WSJT-X/boost/libs/math/doc/distributions/arcsine.qbk

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[section:arcine_dist Arcsine Distribution]
[import ../../example/arcsine_example.cpp] [/ for arcsine snips below]
``#include <boost/math/distributions/arcsine.hpp>``
namespace boost{ namespace math{
template <class RealType = double,
class ``__Policy`` = ``__policy_class`` >
class arcsine_distribution;
typedef arcsine_distribution<double> arcsine; // double precision standard arcsine distribution [0,1].
template <class RealType, class ``__Policy``>
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:
[figspace] [figspace] 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
[figspace] [figspace] 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<double> 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 appplication 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
[sixemspace] 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][space] 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
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).
]