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  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">&lt;</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">&gt;</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">&lt;</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&#160;20.&#160;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&lt;&gt;</a> <span class="special">&gt;</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">&lt;&gt;</span> <span class="identifier">triangular</span><span class="special">;</span>
  37
  38  <span class="keyword">template</span> <span class="special">&lt;</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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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) &#8198;  for a &lt;= x &lt;=
  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) &#8198; for c &lt; x &lt;=
  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 &gt; a (lower).
  87           Parameter <span class="emphasis"><em>c</em></span> (mode) a &lt;= c &lt;= 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 &lt;= x &lt;= 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 &lt;= x &lt;= 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 &lt; 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 &lt; 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 &lt; 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 &gt; 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">Wikpedia
 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 &#169; 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       R&#229;de, 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>
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