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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 15. 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 15. 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 | <p> | |
77 |    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 <= x <= c | |
82 | </li> | |
83 | <li class="listitem"> | |
84 | 2(b-x)/(b-a)(b-c) for c < x <= 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 > a (lower). | |
90 | Parameter <span class="emphasis"><em>c</em></span> (mode) a <= c <= 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 <= x <= 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 <= x <= 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 | ||
251 | </p> | |
252 | </td> | |
253 | <td> | |
254 | <p> | |
255 | Using the relation: pdf = 0 for x < 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 < 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 < 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 > 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> | |
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432 | <td align="right"><div class="copyright-footer">Copyright © 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 Råde, 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> | |
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