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

[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).
]