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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;15.&#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;15.&#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 <p>
  77           &#8192;&#8192; f(x) =
  78         </p>
  79 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
  80 <li class="listitem">
  81               2(x-a)/(b-a) (c-a) for a &lt;= x &lt;= c
  82             </li>
  83 <li class="listitem">
  84               2(b-x)/(b-a)(b-c) for c &lt; x &lt;= b
  85             </li>
  86 </ul></div>
  87 <p>
  88           Parameter <span class="emphasis"><em>a</em></span> (lower) can be any finite value. Parameter
  89           <span class="emphasis"><em>b</em></span> (upper) can be any finite value &gt; a (lower).
  90           Parameter <span class="emphasis"><em>c</em></span> (mode) a &lt;= c &lt;= b. This is the
  91           most probable value.
  92         </p>
  93 <p>
  94           The <a href="http://en.wikipedia.org/wiki/Random_variate" target="_top">random variate</a>
  95           x must also be finite, and is supported lower &lt;= x &lt;= upper.
  96         </p>
  97 <p>
  98           The triangular distribution may be appropriate when an assumption of a
  99           normal distribution is unjustified because uncertainty is caused by rounding
 100           and quantization from analog to digital conversion. Upper and lower limits
 101           are known, and the most probable value lies midway.
 102         </p>
 103 <p>
 104           The distribution simplifies when the 'best guess' is either the lower or
 105           upper limit - a 90 degree angle triangle. The 001 triangular distribution
 106           which expresses an estimate that the lowest value is the most likely; for
 107           example, you believe that the next-day quoted delivery date is most likely
 108           (knowing that a quicker delivery is impossible - the postman only comes
 109           once a day), and that longer delays are decreasingly likely, and delivery
 110           is assumed to never take more than your upper limit.
 111         </p>
 112 <p>
 113           The following graph illustrates how the <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">probability
 114           density function PDF</a> varies with the various parameters:
 115         </p>
 116 <p>
 117           <span class="inlinemediaobject"><img src="../../../../graphs/triangular_pdf.svg" align="middle"></span>
 118         </p>
 119 <p>
 120           and cumulative distribution function
 121         </p>
 122 <p>
 123           <span class="inlinemediaobject"><img src="../../../../graphs/triangular_cdf.svg" align="middle"></span>
 124         </p>
 125 <h5>
 126 <a name="math_toolkit.dist_ref.dists.triangular_dist.h0"></a>
 127           <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
 128           Functions</a>
 129         </h5>
 130 <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>
 131 </pre>
 132 <p>
 133           Constructs a <a href="http://en.wikipedia.org/wiki/triangular_distribution" target="_top">triangular
 134           distribution</a> with lower <span class="emphasis"><em>lower</em></span> (a) and upper
 135           <span class="emphasis"><em>upper</em></span> (b).
 136         </p>
 137 <p>
 138           Requires that the <span class="emphasis"><em>lower</em></span>, <span class="emphasis"><em>mode</em></span>
 139           and <span class="emphasis"><em>upper</em></span> parameters are all finite, otherwise calls
 140           <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
 141         </p>
 142 <div class="warning"><table border="0" summary="Warning">
 143 <tr>
 144 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Warning]" src="../../../../../../../doc/src/images/warning.png"></td>
 145 <th align="left">Warning</th>
 146 </tr>
 147 <tr><td align="left" valign="top">
 148 <p>
 149             These constructors are slightly different from the analogs provided by
 150             <a href="http://mathworld.wolfram.com" target="_top">Wolfram MathWorld</a>
 151             <a href="http://reference.wolfram.com/language/ref/TriangularDistribution.html" target="_top">Triangular
 152             distribution</a>, where
 153           </p>
 154 <p>
 155             <code class="literal">TriangularDistribution[{min, max}]</code> represents a <span class="bold"><strong>symmetric</strong></span> triangular statistical distribution
 156             giving values between min and max.<br> <code class="literal">TriangularDistribution[]</code>
 157             represents a <span class="bold"><strong>symmetric</strong></span> triangular statistical
 158             distribution giving values between 0 and 1.<br> <code class="literal">TriangularDistribution[{min,
 159             max}, c]</code> represents a triangular distribution with mode at
 160             c (usually <span class="bold"><strong>asymmetric</strong></span>).<br>
 161           </p>
 162 <p>
 163             So, for example, to compute a variance using <a href="http://www.wolframalpha.com/" target="_top">Wolfram
 164             Alpha</a>, use <code class="literal">N[variance[TriangularDistribution{1, +2}],
 165             50]</code>
 166           </p>
 167 </td></tr>
 168 </table></div>
 169 <p>
 170           The parameters of a distribution can be obtained using these member functions:
 171         </p>
 172 <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>
 173 </pre>
 174 <p>
 175           Returns the <span class="emphasis"><em>lower</em></span> parameter of this distribution (default
 176           -1).
 177         </p>
 178 <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>
 179 </pre>
 180 <p>
 181           Returns the <span class="emphasis"><em>mode</em></span> parameter of this distribution (default
 182           0).
 183         </p>
 184 <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>
 185 </pre>
 186 <p>
 187           Returns the <span class="emphasis"><em>upper</em></span> parameter of this distribution (default+1).
 188         </p>
 189 <h5>
 190 <a name="math_toolkit.dist_ref.dists.triangular_dist.h1"></a>
 191           <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
 192           Accessors</a>
 193         </h5>
 194 <p>
 195           All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
 196           functions</a> that are generic to all distributions are supported:
 197           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
 198           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
 199           <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>,
 200           <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>,
 201           <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>,
 202           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
 203           <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>,
 204           <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>.
 205         </p>
 206 <p>
 207           The domain of the random variable is \lowerto \upper, and the supported
 208           range is lower &lt;= x &lt;= upper.
 209         </p>
 210 <h5>
 211 <a name="math_toolkit.dist_ref.dists.triangular_dist.h2"></a>
 212           <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>
 213         </h5>
 214 <p>
 215           The triangular distribution is implemented with simple arithmetic operators
 216           and so should have errors within an epsilon or two, except quantiles with
 217           arguments nearing the extremes of zero and unity.
 218         </p>
 219 <h5>
 220 <a name="math_toolkit.dist_ref.dists.triangular_dist.h3"></a>
 221           <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>
 222         </h5>
 223 <p>
 224           In the following table, a is the <span class="emphasis"><em>lower</em></span> parameter of
 225           the distribution, c is the <span class="emphasis"><em>mode</em></span> parameter, b is the
 226           <span class="emphasis"><em>upper</em></span> parameter, <span class="emphasis"><em>x</em></span> is the random
 227           variate, <span class="emphasis"><em>p</em></span> is the probability and <span class="emphasis"><em>q = 1-p</em></span>.
 228         </p>
 229 <div class="informaltable"><table class="table">
 230 <colgroup>
 231 <col>
 232 <col>
 233 </colgroup>
 234 <thead><tr>
 235 <th>
 236                   <p>
 237                     Function
 238                   </p>
 239                 </th>
 240 <th>
 241                   <p>
 242                     Implementation Notes
 243                   </p>
 244                 </th>
 245 </tr></thead>
 246 <tbody>
 247 <tr>
 248 <td>
 249                   <p>
 250                     pdf
 251                   </p>
 252                 </td>
 253 <td>
 254                   <p>
 255                     Using the relation: pdf = 0 for x &lt; mode, 2(x-a)/(b-a)(c-a)
 256                     else 2*(b-x)/((b-a)(b-c))
 257                   </p>
 258                 </td>
 259 </tr>
 260 <tr>
 261 <td>
 262                   <p>
 263                     cdf
 264                   </p>
 265                 </td>
 266 <td>
 267                   <p>
 268                     Using the relation: cdf = 0 for x &lt; mode (x-a)<sup>2</sup>/((b-a)(c-a))
 269                     else 1 - (b-x)<sup>2</sup>/((b-a)(b-c))
 270                   </p>
 271                 </td>
 272 </tr>
 273 <tr>
 274 <td>
 275                   <p>
 276                     cdf complement
 277                   </p>
 278                 </td>
 279 <td>
 280                   <p>
 281                     Using the relation: q = 1 - p
 282                   </p>
 283                 </td>
 284 </tr>
 285 <tr>
 286 <td>
 287                   <p>
 288                     quantile
 289                   </p>
 290                 </td>
 291 <td>
 292                   <p>
 293                     let p0 = (c-a)/(b-a) the point of inflection on the cdf, then
 294                     given probability p and q = 1-p:
 295                   </p>
 296                   <p>
 297                     x = sqrt((b-a)(c-a)p) + a ; for p &lt; p0
 298                   </p>
 299                   <p>
 300                     x = c ; for p == p0
 301                   </p>
 302                   <p>
 303                     x = b - sqrt((b-a)(b-c)q) ; for p &gt; p0
 304                   </p>
 305                   <p>
 306                     (See <a href="../../../../../../../boost/math/distributions/triangular.hpp" target="_top">/boost/math/distributions/triangular.hpp</a>
 307                     for details.)
 308                   </p>
 309                 </td>
 310 </tr>
 311 <tr>
 312 <td>
 313                   <p>
 314                     quantile from the complement
 315                   </p>
 316                 </td>
 317 <td>
 318                   <p>
 319                     As quantile (See <a href="../../../../../../../boost/math/distributions/triangular.hpp" target="_top">/boost/math/distributions/triangular.hpp</a>
 320                     for details.)
 321                   </p>
 322                 </td>
 323 </tr>
 324 <tr>
 325 <td>
 326                   <p>
 327                     mean
 328                   </p>
 329                 </td>
 330 <td>
 331                   <p>
 332                     (a + b + 3) / 3
 333                   </p>
 334                 </td>
 335 </tr>
 336 <tr>
 337 <td>
 338                   <p>
 339                     variance
 340                   </p>
 341                 </td>
 342 <td>
 343                   <p>
 344                     (a<sup>2</sup>+b<sup>2</sup>+c<sup>2</sup> - ab - ac - bc)/18
 345                   </p>
 346                 </td>
 347 </tr>
 348 <tr>
 349 <td>
 350                   <p>
 351                     mode
 352                   </p>
 353                 </td>
 354 <td>
 355                   <p>
 356                     c
 357                   </p>
 358                 </td>
 359 </tr>
 360 <tr>
 361 <td>
 362                   <p>
 363                     skewness
 364                   </p>
 365                 </td>
 366 <td>
 367                   <p>
 368                     (See <a href="../../../../../../../boost/math/distributions/triangular.hpp" target="_top">/boost/math/distributions/triangular.hpp</a>
 369                     for details).
 370                   </p>
 371                 </td>
 372 </tr>
 373 <tr>
 374 <td>
 375                   <p>
 376                     kurtosis
 377                   </p>
 378                 </td>
 379 <td>
 380                   <p>
 381                     12/5
 382                   </p>
 383                 </td>
 384 </tr>
 385 <tr>
 386 <td>
 387                   <p>
 388                     kurtosis excess
 389                   </p>
 390                 </td>
 391 <td>
 392                   <p>
 393                     -3/5
 394                   </p>
 395                 </td>
 396 </tr>
 397 </tbody>
 398 </table></div>
 399 <p>
 400           Some 'known good' test values were obtained using <a href="http://www.wolframalpha.com/" target="_top">Wolfram
 401           Alpha</a>.
 402         </p>
 403 <h5>
 404 <a name="math_toolkit.dist_ref.dists.triangular_dist.h4"></a>
 405           <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>
 406         </h5>
 407 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 408 <li class="listitem">
 409               <a href="http://en.wikipedia.org/wiki/Triangular_distribution" target="_top">Wikpedia
 410               triangular distribution</a>
 411             </li>
 412 <li class="listitem">
 413               <a href="http://mathworld.wolfram.com/TriangularDistribution.html" target="_top">Weisstein,
 414               Eric W. "Triangular Distribution." From MathWorld--A Wolfram
 415               Web Resource.</a>
 416             </li>
 417 <li class="listitem">
 418               Evans, M.; Hastings, N.; and Peacock, B. "Triangular Distribution."
 419               Ch. 40 in Statistical Distributions, 3rd ed. New York: Wiley, pp. 187-188,
 420               2000, ISBN - 0471371246.
 421             </li>
 422 <li class="listitem">
 423               <a href="http://www.measurement.sk/2002/S1/Wimmer2.pdf" target="_top">Gejza Wimmer,
 424               Viktor Witkovsky and Tomas Duby, Measurement Science Review, Volume
 425               2, Section 1, 2002, Proper Rounding Of The Measurement Results Under
 426               The Assumption Of Triangular Distribution.</a>
 427             </li>
 428 </ul></div>
 429 </div>
 430 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 431 <td align="left"></td>
 432 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 433       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 434       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
 435       Benjamin Sobotta, Thijs van den Berg, 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>
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