1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Triangular Distribution</title> 5<link rel="stylesheet" href="../../../math.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="../../../index.html" title="Math Toolkit 2.12.0"> 8<link rel="up" href="../dists.html" title="Distributions"> 9<link rel="prev" href="students_t_dist.html" title="Students t Distribution"> 10<link rel="next" href="uniform_dist.html" title="Uniform Distribution"> 11</head> 12<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> 13<table cellpadding="2" width="100%"><tr> 14<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../boost.png"></td> 15<td align="center"><a href="../../../../../../../index.html">Home</a></td> 16<td align="center"><a href="../../../../../../../libs/libraries.htm">Libraries</a></td> 17<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> 18<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> 19<td align="center"><a href="../../../../../../../more/index.htm">More</a></td> 20</tr></table> 21<hr> 22<div class="spirit-nav"> 23<a accesskey="p" href="students_t_dist.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dists.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="uniform_dist.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a> 24</div> 25<div class="section"> 26<div class="titlepage"><div><div><h4 class="title"> 27<a name="math_toolkit.dist_ref.dists.triangular_dist"></a><a class="link" href="triangular_dist.html" title="Triangular Distribution">Triangular 28 Distribution</a> 29</h4></div></div></div> 30<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">triangular</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span></pre> 31<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span> 32 <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span> 33 <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a> <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy<></a> <span class="special">></span> 34 <span class="keyword">class</span> <span class="identifier">triangular_distribution</span><span class="special">;</span> 35 36 <span class="keyword">typedef</span> <span class="identifier">triangular_distribution</span><span class="special"><></span> <span class="identifier">triangular</span><span class="special">;</span> 37 38 <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 39 <span class="keyword">class</span> <span class="identifier">triangular_distribution</span> 40 <span class="special">{</span> 41 <span class="keyword">public</span><span class="special">:</span> 42 <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span> 43 <span class="keyword">typedef</span> <span class="identifier">Policy</span> <span class="identifier">policy_type</span><span class="special">;</span> 44 45 <span class="identifier">triangular_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">lower</span> <span class="special">=</span> <span class="special">-</span><span class="number">1</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">mode</span> <span class="special">=</span> <span class="number">0</span><span class="special">)</span> <span class="identifier">RealType</span> <span class="identifier">upper</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span> <span class="comment">// Constructor.</span> 46 <span class="special">:</span> <span class="identifier">m_lower</span><span class="special">(</span><span class="identifier">lower</span><span class="special">),</span> <span class="identifier">m_mode</span><span class="special">(</span><span class="identifier">mode</span><span class="special">),</span> <span class="identifier">m_upper</span><span class="special">(</span><span class="identifier">upper</span><span class="special">)</span> <span class="comment">// Default is -1, 0, +1 symmetric triangular distribution.</span> 47 <span class="comment">// Accessor functions.</span> 48 <span class="identifier">RealType</span> <span class="identifier">lower</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> 49 <span class="identifier">RealType</span> <span class="identifier">mode</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> 50 <span class="identifier">RealType</span> <span class="identifier">upper</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> 51 <span class="special">};</span> <span class="comment">// class triangular_distribution</span> 52 53<span class="special">}}</span> <span class="comment">// namespaces</span> 54</pre> 55<p> 56 The <a href="http://en.wikipedia.org/wiki/Triangular_distribution" target="_top">triangular 57 distribution</a> is a <a href="http://en.wikipedia.org/wiki/Continuous_distribution" target="_top">continuous</a> 58 <a href="http://en.wikipedia.org/wiki/Probability_distribution" target="_top">probability 59 distribution</a> with a lower limit a, <a href="http://en.wikipedia.org/wiki/Mode_%28statistics%29" target="_top">mode 60 c</a>, and upper limit b. 61 </p> 62<p> 63 The triangular distribution is often used where the distribution is only 64 vaguely known, but, like the <a href="http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29" target="_top">uniform 65 distribution</a>, upper and limits are 'known', but a 'best guess', 66 the mode or center point, is also added. It has been recommended as a 67 <a href="http://www.worldscibooks.com/mathematics/etextbook/5720/5720_chap1.pdf" target="_top">proxy 68 for the beta distribution.</a> The distribution is used in business 69 decision making and project planning. 70 </p> 71<p> 72 The <a href="http://en.wikipedia.org/wiki/Triangular_distribution" target="_top">triangular 73 distribution</a> is a distribution with the <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">probability 74 density function</a>: 75 </p> 76<div class="blockquote"><blockquote class="blockquote"><p> 77 <span class="serif_italic">f(x) = 2(x-a)/(b-a) (c-a) for a <= x <= 78 c</span> 79 </p></blockquote></div> 80<div class="blockquote"><blockquote class="blockquote"><p> 81 <span class="serif_italic">f(x) = 2(b-x)/(b-a) (b-c) for c < x <= 82 b</span> 83 </p></blockquote></div> 84<p> 85 Parameter <span class="emphasis"><em>a</em></span> (lower) can be any finite value. Parameter 86 <span class="emphasis"><em>b</em></span> (upper) can be any finite value > a (lower). 87 Parameter <span class="emphasis"><em>c</em></span> (mode) a <= c <= b. This is the 88 most probable value. 89 </p> 90<p> 91 The <a href="http://en.wikipedia.org/wiki/Random_variate" target="_top">random variate</a> 92 x must also be finite, and is supported lower <= x <= upper. 93 </p> 94<p> 95 The triangular distribution may be appropriate when an assumption of a 96 normal distribution is unjustified because uncertainty is caused by rounding 97 and quantization from analog to digital conversion. Upper and lower limits 98 are known, and the most probable value lies midway. 99 </p> 100<p> 101 The distribution simplifies when the 'best guess' is either the lower or 102 upper limit - a 90 degree angle triangle. The 001 triangular distribution 103 which expresses an estimate that the lowest value is the most likely; for 104 example, you believe that the next-day quoted delivery date is most likely 105 (knowing that a quicker delivery is impossible - the postman only comes 106 once a day), and that longer delays are decreasingly likely, and delivery 107 is assumed to never take more than your upper limit. 108 </p> 109<p> 110 The following graph illustrates how the <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">probability 111 density function PDF</a> varies with the various parameters: 112 </p> 113<div class="blockquote"><blockquote class="blockquote"><p> 114 <span class="inlinemediaobject"><img src="../../../../graphs/triangular_pdf.svg" align="middle"></span> 115 116 </p></blockquote></div> 117<p> 118 and cumulative distribution function 119 </p> 120<div class="blockquote"><blockquote class="blockquote"><p> 121 <span class="inlinemediaobject"><img src="../../../../graphs/triangular_cdf.svg" align="middle"></span> 122 123 </p></blockquote></div> 124<h5> 125<a name="math_toolkit.dist_ref.dists.triangular_dist.h0"></a> 126 <span class="phrase"><a name="math_toolkit.dist_ref.dists.triangular_dist.member_functions"></a></span><a class="link" href="triangular_dist.html#math_toolkit.dist_ref.dists.triangular_dist.member_functions">Member 127 Functions</a> 128 </h5> 129<pre class="programlisting"><span class="identifier">triangular_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">lower</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">mode</span> <span class="special">=</span> <span class="number">0</span> <span class="identifier">RealType</span> <span class="identifier">upper</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span> 130</pre> 131<p> 132 Constructs a <a href="http://en.wikipedia.org/wiki/triangular_distribution" target="_top">triangular 133 distribution</a> with lower <span class="emphasis"><em>lower</em></span> (a) and upper 134 <span class="emphasis"><em>upper</em></span> (b). 135 </p> 136<p> 137 Requires that the <span class="emphasis"><em>lower</em></span>, <span class="emphasis"><em>mode</em></span> 138 and <span class="emphasis"><em>upper</em></span> parameters are all finite, otherwise calls 139 <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>. 140 </p> 141<div class="warning"><table border="0" summary="Warning"> 142<tr> 143<td rowspan="2" align="center" valign="top" width="25"><img alt="[Warning]" src="../../../../../../../doc/src/images/warning.png"></td> 144<th align="left">Warning</th> 145</tr> 146<tr><td align="left" valign="top"> 147<p> 148 These constructors are slightly different from the analogs provided by 149 <a href="http://mathworld.wolfram.com" target="_top">Wolfram MathWorld</a> 150 <a href="http://reference.wolfram.com/language/ref/TriangularDistribution.html" target="_top">Triangular 151 distribution</a>, where 152 </p> 153<p> 154 <code class="literal">TriangularDistribution[{min, max}]</code> represents a <span class="bold"><strong>symmetric</strong></span> triangular statistical distribution 155 giving values between min and max.<br> <code class="literal">TriangularDistribution[]</code> 156 represents a <span class="bold"><strong>symmetric</strong></span> triangular statistical 157 distribution giving values between 0 and 1.<br> <code class="literal">TriangularDistribution[{min, 158 max}, c]</code> represents a triangular distribution with mode at 159 c (usually <span class="bold"><strong>asymmetric</strong></span>).<br> 160 </p> 161<p> 162 So, for example, to compute a variance using <a href="http://www.wolframalpha.com/" target="_top">Wolfram 163 Alpha</a>, use <code class="literal">N[variance[TriangularDistribution{1, +2}], 164 50]</code> 165 </p> 166</td></tr> 167</table></div> 168<p> 169 The parameters of a distribution can be obtained using these member functions: 170 </p> 171<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">lower</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> 172</pre> 173<p> 174 Returns the <span class="emphasis"><em>lower</em></span> parameter of this distribution (default 175 -1). 176 </p> 177<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">mode</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> 178</pre> 179<p> 180 Returns the <span class="emphasis"><em>mode</em></span> parameter of this distribution (default 181 0). 182 </p> 183<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">upper</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> 184</pre> 185<p> 186 Returns the <span class="emphasis"><em>upper</em></span> parameter of this distribution (default+1). 187 </p> 188<h5> 189<a name="math_toolkit.dist_ref.dists.triangular_dist.h1"></a> 190 <span class="phrase"><a name="math_toolkit.dist_ref.dists.triangular_dist.non_member_accessors"></a></span><a class="link" href="triangular_dist.html#math_toolkit.dist_ref.dists.triangular_dist.non_member_accessors">Non-member 191 Accessors</a> 192 </h5> 193<p> 194 All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor 195 functions</a> that are generic to all distributions are supported: 196 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>, 197 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>, 198 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>, 199 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.median">median</a>, 200 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>, 201 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>, 202 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>, 203 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>. 204 </p> 205<p> 206 The domain of the random variable is \lowerto \upper, and the supported 207 range is lower <= x <= upper. 208 </p> 209<h5> 210<a name="math_toolkit.dist_ref.dists.triangular_dist.h2"></a> 211 <span class="phrase"><a name="math_toolkit.dist_ref.dists.triangular_dist.accuracy"></a></span><a class="link" href="triangular_dist.html#math_toolkit.dist_ref.dists.triangular_dist.accuracy">Accuracy</a> 212 </h5> 213<p> 214 The triangular distribution is implemented with simple arithmetic operators 215 and so should have errors within an epsilon or two, except quantiles with 216 arguments nearing the extremes of zero and unity. 217 </p> 218<h5> 219<a name="math_toolkit.dist_ref.dists.triangular_dist.h3"></a> 220 <span class="phrase"><a name="math_toolkit.dist_ref.dists.triangular_dist.implementation"></a></span><a class="link" href="triangular_dist.html#math_toolkit.dist_ref.dists.triangular_dist.implementation">Implementation</a> 221 </h5> 222<p> 223 In the following table, a is the <span class="emphasis"><em>lower</em></span> parameter of 224 the distribution, c is the <span class="emphasis"><em>mode</em></span> parameter, b is the 225 <span class="emphasis"><em>upper</em></span> parameter, <span class="emphasis"><em>x</em></span> is the random 226 variate, <span class="emphasis"><em>p</em></span> is the probability and <span class="emphasis"><em>q = 1-p</em></span>. 227 </p> 228<div class="informaltable"><table class="table"> 229<colgroup> 230<col> 231<col> 232</colgroup> 233<thead><tr> 234<th> 235 <p> 236 Function 237 </p> 238 </th> 239<th> 240 <p> 241 Implementation Notes 242 </p> 243 </th> 244</tr></thead> 245<tbody> 246<tr> 247<td> 248 <p> 249 pdf 250 </p> 251 </td> 252<td> 253 <p> 254 Using the relation: pdf = 0 for x < mode, 2(x-a)/(b-a)(c-a) 255 else 2*(b-x)/((b-a)(b-c)) 256 </p> 257 </td> 258</tr> 259<tr> 260<td> 261 <p> 262 cdf 263 </p> 264 </td> 265<td> 266 <p> 267 Using the relation: cdf = 0 for x < mode (x-a)<sup>2</sup>/((b-a)(c-a)) 268 else 1 - (b-x)<sup>2</sup>/((b-a)(b-c)) 269 </p> 270 </td> 271</tr> 272<tr> 273<td> 274 <p> 275 cdf complement 276 </p> 277 </td> 278<td> 279 <p> 280 Using the relation: q = 1 - p 281 </p> 282 </td> 283</tr> 284<tr> 285<td> 286 <p> 287 quantile 288 </p> 289 </td> 290<td> 291 <p> 292 let p0 = (c-a)/(b-a) the point of inflection on the cdf, then 293 given probability p and q = 1-p: 294 </p> 295 <p> 296 x = sqrt((b-a)(c-a)p) + a ; for p < p0 297 </p> 298 <p> 299 x = c ; for p == p0 300 </p> 301 <p> 302 x = b - sqrt((b-a)(b-c)q) ; for p > p0 303 </p> 304 <p> 305 (See <a href="../../../../../../../boost/math/distributions/triangular.hpp" target="_top">/boost/math/distributions/triangular.hpp</a> 306 for details.) 307 </p> 308 </td> 309</tr> 310<tr> 311<td> 312 <p> 313 quantile from the complement 314 </p> 315 </td> 316<td> 317 <p> 318 As quantile (See <a href="../../../../../../../boost/math/distributions/triangular.hpp" target="_top">/boost/math/distributions/triangular.hpp</a> 319 for details.) 320 </p> 321 </td> 322</tr> 323<tr> 324<td> 325 <p> 326 mean 327 </p> 328 </td> 329<td> 330 <p> 331 (a + b + 3) / 3 332 </p> 333 </td> 334</tr> 335<tr> 336<td> 337 <p> 338 variance 339 </p> 340 </td> 341<td> 342 <p> 343 (a<sup>2</sup>+b<sup>2</sup>+c<sup>2</sup> - ab - ac - bc)/18 344 </p> 345 </td> 346</tr> 347<tr> 348<td> 349 <p> 350 mode 351 </p> 352 </td> 353<td> 354 <p> 355 c 356 </p> 357 </td> 358</tr> 359<tr> 360<td> 361 <p> 362 skewness 363 </p> 364 </td> 365<td> 366 <p> 367 (See <a href="../../../../../../../boost/math/distributions/triangular.hpp" target="_top">/boost/math/distributions/triangular.hpp</a> 368 for details). 369 </p> 370 </td> 371</tr> 372<tr> 373<td> 374 <p> 375 kurtosis 376 </p> 377 </td> 378<td> 379 <p> 380 12/5 381 </p> 382 </td> 383</tr> 384<tr> 385<td> 386 <p> 387 kurtosis excess 388 </p> 389 </td> 390<td> 391 <p> 392 -3/5 393 </p> 394 </td> 395</tr> 396</tbody> 397</table></div> 398<p> 399 Some 'known good' test values were obtained using <a href="http://www.wolframalpha.com/" target="_top">Wolfram 400 Alpha</a>. 401 </p> 402<h5> 403<a name="math_toolkit.dist_ref.dists.triangular_dist.h4"></a> 404 <span class="phrase"><a name="math_toolkit.dist_ref.dists.triangular_dist.references"></a></span><a class="link" href="triangular_dist.html#math_toolkit.dist_ref.dists.triangular_dist.references">References</a> 405 </h5> 406<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> 407<li class="listitem"> 408 <a href="http://en.wikipedia.org/wiki/Triangular_distribution" target="_top">Wikipedia 409 triangular distribution</a> 410 </li> 411<li class="listitem"> 412 <a href="http://mathworld.wolfram.com/TriangularDistribution.html" target="_top">Weisstein, 413 Eric W. "Triangular Distribution." From MathWorld--A Wolfram 414 Web Resource.</a> 415 </li> 416<li class="listitem"> 417 Evans, M.; Hastings, N.; and Peacock, B. "Triangular Distribution." 418 Ch. 40 in Statistical Distributions, 3rd ed. New York: Wiley, pp. 187-188, 419 2000, ISBN - 0471371246. 420 </li> 421<li class="listitem"> 422 <a href="http://www.measurement.sk/2002/S1/Wimmer2.pdf" target="_top">Gejza Wimmer, 423 Viktor Witkovsky and Tomas Duby, Measurement Science Review, Volume 424 2, Section 1, 2002, Proper Rounding Of The Measurement Results Under 425 The Assumption Of Triangular Distribution.</a> 426 </li> 427</ul></div> 428</div> 429<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 430<td align="left"></td> 431<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 432 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 433 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 434 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 435 Daryle Walker and Xiaogang Zhang<p> 436 Distributed under the Boost Software License, Version 1.0. (See accompanying 437 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>) 438 </p> 439</div></td> 440</tr></table> 441<hr> 442<div class="spirit-nav"> 443<a accesskey="p" href="students_t_dist.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dists.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="uniform_dist.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a> 444</div> 445</body> 446</html> 447
