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diff --git a/libs/math/doc/html/math_toolkit/dist_ref/dists/arcine_dist.html b/libs/math/doc/html/math_toolkit/dist_ref/dists/arcine_dist.html new file mode 100644 index 000000000..21f27354d --- /dev/null +++ b/libs/math/doc/html/math_toolkit/dist_ref/dists/arcine_dist.html @@ -0,0 +1,642 @@ +<html> +<head> +<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> +<title>Arcsine Distribution</title> +<link rel="stylesheet" href="../../../math.css" type="text/css"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../../index.html" title="Math Toolkit 2.2.0"> +<link rel="up" href="../dists.html" title="Distributions"> +<link rel="prev" href="../dists.html" title="Distributions"> +<link rel="next" href="bernoulli_dist.html" title="Bernoulli Distribution"> +</head> +<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> +<table cellpadding="2" width="100%"><tr> +<td valign="top"><img alt="Boost C++ 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class="section"> +<div class="titlepage"><div><div><h4 class="title"> +<a name="math_toolkit.dist_ref.dists.arcine_dist"></a><a class="link" href="arcine_dist.html" title="Arcsine Distribution">Arcsine Distribution</a> +</h4></div></div></div> +<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">arcsine</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span></pre> +<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span> + + <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span> + <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a> <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy<></a> <span class="special">></span> +<span class="keyword">class</span> <span class="identifier">arcsine_distribution</span><span class="special">;</span> + +<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> + +<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> +<span class="keyword">class</span> <span class="identifier">arcsine_distribution</span> +<span class="special">{</span> +<span class="keyword">public</span><span class="special">:</span> + <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span> + <span class="keyword">typedef</span> <span class="identifier">Policy</span> <span class="identifier">policy_type</span><span class="special">;</span> + + <span class="comment">// Constructor from two range parameters, x_min and x_max:</span> + <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> + + <span class="comment">// Range Parameter accessors:</span> + <span class="identifier">RealType</span> <span class="identifier">x_min</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span> + <span class="identifier">RealType</span> <span class="identifier">x_max</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span> +<span class="special">};</span> +<span class="special">}}</span> <span class="comment">// namespaces</span> +</pre> +<p> + The class type <code class="computeroutput"><span class="identifier">arcsine_distribution</span></code> + represents an <a href="http://en.wikipedia.org/wiki/arcsine_distribution" target="_top">arcsine</a> + <a href="http://en.wikipedia.org/wiki/Probability_distribution" target="_top">probability + distribution function</a>. The arcsine distribution is named because + its CDF uses the inverse sin<sup>-1</sup> or arcsine. + </p> +<p> + This is implemented as a generalized version with support from <span class="emphasis"><em>x_min</em></span> + to <span class="emphasis"><em>x_max</em></span> providing the 'standard arcsine distribution' + as default with <span class="emphasis"><em>x_min = 0</em></span> and <span class="emphasis"><em>x_max = 1</em></span>. + (A few make other choices for 'standard'). + </p> +<p> + The arcsine distribution is generalized to include any bounded support + <span class="emphasis"><em>a <= x <= b</em></span> by <a href="http://reference.wolfram.com/language/ref/ArcSinDistribution.html" target="_top">Wolfram</a> + and <a href="http://en.wikipedia.org/wiki/arcsine_distribution" target="_top">Wikipedia</a>, + but also using <span class="emphasis"><em>location</em></span> and <span class="emphasis"><em>scale</em></span> + parameters by <a href="http://www.math.uah.edu/stat/index.html" target="_top">Virtual + Laboratories in Probability and Statistics</a> <a href="http://www.math.uah.edu/stat/special/Arcsine.html" target="_top">Arcsine + distribution</a>. The end-point version is simpler and more obvious, + so we implement that. If desired, <a href="http://en.wikipedia.org/wiki/arcsine_distribution" target="_top">this</a> + outlines how the <a class="link" href="beta_dist.html" title="Beta Distribution">Beta + Distribution</a> can be used to add a shape factor. + </p> +<p> + The <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">probability + density function PDF</a> for the <a href="http://en.wikipedia.org/wiki/arcsine_distribution" target="_top">arcsine + distribution</a> defined on the interval [<span class="emphasis"><em>x_min, x_max</em></span>] + is given by: + </p> +<p> +     f(x; x_min, x_max) = 1 /(π⋅√((x - x_min)⋅(x_max - x)) + </p> +<p> + For example, <a href="http://www.wolframalpha.com/" target="_top">Wolfram Alpha</a> + arcsine distribution, from input of + </p> +<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> +</pre> +<p> + computes the PDF value + </p> +<pre class="programlisting"><span class="number">0.63661977236758134307553505349005744813783858296183</span> +</pre> +<p> + The Probability Density Functions (PDF) of generalized arcsine distributions + are symmetric U-shaped curves, centered on <span class="emphasis"><em>(x_max - x_min)/2</em></span>, + highest (infinite) near the two extrema, and quite flat over the central + region. + </p> +<p> + If random variate <span class="emphasis"><em>x</em></span> is <span class="emphasis"><em>x_min</em></span> + or <span class="emphasis"><em>x_max</em></span>, then the PDF is infinity. If random variate + <span class="emphasis"><em>x</em></span> is <span class="emphasis"><em>x_min</em></span> then the CDF is zero. + If random variate <span class="emphasis"><em>x</em></span> is <span class="emphasis"><em>x_max</em></span> + then the CDF is unity. + </p> +<p> + The 'Standard' (0, 1) arcsine distribution is shown in blue and some generalized + examples with other <span class="emphasis"><em>x</em></span> ranges. + </p> +<p> + <span class="inlinemediaobject"><img src="../../../../graphs/arcsine_pdf.svg" align="middle"></span> + </p> +<p> + The Cumulative Distribution Function CDF is defined as + </p> +<p> +     F(x) = 2⋅arcsin(√((x-x_min)/(x_max - x))) / π + </p> +<p> + <span class="inlinemediaobject"><img src="../../../../graphs/arcsine_cdf.svg" align="middle"></span> + </p> +<h6> +<a name="math_toolkit.dist_ref.dists.arcine_dist.h0"></a> + <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> + </h6> +<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> +</pre> +<p> + constructs an arcsine distribution with range parameters <span class="emphasis"><em>x_min</em></span> + and <span class="emphasis"><em>x_max</em></span>. + </p> +<p> + 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> + is called. + </p> +<p> + For example: + </p> +<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> +</pre> +<p> + constructs an arcsine distribution with <span class="emphasis"><em>x_min = -2</em></span> + and <span class="emphasis"><em>x_max = 4</em></span>. + </p> +<p> + Default values of <span class="emphasis"><em>x_min = 0</em></span> and <span class="emphasis"><em>x_max = + 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> + mean that + </p> +<pre class="programlisting"><span class="identifier">arcsine</span> <span class="identifier">as</span><span class="special">;</span> +</pre> +<p> + constructs a 'Standard 01' arcsine distribution. + </p> +<h6> +<a name="math_toolkit.dist_ref.dists.arcine_dist.h1"></a> + <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 + Accessors</a> + </h6> +<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> +<span class="identifier">RealType</span> <span class="identifier">x_max</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span> +</pre> +<p> + Return the parameter <span class="emphasis"><em>x_min</em></span> or <span class="emphasis"><em>x_max</em></span> + from which this distribution was constructed. + </p> +<p> + So, for example: + </p> +<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> + +<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> +<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> +<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> +</pre> +<h5> +<a name="math_toolkit.dist_ref.dists.arcine_dist.h2"></a> + <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 + Accessor Functions</a> + </h5> +<p> + All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor + functions</a> that are generic to all distributions are supported: + <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>, + <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>, + <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>, + <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.median">median</a>, + <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>, + <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>, + <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>, + <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>. + </p> +<p> + The formulae for calculating these are shown in the table below, and at + <a href="http://mathworld.wolfram.com/arcsineDistribution.html" target="_top">Wolfram + Mathworld</a>. + </p> +<div class="note"><table border="0" summary="Note"> +<tr> +<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td> +<th align="left">Note</th> +</tr> +<tr><td align="left" valign="top"><p> + There are always <span class="bold"><strong>two</strong></span> values for the + <span class="bold"><strong>mode</strong></span>, at <span class="emphasis"><em>x_min</em></span> + and at <span class="emphasis"><em>x_max</em></span>, default 0 and 1, so instead we raise + the exception <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>. + At these extrema, the PDFs are infinite, and the CDFs zero or unity. + </p></td></tr> +</table></div> +<h5> +<a name="math_toolkit.dist_ref.dists.arcine_dist.h3"></a> + <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> + </h5> +<p> + The arcsine distribution is useful to describe <a href="http://en.wikipedia.org/wiki/Random_walk" target="_top">Random + walks</a>, (including drunken walks) <a href="http://en.wikipedia.org/wiki/Brownian_motion" target="_top">Brownian + motion</a>, <a href="http://en.wikipedia.org/wiki/Wiener_process" target="_top">Weiner + processes</a>, <a href="http://en.wikipedia.org/wiki/Bernoulli_trial" target="_top">Bernoulli + trials</a>, and their appplication to solve stock market and other + <a href="http://en.wikipedia.org/wiki/Gambler%27s_ruin" target="_top">ruinous gambling + games</a>. + </p> +<p> + The random variate <span class="emphasis"><em>x</em></span> is constrained to <span class="emphasis"><em>x_min</em></span> + and <span class="emphasis"><em>x_max</em></span>, (for our 'standard' distribution, 0 and + 1), and is usually some fraction. For any other <span class="emphasis"><em>x_min</em></span> + and <span class="emphasis"><em>x_max</em></span> a fraction can be obtained from <span class="emphasis"><em>x</em></span> + using + </p> +<p> +   fraction = (x - x_min) / (x_max - x_min) + </p> +<p> + The simplest example is tossing heads and tails with a fair coin and modelling + the risk of losing, or winning. Walkers (molecules, drunks...) moving left + or right of a centre line are another common example. + </p> +<p> + The random variate <span class="emphasis"><em>x</em></span> is the fraction of time spent + on the 'winning' side. If half the time is spent on the 'winning' side + (and so the other half on the 'losing' side) then <span class="emphasis"><em>x = 1/2</em></span>. + </p> +<p> + For large numbers of tosses, this is modelled by the (standard [0,1]) arcsine + distribution, and the PDF can be calculated thus: + </p> +<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> +<span class="comment">// pdf has a minimum at x = 0.5</span> +</pre> +<p> + From the plot of PDF, it is clear that <span class="emphasis"><em>x</em></span> = ½ is the + <span class="bold"><strong>minimum</strong></span> of the curve, so this is the + <span class="bold"><strong>least likely</strong></span> scenario. (This is highly + counter-intuitive, considering that fair tosses must <span class="bold"><strong>eventually</strong></span> + become equal. It turns out that <span class="emphasis"><em>eventually</em></span> is not + just very long, but <span class="bold"><strong>infinite</strong></span>!). + </p> +<p> + The <span class="bold"><strong>most likely</strong></span> scenarios are towards + the extrema where <span class="emphasis"><em>x</em></span> = 0 or <span class="emphasis"><em>x</em></span> + = 1. + </p> +<p> + If fraction of time on the left is a ¼, it is only slightly more likely + because the curve is quite flat bottomed. + </p> +<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> +</pre> +<p> + If we consider fair coin-tossing games being played for 100 days (hypothetically + continuously to be 'at-limit') the person winning after day 5 will not + change in fraction 0.144 of the cases. + </p> +<p> + We can easily compute this setting <span class="emphasis"><em>x</em></span> = 5./100 = 0.05 + </p> +<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> +</pre> +<p> + Similarly, we can compute from a fraction of 0.05 /2 = 0.025 (halved because + we are considering both winners and losers) corresponding to 1 - 0.025 + or 97.5% of the gamblers, (walkers, particles...) on the <span class="bold"><strong>same + side</strong></span> of the origin + </p> +<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> +</pre> +<p> + (use of the complement gives a bit more clarity, and avoids potential loss + 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 + complements?</a>). + </p> +<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> +</pre> +<p> + or we can reverse the calculation by assuming a fraction of time on one + side, say fraction 0.2, + </p> +<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> + +<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> +</pre> +<p> + <span class="bold"><strong>Summary</strong></span>: Every time we toss, the odds + are equal, so on average we have the same change of winning and losing. + </p> +<p> + But this is <span class="bold"><strong>not true</strong></span> for an an individual + game where one will be <span class="bold"><strong>mostly in a bad or good patch</strong></span>. + </p> +<p> + This is quite counter-intuitive to most people, but the mathematics is + clear, and gamblers continue to provide proof. + </p> +<p> + <span class="bold"><strong>Moral</strong></span>: if you in a losing patch, leave + the game. (Because the odds to recover to a good patch are poor). + </p> +<p> + <span class="bold"><strong>Corollary</strong></span>: Quit while you are ahead? + </p> +<p> + A working example is at <a href="../../../../../example/arcsine_example.cpp" target="_top">arcsine_example.cpp</a> + including sample output . + </p> +<h5> +<a name="math_toolkit.dist_ref.dists.arcine_dist.h4"></a> + <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 + distributions</a> + </h5> +<p> + The arcsine distribution with <span class="emphasis"><em>x_min = 0</em></span> and <span class="emphasis"><em>x_max + = 1</em></span> is special case of the <a class="link" href="beta_dist.html" title="Beta Distribution">Beta + Distribution</a> with α = 1/2 and β = 1/2. + </p> +<h5> +<a name="math_toolkit.dist_ref.dists.arcine_dist.h5"></a> + <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> + </h5> +<p> + This distribution is implemented using sqrt, sine, cos and arc sine and + cos trigonometric functions which are normally accurate to a few <a href="http://en.wikipedia.org/wiki/Machine_epsilon" target="_top">machine epsilon</a>. + But all values suffer from <a href="http://en.wikipedia.org/wiki/Loss_of_significance" target="_top">loss + of significance or cancellation error</a> for values of <span class="emphasis"><em>x</em></span> + close to <span class="emphasis"><em>x_max</em></span>. For example, for a standard [0, 1] + arcsine distribution <span class="emphasis"><em>as</em></span>, the pdf is symmetric about + 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> + <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 + 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 + of significance</a>. To counteract this, the complement versions of + CDF and quantile are implemented with alternative expressions using <span class="emphasis"><em>cos<sup>-1</sup></em></span> + 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 + complements?</a> for guidance on when to avoid loss of accuracy by using + complements. + </p> +<h5> +<a name="math_toolkit.dist_ref.dists.arcine_dist.h6"></a> + <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> + </h5> +<p> + The results were tested against a few accurate spot values computed by + <a href="http://www.wolframalpha.com/" target="_top">Wolfram Alpha</a>, for example: + </p> +<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> + <span class="number">0.63661977236758134307553505349005744813783858296183</span> +</pre> +<h5> +<a name="math_toolkit.dist_ref.dists.arcine_dist.h7"></a> + <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> + </h5> +<p> + In the following table <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span> + are the parameters <span class="emphasis"><em>x_min</em></span>   and <span class="emphasis"><em>x_max</em></span>, + <span class="emphasis"><em>x</em></span> is the random variable, <span class="emphasis"><em>p</em></span> is + the probability and its complement <span class="emphasis"><em>q = 1-p</em></span>. + </p> +<div class="informaltable"><table class="table"> +<colgroup> +<col> +<col> +</colgroup> +<thead><tr> +<th> + <p> + Function + </p> + </th> +<th> + <p> + Implementation Notes + </p> + </th> +</tr></thead> +<tbody> +<tr> +<td> + <p> + support + </p> + </td> +<td> + <p> + x ∈ [a, b], default x ∈ [0, 1] + </p> + </td> +</tr> +<tr> +<td> + <p> + pdf + </p> + </td> +<td> + <p> + f(x; a, b) = 1/(π⋅√(x - a)⋅(b - x)) + </p> + </td> +</tr> +<tr> +<td> + <p> + cdf + </p> + </td> +<td> + <p> + F(x) = 2/π⋅sin<sup>-1</sup>(√(x - a) / (b - a) ) + </p> + </td> +</tr> +<tr> +<td> + <p> + cdf of complement + </p> + </td> +<td> + <p> + 2/(π⋅cos<sup>-1</sup>(√(x - a) / (b - a))) + </p> + </td> +</tr> +<tr> +<td> + <p> + quantile + </p> + </td> +<td> + <p> + -a⋅sin<sup>2</sup>(½π⋅p) + a + b⋅sin<sup>2</sup>(½π⋅p) + </p> + </td> +</tr> +<tr> +<td> + <p> + quantile from the complement + </p> + </td> +<td> + <p> + -a⋅cos<sup>2</sup>(½π⋅p) + a + b⋅cos<sup>2</sup>(½π⋅q) + </p> + </td> +</tr> +<tr> +<td> + <p> + mean + </p> + </td> +<td> + <p> + ½(a+b) + </p> + </td> +</tr> +<tr> +<td> + <p> + median + </p> + </td> +<td> + <p> + ½(a+b) + </p> + </td> +</tr> +<tr> +<td> + <p> + mode + </p> + </td> +<td> + <p> + x ∈ [a, b], so raises domain_error (returning NaN). + </p> + </td> +</tr> +<tr> +<td> + <p> + variance + </p> + </td> +<td> + <p> + (b - a)<sup>2</sup> / 8 + </p> + </td> +</tr> +<tr> +<td> + <p> + skewness + </p> + </td> +<td> + <p> + 0 + </p> + </td> +</tr> +<tr> +<td> + <p> + kurtosis excess + </p> + </td> +<td> + <p> + -3/2 + </p> + </td> +</tr> +<tr> +<td> + <p> + kurtosis + </p> + </td> +<td> + <p> + kurtosis_excess + 3 + </p> + </td> +</tr> +</tbody> +</table></div> +<p> + The quantile was calculated using an expression obtained by using <a href="http://www.wolframalpha.com/" target="_top">Wolfram Alpha</a> to invert the + formula for the CDF thus + </p> +<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> +</pre> +<p> + which was interpreted as + </p> +<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> +</pre> +<p> + and produced the resulting expression + </p> +<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> +</pre> +<p> + Thanks to Wolfram for providing this facility. + </p> +<h5> +<a name="math_toolkit.dist_ref.dists.arcine_dist.h8"></a> + <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> + </h5> +<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> +<li class="listitem"> + <a href="http://en.wikipedia.org/wiki/arcsine_distribution" target="_top">Wikipedia + arcsine distribution</a> + </li> +<li class="listitem"> + <a href="http://en.wikipedia.org/wiki/Beta_distribution" target="_top">Wikipedia + Beta distribution</a> + </li> +<li class="listitem"> + <a href="http://mathworld.wolfram.com/BetaDistribution.html" target="_top">Wolfram + MathWorld</a> + </li> +<li class="listitem"> + <a href="http://www.wolframalpha.com/" target="_top">Wolfram Alpha</a> + </li> +</ul></div> +<h5> +<a name="math_toolkit.dist_ref.dists.arcine_dist.h9"></a> + <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> + </h5> +<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> +<li class="listitem"> + <a href="http://estebanmoro.org/2009/04/the-probability-of-going-through-a-bad-patch" target="_top">The + probability of going through a bad patch</a> Esteban Moro's Blog. + </li> +<li class="listitem"> + <a href="http://www.gotohaggstrom.com/What%20do%20schmucks%20and%20the%20arc%20sine%20law%20have%20in%20common.pdf" target="_top">What + soschumcks and the arc sine have in common</a> Peter Haggstrom. + </li> +<li class="listitem"> + <a href="http://www.math.uah.edu/stat/special/Arcsine.html" target="_top">arcsine + distribution</a>. + </li> +<li class="listitem"> + <a href="http://reference.wolfram.com/language/ref/ArcSinDistribution.html" target="_top">Wolfram + reference arcsine examples</a>. + </li> +<li class="listitem"> + <a href="http://www.math.harvard.edu/library/sternberg/slides/1180908.pdf" target="_top">Shlomo + Sternberg slides</a>. + </li> +</ul></div> +</div> +<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> +<td align="left"></td> +<td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, + Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert + Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin Sobotta, + Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> + Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>) + </p> +</div></td> +</tr></table> +<hr> +<div class="spirit-nav"> +<a accesskey="p" href="../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> +</div> +</body> +</html> |