code>.)</td></tr>
+ <tr><td class="paramname">value</td><td>Value of random variate <img class="formulaInl" alt="$ x_i $" src="form_62.png"/> or <img class="formulaInl" alt="$ y_i $" src="form_60.png"/>. Values of 0 are ignored (i.e., they do not count towards <img class="formulaInl" alt="$ n $" src="form_10.png"/>). </td></tr>
+ <tr><td class="paramname">precision</td><td>The precision <img class="formulaInl" alt="$ \epsilon_i $" src="form_479.png"/> with which value is known. The precision determines the handling of ties. The current value <img class="formulaInl" alt="$ v_i $" src="form_480.png"/> is regarded a tie with the previous value <img class="formulaInl" alt="$ v_{i-1} $" src="form_481.png"/> if <img class="formulaInl" alt="$ v_i - \epsilon_i \leq \max_{j=1, \dots, i-1} v_j + \epsilon_j $" src="form_482.png"/>. If <code>precision</code> is negative, then it will be treated as <code>value * 2^(-52)</code>. (Note that <img class="formulaInl" alt="$ 2^{-52} $" src="form_365.png"/> is the machine epsilon for type <code>DOUBLE PRECISION</code>.)</td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>A composite value:<ul>
-<li><code>statistic FLOAT8</code> - statistic computed as follows. Let \( w^+ = \sum_{i \mid x_i > 0} r_i \) and \( w^- = \sum_{i \mid x_i < 0} r_i \) be the <em>signed rank sums</em> where <p class="formulaDsp">
-\[ r_i = \{ j \mid |x_j| < |x_i| \} + \frac{\{ j \mid |x_j| = |x_i| \} + 1}{2}. \]
+<li><code>statistic FLOAT8</code> - statistic computed as follows. Let <img class="formulaInl" alt="$ w^+ = \sum_{i \mid x_i > 0} r_i $" src="form_483.png"/> and <img class="formulaInl" alt="$ w^- = \sum_{i \mid x_i < 0} r_i $" src="form_484.png"/> be the <em>signed rank sums</em> where <p class="formulaDsp">
+<img class="formulaDsp" alt="\[ r_i = \{ j \mid |x_j| < |x_i| \} + \frac{\{ j \mid |x_j| = |x_i| \} + 1}{2}. \]" src="form_485.png"/>
</p>
- The Wilcoxon signed-rank statistic is \( w = \min \{ w^+, w^- \} \).</li>
-<li><code>rank_sum_pos FLOAT8</code> - rank sum of all positive values, i.e., \( w^+ \)</li>
-<li><code>rank_sum_neg FLOAT8</code> - rank sum of all negative values, i.e., \( w^- \)</li>
-<li><code>num BIGINT</code> - number \( n \) of non-zero values</li>
+ The Wilcoxon signed-rank statistic is <img class="formulaInl" alt="$ w = \min \{ w^+, w^- \} $" src="form_486.png"/>.</li>
+<li><code>rank_sum_pos FLOAT8</code> - rank sum of all positive values, i.e., <img class="formulaInl" alt="$ w^+ $" src="form_487.png"/></li>
+<li><code>rank_sum_neg FLOAT8</code> - rank sum of all negative values, i.e., <img class="formulaInl" alt="$ w^- $" src="form_488.png"/></li>
+<li><code>num BIGINT</code> - number <img class="formulaInl" alt="$ n $" src="form_10.png"/> of non-zero values</li>
<li><code>z_statistic FLOAT8</code> - z-statistic <p class="formulaDsp">
-\[ z = \frac{w^+ - \frac{n(n+1)}{4}} {\sqrt{\frac{n(n+1)(2n+1)}{24} - \sum_{i=1}^n \frac{t_i^2 - 1}{48}}} \]
+<img class="formulaDsp" alt="\[ z = \frac{w^+ - \frac{n(n+1)}{4}} {\sqrt{\frac{n(n+1)(2n+1)}{24} - \sum_{i=1}^n \frac{t_i^2 - 1}{48}}} \]" src="form_489.png"/>
</p>
- where \( t_i \) is the number of values with absolute value equal to \( |x_i| \). The corresponding random variable is approximately standard normally distributed.</li>
-<li><code>p_value_one_sided FLOAT8</code> - One-sided p-value i.e., \( \Pr[Z \geq z \mid \mu \leq 0] \). Computed as <code>(1.0 - <a class="el" href="prob_8sql__in.html#a6c0a499faa80db26c0178f1e69cf7a50">normal_cdf</a>(z_statistic))</code>.</li>
-<li><code>p_value_two_sided FLOAT8</code> - Two-sided p-value, i.e., \( \Pr[ |Z| \geq |z| \mid \mu = 0] \). Computed as <code>(2 * <a class="el" href="prob_8sql__in.html#a6c0a499faa80db26c0178f1e69cf7a50">normal_cdf</a>(-abs(z_statistic)))</code>.</li>
+ where <img class="formulaInl" alt="$ t_i $" src="form_388.png"/> is the number of values with absolute value equal to <img class="formulaInl" alt="$ |x_i| $" src="form_490.png"/>. The corresponding random variable is approximately standard normally distributed.</li>
+<li><code>p_value_one_sided FLOAT8</code> - One-sided p-value i.e., <img class="formulaInl" alt="$ \Pr[Z \geq z \mid \mu \leq 0] $" src="form_491.png"/>. Computed as <code>(1.0 - <a class="el" href="prob_8sql__in.html#a6c0a499faa80db26c0178f1e69cf7a50">normal_cdf</a>(z_statistic))</code>.</li>
+<li><code>p_value_two_sided FLOAT8</code> - Two-sided p-value, i.e., <img class="formulaInl" alt="$ \Pr[ |Z| \geq |z| \mid \mu = 0] $" src="form_492.png"/>. Computed as <code>(2 * <a class="el" href="prob_8sql__in.html#a6c0a499faa80db26c0178f1e69cf7a50">normal_cdf</a>(-abs(z_statistic)))</code>.</li>
</ul>
</dd></dl>
<dl class="section user"><dt>Usage</dt><dd><ul>
-<li>One-sample test: Test null hypothesis that the mean of a sample is at most (or equal to, respectively) \( \mu_0 \): <pre>SELECT (wsr_test(<em>value</em> - <em>mu_0</em> ORDER BY abs(<em>value</em>))).* FROM <em>source</em></pre></li>
-<li>Dependent paired test: Test null hypothesis that the mean difference between the first and second value in a pair is at most (or equal to, respectively) \( \mu_0 \): <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> If correctly determining ties is important (e.g., you may want to do so when comparing to software products that take <code>first</code>, <code>second</code>, and <code>mu_0</code> as individual parameters), supply the precision parameter. This can be done as follows: <pre>SELECT (wsr_test(
+<li>One-sample test: Test null hypothesis that the mean of a sample is at most (or equal to, respectively) <img class="formulaInl" alt="$ \mu_0 $" src="form_412.png"/>: <pre>SELECT (wsr_test(<em>value</em> - <em>mu_0</em> ORDER BY abs(<em>value</em>))).* FROM <em>source</em></pre></li>
+<li>Dependent paired test: Test null hypothesis that the mean difference between the first and second value in a pair is at most (or equal to, respectively) <img class="formulaInl" alt="$ \mu_0 $" src="form_412.png"/>: <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> If correctly determining ties is important (e.g., you may want to do so when comparing to software products that take <code>first</code>, <code>second</code>, and <code>mu_0</code> as individual parameters), supply the precision parameter. This can be done as follows: <pre>SELECT (wsr_test(
<em>first</em> - <em>second</em> - <em>mu_0</em>,
3 * 2^(-52) * greatest(first, second, mu_0)
ORDER BY abs(<em>first</em> - <em>second</em>)
-)).* FROM <em>source</em></pre> Here \( 2^{-52} \) is the machine epsilon, which we scale to the magnitude of the input data and multiply with 3 because we have a sum with three terms.</li>
+)).* FROM <em>source</em></pre> Here <img class="formulaInl" alt="$ 2^{-52} $" src="form_365.png"/> is the machine epsilon, which we scale to the magnitude of the input data and multiply with 3 because we have a sum with three terms.</li>
</ul>
</dd></dl>
<dl class="section note"><dt>Note</dt><dd>This aggregate must be used as an ordered aggregate (<code>ORDER BY abs(<em>value</code></em>)) and will raise an exception if the absolute values are not ordered. </dd></dl>
@@ -844,26 +838,26 @@ FROM (
</tr>
</table>
</div><div class="memdoc">
-<p>Given realizations \( x_1, \dots, x_n \) of i.i.d. random variables \( X_1, \dots, X_n \sim N(\mu, \sigma^2) \) with unknown parameters \( \mu \) and \( \sigma^2 \), test the null hypotheses \( H_0 : \mu \leq 0 \) and \( H_0 : \mu = 0 \).</p>
+<p>Given realizations <img class="formulaInl" alt="$ x_1, \dots, x_n $" src="form_183.png"/> of i.i.d. random variables <img class="formulaInl" alt="$ X_1, \dots, X_n \sim N(\mu, \sigma^2) $" src="form_402.png"/> with unknown parameters <img class="formulaInl" alt="$ \mu $" src="form_286.png"/> and <img class="formulaInl" alt="$ \sigma^2 $" src="form_304.png"/>, test the null hypotheses <img class="formulaInl" alt="$ H_0 : \mu \leq 0 $" src="form_403.png"/> and <img class="formulaInl" alt="$ H_0 : \mu = 0 $" src="form_404.png"/>.</p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
- <tr><td class="paramname">value</td><td>Value of random variate \( x_i \)</td></tr>
+ <tr><td class="paramname">value</td><td>Value of random variate <img class="formulaInl" alt="$ x_i $" src="form_62.png"/></td></tr>
</table>
</dd>
</dl>
-<dl class="section return"><dt>Returns</dt><dd>A composite value as follows. We denote by \( \bar x \) the sample mean and by \( s^2 \) the sample variance.<ul>
+<dl class="section return"><dt>Returns</dt><dd>A composite value as follows. We denote by <img class="formulaInl" alt="$ \bar x $" src="form_405.png"/> the sample mean and by <img class="formulaInl" alt="$ s^2 $" src="form_406.png"/> the sample variance.<ul>
<li><code>statistic FLOAT8</code> - Statistic <p class="formulaDsp">
-\[ t = \frac{\sqrt n \cdot \bar x}{s} \]
+<img class="formulaDsp" alt="\[ t = \frac{\sqrt n \cdot \bar x}{s} \]" src="form_407.png"/>
</p>
- The corresponding random variable is Student-t distributed with \( (n - 1) \) degrees of freedom.</li>
-<li><code>df FLOAT8</code> - Degrees of freedom \( (n - 1) \)</li>
-<li><code>p_value_one_sided FLOAT8</code> - Lower bound on one-sided p-value. In detail, the result is \( \Pr[\bar X \geq \bar x \mid \mu = 0] \), which is a lower bound on \( \Pr[\bar X \geq \bar x \mid \mu \leq 0] \). Computed as <code>(1.0 - <a class="el" href="prob_8sql__in.html#a5322531131074c23a2dbf067ee504ef7">students_t_cdf</a>(statistic))</code>.</li>
-<li><code>p_value_two_sided FLOAT8</code> - Two-sided p-value, i.e., \( \Pr[ |\bar X| \geq |\bar x| \mid \mu = 0] \). Computed as <code>(2 * <a class="el" href="prob_8sql__in.html#a5322531131074c23a2dbf067ee504ef7">students_t_cdf</a>(-abs(statistic)))</code>.</li>
+ The corresponding random variable is Student-t distributed with <img class="formulaInl" alt="$ (n - 1) $" src="form_408.png"/> degrees of freedom.</li>
+<li><code>df FLOAT8</code> - Degrees of freedom <img class="formulaInl" alt="$ (n - 1) $" src="form_408.png"/></li>
+<li><code>p_value_one_sided FLOAT8</code> - Lower bound on one-sided p-value. In detail, the result is <img class="formulaInl" alt="$ \Pr[\bar X \geq \bar x \mid \mu = 0] $" src="form_409.png"/>, which is a lower bound on <img class="formulaInl" alt="$ \Pr[\bar X \geq \bar x \mid \mu \leq 0] $" src="form_410.png"/>. Computed as <code>(1.0 - <a class="el" href="prob_8sql__in.html#a5322531131074c23a2dbf067ee504ef7">students_t_cdf</a>(statistic))</code>.</li>
+<li><code>p_value_two_sided FLOAT8</code> - Two-sided p-value, i.e., <img class="formulaInl" alt="$ \Pr[ |\bar X| \geq |\bar x| \mid \mu = 0] $" src="form_411.png"/>. Computed as <code>(2 * <a class="el" href="prob_8sql__in.html#a5322531131074c23a2dbf067ee504ef7">students_t_cdf</a>(-abs(statistic)))</code>.</li>
</ul>
</dd></dl>
<dl class="section user"><dt>Usage</dt><dd><ul>
-<li>One-sample t-test: Test null hypothesis that the mean of a sample is at most (or equal to, respectively) \( \mu_0 \): <pre>SELECT (t_test_one(<em>value</em> - <em>mu_0</em>)).* FROM <em>source</em></pre></li>
-<li>Dependent paired t-test: Test null hypothesis that the mean difference between the first and second value in each pair is at most (or equal to, respectively) \( \mu_0 \): <pre>SELECT (t_test_one(<em>first</em> - <em>second</em> - <em>mu_0</em>)).*
+<li>One-sample t-test: Test null hypothesis that the mean of a sample is at most (or equal to, respectively) <img class="formulaInl" alt="$ \mu_0 $" src="form_412.png"/>: <pre>SELECT (t_test_one(<em>value</em> - <em>mu_0</em>)).* FROM <em>source</em></pre></li>
+<li>Dependent paired t-test: Test null hypothesis that the mean difference between the first and second value in each pair is at most (or equal to, respectively) <img class="formulaInl" alt="$ \mu_0 $" src="form_412.png"/>: <pre>SELECT (t_test_one(<em>first</em> - <em>second</em> - <em>mu_0</em>)).*
FROM <em>source</em></pre> </li>
</ul>
</dd></dl>
@@ -935,25 +929,25 @@ FROM (
</tr>
</table>
</div><div class="memdoc">
-<p>Given realizations \( x_1, \dots, x_n \) and \( y_1, \dots, y_m \) of i.i.d. random variables \( X_1, \dots, X_n \sim N(\mu_X, \sigma^2) \) and \( Y_1, \dots, Y_m \sim N(\mu_Y, \sigma^2) \) with unknown parameters \( \mu_X, \mu_Y, \) and \( \sigma^2 \), test the null hypotheses \( H_0 : \mu_X \leq \mu_Y \) and \( H_0 : \mu_X = \mu_Y \).</p>
+<p>Given realizations <img class="formulaInl" alt="$ x_1, \dots, x_n $" src="form_183.png"/> and <img class="formulaInl" alt="$ y_1, \dots, y_m $" src="form_413.png"/> of i.i.d. random variables <img class="formulaInl" alt="$ X_1, \dots, X_n \sim N(\mu_X, \sigma^2) $" src="form_414.png"/> and <img class="formulaInl" alt="$ Y_1, \dots, Y_m \sim N(\mu_Y, \sigma^2) $" src="form_415.png"/> with unknown parameters <img class="formulaInl" alt="$ \mu_X, \mu_Y, $" src="form_416.png"/> and <img class="formulaInl" alt="$ \sigma^2 $" src="form_304.png"/>, test the null hypotheses <img class="formulaInl" alt="$ H_0 : \mu_X \leq \mu_Y $" src="form_417.png"/> and <img class="formulaInl" alt="$ H_0 : \mu_X = \mu_Y $" src="form_418.png"/>.</p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
- <tr><td class="paramname">first</td><td>Indicator whether <code>value</code> is from first sample \( x_1, \dots, x_n \) (if <code>TRUE</code>) or from second sample \( y_1, \dots, y_m \) (if <code>FALSE</code>) </td></tr>
- <tr><td class="paramname">value</td><td>Value of random variate \( x_i \) or \( y_i \)</td></tr>
+ <tr><td class="paramname">first</td><td>Indicator whether <code>value</code> is from first sample <img class="formulaInl" alt="$ x_1, \dots, x_n $" src="form_183.png"/> (if <code>TRUE</code>) or from second sample <img class="formulaInl" alt="$ y_1, \dots, y_m $" src="form_413.png"/> (if <code>FALSE</code>) </td></tr>
+ <tr><td class="paramname">value</td><td>Value of random variate <img class="formulaInl" alt="$ x_i $" src="form_62.png"/> or <img class="formulaInl" alt="$ y_i $" src="form_60.png"/></td></tr>
</table>
</dd>
</dl>
-<dl class="section return"><dt>Returns</dt><dd>A composite value as follows. We denote by \( \bar x, \bar y \) the sample means and by \( s_X^2, s_Y^2 \) the sample variances.<ul>
+<dl class="section return"><dt>Returns</dt><dd>A composite value as follows. We denote by <img class="formulaInl" alt="$ \bar x, \bar y $" src="form_419.png"/> the sample means and by <img class="formulaInl" alt="$ s_X^2, s_Y^2 $" src="form_420.png"/> the sample variances.<ul>
<li><code>statistic FLOAT8</code> - Statistic <p class="formulaDsp">
-\[ t = \frac{\bar x - \bar y}{s_p \sqrt{1/n + 1/m}} \]
+<img class="formulaDsp" alt="\[ t = \frac{\bar x - \bar y}{s_p \sqrt{1/n + 1/m}} \]" src="form_421.png"/>
</p>
where <p class="formulaDsp">
-\[ s_p^2 = \frac{\sum_{i=1}^n (x_i - \bar x)^2 + \sum_{i=1}^m (y_i - \bar y)^2} {n + m - 2} \]
+<img class="formulaDsp" alt="\[ s_p^2 = \frac{\sum_{i=1}^n (x_i - \bar x)^2 + \sum_{i=1}^m (y_i - \bar y)^2} {n + m - 2} \]" src="form_422.png"/>
</p>
- is the <em>pooled variance</em>. The corresponding random variable is Student-t distributed with \( (n + m - 2) \) degrees of freedom.</li>
-<li><code>df FLOAT8</code> - Degrees of freedom \( (n + m - 2) \)</li>
-<li><code>p_value_one_sided FLOAT8</code> - Lower bound on one-sided p-value. In detail, the result is \( \Pr[\bar X - \bar Y \geq \bar x - \bar y \mid \mu_X = \mu_Y] \), which is a lower bound on \( \Pr[\bar X - \bar Y \geq \bar x - \bar y \mid \mu_X \leq \mu_Y] \). Computed as <code>(1.0 - <a class="el" href="prob_8sql__in.html#a5322531131074c23a2dbf067ee504ef7">students_t_cdf</a>(statistic))</code>.</li>
-<li><code>p_value_two_sided FLOAT8</code> - Two-sided p-value, i.e., \( \Pr[ |\bar X - \bar Y| \geq |\bar x - \bar y| \mid \mu_X = \mu_Y] \). Computed as <code>(2 * <a class="el" href="prob_8sql__in.html#a5322531131074c23a2dbf067ee504ef7">students_t_cdf</a>(-abs(statistic)))</code>.</li>
+ is the <em>pooled variance</em>. The corresponding random variable is Student-t distributed with <img class="formulaInl" alt="$ (n + m - 2) $" src="form_423.png"/> degrees of freedom.</li>
+<li><code>df FLOAT8</code> - Degrees of freedom <img class="formulaInl" alt="$ (n + m - 2) $" src="form_423.png"/></li>
+<li><code>p_value_one_sided FLOAT8</code> - Lower bound on one-sided p-value. In detail, the result is <img class="formulaInl" alt="$ \Pr[\bar X - \bar Y \geq \bar x - \bar y \mid \mu_X = \mu_Y] $" src="form_424.png"/>, which is a lower bound on <img class="formulaInl" alt="$ \Pr[\bar X - \bar Y \geq \bar x - \bar y \mid \mu_X \leq \mu_Y] $" src="form_425.png"/>. Computed as <code>(1.0 - <a class="el" href="prob_8sql__in.html#a5322531131074c23a2dbf067ee504ef7">students_t_cdf</a>(statistic))</code>.</li>
+<li><code>p_value_two_sided FLOAT8</code> - Two-sided p-value, i.e., <img class="formulaInl" alt="$ \Pr[ |\bar X - \bar Y| \geq |\bar x - \bar y| \mid \mu_X = \mu_Y] $" src="form_426.png"/>. Computed as <code>(2 * <a class="el" href="prob_8sql__in.html#a5322531131074c23a2dbf067ee504ef7">students_t_cdf</a>(-abs(statistic)))</code>.</li>
</ul>
</dd></dl>
<dl class="section user"><dt>Usage</dt><dd><ul>
@@ -1034,25 +1028,25 @@ FROM (
</tr>
</table>
</div><div class="memdoc">
-<p>Given realizations \( x_1, \dots, x_n \) and \( y_1, \dots, y_m \) of i.i.d. random variables \( X_1, \dots, X_n \sim N(\mu_X, \sigma_X^2) \) and \( Y_1, \dots, Y_m \sim N(\mu_Y, \sigma_Y^2) \) with unknown parameters \( \mu_X, \mu_Y, \sigma_X^2, \) and \( \sigma_Y^2 \), test the null hypotheses \( H_0 : \mu_X \leq \mu_Y \) and \( H_0 : \mu_X = \mu_Y \).</p>
+<p>Given realizations <img class="formulaInl" alt="$ x_1, \dots, x_n $" src="form_183.png"/> and <img class="formulaInl" alt="$ y_1, \dots, y_m $" src="form_413.png"/> of i.i.d. random variables <img class="formulaInl" alt="$ X_1, \dots, X_n \sim N(\mu_X, \sigma_X^2) $" src="form_427.png"/> and <img class="formulaInl" alt="$ Y_1, \dots, Y_m \sim N(\mu_Y, \sigma_Y^2) $" src="form_428.png"/> with unknown parameters <img class="formulaInl" alt="$ \mu_X, \mu_Y, \sigma_X^2, $" src="form_429.png"/> and <img class="formulaInl" alt="$ \sigma_Y^2 $" src="form_430.png"/>, test the null hypotheses <img class="formulaInl" alt="$ H_0 : \mu_X \leq \mu_Y $" src="form_417.png"/> and <img class="formulaInl" alt="$ H_0 : \mu_X = \mu_Y $" src="form_418.png"/>.</p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
- <tr><td class="paramname">first</td><td>Indicator whether <code>value</code> is from first sample \( x_1, \dots, x_n \) (if <code>TRUE</code>) or from second sample \( y_1, \dots, y_m \) (if <code>FALSE</code>) </td></tr>
- <tr><td class="paramname">value</td><td>Value of random variate \( x_i \) or \( y_i \)</td></tr>
+ <tr><td class="paramname">first</td><td>Indicator whether <code>value</code> is from first sample <img class="formulaInl" alt="$ x_1, \dots, x_n $" src="form_183.png"/> (if <code>TRUE</code>) or from second sample <img class="formulaInl" alt="$ y_1, \dots, y_m $" src="form_413.png"/> (if <code>FALSE</code>) </td></tr>
+ <tr><td class="paramname">value</td><td>Value of random variate <img class="formulaInl" alt="$ x_i $" src="form_62.png"/> or <img class="formulaInl" alt="$ y_i $" src="form_60.png"/></td></tr>
</table>
</dd>
</dl>
-<dl class="section return"><dt>Returns</dt><dd>A composite value as follows. We denote by \( \bar x, \bar y \) the sample means and by \( s_X^2, s_Y^2 \) the sample variances.<ul>
+<dl class="section return"><dt>Returns</dt><dd>A composite value as follows. We denote by <img class="formulaInl" alt="$ \bar x, \bar y $" src="form_419.png"/> the sample means and by <img class="formulaInl" alt="$ s_X^2, s_Y^2 $" src="form_420.png"/> the sample variances.<ul>
<li><code>statistic FLOAT8</code> - Statistic <p class="formulaDsp">
-\[ t = \frac{\bar x - \bar y}{\sqrt{s_X^2/n + s_Y^2/m}} \]
+<img class="formulaDsp" alt="\[ t = \frac{\bar x - \bar y}{\sqrt{s_X^2/n + s_Y^2/m}} \]" src="form_431.png"/>
</p>
The corresponding random variable is approximately Student-t distributed with <p class="formulaDsp">
-\[ \frac{(s_X^2 / n + s_Y^2 / m)^2}{(s_X^2 / n)^2/(n-1) + (s_Y^2 / m)^2/(m-1)} \]
+<img class="formulaDsp" alt="\[ \frac{(s_X^2 / n + s_Y^2 / m)^2}{(s_X^2 / n)^2/(n-1) + (s_Y^2 / m)^2/(m-1)} \]" src="form_432.png"/>
</p>
degrees of freedom (Welch–Satterthwaite formula).</li>
<li><code>df FLOAT8</code> - Degrees of freedom (as above)</li>
-<li><code>p_value_one_sided FLOAT8</code> - Lower bound on one-sided p-value. In detail, the result is \( \Pr[\bar X - \bar Y \geq \bar x - \bar y \mid \mu_X = \mu_Y] \), which is a lower bound on \( \Pr[\bar X - \bar Y \geq \bar x - \bar y \mid \mu_X \leq \mu_Y] \). Computed as <code>(1.0 - <a class="el" href="prob_8sql__in.html#a5322531131074c23a2dbf067ee504ef7">students_t_cdf</a>(statistic))</code>.</li>
-<li><code>p_value_two_sided FLOAT8</code> - Two-sided p-value, i.e., \( \Pr[ |\bar X - \bar Y| \geq |\bar x - \bar y| \mid \mu_X = \mu_Y] \). Computed as <code>(2 * <a class="el" href="prob_8sql__in.html#a5322531131074c23a2dbf067ee504ef7">students_t_cdf</a>(-abs(statistic)))</code>.</li>
+<li><code>p_value_one_sided FLOAT8</code> - Lower bound on one-sided p-value. In detail, the result is <img class="formulaInl" alt="$ \Pr[\bar X - \bar Y \geq \bar x - \bar y \mid \mu_X = \mu_Y] $" src="form_424.png"/>, which is a lower bound on <img class="formulaInl" alt="$ \Pr[\bar X - \bar Y \geq \bar x - \bar y \mid \mu_X \leq \mu_Y] $" src="form_425.png"/>. Computed as <code>(1.0 - <a class="el" href="prob_8sql__in.html#a5322531131074c23a2dbf067ee504ef7">students_t_cdf</a>(statistic))</code>.</li>
+<li><code>p_value_two_sided FLOAT8</code> - Two-sided p-value, i.e., <img class="formulaInl" alt="$ \Pr[ |\bar X - \bar Y| \geq |\bar x - \bar y| \mid \mu_X = \mu_Y] $" src="form_426.png"/>. Computed as <code>(2 * <a class="el" href="prob_8sql__in.html#a5322531131074c23a2dbf067ee504ef7">students_t_cdf</a>(-abs(statistic)))</code>.</li>
</ul>
</dd></dl>
<dl class="section user"><dt>Usage</dt><dd><ul>
@@ -1123,28 +1117,28 @@ FROM (
</tr>
</table>
</div><div class="memdoc">
-<p>Given realizations \( x_1, \dots, x_m \) and \( y_1, \dots, y_m \) of i.i.d. random variables \( X_1, \dots, X_m \) and i.i.d. \( Y_1, \dots, Y_n \), respectively, test the null hypothesis that the underlying distributions are equal, i.e., \( H_0 : \forall i,j: \Pr[X_i > Y_j] + \frac{\Pr[X_i = Y_j]}{2} = \frac 12 \).</p>
+<p>Given realizations <img class="formulaInl" alt="$ x_1, \dots, x_m $" src="form_433.png"/> and <img class="formulaInl" alt="$ y_1, \dots, y_m $" src="form_413.png"/> of i.i.d. random variables <img class="formulaInl" alt="$ X_1, \dots, X_m $" src="form_458.png"/> and i.i.d. <img class="formulaInl" alt="$ Y_1, \dots, Y_n $" src="form_459.png"/>, respectively, test the null hypothesis that the underlying distributions are equal, i.e., <img class="formulaInl" alt="$ H_0 : \forall i,j: \Pr[X_i > Y_j] + \frac{\Pr[X_i = Y_j]}{2} = \frac 12 $" src="form_469.png"/>.</p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">first</td><td>Determines whether the value belongs to the first (if <code>TRUE</code>) or the second sample (if <code>FALSE</code>) </td></tr>
- <tr><td class="paramname">value</td><td>Value of random variate \( x_i \) or \( y_i \)</td></tr>
+ <tr><td class="paramname">value</td><td>Value of random variate <img class="formulaInl" alt="$ x_i $" src="form_62.png"/> or <img class="formulaInl" alt="$ y_i $" src="form_60.png"/></td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>A composite value.<ul>
<li><code>statistic FLOAT8</code> - Statistic <p class="formulaDsp">
-\[ z = \frac{u - \bar x}{\sqrt{\frac{mn(m+n+1)}{12}}} \]
+<img class="formulaDsp" alt="\[ z = \frac{u - \bar x}{\sqrt{\frac{mn(m+n+1)}{12}}} \]" src="form_470.png"/>
</p>
- where \( u \) is the u-statistic computed as follows. The z-statistic is approximately standard normally distributed.</li>
-<li><code>u_statistic FLOAT8</code> - Statistic \( u = \min \{ u_x, u_y \} \) where <p class="formulaDsp">
-\[ u_x = mn + \binom{m+1}{2} - \sum_{i=1}^m r_{x,i} \]
+ where <img class="formulaInl" alt="$ u $" src="form_471.png"/> is the u-statistic computed as follows. The z-statistic is approximately standard normally distributed.</li>
+<li><code>u_statistic FLOAT8</code> - Statistic <img class="formulaInl" alt="$ u = \min \{ u_x, u_y \} $" src="form_472.png"/> where <p class="formulaDsp">
+<img class="formulaDsp" alt="\[ u_x = mn + \binom{m+1}{2} - \sum_{i=1}^m r_{x,i} \]" src="form_473.png"/>
</p>
where <p class="formulaDsp">
-\[ r_{x,i} = \{ j \mid x_j < x_i \} + \{ j \mid y_j < x_i \} + \frac{\{ j \mid x_j = x_i \} + \{ j \mid y_j = x_i \} + 1}{2} \]
+<img class="formulaDsp" alt="\[ r_{x,i} = \{ j \mid x_j < x_i \} + \{ j \mid y_j < x_i \} + \frac{\{ j \mid x_j = x_i \} + \{ j \mid y_j = x_i \} + 1}{2} \]" src="form_474.png"/>
</p>
- is defined as the rank of \( x_i \) in the combined list of all \( m+n \) observations. For ties, the average rank of all equal values is used.</li>
-<li><code>p_value_one_sided FLOAT8</code> - Approximate one-sided p-value, i.e., an approximate value for \( \Pr[Z \geq z \mid H_0] \). Computed as <code>(1.0 - <a class="el" href="prob_8sql__in.html#a6c0a499faa80db26c0178f1e69cf7a50">normal_cdf</a>(z_statistic))</code>.</li>
-<li><code>p_value_two_sided FLOAT8</code> - Approximate two-sided p-value, i.e., an approximate value for \( \Pr[|Z| \geq |z| \mid H_0] \). Computed as <code>(2 * <a class="el" href="prob_8sql__in.html#a6c0a499faa80db26c0178f1e69cf7a50">normal_cdf</a>(-abs(z_statistic)))</code>.</li>
+ is defined as the rank of <img class="formulaInl" alt="$ x_i $" src="form_62.png"/> in the combined list of all <img class="formulaInl" alt="$ m+n $" src="form_475.png"/> observations. For ties, the average rank of all equal values is used.</li>
+<li><code>p_value_one_sided FLOAT8</code> - Approximate one-sided p-value, i.e., an approximate value for <img class="formulaInl" alt="$ \Pr[Z \geq z \mid H_0] $" src="form_476.png"/>. Computed as <code>(1.0 - <a class="el" href="prob_8sql__in.html#a6c0a499faa80db26c0178f1e69cf7a50">normal_cdf</a>(z_statistic))</code>.</li>
+<li><code>p_value_two_sided FLOAT8</code> - Approximate two-sided p-value, i.e., an approximate value for <img class="formulaInl" alt="$ \Pr[|Z| \geq |z| \mid H_0] $" src="form_477.png"/>. Computed as <code>(2 * <a class="el" href="prob_8sql__in.html#a6c0a499faa80db26c0178f1e69cf7a50">normal_cdf</a>(-abs(z_statistic)))</code>.</li>
</ul>
</dd></dl>
<dl class="section user"><dt>Usage</dt><dd><ul>
@@ -1187,7 +1181,7 @@ FROM (
<div id="nav-path" class="navpath"><!-- id is needed for treeview function! -->
<ul>
<li class="navelem"><a class="el" href="dir_68267d1309a1af8e8297ef4c3efbcdba.html">src</a></li><li class="navelem"><a class="el" href="dir_efbcf68973d247bbf15f9eecae7f24e3.html">ports</a></li><li class="navelem"><a class="el" href="dir_a4a48839224ef8488facbffa8a397967.html">postgres</a></li><li class="navelem"><a class="el" href="dir_dc596537ad427a4d866006d1a3e1fe29.html">modules</a></li><li class="navelem"><a class="el" href="dir_505cd743a8a717435eca324f49291a46.html">stats</a></li><li class="navelem"><a class="el" href="hypothesis__tests_8sql__in.html">hypothesis_tests.sql_in</a></li>
- <li class="footer">Generated on Mon Jul 27 2015 20:37:45 for MADlib by
+ <li class="footer">Generated on Thu Apr 7 2016 14:24:10 for MADlib by
<a href="http://www.doxygen.org/index.html">
<img class="footer" src="doxygen.png" alt="doxygen"/></a> 1.8.10 </li>
</ul>
http://git-wip-us.apache.org/repos/asf/incubator-madlib-site/blob/c506dd05/docs/latest/index.html
----------------------------------------------------------------------
diff --git a/docs/latest/index.html b/docs/latest/index.html
index c2dff64..645fa47 100644
--- a/docs/latest/index.html
+++ b/docs/latest/index.html
@@ -24,14 +24,8 @@
<script type="text/javascript">
$(document).ready(function() { init_search(); });
</script>
-<script type="text/x-mathjax-config">
- MathJax.Hub.Config({
- extensions: ["tex2jax.js", "TeX/AMSmath.js", "TeX/AMSsymbols.js"],
- jax: ["input/TeX","output/HTML-CSS"],
-});
-</script><script src="../mathjax/MathJax.js"></script>
<!-- hack in the navigation tree -->
-<script type="text/javascript" src="navtree_hack.js"></script>
+<script type="text/javascript" src="eigen_navtree_hacks.js"></script>
<link href="doxygen.css" rel="stylesheet" type="text/css" />
<link href="madlib_extra.css" rel="stylesheet" type="text/css"/>
<!-- google analytics -->
@@ -40,7 +34,7 @@
(i[r].q=i[r].q||[]).push(arguments)},i[r].l=1*new Date();a=s.createElement(o),
m=s.getElementsByTagName(o)[0];a.async=1;a.src=g;m.parentNode.insertBefore(a,m)
})(window,document,'script','//www.google-analytics.com/analytics.js','ga');
- ga('create', 'UA-45382226-1', 'auto');
+ ga('create', 'UA-45382226-1', 'madlib.net');
ga('send', 'pageview');
</script>
</head>
@@ -50,10 +44,10 @@
<table cellspacing="0" cellpadding="0">
<tbody>
<tr style="height: 56px;">
- <td id="projectlogo"><a href="http://madlib.incubator.apache.org"><img alt="Logo" src="madlib.png" height="50" style="padding-left:0.5em;" border="0"/ ></a></td>
+ <td id="projectlogo"><a href="http://madlib.net"><img alt="Logo" src="madlib.png" height="50" style="padding-left:0.5em;" border="0"/ ></a></td>
<td style="padding-left: 0.5em;">
<div id="projectname">
- <span id="projectnumber">1.8</span>
+ <span id="projectnumber">1.9</span>
</div>
<div id="projectbrief">User Documentation for MADlib</div>
</td>
@@ -115,23 +109,29 @@ $(document).ready(function(){initNavTree('index.html','');});
<div class="title">MADlib Documentation</div> </div>
</div><!--header-->
<div class="contents">
-<div class="textblock"><p>MADlib is an open-source library for scalable in-database analytics. It provides data-parallel implementations of mathematical, statistical and machine learning methods for structured and unstructured data.</p>
+<div class="textblock"><p>Apache MADlib (incubating) is an open-source library for scalable in-database analytics. It provides data-parallel implementations of mathematical, statistical and machine learning methods for structured and unstructured data.</p>
<p>The MADlib mission: to foster widespread development of scalable analytic skills, by harnessing efforts from commercial practice, academic research, and open-source development.</p>
<p>Useful links: </p><ul>
<li>
-MADlib project site <a href="http://madlib.incubator.apache.org/">http://madlib.incubator.apache.org/</a> </li>
+<a href="http://madlib.incubator.apache.org">MADlib web site</a> </li>
<li>
-MADlib JIRAs <a href="https://issues.apache.org/jira/browse/MADLIB/">https://issues.apache.org/jira/browse/MADLIB/</a> and wiki <a href="https://cwiki.apache.org/confluence/display/MADLIB">https://cwiki.apache.org/confluence/display/MADLIB</a> </li>
+<a href="https://cwiki.apache.org/confluence/display/MADLIB">MADlib wiki</a> </li>
<li>
-User documentation for earlier releases: <a href="../v1.7.1/index.html">v1.7.1</a>, <a href="../v1.7/index.html">v1.7</a>, <a href="../v1.6/index.html">v1.6</a>, <a href="../v1.5/index.html">v1.5</a>, <a href="../v1.4/index.html">v1.4</a>, <a href="../v1.3/index.html">v1.3</a>, <a href="../v1.2/index.html">v1.2</a> </li>
+<a href="https://issues.apache.org/jira/browse/MADLIB/">JIRAs for reporting bugs and reviewing backlog</a> </li>
+<li>
+<a href="https://mail-archives.apache.org/mod_mbox/incubator-madlib-user/">User mailing list</a> </li>
+<li>
+<a href="https://mail-archives.apache.org/mod_mbox/incubator-madlib-dev/">Dev mailing list</a> </li>
+<li>
+User documentation for earlier releases: <a href="../v1.8/index.html">v1.8</a>, <a href="../v1.7.1/index.html">v1.7.1</a>, <a href="../v1.7/index.html">v1.7</a>, <a href="../v1.6/index.html">v1.6</a>, <a href="../v1.5/index.html">v1.5</a>, <a href="../v1.4/index.html">v1.4</a>, <a href="../v1.3/index.html">v1.3</a>, <a href="../v1.2/index.html">v1.2</a> </li>
</ul>
-<p>Please refer to the <a href="https://github.com/madlib/madlib/blob/v1.8/ReadMe.txt">Read-Me</a> file for information about incorporated third-party material. License information regarding MADlib and included third-party libraries can be found inside the <a href="https://github.com/madlib/madlib/tree/v1.8/license">license</a> directory. </p>
+<p>Please refer to the <a href="https://github.com/apache/incubator-madlib/blob/master/ReadMe.txt">Read-Me</a> file for information about incorporated third-party material. License information regarding MADlib and included third-party libraries can be found inside the <a href="https://github.com/apache/incubator-madlib/blob/master/LICENSE">License</a> directory. </p>
</div></div><!-- contents -->
</div><!-- doc-content -->
<!-- start footer part -->
<div id="nav-path" class="navpath"><!-- id is needed for treeview function! -->
<ul>
- <li class="footer">Generated on Mon Jul 27 2015 20:37:46 for MADlib by
+ <li class="footer">Generated on Thu Apr 7 2016 14:24:11 for MADlib by
<a href="http://www.doxygen.org/index.html">
<img class="footer" src="doxygen.png" alt="doxygen"/></a> 1.8.10 </li>
</ul>
http://git-wip-us.apache.org/repos/asf/incubator-madlib-site/blob/c506dd05/docs/latest/kmeans_8sql__in.html
----------------------------------------------------------------------
diff --git a/docs/latest/kmeans_8sql__in.html b/docs/latest/kmeans_8sql__in.html
index 08f057f..9b215f2 100644
--- a/docs/latest/kmeans_8sql__in.html
+++ b/docs/latest/kmeans_8sql__in.html
@@ -24,14 +24,8 @@
<script type="text/javascript">
$(document).ready(function() { init_search(); });
</script>
-<script type="text/x-mathjax-config">
- MathJax.Hub.Config({
- extensions: ["tex2jax.js", "TeX/AMSmath.js", "TeX/AMSsymbols.js"],
- jax: ["input/TeX","output/HTML-CSS"],
-});
-</script><script src="../mathjax/MathJax.js"></script>
<!-- hack in the navigation tree -->
-<script type="text/javascript" src="navtree_hack.js"></script>
+<script type="text/javascript" src="eigen_navtree_hacks.js"></script>
<link href="doxygen.css" rel="stylesheet" type="text/css" />
<link href="madlib_extra.css" rel="stylesheet" type="text/css"/>
<!-- google analytics -->
@@ -40,7 +34,7 @@
(i[r].q=i[r].q||[]).push(arguments)},i[r].l=1*new Date();a=s.createElement(o),
m=s.getElementsByTagName(o)[0];a.async=1;a.src=g;m.parentNode.insertBefore(a,m)
})(window,document,'script','//www.google-analytics.com/analytics.js','ga');
- ga('create', 'UA-45382226-1', 'auto');
+ ga('create', 'UA-45382226-1', 'madlib.net');
ga('send', 'pageview');
</script>
</head>
@@ -50,10 +44,10 @@
<table cellspacing="0" cellpadding="0">
<tbody>
<tr style="height: 56px;">
- <td id="projectlogo"><a href="http://madlib.incubator.apache.org"><img alt="Logo" src="madlib.png" height="50" style="padding-left:0.5em;" border="0"/ ></a></td>
+ <td id="projectlogo"><a href="http://madlib.net"><img alt="Logo" src="madlib.png" height="50" style="padding-left:0.5em;" border="0"/ ></a></td>
<td style="padding-left: 0.5em;">
<div id="projectname">
- <span id="projectnumber">1.8</span>
+ <span id="projectnumber">1.9</span>
</div>
<div id="projectbrief">User Documentation for MADlib</div>
</td>
@@ -249,7 +243,7 @@ Functions</h2></td></tr>
<div id="nav-path" class="navpath"><!-- id is needed for treeview function! -->
<ul>
<li class="navelem"><a class="el" href="dir_68267d1309a1af8e8297ef4c3efbcdba.html">src</a></li><li class="navelem"><a class="el" href="dir_efbcf68973d247bbf15f9eecae7f24e3.html">ports</a></li><li class="navelem"><a class="el" href="dir_a4a48839224ef8488facbffa8a397967.html">postgres</a></li><li class="navelem"><a class="el" href="dir_dc596537ad427a4d866006d1a3e1fe29.html">modules</a></li><li class="navelem"><a class="el" href="dir_73ccba3aa44ce35463f879b4ebbd3f46.html">kmeans</a></li><li class="navelem"><a class="el" href="kmeans_8sql__in.html">kmeans.sql_in</a></li>
- <li class="footer">Generated on Mon Jul 27 2015 20:37:45 for MADlib by
+ <li class="footer">Generated on Thu Apr 7 2016 14:24:10 for MADlib by
<a href="http://www.doxygen.org/index.html">
<img class="footer" src="doxygen.png" alt="doxygen"/></a> 1.8.10 </li>
</ul>
http://git-wip-us.apache.org/repos/asf/incubator-madlib-site/blob/c506dd05/docs/latest/lda_8sql__in.html
----------------------------------------------------------------------
diff --git a/docs/latest/lda_8sql__in.html b/docs/latest/lda_8sql__in.html
index 3977d5c..da80cce 100644
--- a/docs/latest/lda_8sql__in.html
+++ b/docs/latest/lda_8sql__in.html
@@ -24,14 +24,8 @@
<script type="text/javascript">
$(document).ready(function() { init_search(); });
</script>
-<script type="text/x-mathjax-config">
- MathJax.Hub.Config({
- extensions: ["tex2jax.js", "TeX/AMSmath.js", "TeX/AMSsymbols.js"],
- jax: ["input/TeX","output/HTML-CSS"],
-});
-</script><script src="../mathjax/MathJax.js"></script>
<!-- hack in the navigation tree -->
-<script type="text/javascript" src="navtree_hack.js"></script>
+<script type="text/javascript" src="eigen_navtree_hacks.js"></script>
<link href="doxygen.css" rel="stylesheet" type="text/css" />
<link href="madlib_extra.css" rel="stylesheet" type="text/css"/>
<!-- google analytics -->
@@ -40,7 +34,7 @@
(i[r].q=i[r].q||[]).push(arguments)},i[r].l=1*new Date();a=s.createElement(o),
m=s.getElementsByTagName(o)[0];a.async=1;a.src=g;m.parentNode.insertBefore(a,m)
})(window,document,'script','//www.google-analytics.com/analytics.js','ga');
- ga('create', 'UA-45382226-1', 'auto');
+ ga('create', 'UA-45382226-1', 'madlib.net');
ga('send', 'pageview');
</script>
</head>
@@ -50,10 +44,10 @@
<table cellspacing="0" cellpadding="0">
<tbody>
<tr style="height: 56px;">
- <td id="projectlogo"><a href="http://madlib.incubator.apache.org"><img alt="Logo" src="madlib.png" height="50" style="padding-left:0.5em;" border="0"/ ></a></td>
+ <td id="projectlogo"><a href="http://madlib.net"><img alt="Logo" src="madlib.png" height="50" style="padding-left:0.5em;" border="0"/ ></a></td>
<td style="padding-left: 0.5em;">
<div id="projectname">
- <span id="projectnumber">1.8</span>
+ <span id="projectnumber">1.9</span>
</div>
<div id="projectbrief">User Documentation for MADlib</div>
</td>
@@ -195,6 +189,8 @@ Functions</h2></td></tr>
<tr class="memitem:aa4fd0a274f1c400014f2ea9549507436"><td class="memItemLeft" align="right" valign="top">set< lda_result > </td><td class="memItemRight" valign="bottom"><a class="el" href="lda_8sql__in.html#aa4fd0a274f1c400014f2ea9549507436">__lda_util_conorm_data</a> (text data_table, text vocab_table, text output_data_table, text output_vocab_table)</td></tr>
<tr class="memdesc:aa4fd0a274f1c400014f2ea9549507436"><td class="mdescLeft"> </td><td class="mdescRight">This UDF extracts the list of wordids from the data table and joins it with the vocabulary table to get the list of common wordids, next it will normalize the vocabulary based on the common wordids and then normalize the data table based on the normalized vocabulary. <a href="#aa4fd0a274f1c400014f2ea9549507436">More...</a><br /></td></tr>
<tr class="separator:aa4fd0a274f1c400014f2ea9549507436"><td class="memSeparator" colspan="2"> </td></tr>
+<tr class="memitem:a3ea35eb8ae6670a07948a56b85c1710c"><td class="memItemLeft" align="right" valign="top">_pivotalr_lda_model </td><td class="memItemRight" valign="bottom"><a class="el" href="lda_8sql__in.html#a3ea35eb8ae6670a07948a56b85c1710c">lda_parse_model</a> (bigint[] lda_model, integer voc_size, integer topic_num)</td></tr>
+<tr class="separator:a3ea35eb8ae6670a07948a56b85c1710c"><td class="memSeparator" colspan="2"> </td></tr>
</table>
<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
<div class="textblock"><dl class="section date"><dt>Date</dt><dd>Dec 2012</dd></dl>
@@ -1120,6 +1116,38 @@ Functions</h2></td></tr>
</div>
</div>
+<a class="anchor" id="a3ea35eb8ae6670a07948a56b85c1710c"></a>
+<div class="memitem">
+<div class="memproto">
+ <table class="memname">
+ <tr>
+ <td class="memname">_pivotalr_lda_model lda_parse_model </td>
+ <td>(</td>
+ <td class="paramtype">bigint[] </td>
+ <td class="paramname"><em>lda_model</em>, </td>
+ </tr>
+ <tr>
+ <td class="paramkey"></td>
+ <td></td>
+ <td class="paramtype">integer </td>
+ <td class="paramname"><em>voc_size</em>, </td>
+ </tr>
+ <tr>
+ <td class="paramkey"></td>
+ <td></td>
+ <td class="paramtype">integer </td>
+ <td class="paramname"><em>topic_num</em> </td>
+ </tr>
+ <tr>
+ <td></td>
+ <td>)</td>
+ <td></td><td></td>
+ </tr>
+ </table>
+</div><div class="memdoc">
+
+</div>
+</div>
<a class="anchor" id="af1fde06c39dd12bb9e5544997f815323"></a>
<div class="memitem">
<div class="memproto">
@@ -1283,7 +1311,7 @@ Functions</h2></td></tr>
<div id="nav-path" class="navpath"><!-- id is needed for treeview function! -->
<ul>
<li class="navelem"><a class="el" href="dir_68267d1309a1af8e8297ef4c3efbcdba.html">src</a></li><li class="navelem"><a class="el" href="dir_efbcf68973d247bbf15f9eecae7f24e3.html">ports</a></li><li class="navelem"><a class="el" href="dir_a4a48839224ef8488facbffa8a397967.html">postgres</a></li><li class="navelem"><a class="el" href="dir_dc596537ad427a4d866006d1a3e1fe29.html">modules</a></li><li class="navelem"><a class="el" href="dir_6ff79b0655deb26abf8f86290b84a97c.html">lda</a></li><li class="navelem"><a class="el" href="lda_8sql__in.html">lda.sql_in</a></li>
- <li class="footer">Generated on Mon Jul 27 2015 20:37:45 for MADlib by
+ <li class="footer">Generated on Thu Apr 7 2016 14:24:10 for MADlib by
<a href="http://www.doxygen.org/index.html">
<img class="footer" src="doxygen.png" alt="doxygen"/></a> 1.8.10 </li>
</ul>
http://git-wip-us.apache.org/repos/asf/incubator-madlib-site/blob/c506dd05/docs/latest/linalg_8sql__in.html
----------------------------------------------------------------------
diff --git a/docs/latest/linalg_8sql__in.html b/docs/latest/linalg_8sql__in.html
index 2b5f0b1..64cabeb 100644
--- a/docs/latest/linalg_8sql__in.html
+++ b/docs/latest/linalg_8sql__in.html
@@ -24,14 +24,8 @@
<script type="text/javascript">
$(document).ready(function() { init_search(); });
</script>
-<script type="text/x-mathjax-config">
- MathJax.Hub.Config({
- extensions: ["tex2jax.js", "TeX/AMSmath.js", "TeX/AMSsymbols.js"],
- jax: ["input/TeX","output/HTML-CSS"],
-});
-</script><script src="../mathjax/MathJax.js"></script>
<!-- hack in the navigation tree -->
-<script type="text/javascript" src="navtree_hack.js"></script>
+<script type="text/javascript" src="eigen_navtree_hacks.js"></script>
<link href="doxygen.css" rel="stylesheet" type="text/css" />
<link href="madlib_extra.css" rel="stylesheet" type="text/css"/>
<!-- google analytics -->
@@ -40,7 +34,7 @@
(i[r].q=i[r].q||[]).push(arguments)},i[r].l=1*new Date();a=s.createElement(o),
m=s.getElementsByTagName(o)[0];a.async=1;a.src=g;m.parentNode.insertBefore(a,m)
})(window,document,'script','//www.google-analytics.com/analytics.js','ga');
- ga('create', 'UA-45382226-1', 'auto');
+ ga('create', 'UA-45382226-1', 'madlib.net');
ga('send', 'pageview');
</script>
</head>
@@ -50,10 +44,10 @@
<table cellspacing="0" cellpadding="0">
<tbody>
<tr style="height: 56px;">
- <td id="projectlogo"><a href="http://madlib.incubator.apache.org"><img alt="Logo" src="madlib.png" height="50" style="padding-left:0.5em;" border="0"/ ></a></td>
+ <td id="projectlogo"><a href="http://madlib.net"><img alt="Logo" src="madlib.png" height="50" style="padding-left:0.5em;" border="0"/ ></a></td>
<td style="padding-left: 0.5em;">
<div id="projectname">
- <span id="projectnumber">1.8</span>
+ <span id="projectnumber">1.9</span>
</div>
<div id="projectbrief">User Documentation for MADlib</div>
</td>
@@ -159,12 +153,12 @@ Functions</h2></td></tr>
<tr class="memitem:a8239fac12096a5dc2720f6cb35b011e5"><td class="memItemLeft" align="right" valign="top">closest_column_result </td><td class="memItemRight" valign="bottom"><a class="el" href="linalg_8sql__in.html#a8239fac12096a5dc2720f6cb35b011e5">_closest_column</a> (float8[] m, float8[] x, regproc dist, text dist_dn)</td></tr>
<tr class="separator:a8239fac12096a5dc2720f6cb35b011e5"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:acf6628dfa4d73dfce65a582aa5c5a3db"><td class="memItemLeft" align="right" valign="top">closest_column_result </td><td class="memItemRight" valign="bottom"><a class="el" href="linalg_8sql__in.html#acf6628dfa4d73dfce65a582aa5c5a3db">closest_column</a> (float8[] m, float8[] x, regproc dist="squared_dist_norm2")</td></tr>
-<tr class="memdesc:acf6628dfa4d73dfce65a582aa5c5a3db"><td class="mdescLeft"> </td><td class="mdescRight">Given matrix \( M \) and vector \( \vec x \) compute the column of \( M \) that is closest to \( \vec x \). <a href="#acf6628dfa4d73dfce65a582aa5c5a3db">More...</a><br /></td></tr>
+<tr class="memdesc:acf6628dfa4d73dfce65a582aa5c5a3db"><td class="mdescLeft"> </td><td class="mdescRight">Given matrix <img class="formulaInl" alt="$ M $" src="form_174.png"/> and vector <img class="formulaInl" alt="$ \vec x $" src="form_175.png"/> compute the column of <img class="formulaInl" alt="$ M $" src="form_174.png"/> that is closest to <img class="formulaInl" alt="$ \vec x $" src="form_175.png"/>. <a href="#acf6628dfa4d73dfce65a582aa5c5a3db">More...</a><br /></td></tr>
<tr class="separator:acf6628dfa4d73dfce65a582aa5c5a3db"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a31c8d60f9a631c27f5f91964e0108da9"><td class="memItemLeft" align="right" valign="top">closest_column_result </td><td class="memItemRight" valign="bottom"><a class="el" href="linalg_8sql__in.html#a31c8d60f9a631c27f5f91964e0108da9">closest_column</a> (float8[] m, float8[] x)</td></tr>
<tr class="separator:a31c8d60f9a631c27f5f91964e0108da9"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a37fd07274dbc9a7f779346b8572ec989"><td class="memItemLeft" align="right" valign="top">closest_columns_result </td><td class="memItemRight" valign="bottom"><a class="el" href="linalg_8sql__in.html#a37fd07274dbc9a7f779346b8572ec989">_closest_columns</a> (float8[] m, float8[] x, integer num, regproc dist, text dist_dn)</td></tr>
-<tr class="memdesc:a37fd07274dbc9a7f779346b8572ec989"><td class="mdescLeft"> </td><td class="mdescRight">Given matrix \( M \) and vector \( \vec x \) compute the columns of \( M \) that are closest to \( \vec x \). <a href="#a37fd07274dbc9a7f779346b8572ec989">More...</a><br /></td></tr>
+<tr class="memdesc:a37fd07274dbc9a7f779346b8572ec989"><td class="mdescLeft"> </td><td class="mdescRight">Given matrix <img class="formulaInl" alt="$ M $" src="form_174.png"/> and vector <img class="formulaInl" alt="$ \vec x $" src="form_175.png"/> compute the columns of <img class="formulaInl" alt="$ M $" src="form_174.png"/> that are closest to <img class="formulaInl" alt="$ \vec x $" src="form_175.png"/>. <a href="#a37fd07274dbc9a7f779346b8572ec989">More...</a><br /></td></tr>
<tr class="separator:a37fd07274dbc9a7f779346b8572ec989"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ad864339591086b635d12015db993b5bc"><td class="memItemLeft" align="right" valign="top">closest_columns_result </td><td class="memItemRight" valign="bottom"><a class="el" href="linalg_8sql__in.html#ad864339591086b635d12015db993b5bc">closest_columns</a> (float8[] m, float8[] x, integer num, regproc dist)</td></tr>
<tr class="separator:ad864339591086b635d12015db993b5bc"><td class="memSeparator" colspan="2"> </td></tr>
@@ -323,8 +317,8 @@ Functions</h2></td></tr>
</table>
</div><div class="memdoc">
<p>This function does essentially the same as <a class="el" href="linalg_8sql__in.html#acf6628dfa4d73dfce65a582aa5c5a3db">closest_column()</a>, except that it allows to specify the number of closest columns to return. The return value is a composite value:</p><ul>
-<li><code>columns_ids INTEGER[]</code> - The 0-based indices of the <code>num</code> columns of \( M \) that are closest to \( x \). In case of ties, the first such indices are returned.</li>
-<li><code>distances DOUBLE PRECISION[]</code> - The distances between the columns of \( M \) with indices in <code>columns_ids</code> and \( x \). That is, <code>distances[i]</code> contains \( \operatorname{dist}(\vec{m_j}, \vec x) \), where \( j = \) <code>columns_ids[i]</code>. </li>
+<li><code>columns_ids INTEGER[]</code> - The 0-based indices of the <code>num</code> columns of <img class="formulaInl" alt="$ M $" src="form_174.png"/> that are closest to <img class="formulaInl" alt="$ x $" src="form_178.png"/>. In case of ties, the first such indices are returned.</li>
+<li><code>distances DOUBLE PRECISION[]</code> - The distances between the columns of <img class="formulaInl" alt="$ M $" src="form_174.png"/> with indices in <code>columns_ids</code> and <img class="formulaInl" alt="$ x $" src="form_178.png"/>. That is, <code>distances[i]</code> contains <img class="formulaInl" alt="$ \operatorname{dist}(\vec{m_j}, \vec x) $" src="form_181.png"/>, where <img class="formulaInl" alt="$ j = $" src="form_182.png"/> <code>columns_ids[i]</code>. </li>
</ul>
</div>
@@ -386,14 +380,14 @@ Functions</h2></td></tr>
</tr>
</table>
</div><div class="memdoc">
-<p>Given vectors \( x_1, \dots, x_n \), compute the average \( \frac 1n \sum_{i=1}^n x_i \).</p>
+<p>Given vectors <img class="formulaInl" alt="$ x_1, \dots, x_n $" src="form_183.png"/>, compute the average <img class="formulaInl" alt="$ \frac 1n \sum_{i=1}^n x_i $" src="form_184.png"/>.</p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
- <tr><td class="paramname">x</td><td>Point \( x_i \) </td></tr>
+ <tr><td class="paramname">x</td><td>Point <img class="formulaInl" alt="$ x_i $" src="form_62.png"/> </td></tr>
</table>
</dd>
</dl>
-<dl class="section return"><dt>Returns</dt><dd>Average \( \frac 1n \sum_{i=1}^n x_i \) </dd></dl>
+<dl class="section return"><dt>Returns</dt><dd>Average <img class="formulaInl" alt="$ \frac 1n \sum_{i=1}^n x_i $" src="form_184.png"/> </dd></dl>
</div>
</div>
@@ -496,15 +490,15 @@ Functions</h2></td></tr>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
- <tr><td class="paramname">M</td><td>Matrix \( M = (\vec{m_0} \dots \vec{m_{l-1}}) \in \mathbb{R}^{k \times l} \) </td></tr>
- <tr><td class="paramname">x</td><td>Vector \( \vec x \in \mathbb R^k \) </td></tr>
- <tr><td class="paramname">dist</td><td>The metric \( \operatorname{dist} \). This needs to be a function with signature <code>DOUBLE PRECISION[] x DOUBLE PRECISION[] -> DOUBLE PRECISION</code>.</td></tr>
+ <tr><td class="paramname">M</td><td>Matrix <img class="formulaInl" alt="$ M = (\vec{m_0} \dots \vec{m_{l-1}}) \in \mathbb{R}^{k \times l} $" src="form_176.png"/> </td></tr>
+ <tr><td class="paramname">x</td><td>Vector <img class="formulaInl" alt="$ \vec x \in \mathbb R^k $" src="form_177.png"/> </td></tr>
+ <tr><td class="paramname">dist</td><td>The metric <img class="formulaInl" alt="$ \operatorname{dist} $" src="form_141.png"/>. This needs to be a function with signature <code>DOUBLE PRECISION[] x DOUBLE PRECISION[] -> DOUBLE PRECISION</code>.</td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>A composite value:<ul>
-<li><code>columns_id INTEGER</code> - The 0-based index of the column of \( M \) that is closest to \( x \). In case of ties, the first such index is returned. That is, <code>columns_id</code> is the minimum element in the set \( \arg\min_{i=0,\dots,l-1} \operatorname{dist}(\vec{m_i}, \vec x) \).</li>
-<li><code>distance DOUBLE PRECISION</code> - The minimum distance between any column of \( M \) and \( x \). That is, \( \min_{i=0,\dots,l-1} \operatorname{dist}(\vec{m_i}, \vec x) \). </li>
+<li><code>columns_id INTEGER</code> - The 0-based index of the column of <img class="formulaInl" alt="$ M $" src="form_174.png"/> that is closest to <img class="formulaInl" alt="$ x $" src="form_178.png"/>. In case of ties, the first such index is returned. That is, <code>columns_id</code> is the minimum element in the set <img class="formulaInl" alt="$ \arg\min_{i=0,\dots,l-1} \operatorname{dist}(\vec{m_i}, \vec x) $" src="form_179.png"/>.</li>
+<li><code>distance DOUBLE PRECISION</code> - The minimum distance between any column of <img class="formulaInl" alt="$ M $" src="form_174.png"/> and <img class="formulaInl" alt="$ x $" src="form_178.png"/>. That is, <img class="formulaInl" alt="$ \min_{i=0,\dots,l-1} \operatorname{dist}(\vec{m_i}, \vec x) $" src="form_180.png"/>. </li>
</ul>
</dd></dl>
@@ -631,12 +625,12 @@ Functions</h2></td></tr>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
- <tr><td class="paramname">x</td><td>Vector \( \vec x = (x_1, \dots, x_n) \) </td></tr>
- <tr><td class="paramname">y</td><td>Vector \( \vec y = (y_1, \dots, y_n) \) </td></tr>
+ <tr><td class="paramname">x</td><td>Vector <img class="formulaInl" alt="$ \vec x = (x_1, \dots, x_n) $" src="form_159.png"/> </td></tr>
+ <tr><td class="paramname">y</td><td>Vector <img class="formulaInl" alt="$ \vec y = (y_1, \dots, y_n) $" src="form_162.png"/> </td></tr>
</table>
</dd>
</dl>
-<dl class="section return"><dt>Returns</dt><dd>\( \frac{\langle \vec x, \vec y \rangle} {\| \vec x \| \cdot \| \vec y \|} \) </dd></dl>
+<dl class="section return"><dt>Returns</dt><dd><img class="formulaInl" alt="$ \frac{\langle \vec x, \vec y \rangle} {\| \vec x \| \cdot \| \vec y \|} $" src="form_168.png"/> </dd></dl>
</div>
</div>
@@ -687,12 +681,12 @@ Functions</h2></td></tr>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
- <tr><td class="paramname">x</td><td>Vector \( \vec x = (x_1, \dots, x_n) \) </td></tr>
- <tr><td class="paramname">y</td><td>Vector \( \vec y = (y_1, \dots, y_n) \) </td></tr>
+ <tr><td class="paramname">x</td><td>Vector <img class="formulaInl" alt="$ \vec x = (x_1, \dots, x_n) $" src="form_159.png"/> </td></tr>
+ <tr><td class="paramname">y</td><td>Vector <img class="formulaInl" alt="$ \vec y = (y_1, \dots, y_n) $" src="form_162.png"/> </td></tr>
</table>
</dd>
</dl>
-<dl class="section return"><dt>Returns</dt><dd>\( \arccos\left(\frac{\langle \vec x, \vec y \rangle} {\| \vec x \| \cdot \| \vec y \|}\right) \) </dd></dl>
+<dl class="section return"><dt>Returns</dt><dd><img class="formulaInl" alt="$ \arccos\left(\frac{\langle \vec x, \vec y \rangle} {\| \vec x \| \cdot \| \vec y \|}\right) $" src="form_170.png"/> </dd></dl>
</div>
</div>
@@ -721,12 +715,12 @@ Functions</h2></td></tr>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
- <tr><td class="paramname">x</td><td>Vector \( \vec x = (x_1, \dots, x_n) \) </td></tr>
- <tr><td class="paramname">y</td><td>Vector \( \vec y = (y_1, \dots, y_n) \) </td></tr>
+ <tr><td class="paramname">x</td><td>Vector <img class="formulaInl" alt="$ \vec x = (x_1, \dots, x_n) $" src="form_159.png"/> </td></tr>
+ <tr><td class="paramname">y</td><td>Vector <img class="formulaInl" alt="$ \vec y = (y_1, \dots, y_n) $" src="form_162.png"/> </td></tr>
</table>
</dd>
</dl>
-<dl class="section return"><dt>Returns</dt><dd>\( \| x - y \|_\infty = \max_{i=1}^n \|x_i - y_i\| \) </dd></dl>
+<dl class="section return"><dt>Returns</dt><dd><img class="formulaInl" alt="$ \| x - y \|_\infty = \max_{i=1}^n \|x_i - y_i\| $" src="form_163.png"/> </dd></dl>
</div>
</div>
@@ -755,12 +749,12 @@ Functions</h2></td></tr>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
- <tr><td class="paramname">x</td><td>Vector \( \vec x = (x_1, \dots, x_m) \) </td></tr>
- <tr><td class="paramname">y</td><td>Vector \( \vec y = (y_1, \dots, y_n) \) </td></tr>
+ <tr><td class="paramname">x</td><td>Vector <img class="formulaInl" alt="$ \vec x = (x_1, \dots, x_m) $" src="form_172.png"/> </td></tr>
+ <tr><td class="paramname">y</td><td>Vector <img class="formulaInl" alt="$ \vec y = (y_1, \dots, y_n) $" src="form_162.png"/> </td></tr>
</table>
</dd>
</dl>
-<dl class="section return"><dt>Returns</dt><dd>\( 1 - \frac{|x \cap y|}{|x \cup y|} \) </dd></dl>
+<dl class="section return"><dt>Returns</dt><dd><img class="formulaInl" alt="$ 1 - \frac{|x \cap y|}{|x \cup y|} $" src="form_173.png"/> </dd></dl>
</div>
</div>
@@ -789,12 +783,12 @@ Functions</h2></td></tr>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
- <tr><td class="paramname">x</td><td>Vector \( \vec x = (x_1, \dots, x_n) \) </td></tr>
- <tr><td class="paramname">y</td><td>Vector \( \vec y = (y_1, \dots, y_n) \) </td></tr>
+ <tr><td class="paramname">x</td><td>Vector <img class="formulaInl" alt="$ \vec x = (x_1, \dots, x_n) $" src="form_159.png"/> </td></tr>
+ <tr><td class="paramname">y</td><td>Vector <img class="formulaInl" alt="$ \vec y = (y_1, \dots, y_n) $" src="form_162.png"/> </td></tr>
</table>
</dd>
</dl>
-<dl class="section return"><dt>Returns</dt><dd>\( \| x - y \|_1 = \sum_{i=1}^n |x_i - y_i| \) </dd></dl>
+<dl class="section return"><dt>Returns</dt><dd><img class="formulaInl" alt="$ \| x - y \|_1 = \sum_{i=1}^n |x_i - y_i| $" src="form_166.png"/> </dd></dl>
</div>
</div>
@@ -823,12 +817,12 @@ Functions</h2></td></tr>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
- <tr><td class="paramname">x</td><td>Vector \( \vec x = (x_1, \dots, x_n) \) </td></tr>
- <tr><td class="paramname">y</td><td>Vector \( \vec y = (y_1, \dots, y_n) \) </td></tr>
+ <tr><td class="paramname">x</td><td>Vector <img class="formulaInl" alt="$ \vec x = (x_1, \dots, x_n) $" src="form_159.png"/> </td></tr>
+ <tr><td class="paramname">y</td><td>Vector <img class="formulaInl" alt="$ \vec y = (y_1, \dots, y_n) $" src="form_162.png"/> </td></tr>
</table>
</dd>
</dl>
-<dl class="section return"><dt>Returns</dt><dd>\( \| x - y \|_2 = \sqrt{\sum_{i=1}^n (x_i - y_i)^2} \) </dd></dl>
+<dl class="section return"><dt>Returns</dt><dd><img class="formulaInl" alt="$ \| x - y \|_2 = \sqrt{\sum_{i=1}^n (x_i - y_i)^2} $" src="form_167.png"/> </dd></dl>
</div>
</div>
@@ -863,13 +857,13 @@ Functions</h2></td></tr>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
- <tr><td class="paramname">x</td><td>Vector \( \vec x = (x_1, \dots, x_n) \) </td></tr>
- <tr><td class="paramname">y</td><td>Vector \( \vec y = (y_1, \dots, y_n) \) </td></tr>
- <tr><td class="paramname">p</td><td>Scalar \( p > 0 \) </td></tr>
+ <tr><td class="paramname">x</td><td>Vector <img class="formulaInl" alt="$ \vec x = (x_1, \dots, x_n) $" src="form_159.png"/> </td></tr>
+ <tr><td class="paramname">y</td><td>Vector <img class="formulaInl" alt="$ \vec y = (y_1, \dots, y_n) $" src="form_162.png"/> </td></tr>
+ <tr><td class="paramname">p</td><td>Scalar <img class="formulaInl" alt="$ p > 0 $" src="form_164.png"/> </td></tr>
</table>
</dd>
</dl>
-<dl class="section return"><dt>Returns</dt><dd>\( \| x - y \|_p = (\sum_{i=1}^n \|x_i - y_i\|^p)^{\frac{1}{p}} \) </dd></dl>
+<dl class="section return"><dt>Returns</dt><dd><img class="formulaInl" alt="$ \| x - y \|_p = (\sum_{i=1}^n \|x_i - y_i\|^p)^{\frac{1}{p}} $" src="form_165.png"/> </dd></dl>
</div>
</div>
@@ -898,12 +892,12 @@ Functions</h2></td></tr>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
- <tr><td class="paramname">x</td><td>Vector \( \vec x = (x_1, \dots, x_n) \) </td></tr>
- <tr><td class="paramname">y</td><td>Vector \( \vec y = (y_1, \dots, y_n) \) </td></tr>
+ <tr><td class="paramname">x</td><td>Vector <img class="formulaInl" alt="$ \vec x = (x_1, \dots, x_n) $" src="form_159.png"/> </td></tr>
+ <tr><td class="paramname">y</td><td>Vector <img class="formulaInl" alt="$ \vec y = (y_1, \dots, y_n) $" src="form_162.png"/> </td></tr>
</table>
</dd>
</dl>
-<dl class="section return"><dt>Returns</dt><dd>\( 1 - \frac{\langle \vec x, \vec y \rangle} {\| \vec x \|^2 \cdot \| \vec y \|^2 - \langle \vec x, \vec y \rangle} \) </dd></dl>
+<dl class="section return"><dt>Returns</dt><dd><img class="formulaInl" alt="$ 1 - \frac{\langle \vec x, \vec y \rangle} {\| \vec x \|^2 \cdot \| \vec y \|^2 - \langle \vec x, \vec y \rangle} $" src="form_171.png"/> </dd></dl>
</div>
</div>
@@ -1019,14 +1013,14 @@ Functions</h2></td></tr>
</tr>
</table>
</div><div class="memdoc">
-<p>Given vectors \( \vec x_1, \dots, \vec x_n \in \mathbb R^m \), return matrix \( ( \vec x_1 \dots \vec x_n ) \in \mathbb R^{m \times n}\).</p>
+<p>Given vectors <img class="formulaInl" alt="$ \vec x_1, \dots, \vec x_n \in \mathbb R^m $" src="form_187.png"/>, return matrix <img class="formulaInl" alt="$ ( \vec x_1 \dots \vec x_n ) \in \mathbb R^{m \times n}$" src="form_188.png"/>.</p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
- <tr><td class="paramname">x</td><td>Vector \( x_i \) </td></tr>
+ <tr><td class="paramname">x</td><td>Vector <img class="formulaInl" alt="$ x_i $" src="form_62.png"/> </td></tr>
</table>
</dd>
</dl>
-<dl class="section return"><dt>Returns</dt><dd>Matrix with columns \( x_1, \dots, x_n \) </dd></dl>
+<dl class="section return"><dt>Returns</dt><dd>Matrix with columns <img class="formulaInl" alt="$ x_1, \dots, x_n $" src="form_183.png"/> </dd></dl>
</div>
</div>
@@ -1120,11 +1114,11 @@ Functions</h2></td></tr>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
- <tr><td class="paramname">x</td><td>Vector \( \vec x = (x_1, \dots, x_n) \) </td></tr>
+ <tr><td class="paramname">x</td><td>Vector <img class="formulaInl" alt="$ \vec x = (x_1, \dots, x_n) $" src="form_159.png"/> </td></tr>
</table>
</dd>
</dl>
-<dl class="section return"><dt>Returns</dt><dd>\( \| x \|_1 = \sum_{i=1}^n |x_i| \) </dd></dl>
+<dl class="section return"><dt>Returns</dt><dd><img class="formulaInl" alt="$ \| x \|_1 = \sum_{i=1}^n |x_i| $" src="form_160.png"/> </dd></dl>
</div>
</div>
@@ -1143,11 +1137,11 @@ Functions</h2></td></tr>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
- <tr><td class="paramname">x</td><td>Vector \( \vec x = (x_1, \dots, x_n) \) </td></tr>
+ <tr><td class="paramname">x</td><td>Vector <img class="formulaInl" alt="$ \vec x = (x_1, \dots, x_n) $" src="form_159.png"/> </td></tr>
</table>
</dd>
</dl>
-<dl class="section return"><dt>Returns</dt><dd>\( \| x \|_2 = \sqrt{\sum_{i=1}^n x_i^2} \) </dd></dl>
+<dl class="section return"><dt>Returns</dt><dd><img class="formulaInl" alt="$ \| x \|_2 = \sqrt{\sum_{i=1}^n x_i^2} $" src="form_161.png"/> </dd></dl>
</div>
</div>
@@ -1164,14 +1158,14 @@ Functions</h2></td></tr>
</tr>
</table>
</div><div class="memdoc">
-<p>Given vectors \( x_1, \dots, x_n \), define \( \widetilde{x} := \frac 1n \sum_{i=1}^n \frac{x_i}{\| x_i \|} \), and compute the normalized average \( \frac{\widetilde{x}}{\| \widetilde{x} \|} \).</p>
+<p>Given vectors <img class="formulaInl" alt="$ x_1, \dots, x_n $" src="form_183.png"/>, define <img class="formulaInl" alt="$ \widetilde{x} := \frac 1n \sum_{i=1}^n \frac{x_i}{\| x_i \|} $" src="form_185.png"/>, and compute the normalized average <img class="formulaInl" alt="$ \frac{\widetilde{x}}{\| \widetilde{x} \|} $" src="form_186.png"/>.</p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
- <tr><td class="paramname">x</td><td>Point \( x_i \) </td></tr>
+ <tr><td class="paramname">x</td><td>Point <img class="formulaInl" alt="$ x_i $" src="form_62.png"/> </td></tr>
</table>
</dd>
</dl>
-<dl class="section return"><dt>Returns</dt><dd>Normalized average \( \frac{\widetilde{x}}{\| \widetilde{x} \|} \) </dd></dl>
+<dl class="section return"><dt>Returns</dt><dd>Normalized average <img class="formulaInl" alt="$ \frac{\widetilde{x}}{\| \widetilde{x} \|} $" src="form_186.png"/> </dd></dl>
</div>
</div>
@@ -1242,12 +1236,12 @@ Functions</h2></td></tr>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
- <tr><td class="paramname">x</td><td>Vector \( \vec x = (x_1, \dots, x_n) \) </td></tr>
- <tr><td class="paramname">y</td><td>Vector \( \vec y = (y_1, \dots, y_n) \) </td></tr>
+ <tr><td class="paramname">x</td><td>Vector <img class="formulaInl" alt="$ \vec x = (x_1, \dots, x_n) $" src="form_159.png"/> </td></tr>
+ <tr><td class="paramname">y</td><td>Vector <img class="formulaInl" alt="$ \vec y = (y_1, \dots, y_n) $" src="form_162.png"/> </td></tr>
</table>
</dd>
</dl>
-<dl class="section return"><dt>Returns</dt><dd>\( \| x - y \|_2^2 = \sum_{i=1}^n (x_i - y_i)^2 \) </dd></dl>
+<dl class="section return"><dt>Returns</dt><dd><img class="formulaInl" alt="$ \| x - y \|_2^2 = \sum_{i=1}^n (x_i - y_i)^2 $" src="form_169.png"/> </dd></dl>
</div>
</div>
@@ -1257,7 +1251,7 @@ Functions</h2></td></tr>
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