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  1. <html>
  2. <head>
  3. <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
  4. <title>Lists and Tables</title>
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  24. </div>
  25. <div class="section">
  26. <div class="titlepage"><div><div><h2 class="title" style="clear: both">
  27. <a name="document_to_test_formatting.lists_and_tables"></a><a class="link" href="lists_and_tables.html" title="Lists and Tables">Lists and
  28. Tables</a>
  29. </h2></div></div></div>
  30. <div class="toc"><dl>
  31. <dt><span class="section"><a href="lists_and_tables.html#document_to_test_formatting.lists_and_tables.lists">Lists</a></span></dt>
  32. <dt><span class="section"><a href="lists_and_tables.html#document_to_test_formatting.lists_and_tables.variable_lists">Variable
  33. Lists</a></span></dt>
  34. <dt><span class="section"><a href="lists_and_tables.html#document_to_test_formatting.lists_and_tables.tables">Tables</a></span></dt>
  35. </dl></div>
  36. <div class="section">
  37. <div class="titlepage"><div><div><h3 class="title">
  38. <a name="document_to_test_formatting.lists_and_tables.lists"></a><a class="link" href="lists_and_tables.html#document_to_test_formatting.lists_and_tables.lists" title="Lists">Lists</a>
  39. </h3></div></div></div>
  40. <p>
  41. A numbered list:
  42. </p>
  43. <div class="orderedlist"><ol class="orderedlist" type="1">
  44. <li class="listitem">
  45. One
  46. </li>
  47. <li class="listitem">
  48. Two
  49. </li>
  50. <li class="listitem">
  51. Three
  52. <div class="orderedlist"><ol class="orderedlist" type="a">
  53. <li class="listitem">
  54. Three.a
  55. </li>
  56. <li class="listitem">
  57. Three.b
  58. </li>
  59. <li class="listitem">
  60. Three.c
  61. </li>
  62. </ol></div>
  63. </li>
  64. <li class="listitem">
  65. Four
  66. <div class="orderedlist"><ol class="orderedlist" type="a"><li class="listitem">
  67. Four.a
  68. <div class="orderedlist"><ol class="orderedlist" type="i">
  69. <li class="listitem">
  70. Four.a.i
  71. </li>
  72. <li class="listitem">
  73. Four.a.ii
  74. </li>
  75. </ol></div>
  76. </li></ol></div>
  77. </li>
  78. <li class="listitem">
  79. Five
  80. </li>
  81. </ol></div>
  82. <p>
  83. An unordered list:
  84. </p>
  85. <div class="itemizedlist"><ul class="itemizedlist" type="disc">
  86. <li class="listitem">
  87. First
  88. </li>
  89. <li class="listitem">
  90. Second
  91. </li>
  92. <li class="listitem">
  93. Third
  94. </li>
  95. </ul></div>
  96. <p>
  97. A mixture of the two:
  98. </p>
  99. <div class="orderedlist"><ol class="orderedlist" type="1">
  100. <li class="listitem">
  101. 1
  102. <div class="itemizedlist"><ul class="itemizedlist" type="disc">
  103. <li class="listitem">
  104. 1.a
  105. <div class="orderedlist"><ol class="orderedlist" type="a">
  106. <li class="listitem">
  107. 1.a.1
  108. </li>
  109. <li class="listitem">
  110. 1.a.2
  111. </li>
  112. </ol></div>
  113. </li>
  114. <li class="listitem">
  115. 1.b
  116. </li>
  117. </ul></div>
  118. </li>
  119. <li class="listitem">
  120. 2
  121. <div class="itemizedlist"><ul class="itemizedlist" type="disc">
  122. <li class="listitem">
  123. 2.a
  124. </li>
  125. <li class="listitem">
  126. 2.b
  127. <div class="orderedlist"><ol class="orderedlist" type="a">
  128. <li class="listitem">
  129. 2.b.1
  130. </li>
  131. <li class="listitem">
  132. 2.b.2
  133. <div class="itemizedlist"><ul class="itemizedlist" type="circle">
  134. <li class="listitem">
  135. 2.b.2.a
  136. </li>
  137. <li class="listitem">
  138. 2.b.2.b
  139. </li>
  140. </ul></div>
  141. </li>
  142. </ol></div>
  143. </li>
  144. </ul></div>
  145. </li>
  146. </ol></div>
  147. </div>
  148. <div class="section">
  149. <div class="titlepage"><div><div><h3 class="title">
  150. <a name="document_to_test_formatting.lists_and_tables.variable_lists"></a><a class="link" href="lists_and_tables.html#document_to_test_formatting.lists_and_tables.variable_lists" title="Variable Lists">Variable
  151. Lists</a>
  152. </h3></div></div></div>
  153. <div class="variablelist">
  154. <p class="title"><b>A Variable List</b></p>
  155. <dl>
  156. <dt><span class="term">term 1</span></dt>
  157. <dd><p>
  158. The definition of term 1
  159. </p></dd>
  160. <dt><span class="term">term 2</span></dt>
  161. <dd><p>
  162. The definition of term 2
  163. </p></dd>
  164. <dt><span class="term">term 3</span></dt>
  165. <dd><p>
  166. The definition of term 3
  167. </p></dd>
  168. </dl>
  169. </div>
  170. </div>
  171. <div class="section">
  172. <div class="titlepage"><div><div><h3 class="title">
  173. <a name="document_to_test_formatting.lists_and_tables.tables"></a><a class="link" href="lists_and_tables.html#document_to_test_formatting.lists_and_tables.tables" title="Tables">Tables</a>
  174. </h3></div></div></div>
  175. <p>
  176. Here's a big table with code and other tricky things:
  177. </p>
  178. <div class="table">
  179. <a name="id769721"></a><p class="title"><b>Table&#160;1.&#160;Notes on the Implementation of the Beta Distribution</b></p>
  180. <div class="table-contents"><table class="table" summary="Notes on the Implementation of the Beta Distribution">
  181. <colgroup>
  182. <col>
  183. <col>
  184. </colgroup>
  185. <thead><tr>
  186. <th>
  187. <p>
  188. Function
  189. </p>
  190. </th>
  191. <th>
  192. <p>
  193. Implementation Notes
  194. </p>
  195. </th>
  196. </tr></thead>
  197. <tbody>
  198. <tr>
  199. <td>
  200. <p>
  201. pdf
  202. </p>
  203. </td>
  204. <td>
  205. <p>
  206. f(x;&#945;,&#946;) = x<sup>&#945; - 1</sup> (1 - x)<sup>&#946; -1</sup> / B(&#945;, &#946;)
  207. </p>
  208. <p>
  209. Implemented using ibeta_derivative(a, b, x).
  210. </p>
  211. </td>
  212. </tr>
  213. <tr>
  214. <td>
  215. <p>
  216. cdf
  217. </p>
  218. </td>
  219. <td>
  220. <p>
  221. Using the incomplete beta function ibeta(a, b, x)
  222. </p>
  223. </td>
  224. </tr>
  225. <tr>
  226. <td>
  227. <p>
  228. cdf complement
  229. </p>
  230. </td>
  231. <td>
  232. <p>
  233. ibetac(a, b, x)
  234. </p>
  235. </td>
  236. </tr>
  237. <tr>
  238. <td>
  239. <p>
  240. quantile
  241. </p>
  242. </td>
  243. <td>
  244. <p>
  245. Using the inverse incomplete beta function ibeta_inv(a, b, p)
  246. </p>
  247. </td>
  248. </tr>
  249. <tr>
  250. <td>
  251. <p>
  252. quantile from the complement
  253. </p>
  254. </td>
  255. <td>
  256. <p>
  257. ibetac_inv(a, b, q)
  258. </p>
  259. </td>
  260. </tr>
  261. <tr>
  262. <td>
  263. <p>
  264. mean
  265. </p>
  266. </td>
  267. <td>
  268. <p>
  269. <code class="computeroutput"><span class="identifier">a</span><span class="special">/(</span><span class="identifier">a</span><span class="special">+</span><span class="identifier">b</span><span class="special">)</span></code>
  270. </p>
  271. </td>
  272. </tr>
  273. <tr>
  274. <td>
  275. <p>
  276. variance
  277. </p>
  278. </td>
  279. <td>
  280. <p>
  281. <code class="computeroutput"><span class="identifier">a</span> <span class="special">*</span>
  282. <span class="identifier">b</span> <span class="special">/</span>
  283. <span class="special">(</span><span class="identifier">a</span><span class="special">+</span><span class="identifier">b</span><span class="special">)^</span><span class="number">2</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">+</span>
  284. <span class="identifier">b</span> <span class="special">+</span>
  285. <span class="number">1</span><span class="special">)</span></code>
  286. </p>
  287. </td>
  288. </tr>
  289. <tr>
  290. <td>
  291. <p>
  292. mode
  293. </p>
  294. </td>
  295. <td>
  296. <p>
  297. <code class="computeroutput"><span class="special">(</span><span class="identifier">a</span><span class="special">-</span><span class="number">1</span><span class="special">)</span> <span class="special">/</span> <span class="special">(</span><span class="identifier">a</span>
  298. <span class="special">+</span> <span class="identifier">b</span>
  299. <span class="special">+</span> <span class="number">2</span><span class="special">)</span></code>
  300. </p>
  301. </td>
  302. </tr>
  303. <tr>
  304. <td>
  305. <p>
  306. skewness
  307. </p>
  308. </td>
  309. <td>
  310. <p>
  311. <code class="computeroutput"><span class="number">2</span> <span class="special">(</span><span class="identifier">b</span><span class="special">-</span><span class="identifier">a</span><span class="special">)</span>
  312. <span class="identifier">sqrt</span><span class="special">(</span><span class="identifier">a</span><span class="special">+</span><span class="identifier">b</span><span class="special">+</span><span class="number">1</span><span class="special">)/(</span><span class="identifier">a</span><span class="special">+</span><span class="identifier">b</span><span class="special">+</span><span class="number">2</span><span class="special">)</span> <span class="special">*</span> <span class="identifier">sqrt</span><span class="special">(</span><span class="identifier">a</span>
  313. <span class="special">*</span> <span class="identifier">b</span><span class="special">)</span></code>
  314. </p>
  315. </td>
  316. </tr>
  317. <tr>
  318. <td>
  319. <p>
  320. kurtosis excess
  321. </p>
  322. </td>
  323. <td>
  324. <p>
  325. <span class="inlinemediaobject"><img src="../images/beta_dist_kurtosis.png" alt="beta_dist_kurtosis"></span>
  326. </p>
  327. </td>
  328. </tr>
  329. <tr>
  330. <td>
  331. <p>
  332. kurtosis
  333. </p>
  334. </td>
  335. <td>
  336. <p>
  337. <code class="computeroutput"><span class="identifier">kurtosis</span> <span class="special">+</span>
  338. <span class="number">3</span></code>
  339. </p>
  340. </td>
  341. </tr>
  342. <tr>
  343. <td>
  344. <p>
  345. parameter estimation
  346. </p>
  347. </td>
  348. <td>
  349. </td>
  350. </tr>
  351. <tr>
  352. <td>
  353. <p>
  354. alpha
  355. </p>
  356. <p>
  357. from mean and variance
  358. </p>
  359. </td>
  360. <td>
  361. <p>
  362. <code class="computeroutput"><span class="identifier">mean</span> <span class="special">*</span>
  363. <span class="special">((</span> <span class="special">(</span><span class="identifier">mean</span> <span class="special">*</span>
  364. <span class="special">(</span><span class="number">1</span>
  365. <span class="special">-</span> <span class="identifier">mean</span><span class="special">))</span> <span class="special">/</span> <span class="identifier">variance</span><span class="special">)-</span>
  366. <span class="number">1</span><span class="special">)</span></code>
  367. </p>
  368. </td>
  369. </tr>
  370. <tr>
  371. <td>
  372. <p>
  373. beta
  374. </p>
  375. <p>
  376. from mean and variance
  377. </p>
  378. </td>
  379. <td>
  380. <p>
  381. <code class="computeroutput"><span class="special">(</span><span class="number">1</span>
  382. <span class="special">-</span> <span class="identifier">mean</span><span class="special">)</span> <span class="special">*</span> <span class="special">(((</span><span class="identifier">mean</span>
  383. <span class="special">*</span> <span class="special">(</span><span class="number">1</span> <span class="special">-</span> <span class="identifier">mean</span><span class="special">))</span>
  384. <span class="special">/</span><span class="identifier">variance</span><span class="special">)-</span><span class="number">1</span><span class="special">)</span></code>
  385. </p>
  386. </td>
  387. </tr>
  388. <tr>
  389. <td>
  390. <p>
  391. The member functions <code class="computeroutput"><span class="identifier">estimate_alpha</span></code>
  392. and <code class="computeroutput"><span class="identifier">estimate_beta</span></code>
  393. </p>
  394. <p>
  395. from cdf and probability x
  396. </p>
  397. <p>
  398. and <span class="bold"><strong>either</strong></span> <code class="computeroutput"><span class="identifier">alpha</span></code>
  399. or <code class="computeroutput"><span class="identifier">beta</span></code>
  400. </p>
  401. </td>
  402. <td>
  403. <p>
  404. Implemented in terms of the inverse incomplete beta functions
  405. </p>
  406. <p>
  407. ibeta_inva, and ibeta_invb respectively.
  408. </p>
  409. </td>
  410. </tr>
  411. <tr>
  412. <td>
  413. <p>
  414. <code class="computeroutput"><span class="identifier">estimate_alpha</span></code>
  415. </p>
  416. </td>
  417. <td>
  418. <p>
  419. <code class="computeroutput"><span class="identifier">ibeta_inva</span><span class="special">(</span><span class="identifier">beta</span><span class="special">,</span>
  420. <span class="identifier">x</span><span class="special">,</span>
  421. <span class="identifier">probability</span><span class="special">)</span></code>
  422. </p>
  423. </td>
  424. </tr>
  425. <tr>
  426. <td>
  427. <p>
  428. <code class="computeroutput"><span class="identifier">estimate_beta</span></code>
  429. </p>
  430. </td>
  431. <td>
  432. <p>
  433. <code class="computeroutput"><span class="identifier">ibeta_invb</span><span class="special">(</span><span class="identifier">alpha</span><span class="special">,</span>
  434. <span class="identifier">x</span><span class="special">,</span>
  435. <span class="identifier">probability</span><span class="special">)</span></code>
  436. </p>
  437. </td>
  438. </tr>
  439. </tbody>
  440. </table></div>
  441. </div>
  442. <br class="table-break">
  443. </div>
  444. </div>
  445. <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
  446. <td align="left"></td>
  447. <td align="right"><div class="copyright-footer">Copyright &#169; 2007 John Maddock, Joel de Guzman, Eric Niebler and Matias
  448. Capeletto<p>
  449. Distributed under the Boost Software License, Version 1.0. (See accompanying
  450. file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
  451. </p>
  452. </div></td>
  453. </tr></table>
  454. <hr>
  455. <div class="spirit-nav">
  456. <a accesskey="p" href="blurbs.html"><img src="../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../index.html"><img src="../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../index.html"><img src="../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="images.html"><img src="../../../../doc/src/images/next.png" alt="Next"></a>
  457. </div>
  458. </body>
  459. </html>