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| <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> |
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