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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.arcine_dist"></a><a class="link" href="arcine_dist.html" title="Arcsine Distribution">Arcsine Distribution</a> | |
28 | </h4></div></div></div> | |
29 | <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">arcsine</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span></pre> | |
30 | <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> | |
31 | ||
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">arcsine_distribution</span><span class="special">;</span> | |
35 | ||
36 | <span class="keyword">typedef</span> <span class="identifier">arcsine_distribution</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">arcsine</span><span class="special">;</span> <span class="comment">// double precision standard arcsine distribution [0,1].</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">arcsine_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="comment">// Constructor from two range parameters, x_min and x_max:</span> | |
46 | <span class="identifier">arcsine_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">x_min</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">x_max</span><span class="special">);</span> | |
47 | ||
48 | <span class="comment">// Range Parameter accessors:</span> | |
49 | <span class="identifier">RealType</span> <span class="identifier">x_min</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span> | |
50 | <span class="identifier">RealType</span> <span class="identifier">x_max</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span> | |
51 | <span class="special">};</span> | |
52 | <span class="special">}}</span> <span class="comment">// namespaces</span> | |
53 | </pre> | |
54 | <p> | |
55 | The class type <code class="computeroutput"><span class="identifier">arcsine_distribution</span></code> | |
56 | represents an <a href="http://en.wikipedia.org/wiki/arcsine_distribution" target="_top">arcsine</a> | |
57 | <a href="http://en.wikipedia.org/wiki/Probability_distribution" target="_top">probability | |
58 | distribution function</a>. The arcsine distribution is named because | |
59 | its CDF uses the inverse sin<sup>-1</sup> or arcsine. | |
60 | </p> | |
61 | <p> | |
62 | This is implemented as a generalized version with support from <span class="emphasis"><em>x_min</em></span> | |
63 | to <span class="emphasis"><em>x_max</em></span> providing the 'standard arcsine distribution' | |
64 | as default with <span class="emphasis"><em>x_min = 0</em></span> and <span class="emphasis"><em>x_max = 1</em></span>. | |
65 | (A few make other choices for 'standard'). | |
66 | </p> | |
67 | <p> | |
68 | The arcsine distribution is generalized to include any bounded support | |
69 | <span class="emphasis"><em>a <= x <= b</em></span> by <a href="http://reference.wolfram.com/language/ref/ArcSinDistribution.html" target="_top">Wolfram</a> | |
70 | and <a href="http://en.wikipedia.org/wiki/arcsine_distribution" target="_top">Wikipedia</a>, | |
71 | but also using <span class="emphasis"><em>location</em></span> and <span class="emphasis"><em>scale</em></span> | |
72 | parameters by <a href="http://www.math.uah.edu/stat/index.html" target="_top">Virtual | |
73 | Laboratories in Probability and Statistics</a> <a href="http://www.math.uah.edu/stat/special/Arcsine.html" target="_top">Arcsine | |
74 | distribution</a>. The end-point version is simpler and more obvious, | |
75 | so we implement that. If desired, <a href="http://en.wikipedia.org/wiki/arcsine_distribution" target="_top">this</a> | |
76 | outlines how the <a class="link" href="beta_dist.html" title="Beta Distribution">Beta | |
77 | Distribution</a> can be used to add a shape factor. | |
78 | </p> | |
79 | <p> | |
80 | The <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">probability | |
81 | density function PDF</a> for the <a href="http://en.wikipedia.org/wiki/arcsine_distribution" target="_top">arcsine | |
82 | distribution</a> defined on the interval [<span class="emphasis"><em>x_min, x_max</em></span>] | |
83 | is given by: | |
84 | </p> | |
85 | <p> | |
86 |     f(x; x_min, x_max) = 1 /(π⋅√((x - x_min)⋅(x_max - x_min)) | |
87 | </p> | |
88 | <p> | |
89 | For example, <a href="http://www.wolframalpha.com/" target="_top">Wolfram Alpha</a> | |
90 | arcsine distribution, from input of | |
91 | </p> | |
92 | <pre class="programlisting"><span class="identifier">N</span><span class="special">[</span><span class="identifier">PDF</span><span class="special">[</span><span class="identifier">arcsinedistribution</span><span class="special">[</span><span class="number">0</span><span class="special">,</span> <span class="number">1</span><span class="special">],</span> <span class="number">0.5</span><span class="special">],</span> <span class="number">50</span><span class="special">]</span> | |
93 | </pre> | |
94 | <p> | |
95 | computes the PDF value | |
96 | </p> | |
97 | <pre class="programlisting"><span class="number">0.63661977236758134307553505349005744813783858296183</span> | |
98 | </pre> | |
99 | <p> | |
100 | The Probability Density Functions (PDF) of generalized arcsine distributions | |
101 | are symmetric U-shaped curves, centered on <span class="emphasis"><em>(x_max - x_min)/2</em></span>, | |
102 | highest (infinite) near the two extrema, and quite flat over the central | |
103 | region. | |
104 | </p> | |
105 | <p> | |
106 | If random variate <span class="emphasis"><em>x</em></span> is <span class="emphasis"><em>x_min</em></span> | |
107 | or <span class="emphasis"><em>x_max</em></span>, then the PDF is infinity. If random variate | |
108 | <span class="emphasis"><em>x</em></span> is <span class="emphasis"><em>x_min</em></span> then the CDF is zero. | |
109 | If random variate <span class="emphasis"><em>x</em></span> is <span class="emphasis"><em>x_max</em></span> | |
110 | then the CDF is unity. | |
111 | </p> | |
112 | <p> | |
113 | The 'Standard' (0, 1) arcsine distribution is shown in blue and some generalized | |
114 | examples with other <span class="emphasis"><em>x</em></span> ranges. | |
115 | </p> | |
116 | <p> | |
117 | <span class="inlinemediaobject"><img src="../../../../graphs/arcsine_pdf.svg" align="middle"></span> | |
118 | </p> | |
119 | <p> | |
120 | The Cumulative Distribution Function CDF is defined as | |
121 | </p> | |
122 | <p> | |
123 |     F(x) = 2⋅arcsin(√((x-x_min)/(x_max - x))) / π | |
124 | </p> | |
125 | <p> | |
126 | <span class="inlinemediaobject"><img src="../../../../graphs/arcsine_cdf.svg" align="middle"></span> | |
127 | </p> | |
128 | <h6> | |
129 | <a name="math_toolkit.dist_ref.dists.arcine_dist.h0"></a> | |
130 | <span class="phrase"><a name="math_toolkit.dist_ref.dists.arcine_dist.constructor"></a></span><a class="link" href="arcine_dist.html#math_toolkit.dist_ref.dists.arcine_dist.constructor">Constructor</a> | |
131 | </h6> | |
132 | <pre class="programlisting"><span class="identifier">arcsine_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">x_min</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">x_max</span><span class="special">);</span> | |
133 | </pre> | |
134 | <p> | |
135 | constructs an arcsine distribution with range parameters <span class="emphasis"><em>x_min</em></span> | |
136 | and <span class="emphasis"><em>x_max</em></span>. | |
137 | </p> | |
138 | <p> | |
139 | Requires <span class="emphasis"><em>x_min < x_max</em></span>, otherwise <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a> | |
140 | is called. | |
141 | </p> | |
142 | <p> | |
143 | For example: | |
144 | </p> | |
145 | <pre class="programlisting"><span class="identifier">arcsine_distribution</span><span class="special"><></span> <span class="identifier">myarcsine</span><span class="special">(-</span><span class="number">2</span><span class="special">,</span> <span class="number">4</span><span class="special">);</span> | |
146 | </pre> | |
147 | <p> | |
148 | constructs an arcsine distribution with <span class="emphasis"><em>x_min = -2</em></span> | |
149 | and <span class="emphasis"><em>x_max = 4</em></span>. | |
150 | </p> | |
151 | <p> | |
152 | Default values of <span class="emphasis"><em>x_min = 0</em></span> and <span class="emphasis"><em>x_max = | |
153 | 1</em></span> and a <code class="computeroutput"> <span class="keyword">typedef</span> <span class="identifier">arcsine_distribution</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">arcsine</span><span class="special">;</span></code> | |
154 | mean that | |
155 | </p> | |
156 | <pre class="programlisting"><span class="identifier">arcsine</span> <span class="identifier">as</span><span class="special">;</span> | |
157 | </pre> | |
158 | <p> | |
159 | constructs a 'Standard 01' arcsine distribution. | |
160 | </p> | |
161 | <h6> | |
162 | <a name="math_toolkit.dist_ref.dists.arcine_dist.h1"></a> | |
163 | <span class="phrase"><a name="math_toolkit.dist_ref.dists.arcine_dist.parameter_accessors"></a></span><a class="link" href="arcine_dist.html#math_toolkit.dist_ref.dists.arcine_dist.parameter_accessors">Parameter | |
164 | Accessors</a> | |
165 | </h6> | |
166 | <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">x_min</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span> | |
167 | <span class="identifier">RealType</span> <span class="identifier">x_max</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span> | |
168 | </pre> | |
169 | <p> | |
170 | Return the parameter <span class="emphasis"><em>x_min</em></span> or <span class="emphasis"><em>x_max</em></span> | |
171 | from which this distribution was constructed. | |
172 | </p> | |
173 | <p> | |
174 | So, for example: | |
175 | </p> | |
176 | <pre class="programlisting"><span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">arcsine_distribution</span><span class="special">;</span> | |
177 | ||
178 | <span class="identifier">arcsine_distribution</span><span class="special"><></span> <span class="identifier">as</span><span class="special">(</span><span class="number">2</span><span class="special">,</span> <span class="number">5</span><span class="special">);</span> <span class="comment">// Cconstructs a double arcsine distribution.</span> | |
179 | <span class="identifier">assert</span><span class="special">(</span><span class="identifier">as</span><span class="special">.</span><span class="identifier">x_min</span><span class="special">()</span> <span class="special">==</span> <span class="number">2.</span><span class="special">);</span> <span class="comment">// as.x_min() returns 2.</span> | |
180 | <span class="identifier">assert</span><span class="special">(</span><span class="identifier">as</span><span class="special">.</span><span class="identifier">x_max</span><span class="special">()</span> <span class="special">==</span> <span class="number">5.</span><span class="special">);</span> <span class="comment">// as.x_max() returns 5.</span> | |
181 | </pre> | |
182 | <h5> | |
183 | <a name="math_toolkit.dist_ref.dists.arcine_dist.h2"></a> | |
184 | <span class="phrase"><a name="math_toolkit.dist_ref.dists.arcine_dist.non_member_accessor_functions"></a></span><a class="link" href="arcine_dist.html#math_toolkit.dist_ref.dists.arcine_dist.non_member_accessor_functions">Non-member | |
185 | Accessor Functions</a> | |
186 | </h5> | |
187 | <p> | |
188 | All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor | |
189 | functions</a> that are generic to all distributions are supported: | |
190 | <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>, | |
191 | <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>, | |
192 | <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>, | |
193 | <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>, | |
194 | <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>, | |
195 | <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>, | |
196 | <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>, | |
197 | <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>. | |
198 | </p> | |
199 | <p> | |
200 | The formulae for calculating these are shown in the table below, and at | |
201 | <a href="http://mathworld.wolfram.com/arcsineDistribution.html" target="_top">Wolfram | |
202 | Mathworld</a>. | |
203 | </p> | |
204 | <div class="note"><table border="0" summary="Note"> | |
205 | <tr> | |
206 | <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td> | |
207 | <th align="left">Note</th> | |
208 | </tr> | |
209 | <tr><td align="left" valign="top"><p> | |
210 | There are always <span class="bold"><strong>two</strong></span> values for the | |
211 | <span class="bold"><strong>mode</strong></span>, at <span class="emphasis"><em>x_min</em></span> | |
212 | and at <span class="emphasis"><em>x_max</em></span>, default 0 and 1, so instead we raise | |
213 | the exception <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>. | |
214 | At these extrema, the PDFs are infinite, and the CDFs zero or unity. | |
215 | </p></td></tr> | |
216 | </table></div> | |
217 | <h5> | |
218 | <a name="math_toolkit.dist_ref.dists.arcine_dist.h3"></a> | |
219 | <span class="phrase"><a name="math_toolkit.dist_ref.dists.arcine_dist.applications"></a></span><a class="link" href="arcine_dist.html#math_toolkit.dist_ref.dists.arcine_dist.applications">Applications</a> | |
220 | </h5> | |
221 | <p> | |
222 | The arcsine distribution is useful to describe <a href="http://en.wikipedia.org/wiki/Random_walk" target="_top">Random | |
223 | walks</a>, (including drunken walks) <a href="http://en.wikipedia.org/wiki/Brownian_motion" target="_top">Brownian | |
224 | motion</a>, <a href="http://en.wikipedia.org/wiki/Wiener_process" target="_top">Weiner | |
225 | processes</a>, <a href="http://en.wikipedia.org/wiki/Bernoulli_trial" target="_top">Bernoulli | |
226 | trials</a>, and their appplication to solve stock market and other | |
227 | <a href="http://en.wikipedia.org/wiki/Gambler%27s_ruin" target="_top">ruinous gambling | |
228 | games</a>. | |
229 | </p> | |
230 | <p> | |
231 | The random variate <span class="emphasis"><em>x</em></span> is constrained to <span class="emphasis"><em>x_min</em></span> | |
232 | and <span class="emphasis"><em>x_max</em></span>, (for our 'standard' distribution, 0 and | |
233 | 1), and is usually some fraction. For any other <span class="emphasis"><em>x_min</em></span> | |
234 | and <span class="emphasis"><em>x_max</em></span> a fraction can be obtained from <span class="emphasis"><em>x</em></span> | |
235 | using | |
236 | </p> | |
237 | <p> | |
238 |   fraction = (x - x_min) / (x_max - x_min) | |
239 | </p> | |
240 | <p> | |
241 | The simplest example is tossing heads and tails with a fair coin and modelling | |
242 | the risk of losing, or winning. Walkers (molecules, drunks...) moving left | |
243 | or right of a centre line are another common example. | |
244 | </p> | |
245 | <p> | |
246 | The random variate <span class="emphasis"><em>x</em></span> is the fraction of time spent | |
247 | on the 'winning' side. If half the time is spent on the 'winning' side | |
248 | (and so the other half on the 'losing' side) then <span class="emphasis"><em>x = 1/2</em></span>. | |
249 | </p> | |
250 | <p> | |
251 | For large numbers of tosses, this is modelled by the (standard [0,1]) arcsine | |
252 | distribution, and the PDF can be calculated thus: | |
253 | </p> | |
254 | <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">as</span><span class="special">,</span> <span class="number">1.</span> <span class="special">/</span> <span class="number">2</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.637</span> | |
255 | <span class="comment">// pdf has a minimum at x = 0.5</span> | |
256 | </pre> | |
257 | <p> | |
258 | From the plot of PDF, it is clear that <span class="emphasis"><em>x</em></span> = ½ is the | |
259 | <span class="bold"><strong>minimum</strong></span> of the curve, so this is the | |
260 | <span class="bold"><strong>least likely</strong></span> scenario. (This is highly | |
261 | counter-intuitive, considering that fair tosses must <span class="bold"><strong>eventually</strong></span> | |
262 | become equal. It turns out that <span class="emphasis"><em>eventually</em></span> is not | |
263 | just very long, but <span class="bold"><strong>infinite</strong></span>!). | |
264 | </p> | |
265 | <p> | |
266 | The <span class="bold"><strong>most likely</strong></span> scenarios are towards | |
267 | the extrema where <span class="emphasis"><em>x</em></span> = 0 or <span class="emphasis"><em>x</em></span> | |
268 | = 1. | |
269 | </p> | |
270 | <p> | |
271 | If fraction of time on the left is a ¼, it is only slightly more likely | |
272 | because the curve is quite flat bottomed. | |
273 | </p> | |
274 | <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">as</span><span class="special">,</span> <span class="number">1.</span> <span class="special">/</span> <span class="number">4</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.735</span> | |
275 | </pre> | |
276 | <p> | |
277 | If we consider fair coin-tossing games being played for 100 days (hypothetically | |
278 | continuously to be 'at-limit') the person winning after day 5 will not | |
279 | change in fraction 0.144 of the cases. | |
280 | </p> | |
281 | <p> | |
282 | We can easily compute this setting <span class="emphasis"><em>x</em></span> = 5./100 = 0.05 | |
283 | </p> | |
284 | <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">as</span><span class="special">,</span> <span class="number">0.05</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.144</span> | |
285 | </pre> | |
286 | <p> | |
287 | Similarly, we can compute from a fraction of 0.05 /2 = 0.025 (halved because | |
288 | we are considering both winners and losers) corresponding to 1 - 0.025 | |
289 | or 97.5% of the gamblers, (walkers, particles...) on the <span class="bold"><strong>same | |
290 | side</strong></span> of the origin | |
291 | </p> | |
292 | <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="number">2</span> <span class="special">*</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">as</span><span class="special">,</span> <span class="number">1</span> <span class="special">-</span> <span class="number">0.975</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.202</span> | |
293 | </pre> | |
294 | <p> | |
295 | (use of the complement gives a bit more clarity, and avoids potential loss | |
296 | of accuracy when <span class="emphasis"><em>x</em></span> is close to unity, see <a class="link" href="../../stat_tut/overview/complements.html#why_complements">why | |
297 | complements?</a>). | |
298 | </p> | |
299 | <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="number">2</span> <span class="special">*</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">as</span><span class="special">,</span> <span class="number">0.975</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.202</span> | |
300 | </pre> | |
301 | <p> | |
302 | or we can reverse the calculation by assuming a fraction of time on one | |
303 | side, say fraction 0.2, | |
304 | </p> | |
305 | <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">as</span><span class="special">,</span> <span class="number">1</span> <span class="special">-</span> <span class="number">0.2</span> <span class="special">/</span> <span class="number">2</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.976</span> | |
306 | ||
307 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">as</span><span class="special">,</span> <span class="number">0.2</span> <span class="special">/</span> <span class="number">2</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.976</span> | |
308 | </pre> | |
309 | <p> | |
310 | <span class="bold"><strong>Summary</strong></span>: Every time we toss, the odds | |
311 | are equal, so on average we have the same change of winning and losing. | |
312 | </p> | |
313 | <p> | |
314 | But this is <span class="bold"><strong>not true</strong></span> for an an individual | |
315 | game where one will be <span class="bold"><strong>mostly in a bad or good patch</strong></span>. | |
316 | </p> | |
317 | <p> | |
318 | This is quite counter-intuitive to most people, but the mathematics is | |
319 | clear, and gamblers continue to provide proof. | |
320 | </p> | |
321 | <p> | |
322 | <span class="bold"><strong>Moral</strong></span>: if you in a losing patch, leave | |
323 | the game. (Because the odds to recover to a good patch are poor). | |
324 | </p> | |
325 | <p> | |
326 | <span class="bold"><strong>Corollary</strong></span>: Quit while you are ahead? | |
327 | </p> | |
328 | <p> | |
329 | A working example is at <a href="../../../../../example/arcsine_example.cpp" target="_top">arcsine_example.cpp</a> | |
330 | including sample output . | |
331 | </p> | |
332 | <h5> | |
333 | <a name="math_toolkit.dist_ref.dists.arcine_dist.h4"></a> | |
334 | <span class="phrase"><a name="math_toolkit.dist_ref.dists.arcine_dist.related_distributions"></a></span><a class="link" href="arcine_dist.html#math_toolkit.dist_ref.dists.arcine_dist.related_distributions">Related | |
335 | distributions</a> | |
336 | </h5> | |
337 | <p> | |
338 | The arcsine distribution with <span class="emphasis"><em>x_min = 0</em></span> and <span class="emphasis"><em>x_max | |
339 | = 1</em></span> is special case of the <a class="link" href="beta_dist.html" title="Beta Distribution">Beta | |
340 | Distribution</a> with α = 1/2 and β = 1/2. | |
341 | </p> | |
342 | <h5> | |
343 | <a name="math_toolkit.dist_ref.dists.arcine_dist.h5"></a> | |
344 | <span class="phrase"><a name="math_toolkit.dist_ref.dists.arcine_dist.accuracy"></a></span><a class="link" href="arcine_dist.html#math_toolkit.dist_ref.dists.arcine_dist.accuracy">Accuracy</a> | |
345 | </h5> | |
346 | <p> | |
347 | This distribution is implemented using sqrt, sine, cos and arc sine and | |
348 | cos trigonometric functions which are normally accurate to a few <a href="http://en.wikipedia.org/wiki/Machine_epsilon" target="_top">machine epsilon</a>. | |
349 | But all values suffer from <a href="http://en.wikipedia.org/wiki/Loss_of_significance" target="_top">loss | |
350 | of significance or cancellation error</a> for values of <span class="emphasis"><em>x</em></span> | |
351 | close to <span class="emphasis"><em>x_max</em></span>. For example, for a standard [0, 1] | |
352 | arcsine distribution <span class="emphasis"><em>as</em></span>, the pdf is symmetric about | |
353 | random variate <span class="emphasis"><em>x = 0.5</em></span> so that one would expect <code class="computeroutput"><span class="identifier">pdf</span><span class="special">(</span><span class="identifier">as</span><span class="special">,</span> <span class="number">0.01</span><span class="special">)</span> <span class="special">==</span> | |
354 | <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">as</span><span class="special">,</span> <span class="number">0.99</span><span class="special">)</span></code>. But | |
355 | as <span class="emphasis"><em>x</em></span> nears unity, there is increasing <a href="http://en.wikipedia.org/wiki/Loss_of_significance" target="_top">loss | |
356 | of significance</a>. To counteract this, the complement versions of | |
357 | CDF and quantile are implemented with alternative expressions using <span class="emphasis"><em>cos<sup>-1</sup></em></span> | |
358 | instead of <span class="emphasis"><em>sin<sup>-1</sup></em></span>. Users should see <a class="link" href="../../stat_tut/overview/complements.html#why_complements">why | |
359 | complements?</a> for guidance on when to avoid loss of accuracy by using | |
360 | complements. | |
361 | </p> | |
362 | <h5> | |
363 | <a name="math_toolkit.dist_ref.dists.arcine_dist.h6"></a> | |
364 | <span class="phrase"><a name="math_toolkit.dist_ref.dists.arcine_dist.testing"></a></span><a class="link" href="arcine_dist.html#math_toolkit.dist_ref.dists.arcine_dist.testing">Testing</a> | |
365 | </h5> | |
366 | <p> | |
367 | The results were tested against a few accurate spot values computed by | |
368 | <a href="http://www.wolframalpha.com/" target="_top">Wolfram Alpha</a>, for example: | |
369 | </p> | |
370 | <pre class="programlisting"><span class="identifier">N</span><span class="special">[</span><span class="identifier">PDF</span><span class="special">[</span><span class="identifier">arcsinedistribution</span><span class="special">[</span><span class="number">0</span><span class="special">,</span> <span class="number">1</span><span class="special">],</span> <span class="number">0.5</span><span class="special">],</span> <span class="number">50</span><span class="special">]</span> | |
371 | <span class="number">0.63661977236758134307553505349005744813783858296183</span> | |
372 | </pre> | |
373 | <h5> | |
374 | <a name="math_toolkit.dist_ref.dists.arcine_dist.h7"></a> | |
375 | <span class="phrase"><a name="math_toolkit.dist_ref.dists.arcine_dist.implementation"></a></span><a class="link" href="arcine_dist.html#math_toolkit.dist_ref.dists.arcine_dist.implementation">Implementation</a> | |
376 | </h5> | |
377 | <p> | |
378 | In the following table <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span> | |
379 | are the parameters <span class="emphasis"><em>x_min</em></span>   and <span class="emphasis"><em>x_max</em></span>, | |
380 | <span class="emphasis"><em>x</em></span> is the random variable, <span class="emphasis"><em>p</em></span> is | |
381 | the probability and its complement <span class="emphasis"><em>q = 1-p</em></span>. | |
382 | </p> | |
383 | <div class="informaltable"><table class="table"> | |
384 | <colgroup> | |
385 | <col> | |
386 | <col> | |
387 | </colgroup> | |
388 | <thead><tr> | |
389 | <th> | |
390 | <p> | |
391 | Function | |
392 | </p> | |
393 | </th> | |
394 | <th> | |
395 | <p> | |
396 | Implementation Notes | |
397 | </p> | |
398 | </th> | |
399 | </tr></thead> | |
400 | <tbody> | |
401 | <tr> | |
402 | <td> | |
403 | <p> | |
404 | support | |
405 | </p> | |
406 | </td> | |
407 | <td> | |
408 | <p> | |
409 | x ∈ [a, b], default x ∈ [0, 1] | |
410 | </p> | |
411 | </td> | |
412 | </tr> | |
413 | <tr> | |
414 | <td> | |
415 | <p> | |
416 | ||
417 | </p> | |
418 | </td> | |
419 | <td> | |
420 | <p> | |
421 | f(x; a, b) = 1/(π⋅√(x - a)⋅(b - x)) | |
422 | </p> | |
423 | </td> | |
424 | </tr> | |
425 | <tr> | |
426 | <td> | |
427 | <p> | |
428 | cdf | |
429 | </p> | |
430 | </td> | |
431 | <td> | |
432 | <p> | |
433 | F(x) = 2/π⋅sin<sup>-1</sup>(√(x - a) / (b - a) ) | |
434 | </p> | |
435 | </td> | |
436 | </tr> | |
437 | <tr> | |
438 | <td> | |
439 | <p> | |
440 | cdf of complement | |
441 | </p> | |
442 | </td> | |
443 | <td> | |
444 | <p> | |
445 | 2/(π⋅cos<sup>-1</sup>(√(x - a) / (b - a))) | |
446 | </p> | |
447 | </td> | |
448 | </tr> | |
449 | <tr> | |
450 | <td> | |
451 | <p> | |
452 | quantile | |
453 | </p> | |
454 | </td> | |
455 | <td> | |
456 | <p> | |
457 | -a⋅sin<sup>2</sup>(½π⋅p) + a + b⋅sin<sup>2</sup>(½π⋅p) | |
458 | </p> | |
459 | </td> | |
460 | </tr> | |
461 | <tr> | |
462 | <td> | |
463 | <p> | |
464 | quantile from the complement | |
465 | </p> | |
466 | </td> | |
467 | <td> | |
468 | <p> | |
469 | -a⋅cos<sup>2</sup>(½π⋅p) + a + b⋅cos<sup>2</sup>(½π⋅q) | |
470 | </p> | |
471 | </td> | |
472 | </tr> | |
473 | <tr> | |
474 | <td> | |
475 | <p> | |
476 | mean | |
477 | </p> | |
478 | </td> | |
479 | <td> | |
480 | <p> | |
481 | ½(a+b) | |
482 | </p> | |
483 | </td> | |
484 | </tr> | |
485 | <tr> | |
486 | <td> | |
487 | <p> | |
488 | median | |
489 | </p> | |
490 | </td> | |
491 | <td> | |
492 | <p> | |
493 | ½(a+b) | |
494 | </p> | |
495 | </td> | |
496 | </tr> | |
497 | <tr> | |
498 | <td> | |
499 | <p> | |
500 | mode | |
501 | </p> | |
502 | </td> | |
503 | <td> | |
504 | <p> | |
505 | x ∈ [a, b], so raises domain_error (returning NaN). | |
506 | </p> | |
507 | </td> | |
508 | </tr> | |
509 | <tr> | |
510 | <td> | |
511 | <p> | |
512 | variance | |
513 | </p> | |
514 | </td> | |
515 | <td> | |
516 | <p> | |
517 | (b - a)<sup>2</sup> / 8 | |
518 | </p> | |
519 | </td> | |
520 | </tr> | |
521 | <tr> | |
522 | <td> | |
523 | <p> | |
524 | skewness | |
525 | </p> | |
526 | </td> | |
527 | <td> | |
528 | <p> | |
529 | 0 | |
530 | </p> | |
531 | </td> | |
532 | </tr> | |
533 | <tr> | |
534 | <td> | |
535 | <p> | |
536 | kurtosis excess | |
537 | </p> | |
538 | </td> | |
539 | <td> | |
540 | <p> | |
541 | -3/2 | |
542 | </p> | |
543 | </td> | |
544 | </tr> | |
545 | <tr> | |
546 | <td> | |
547 | <p> | |
548 | kurtosis | |
549 | </p> | |
550 | </td> | |
551 | <td> | |
552 | <p> | |
553 | kurtosis_excess + 3 | |
554 | </p> | |
555 | </td> | |
556 | </tr> | |
557 | </tbody> | |
558 | </table></div> | |
559 | <p> | |
560 | The quantile was calculated using an expression obtained by using <a href="http://www.wolframalpha.com/" target="_top">Wolfram Alpha</a> to invert the | |
561 | formula for the CDF thus | |
562 | </p> | |
563 | <pre class="programlisting"><span class="identifier">solve</span> <span class="special">[</span><span class="identifier">p</span> <span class="special">-</span> <span class="number">2</span><span class="special">/</span><span class="identifier">pi</span> <span class="identifier">sin</span><span class="special">^-</span><span class="number">1</span><span class="special">(</span><span class="identifier">sqrt</span><span class="special">((</span><span class="identifier">x</span><span class="special">-</span><span class="identifier">a</span><span class="special">)/(</span><span class="identifier">b</span><span class="special">-</span><span class="identifier">a</span><span class="special">)))</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">x</span><span class="special">]</span> | |
564 | </pre> | |
565 | <p> | |
566 | which was interpreted as | |
567 | </p> | |
568 | <pre class="programlisting"><span class="identifier">Solve</span><span class="special">[</span><span class="identifier">p</span> <span class="special">-</span> <span class="special">(</span><span class="number">2</span> <span class="identifier">ArcSin</span><span class="special">[</span><span class="identifier">Sqrt</span><span class="special">[(-</span><span class="identifier">a</span> <span class="special">+</span> <span class="identifier">x</span><span class="special">)/(-</span><span class="identifier">a</span> <span class="special">+</span> <span class="identifier">b</span><span class="special">)]])/</span><span class="identifier">Pi</span> <span class="special">==</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">MaxExtraConditions</span> <span class="special">-></span> <span class="identifier">Automatic</span><span class="special">]</span> | |
569 | </pre> | |
570 | <p> | |
571 | and produced the resulting expression | |
572 | </p> | |
573 | <pre class="programlisting"><span class="identifier">x</span> <span class="special">=</span> <span class="special">-</span><span class="identifier">a</span> <span class="identifier">sin</span><span class="special">^</span><span class="number">2</span><span class="special">((</span><span class="identifier">pi</span> <span class="identifier">p</span><span class="special">)/</span><span class="number">2</span><span class="special">)+</span><span class="identifier">a</span><span class="special">+</span><span class="identifier">b</span> <span class="identifier">sin</span><span class="special">^</span><span class="number">2</span><span class="special">((</span><span class="identifier">pi</span> <span class="identifier">p</span><span class="special">)/</span><span class="number">2</span><span class="special">)</span> | |
574 | </pre> | |
575 | <p> | |
576 | Thanks to Wolfram for providing this facility. | |
577 | </p> | |
578 | <h5> | |
579 | <a name="math_toolkit.dist_ref.dists.arcine_dist.h8"></a> | |
580 | <span class="phrase"><a name="math_toolkit.dist_ref.dists.arcine_dist.references"></a></span><a class="link" href="arcine_dist.html#math_toolkit.dist_ref.dists.arcine_dist.references">References</a> | |
581 | </h5> | |
582 | <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> | |
583 | <li class="listitem"> | |
584 | <a href="http://en.wikipedia.org/wiki/arcsine_distribution" target="_top">Wikipedia | |
585 | arcsine distribution</a> | |
586 | </li> | |
587 | <li class="listitem"> | |
588 | <a href="http://en.wikipedia.org/wiki/Beta_distribution" target="_top">Wikipedia | |
589 | Beta distribution</a> | |
590 | </li> | |
591 | <li class="listitem"> | |
592 | <a href="http://mathworld.wolfram.com/BetaDistribution.html" target="_top">Wolfram | |
593 | MathWorld</a> | |
594 | </li> | |
595 | <li class="listitem"> | |
596 | <a href="http://www.wolframalpha.com/" target="_top">Wolfram Alpha</a> | |
597 | </li> | |
598 | </ul></div> | |
599 | <h5> | |
600 | <a name="math_toolkit.dist_ref.dists.arcine_dist.h9"></a> | |
601 | <span class="phrase"><a name="math_toolkit.dist_ref.dists.arcine_dist.sources"></a></span><a class="link" href="arcine_dist.html#math_toolkit.dist_ref.dists.arcine_dist.sources">Sources</a> | |
602 | </h5> | |
603 | <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> | |
604 | <li class="listitem"> | |
605 | <a href="http://estebanmoro.org/2009/04/the-probability-of-going-through-a-bad-patch" target="_top">The | |
606 | probability of going through a bad patch</a> Esteban Moro's Blog. | |
607 | </li> | |
608 | <li class="listitem"> | |
609 | <a href="http://www.gotohaggstrom.com/What%20do%20schmucks%20and%20the%20arc%20sine%20law%20have%20in%20common.pdf" target="_top">What | |
610 | soschumcks and the arc sine have in common</a> Peter Haggstrom. | |
611 | </li> | |
612 | <li class="listitem"> | |
613 | <a href="http://www.math.uah.edu/stat/special/Arcsine.html" target="_top">arcsine | |
614 | distribution</a>. | |
615 | </li> | |
616 | <li class="listitem"> | |
617 | <a href="http://reference.wolfram.com/language/ref/ArcSinDistribution.html" target="_top">Wolfram | |
618 | reference arcsine examples</a>. | |
619 | </li> | |
620 | <li class="listitem"> | |
621 | <a href="http://www.math.harvard.edu/library/sternberg/slides/1180908.pdf" target="_top">Shlomo | |
622 | Sternberg slides</a>. | |
623 | </li> | |
624 | </ul></div> | |
625 | </div> | |
626 | <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> | |
627 | <td align="left"></td> | |
628 | <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, | |
629 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
630 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
631 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
632 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
633 | 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>) | |
634 | </p> | |
635 | </div></td> | |
636 | </tr></table> | |
637 | <hr> | |
638 | <div class="spirit-nav"> | |
639 | <a accesskey="p" href="../dists.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="bernoulli_dist.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a> | |
640 | </div> | |
641 | </body> | |
642 | </html> |