1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Arcsine Distribution</title> 5<link rel="stylesheet" href="../../../math.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="../../../index.html" title="Math Toolkit 2.12.0"> 8<link rel="up" href="../dists.html" title="Distributions"> 9<link rel="prev" href="../dists.html" title="Distributions"> 10<link rel="next" href="bernoulli_dist.html" title="Bernoulli Distribution"> 11</head> 12<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> 13<table cellpadding="2" width="100%"><tr> 14<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../boost.png"></td> 15<td align="center"><a href="../../../../../../../index.html">Home</a></td> 16<td align="center"><a href="../../../../../../../libs/libraries.htm">Libraries</a></td> 17<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> 18<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> 19<td align="center"><a href="../../../../../../../more/index.htm">More</a></td> 20</tr></table> 21<hr> 22<div class="spirit-nav"> 23<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> 24</div> 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 21. 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 21. 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<div class="blockquote"><blockquote class="blockquote"><p> 86 <span class="serif_italic">f(x; x_min, x_max) = 1 /(π⋅√((x - x_min)⋅(x_max 87 - x_min))</span> 88 </p></blockquote></div> 89<p> 90 For example, <a href="http://www.wolframalpha.com/" target="_top">Wolfram Alpha</a> 91 arcsine distribution, from input of 92 </p> 93<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> 94</pre> 95<p> 96 computes the PDF value 97 </p> 98<pre class="programlisting"><span class="number">0.63661977236758134307553505349005744813783858296183</span> 99</pre> 100<p> 101 The Probability Density Functions (PDF) of generalized arcsine distributions 102 are symmetric U-shaped curves, centered on <span class="emphasis"><em>(x_max - x_min)/2</em></span>, 103 highest (infinite) near the two extrema, and quite flat over the central 104 region. 105 </p> 106<p> 107 If random variate <span class="emphasis"><em>x</em></span> is <span class="emphasis"><em>x_min</em></span> 108 or <span class="emphasis"><em>x_max</em></span>, then the PDF is infinity. If random variate 109 <span class="emphasis"><em>x</em></span> is <span class="emphasis"><em>x_min</em></span> then the CDF is zero. 110 If random variate <span class="emphasis"><em>x</em></span> is <span class="emphasis"><em>x_max</em></span> 111 then the CDF is unity. 112 </p> 113<p> 114 The 'Standard' (0, 1) arcsine distribution is shown in blue and some generalized 115 examples with other <span class="emphasis"><em>x</em></span> ranges. 116 </p> 117<div class="blockquote"><blockquote class="blockquote"><p> 118 <span class="inlinemediaobject"><img src="../../../../graphs/arcsine_pdf.svg" align="middle"></span> 119 120 </p></blockquote></div> 121<p> 122 The Cumulative Distribution Function CDF is defined as 123 </p> 124<div class="blockquote"><blockquote class="blockquote"><p> 125 <span class="serif_italic">F(x) = 2⋅arcsin(√((x-x_min)/(x_max - x))) / 126 π</span> 127 </p></blockquote></div> 128<div class="blockquote"><blockquote class="blockquote"><p> 129 <span class="inlinemediaobject"><img src="../../../../graphs/arcsine_cdf.svg" align="middle"></span> 130 131 </p></blockquote></div> 132<h6> 133<a name="math_toolkit.dist_ref.dists.arcine_dist.h0"></a> 134 <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> 135 </h6> 136<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> 137</pre> 138<p> 139 constructs an arcsine distribution with range parameters <span class="emphasis"><em>x_min</em></span> 140 and <span class="emphasis"><em>x_max</em></span>. 141 </p> 142<p> 143 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> 144 is called. 145 </p> 146<p> 147 For example: 148 </p> 149<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> 150</pre> 151<p> 152 constructs an arcsine distribution with <span class="emphasis"><em>x_min = -2</em></span> 153 and <span class="emphasis"><em>x_max = 4</em></span>. 154 </p> 155<p> 156 Default values of <span class="emphasis"><em>x_min = 0</em></span> and <span class="emphasis"><em>x_max = 157 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> 158 mean that 159 </p> 160<pre class="programlisting"><span class="identifier">arcsine</span> <span class="identifier">as</span><span class="special">;</span> 161</pre> 162<p> 163 constructs a 'Standard 01' arcsine distribution. 164 </p> 165<h6> 166<a name="math_toolkit.dist_ref.dists.arcine_dist.h1"></a> 167 <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 168 Accessors</a> 169 </h6> 170<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> 171<span class="identifier">RealType</span> <span class="identifier">x_max</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span> 172</pre> 173<p> 174 Return the parameter <span class="emphasis"><em>x_min</em></span> or <span class="emphasis"><em>x_max</em></span> 175 from which this distribution was constructed. 176 </p> 177<p> 178 So, for example: 179 </p> 180<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> 181 182<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">// Constructs a double arcsine distribution.</span> 183<span class="identifier">BOOST_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> 184<span class="identifier">BOOST_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> 185</pre> 186<h5> 187<a name="math_toolkit.dist_ref.dists.arcine_dist.h2"></a> 188 <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 189 Accessor Functions</a> 190 </h5> 191<p> 192 All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor 193 functions</a> that are generic to all distributions are supported: 194 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>, 195 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>, 196 <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>, 197 <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>, 198 <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>, 199 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>, 200 <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>, 201 <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>. 202 </p> 203<p> 204 The formulae for calculating these are shown in the table below, and at 205 <a href="http://mathworld.wolfram.com/arcsineDistribution.html" target="_top">Wolfram 206 Mathworld</a>. 207 </p> 208<div class="note"><table border="0" summary="Note"> 209<tr> 210<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td> 211<th align="left">Note</th> 212</tr> 213<tr><td align="left" valign="top"><p> 214 There are always <span class="bold"><strong>two</strong></span> values for the 215 <span class="bold"><strong>mode</strong></span>, at <span class="emphasis"><em>x_min</em></span> 216 and at <span class="emphasis"><em>x_max</em></span>, default 0 and 1, so instead we raise 217 the exception <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>. 218 At these extrema, the PDFs are infinite, and the CDFs zero or unity. 219 </p></td></tr> 220</table></div> 221<h5> 222<a name="math_toolkit.dist_ref.dists.arcine_dist.h3"></a> 223 <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> 224 </h5> 225<p> 226 The arcsine distribution is useful to describe <a href="http://en.wikipedia.org/wiki/Random_walk" target="_top">Random 227 walks</a>, (including drunken walks) <a href="http://en.wikipedia.org/wiki/Brownian_motion" target="_top">Brownian 228 motion</a>, <a href="http://en.wikipedia.org/wiki/Wiener_process" target="_top">Weiner 229 processes</a>, <a href="http://en.wikipedia.org/wiki/Bernoulli_trial" target="_top">Bernoulli 230 trials</a>, and their application to solve stock market and other 231 <a href="http://en.wikipedia.org/wiki/Gambler%27s_ruin" target="_top">ruinous gambling 232 games</a>. 233 </p> 234<p> 235 The random variate <span class="emphasis"><em>x</em></span> is constrained to <span class="emphasis"><em>x_min</em></span> 236 and <span class="emphasis"><em>x_max</em></span>, (for our 'standard' distribution, 0 and 237 1), and is usually some fraction. For any other <span class="emphasis"><em>x_min</em></span> 238 and <span class="emphasis"><em>x_max</em></span> a fraction can be obtained from <span class="emphasis"><em>x</em></span> 239 using 240 </p> 241<div class="blockquote"><blockquote class="blockquote"><p> 242 <span class="serif_italic">fraction = (x - x_min) / (x_max - x_min)</span> 243 </p></blockquote></div> 244<p> 245 The simplest example is tossing heads and tails with a fair coin and modelling 246 the risk of losing, or winning. Walkers (molecules, drunks...) moving left 247 or right of a centre line are another common example. 248 </p> 249<p> 250 The random variate <span class="emphasis"><em>x</em></span> is the fraction of time spent 251 on the 'winning' side. If half the time is spent on the 'winning' side 252 (and so the other half on the 'losing' side) then <span class="emphasis"><em>x = 1/2</em></span>. 253 </p> 254<p> 255 For large numbers of tosses, this is modelled by the (standard [0,1]) arcsine 256 distribution, and the PDF can be calculated thus: 257 </p> 258<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> 259<span class="comment">// pdf has a minimum at x = 0.5</span> 260</pre> 261<p> 262 From the plot of PDF, it is clear that <span class="emphasis"><em>x</em></span> = ½ is the 263 <span class="bold"><strong>minimum</strong></span> of the curve, so this is the 264 <span class="bold"><strong>least likely</strong></span> scenario. (This is highly 265 counter-intuitive, considering that fair tosses must <span class="bold"><strong>eventually</strong></span> 266 become equal. It turns out that <span class="emphasis"><em>eventually</em></span> is not 267 just very long, but <span class="bold"><strong>infinite</strong></span>!). 268 </p> 269<p> 270 The <span class="bold"><strong>most likely</strong></span> scenarios are towards 271 the extrema where <span class="emphasis"><em>x</em></span> = 0 or <span class="emphasis"><em>x</em></span> 272 = 1. 273 </p> 274<p> 275 If fraction of time on the left is a ¼, it is only slightly more likely 276 because the curve is quite flat bottomed. 277 </p> 278<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> 279</pre> 280<p> 281 If we consider fair coin-tossing games being played for 100 days (hypothetically 282 continuously to be 'at-limit') the person winning after day 5 will not 283 change in fraction 0.144 of the cases. 284 </p> 285<p> 286 We can easily compute this setting <span class="emphasis"><em>x</em></span> = 5./100 = 0.05 287 </p> 288<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> 289</pre> 290<p> 291 Similarly, we can compute from a fraction of 0.05 /2 = 0.025 (halved because 292 we are considering both winners and losers) corresponding to 1 - 0.025 293 or 97.5% of the gamblers, (walkers, particles...) on the <span class="bold"><strong>same 294 side</strong></span> of the origin 295 </p> 296<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> 297</pre> 298<p> 299 (use of the complement gives a bit more clarity, and avoids potential loss 300 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 301 complements?</a>). 302 </p> 303<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> 304</pre> 305<p> 306 or we can reverse the calculation by assuming a fraction of time on one 307 side, say fraction 0.2, 308 </p> 309<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> 310 311<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> 312</pre> 313<p> 314 <span class="bold"><strong>Summary</strong></span>: Every time we toss, the odds 315 are equal, so on average we have the same change of winning and losing. 316 </p> 317<p> 318 But this is <span class="bold"><strong>not true</strong></span> for an an individual 319 game where one will be <span class="bold"><strong>mostly in a bad or good patch</strong></span>. 320 </p> 321<p> 322 This is quite counter-intuitive to most people, but the mathematics is 323 clear, and gamblers continue to provide proof. 324 </p> 325<p> 326 <span class="bold"><strong>Moral</strong></span>: if you in a losing patch, leave 327 the game. (Because the odds to recover to a good patch are poor). 328 </p> 329<p> 330 <span class="bold"><strong>Corollary</strong></span>: Quit while you are ahead? 331 </p> 332<p> 333 A working example is at <a href="../../../../../example/arcsine_example.cpp" target="_top">arcsine_example.cpp</a> 334 including sample output . 335 </p> 336<h5> 337<a name="math_toolkit.dist_ref.dists.arcine_dist.h4"></a> 338 <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 339 distributions</a> 340 </h5> 341<p> 342 The arcsine distribution with <span class="emphasis"><em>x_min = 0</em></span> and <span class="emphasis"><em>x_max 343 = 1</em></span> is special case of the <a class="link" href="beta_dist.html" title="Beta Distribution">Beta 344 Distribution</a> with α = 1/2 and β = 1/2. 345 </p> 346<h5> 347<a name="math_toolkit.dist_ref.dists.arcine_dist.h5"></a> 348 <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> 349 </h5> 350<p> 351 This distribution is implemented using sqrt, sine, cos and arc sine and 352 cos trigonometric functions which are normally accurate to a few <a href="http://en.wikipedia.org/wiki/Machine_epsilon" target="_top">machine epsilon</a>. 353 But all values suffer from <a href="http://en.wikipedia.org/wiki/Loss_of_significance" target="_top">loss 354 of significance or cancellation error</a> for values of <span class="emphasis"><em>x</em></span> 355 close to <span class="emphasis"><em>x_max</em></span>. For example, for a standard [0, 1] 356 arcsine distribution <span class="emphasis"><em>as</em></span>, the pdf is symmetric about 357 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> 358 <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 359 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 360 of significance</a>. To counteract this, the complement versions of 361 CDF and quantile are implemented with alternative expressions using <span class="emphasis"><em>cos<sup>-1</sup></em></span> 362 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 363 complements?</a> for guidance on when to avoid loss of accuracy by using 364 complements. 365 </p> 366<h5> 367<a name="math_toolkit.dist_ref.dists.arcine_dist.h6"></a> 368 <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> 369 </h5> 370<p> 371 The results were tested against a few accurate spot values computed by 372 <a href="http://www.wolframalpha.com/" target="_top">Wolfram Alpha</a>, for example: 373 </p> 374<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> 375<span class="number">0.63661977236758134307553505349005744813783858296183</span> 376</pre> 377<h5> 378<a name="math_toolkit.dist_ref.dists.arcine_dist.h7"></a> 379 <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> 380 </h5> 381<p> 382 In the following table <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span> 383 are the parameters <span class="emphasis"><em>x_min</em></span> and <span class="emphasis"><em>x_max</em></span>, 384 <span class="emphasis"><em>x</em></span> is the random variable, <span class="emphasis"><em>p</em></span> is 385 the probability and its complement <span class="emphasis"><em>q = 1-p</em></span>. 386 </p> 387<div class="informaltable"><table class="table"> 388<colgroup> 389<col> 390<col> 391</colgroup> 392<thead><tr> 393<th> 394 <p> 395 Function 396 </p> 397 </th> 398<th> 399 <p> 400 Implementation Notes 401 </p> 402 </th> 403</tr></thead> 404<tbody> 405<tr> 406<td> 407 <p> 408 support 409 </p> 410 </td> 411<td> 412 <p> 413 x ∈ [a, b], default x ∈ [0, 1] 414 </p> 415 </td> 416</tr> 417<tr> 418<td> 419 <p> 420 pdf 421 </p> 422 </td> 423<td> 424 <p> 425 f(x; a, b) = 1/(π⋅√(x - a)⋅(b - x)) 426 </p> 427 </td> 428</tr> 429<tr> 430<td> 431 <p> 432 cdf 433 </p> 434 </td> 435<td> 436 <p> 437 F(x) = 2/π⋅sin<sup>-1</sup>(√(x - a) / (b - a) ) 438 </p> 439 </td> 440</tr> 441<tr> 442<td> 443 <p> 444 cdf of complement 445 </p> 446 </td> 447<td> 448 <p> 449 2/(π⋅cos<sup>-1</sup>(√(x - a) / (b - a))) 450 </p> 451 </td> 452</tr> 453<tr> 454<td> 455 <p> 456 quantile 457 </p> 458 </td> 459<td> 460 <p> 461 -a⋅sin<sup>2</sup>(½π⋅p) + a + b⋅sin<sup>2</sup>(½π⋅p) 462 </p> 463 </td> 464</tr> 465<tr> 466<td> 467 <p> 468 quantile from the complement 469 </p> 470 </td> 471<td> 472 <p> 473 -a⋅cos<sup>2</sup>(½π⋅p) + a + b⋅cos<sup>2</sup>(½π⋅q) 474 </p> 475 </td> 476</tr> 477<tr> 478<td> 479 <p> 480 mean 481 </p> 482 </td> 483<td> 484 <p> 485 ½(a+b) 486 </p> 487 </td> 488</tr> 489<tr> 490<td> 491 <p> 492 median 493 </p> 494 </td> 495<td> 496 <p> 497 ½(a+b) 498 </p> 499 </td> 500</tr> 501<tr> 502<td> 503 <p> 504 mode 505 </p> 506 </td> 507<td> 508 <p> 509 x ∈ [a, b], so raises domain_error (returning NaN). 510 </p> 511 </td> 512</tr> 513<tr> 514<td> 515 <p> 516 variance 517 </p> 518 </td> 519<td> 520 <p> 521 (b - a)<sup>2</sup> / 8 522 </p> 523 </td> 524</tr> 525<tr> 526<td> 527 <p> 528 skewness 529 </p> 530 </td> 531<td> 532 <p> 533 0 534 </p> 535 </td> 536</tr> 537<tr> 538<td> 539 <p> 540 kurtosis excess 541 </p> 542 </td> 543<td> 544 <p> 545 -3/2 546 </p> 547 </td> 548</tr> 549<tr> 550<td> 551 <p> 552 kurtosis 553 </p> 554 </td> 555<td> 556 <p> 557 kurtosis_excess + 3 558 </p> 559 </td> 560</tr> 561</tbody> 562</table></div> 563<p> 564 The quantile was calculated using an expression obtained by using <a href="http://www.wolframalpha.com/" target="_top">Wolfram Alpha</a> to invert the 565 formula for the CDF thus 566 </p> 567<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> 568</pre> 569<p> 570 which was interpreted as 571 </p> 572<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> 573</pre> 574<p> 575 and produced the resulting expression 576 </p> 577<div class="blockquote"><blockquote class="blockquote"><p> 578 <span class="serif_italic">x = -a sin^2((pi p)/2)+a+b sin^2((pi p)/2)</span> 579 </p></blockquote></div> 580<p> 581 Thanks to Wolfram for providing this facility. 582 </p> 583<h5> 584<a name="math_toolkit.dist_ref.dists.arcine_dist.h8"></a> 585 <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> 586 </h5> 587<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> 588<li class="listitem"> 589 <a href="http://en.wikipedia.org/wiki/arcsine_distribution" target="_top">Wikipedia 590 arcsine distribution</a> 591 </li> 592<li class="listitem"> 593 <a href="http://en.wikipedia.org/wiki/Beta_distribution" target="_top">Wikipedia 594 Beta distribution</a> 595 </li> 596<li class="listitem"> 597 <a href="http://mathworld.wolfram.com/BetaDistribution.html" target="_top">Wolfram 598 MathWorld</a> 599 </li> 600<li class="listitem"> 601 <a href="http://www.wolframalpha.com/" target="_top">Wolfram Alpha</a> 602 </li> 603</ul></div> 604<h5> 605<a name="math_toolkit.dist_ref.dists.arcine_dist.h9"></a> 606 <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> 607 </h5> 608<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> 609<li class="listitem"> 610 <a href="http://estebanmoro.org/2009/04/the-probability-of-going-through-a-bad-patch" target="_top">The 611 probability of going through a bad patch</a> Esteban Moro's Blog. 612 </li> 613<li class="listitem"> 614 <a href="http://www.gotohaggstrom.com/What%20do%20schmucks%20and%20the%20arc%20sine%20law%20have%20in%20common.pdf" target="_top">What 615 soschumcks and the arc sine have in common</a> Peter Haggstrom. 616 </li> 617<li class="listitem"> 618 <a href="http://www.math.uah.edu/stat/special/Arcsine.html" target="_top">arcsine 619 distribution</a>. 620 </li> 621<li class="listitem"> 622 <a href="http://reference.wolfram.com/language/ref/ArcSinDistribution.html" target="_top">Wolfram 623 reference arcsine examples</a>. 624 </li> 625<li class="listitem"> 626 <a href="http://www.math.harvard.edu/library/sternberg/slides/1180908.pdf" target="_top">Shlomo 627 Sternberg slides</a>. 628 </li> 629</ul></div> 630</div> 631<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 632<td align="left"></td> 633<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 634 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 635 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 636 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 637 Daryle Walker and Xiaogang Zhang<p> 638 Distributed under the Boost Software License, Version 1.0. (See accompanying 639 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>) 640 </p> 641</div></td> 642</tr></table> 643<hr> 644<div class="spirit-nav"> 645<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> 646</div> 647</body> 648</html> 649
