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| 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 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> | |
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