[section:arcine_dist Arcsine Distribution] [import ../../example/arcsine_example.cpp] [/ for arcsine snips below] ``#include `` namespace boost{ namespace math{ template class arcsine_distribution; typedef arcsine_distribution arcsine; // double precision standard arcsine distribution [0,1]. template class arcsine_distribution { public: typedef RealType value_type; typedef Policy policy_type; // Constructor from two range parameters, x_min and x_max: arcsine_distribution(RealType x_min, RealType x_max); // Range Parameter accessors: RealType x_min() const; RealType x_max() const; }; }} // namespaces The class type `arcsine_distribution` represents an [@http://en.wikipedia.org/wiki/arcsine_distribution arcsine] [@http://en.wikipedia.org/wiki/Probability_distribution probability distribution function]. The arcsine distribution is named because its CDF uses the inverse sin[super -1] or arcsine. This is implemented as a generalized version with support from ['x_min] to ['x_max] providing the 'standard arcsine distribution' as default with ['x_min = 0] and ['x_max = 1]. (A few make other choices for 'standard'). The arcsine distribution is generalized to include any bounded support ['a <= x <= b] by [@http://reference.wolfram.com/language/ref/ArcSinDistribution.html Wolfram] and [@http://en.wikipedia.org/wiki/arcsine_distribution Wikipedia], but also using ['location] and ['scale] parameters by [@http://www.math.uah.edu/stat/index.html Virtual Laboratories in Probability and Statistics] [@http://www.math.uah.edu/stat/special/Arcsine.html Arcsine distribution]. The end-point version is simpler and more obvious, so we implement that. If desired, [@http://en.wikipedia.org/wiki/arcsine_distribution this] outlines how the __beta_distrib can be used to add a shape factor. The [@http://en.wikipedia.org/wiki/Probability_density_function probability density function PDF] for the [@http://en.wikipedia.org/wiki/arcsine_distribution arcsine distribution] defined on the interval \[['x_min, x_max]\] is given by: [expression f(x; x_min, x_max) = 1 /([pi][sdot][sqrt]((x - x_min)[sdot](x_max - x_min))] For example, __WolframAlpha arcsine distribution, from input of N[PDF[arcsinedistribution[0, 1], 0.5], 50] computes the PDF value 0.63661977236758134307553505349005744813783858296183 The Probability Density Functions (PDF) of generalized arcsine distributions are symmetric U-shaped curves, centered on ['(x_max - x_min)/2], highest (infinite) near the two extrema, and quite flat over the central region. If random variate ['x] is ['x_min] or ['x_max], then the PDF is infinity. If random variate ['x] is ['x_min] then the CDF is zero. If random variate ['x] is ['x_max] then the CDF is unity. The 'Standard' (0, 1) arcsine distribution is shown in blue and some generalized examples with other ['x] ranges. [graph arcsine_pdf] The Cumulative Distribution Function CDF is defined as [expression F(x) = 2[sdot]arcsin([sqrt]((x-x_min)/(x_max - x))) / [pi]] [graph arcsine_cdf] [h5 Constructor] arcsine_distribution(RealType x_min, RealType x_max); constructs an arcsine distribution with range parameters ['x_min] and ['x_max]. Requires ['x_min < x_max], otherwise __domain_error is called. For example: arcsine_distribution<> myarcsine(-2, 4); constructs an arcsine distribution with ['x_min = -2] and ['x_max = 4]. Default values of ['x_min = 0] and ['x_max = 1] and a ` typedef arcsine_distribution arcsine;` mean that arcsine as; constructs a 'Standard 01' arcsine distribution. [h5 Parameter Accessors] RealType x_min() const; RealType x_max() const; Return the parameter ['x_min] or ['x_max] from which this distribution was constructed. So, for example: [arcsine_snip_8] [h4 Non-member Accessor Functions] All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions] that are generic to all distributions are supported: __usual_accessors. The formulae for calculating these are shown in the table below, and at [@http://mathworld.wolfram.com/arcsineDistribution.html Wolfram Mathworld]. [note There are always [*two] values for the [*mode], at ['x_min] and at ['x_max], default 0 and 1, so instead we raise the exception __domain_error. At these extrema, the PDFs are infinite, and the CDFs zero or unity.] [h4 Applications] The arcsine distribution is useful to describe [@http://en.wikipedia.org/wiki/Random_walk Random walks], (including drunken walks) [@http://en.wikipedia.org/wiki/Brownian_motion Brownian motion], [@http://en.wikipedia.org/wiki/Wiener_process Weiner processes], [@http://en.wikipedia.org/wiki/Bernoulli_trial Bernoulli trials], and their application to solve stock market and other [@http://en.wikipedia.org/wiki/Gambler%27s_ruin ruinous gambling games]. The random variate ['x] is constrained to ['x_min] and ['x_max], (for our 'standard' distribution, 0 and 1), and is usually some fraction. For any other ['x_min] and ['x_max] a fraction can be obtained from ['x] using [expression fraction = (x - x_min) / (x_max - x_min)] 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. The random variate ['x] 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 ['x = 1/2]. For large numbers of tosses, this is modelled by the (standard \[0,1\]) arcsine distribution, and the PDF can be calculated thus: [arcsine_snip_2] From the plot of PDF, it is clear that ['x] = [frac12] is the [*minimum] of the curve, so this is the [*least likely] scenario. (This is highly counter-intuitive, considering that fair tosses must [*eventually] become equal. It turns out that ['eventually] is not just very long, but [*infinite]!). The [*most likely] scenarios are towards the extrema where ['x] = 0 or ['x] = 1. If fraction of time on the left is a [frac14], it is only slightly more likely because the curve is quite flat bottomed. [arcsine_snip_3] 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. We can easily compute this setting ['x] = 5./100 = 0.05 [arcsine_snip_4] 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 [*same side] of the origin [arcsine_snip_5] (use of the complement gives a bit more clarity, and avoids potential loss of accuracy when ['x] is close to unity, see __why_complements). [arcsine_snip_6] or we can reverse the calculation by assuming a fraction of time on one side, say fraction 0.2, [arcsine_snip_7] [*Summary]: Every time we toss, the odds are equal, so on average we have the same change of winning and losing. But this is [*not true] for an an individual game where one will be [*mostly in a bad or good patch]. This is quite counter-intuitive to most people, but the mathematics is clear, and gamblers continue to provide proof. [*Moral]: if you in a losing patch, leave the game. (Because the odds to recover to a good patch are poor). [*Corollary]: Quit while you are ahead? A working example is at [@../../example/arcsine_example.cpp arcsine_example.cpp] including sample output . [h4 Related distributions] The arcsine distribution with ['x_min = 0] and ['x_max = 1] is special case of the __beta_distrib with [alpha] = 1/2 and [beta] = 1/2. [h4 Accuracy] This distribution is implemented using sqrt, sine, cos and arc sine and cos trigonometric functions which are normally accurate to a few __epsilon. But all values suffer from [@http://en.wikipedia.org/wiki/Loss_of_significance loss of significance or cancellation error] for values of ['x] close to ['x_max]. For example, for a standard [0, 1] arcsine distribution ['as], the pdf is symmetric about random variate ['x = 0.5] so that one would expect `pdf(as, 0.01) == pdf(as, 0.99)`. But as ['x] nears unity, there is increasing [@http://en.wikipedia.org/wiki/Loss_of_significance loss of significance]. To counteract this, the complement versions of CDF and quantile are implemented with alternative expressions using ['cos[super -1]] instead of ['sin[super -1]]. Users should see __why_complements for guidance on when to avoid loss of accuracy by using complements. [h4 Testing] The results were tested against a few accurate spot values computed by __WolframAlpha, for example: N[PDF[arcsinedistribution[0, 1], 0.5], 50] 0.63661977236758134307553505349005744813783858296183 [h4 Implementation] In the following table ['a] and ['b] are the parameters ['x_min] and ['x_max], ['x] is the random variable, ['p] is the probability and its complement ['q = 1-p]. [table [[Function][Implementation Notes]] [[support] [x [isin] \[a, b\], default x [isin] \[0, 1\] ]] [[pdf] [f(x; a, b) = 1/([pi][sdot][sqrt](x - a)[sdot](b - x))]] [[cdf] [F(x) = 2/[pi][sdot]sin[super-1]([sqrt](x - a) / (b - a) ) ]] [[cdf of complement] [2/([pi][sdot]cos[super-1]([sqrt](x - a) / (b - a)))]] [[quantile] [-a[sdot]sin[super 2]([frac12][pi][sdot]p) + a + b[sdot]sin[super 2]([frac12][pi][sdot]p)]] [[quantile from the complement] [-a[sdot]cos[super 2]([frac12][pi][sdot]p) + a + b[sdot]cos[super 2]([frac12][pi][sdot]q)]] [[mean] [[frac12](a+b)]] [[median] [[frac12](a+b)]] [[mode] [ x [isin] \[a, b\], so raises domain_error (returning NaN).]] [[variance] [(b - a)[super 2] / 8]] [[skewness] [0]] [[kurtosis excess] [ -3/2 ]] [[kurtosis] [kurtosis_excess + 3]] ] The quantile was calculated using an expression obtained by using __WolframAlpha to invert the formula for the CDF thus solve [p - 2/pi sin^-1(sqrt((x-a)/(b-a))) = 0, x] which was interpreted as Solve[p - (2 ArcSin[Sqrt[(-a + x)/(-a + b)]])/Pi == 0, x, MaxExtraConditions -> Automatic] and produced the resulting expression [expression x = -a sin^2((pi p)/2)+a+b sin^2((pi p)/2)] Thanks to Wolfram for providing this facility. [h4 References] * [@http://en.wikipedia.org/wiki/arcsine_distribution Wikipedia arcsine distribution] * [@http://en.wikipedia.org/wiki/Beta_distribution Wikipedia Beta distribution] * [@http://mathworld.wolfram.com/BetaDistribution.html Wolfram MathWorld] * [@http://www.wolframalpha.com/ Wolfram Alpha] [h4 Sources] *[@http://estebanmoro.org/2009/04/the-probability-of-going-through-a-bad-patch The probability of going through a bad patch] Esteban Moro's Blog. *[@http://www.gotohaggstrom.com/What%20do%20schmucks%20and%20the%20arc%20sine%20law%20have%20in%20common.pdf What soschumcks and the arc sine have in common] Peter Haggstrom. *[@http://www.math.uah.edu/stat/special/Arcsine.html arcsine distribution]. *[@http://reference.wolfram.com/language/ref/ArcSinDistribution.html Wolfram reference arcsine examples]. *[@http://www.math.harvard.edu/library/sternberg/slides/1180908.pdf Shlomo Sternberg slides]. [endsect] [/section:arcsine_dist arcsine] [/ arcsine.qbk Copyright 2014 John Maddock and Paul A. Bristow. Distributed under the Boost Software License, Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt). ]