| <html> |
| <head> |
| <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> |
| <title>Triangular Distribution</title> |
| <link rel="stylesheet" href="../../../math.css" type="text/css"> |
| <meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> |
| <link rel="home" href="../../../index.html" title="Math Toolkit 2.2.0"> |
| <link rel="up" href="../dists.html" title="Distributions"> |
| <link rel="prev" href="students_t_dist.html" title="Students t Distribution"> |
| <link rel="next" href="uniform_dist.html" title="Uniform Distribution"> |
| </head> |
| <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> |
| <table cellpadding="2" width="100%"><tr> |
| <td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../boost.png"></td> |
| <td align="center"><a href="../../../../../../../index.html">Home</a></td> |
| <td align="center"><a href="../../../../../../../libs/libraries.htm">Libraries</a></td> |
| <td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> |
| <td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> |
| <td align="center"><a href="../../../../../../../more/index.htm">More</a></td> |
| </tr></table> |
| <hr> |
| <div class="spirit-nav"> |
| <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> |
| </div> |
| <div class="section"> |
| <div class="titlepage"><div><div><h4 class="title"> |
| <a name="math_toolkit.dist_ref.dists.triangular_dist"></a><a class="link" href="triangular_dist.html" title="Triangular Distribution">Triangular |
| Distribution</a> |
| </h4></div></div></div> |
| <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> |
| <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> |
| <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> |
| <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 14. 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> |
| <span class="keyword">class</span> <span class="identifier">triangular_distribution</span><span class="special">;</span> |
| |
| <span class="keyword">typedef</span> <span class="identifier">triangular_distribution</span><span class="special"><></span> <span class="identifier">triangular</span><span class="special">;</span> |
| |
| <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 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> |
| <span class="keyword">class</span> <span class="identifier">triangular_distribution</span> |
| <span class="special">{</span> |
| <span class="keyword">public</span><span class="special">:</span> |
| <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span> |
| <span class="keyword">typedef</span> <span class="identifier">Policy</span> <span class="identifier">policy_type</span><span class="special">;</span> |
| |
| <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> |
| <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 triangular distribution.</span> |
| <span class="comment">// Accessor functions.</span> |
| <span class="identifier">RealType</span> <span class="identifier">lower</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> |
| <span class="identifier">RealType</span> <span class="identifier">mode</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> |
| <span class="identifier">RealType</span> <span class="identifier">upper</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> |
| <span class="special">};</span> <span class="comment">// class triangular_distribution</span> |
| |
| <span class="special">}}</span> <span class="comment">// namespaces</span> |
| </pre> |
| <p> |
| The <a href="http://en.wikipedia.org/wiki/Triangular_distribution" target="_top">triangular |
| distribution</a> is a <a href="http://en.wikipedia.org/wiki/Continuous_distribution" target="_top">continuous</a> |
| <a href="http://en.wikipedia.org/wiki/Probability_distribution" target="_top">probability |
| distribution</a> with a lower limit a, <a href="http://en.wikipedia.org/wiki/Mode_%28statistics%29" target="_top">mode |
| c</a>, and upper limit b. |
| </p> |
| <p> |
| The triangular distribution is often used where the distribution is only |
| vaguely known, but, like the <a href="http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29" target="_top">uniform |
| distribution</a>, upper and limits are 'known', but a 'best guess', |
| the mode or center point, is also added. It has been recommended as a |
| <a href="http://www.worldscibooks.com/mathematics/etextbook/5720/5720_chap1.pdf" target="_top">proxy |
| for the beta distribution.</a> The distribution is used in business |
| decision making and project planning. |
| </p> |
| <p> |
| The <a href="http://en.wikipedia.org/wiki/Triangular_distribution" target="_top">triangular |
| distribution</a> is a distribution with the <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">probability |
| density function</a>: |
| </p> |
| <p> |
| f(x) = |
| </p> |
| <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> |
| <li class="listitem"> |
| 2(x-a)/(b-a) (c-a) for a <= x <= c |
| </li> |
| <li class="listitem"> |
| 2(b-x)/(b-a)(b-c) for c < x <= b |
| </li> |
| </ul></div> |
| <p> |
| Parameter a (lower) can be any finite value. Parameter b (upper) can be |
| any finite value > a (lower). Parameter c (mode) a <= c <= b. |
| This is the most probable value. |
| </p> |
| <p> |
| The <a href="http://en.wikipedia.org/wiki/Random_variate" target="_top">random variate</a> |
| x must also be finite, and is supported lower <= x <= upper. |
| </p> |
| <p> |
| The triangular distribution may be appropriate when an assumption of a |
| normal distribution is unjustified because uncertainty is caused by rounding |
| and quantization from analog to digital conversion. Upper and lower limits |
| are known, and the most probable value lies midway. |
| </p> |
| <p> |
| The distribution simplifies when the 'best guess' is either the lower or |
| upper limit - a 90 degree angle triangle. The default chosen is the 001 |
| triangular distribution which expresses an estimate that the lowest value |
| is the most likely; for example, you believe that the next-day quoted delivery |
| date is most likely (knowing that a quicker delivery is impossible - the |
| postman only comes once a day), and that longer delays are decreasingly |
| likely, and delivery is assumed to never take more than your upper limit. |
| </p> |
| <p> |
| The following graph illustrates how the <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">probability |
| density function PDF</a> varies with the various parameters: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../graphs/triangular_pdf.svg" align="middle"></span> |
| </p> |
| <p> |
| and cumulative distribution function |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../graphs/triangular_cdf.svg" align="middle"></span> |
| </p> |
| <h5> |
| <a name="math_toolkit.dist_ref.dists.triangular_dist.h0"></a> |
| <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 |
| Functions</a> |
| </h5> |
| <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> |
| </pre> |
| <p> |
| Constructs a <a href="http://en.wikipedia.org/wiki/triangular_distribution" target="_top">triangular |
| distribution</a> with lower <span class="emphasis"><em>lower</em></span> (a) and upper |
| <span class="emphasis"><em>upper</em></span> (b). |
| </p> |
| <p> |
| Requires that the <span class="emphasis"><em>lower</em></span>, <span class="emphasis"><em>mode</em></span> |
| and <span class="emphasis"><em>upper</em></span> parameters are all finite, otherwise calls |
| <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>. |
| </p> |
| <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> |
| </pre> |
| <p> |
| Returns the <span class="emphasis"><em>lower</em></span> parameter of this distribution (default |
| -1). |
| </p> |
| <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> |
| </pre> |
| <p> |
| Returns the <span class="emphasis"><em>mode</em></span> parameter of this distribution (default |
| 0). |
| </p> |
| <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> |
| </pre> |
| <p> |
| Returns the <span class="emphasis"><em>upper</em></span> parameter of this distribution (default+1). |
| </p> |
| <h5> |
| <a name="math_toolkit.dist_ref.dists.triangular_dist.h1"></a> |
| <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 |
| Accessors</a> |
| </h5> |
| <p> |
| All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor |
| functions</a> that are generic to all distributions are supported: |
| <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>, |
| <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>, |
| <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>, |
| <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>, |
| <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>, |
| <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>, |
| <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>, |
| <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>. |
| </p> |
| <p> |
| The domain of the random variable is \lowerto \upper, and the supported |
| range is lower <= x <= upper. |
| </p> |
| <h5> |
| <a name="math_toolkit.dist_ref.dists.triangular_dist.h2"></a> |
| <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> |
| </h5> |
| <p> |
| The triangular distribution is implemented with simple arithmetic operators |
| and so should have errors within an epsilon or two, except quantiles with |
| arguments nearing the extremes of zero and unity. |
| </p> |
| <h5> |
| <a name="math_toolkit.dist_ref.dists.triangular_dist.h3"></a> |
| <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> |
| </h5> |
| <p> |
| In the following table, a is the <span class="emphasis"><em>lower</em></span> parameter of |
| the distribution, c is the <span class="emphasis"><em>mode</em></span> parameter, b is the |
| <span class="emphasis"><em>upper</em></span> parameter, <span class="emphasis"><em>x</em></span> is the random |
| variate, <span class="emphasis"><em>p</em></span> is the probability and <span class="emphasis"><em>q = 1-p</em></span>. |
| </p> |
| <div class="informaltable"><table class="table"> |
| <colgroup> |
| <col> |
| <col> |
| </colgroup> |
| <thead><tr> |
| <th> |
| <p> |
| Function |
| </p> |
| </th> |
| <th> |
| <p> |
| Implementation Notes |
| </p> |
| </th> |
| </tr></thead> |
| <tbody> |
| <tr> |
| <td> |
| <p> |
| pdf |
| </p> |
| </td> |
| <td> |
| <p> |
| Using the relation: pdf = 0 for x < mode, 2(x-a)/(b-a)(c-a) |
| else 2*(b-x)/((b-a)(b-c)) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| cdf |
| </p> |
| </td> |
| <td> |
| <p> |
| Using the relation: cdf = 0 for x < mode (x-a)<sup>2</sup>/((b-a)(c-a)) |
| else 1 - (b-x)<sup>2</sup>/((b-a)(b-c)) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| cdf complement |
| </p> |
| </td> |
| <td> |
| <p> |
| Using the relation: q = 1 - p |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| quantile |
| </p> |
| </td> |
| <td> |
| <p> |
| let p0 = (c-a)/(b-a) the point of inflection on the cdf, then |
| given probability p and q = 1-p: |
| </p> |
| <p> |
| x = sqrt((b-a)(c-a)p) + a ; for p < p0 |
| </p> |
| <p> |
| x = c ; for p == p0 |
| </p> |
| <p> |
| x = b - sqrt((b-a)(b-c)q) ; for p > p0 |
| </p> |
| <p> |
| (See <a href="../../../../../../../boost/math/distributions/triangular.hpp" target="_top">/boost/math/distributions/triangular.hpp</a> |
| for details.) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| quantile from the complement |
| </p> |
| </td> |
| <td> |
| <p> |
| As quantile (See <a href="../../../../../../../boost/math/distributions/triangular.hpp" target="_top">/boost/math/distributions/triangular.hpp</a> |
| for details.) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| mean |
| </p> |
| </td> |
| <td> |
| <p> |
| (a + b + 3) / 3 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| variance |
| </p> |
| </td> |
| <td> |
| <p> |
| (a<sup>2</sup>+b<sup>2</sup>+c<sup>2</sup> - ab - ac - bc)/18 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| mode |
| </p> |
| </td> |
| <td> |
| <p> |
| c |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| skewness |
| </p> |
| </td> |
| <td> |
| <p> |
| (See <a href="../../../../../../../boost/math/distributions/triangular.hpp" target="_top">/boost/math/distributions/triangular.hpp</a> |
| for details). |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| kurtosis |
| </p> |
| </td> |
| <td> |
| <p> |
| 12/5 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| kurtosis excess |
| </p> |
| </td> |
| <td> |
| <p> |
| -3/5 |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| <p> |
| Some 'known good' test values were obtained from <a href="http://espse.ed.psu.edu/edpsych/faculty/rhale/hale/507Mat/statlets/free/pdist.htm" target="_top">Statlet: |
| Calculate and plot probability distributions</a> |
| </p> |
| <h5> |
| <a name="math_toolkit.dist_ref.dists.triangular_dist.h4"></a> |
| <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> |
| </h5> |
| <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> |
| <li class="listitem"> |
| <a href="http://en.wikipedia.org/wiki/Triangular_distribution" target="_top">Wikpedia |
| triangular distribution</a> |
| </li> |
| <li class="listitem"> |
| <a href="http://mathworld.wolfram.com/TriangularDistribution.html" target="_top">Weisstein, |
| Eric W. "Triangular Distribution." From MathWorld--A Wolfram |
| Web Resource.</a> |
| </li> |
| <li class="listitem"> |
| Evans, M.; Hastings, N.; and Peacock, B. "Triangular Distribution." |
| Ch. 40 in Statistical Distributions, 3rd ed. New York: Wiley, pp. 187-188, |
| 2000, ISBN - 0471371246. |
| </li> |
| <li class="listitem"> |
| <a href="http://www.brighton-webs.co.uk/distributions/triangular.asp" target="_top">Brighton |
| Webs Ltd. BW D-Calc 1.0 Distribution Calculator</a> |
| </li> |
| <li class="listitem"> |
| <a href="http://www.worldscibooks.com/mathematics/etextbook/5720/5720_chap1.pdf" target="_top">The |
| Triangular Distribution including its history.</a> |
| </li> |
| <li class="listitem"> |
| <a href="http://www.measurement.sk/2002/S1/Wimmer2.pdf" target="_top">Gejza Wimmer, |
| Viktor Witkovsky and Tomas Duby, Measurement Science Review, Volume |
| 2, Section 1, 2002, Proper Rounding Of The Measurement Results Under |
| The Assumption Of Triangular Distribution.</a> |
| </li> |
| </ul></div> |
| </div> |
| <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> |
| <td align="left"></td> |
| <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, |
| Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert |
| Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin Sobotta, |
| Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> |
| Distributed under the Boost Software License, Version 1.0. (See accompanying |
| 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>) |
| </p> |
| </div></td> |
| </tr></table> |
| <hr> |
| <div class="spirit-nav"> |
| <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> |
| </div> |
| </body> |
| </html> |
