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180 lines
7.4 KiB
Plaintext
180 lines
7.4 KiB
Plaintext
[section:triangular_dist Triangular Distribution]
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``#include <boost/math/distributions/triangular.hpp>``
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namespace boost{ namespace math{
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template <class RealType = double,
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class ``__Policy`` = ``__policy_class`` >
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class triangular_distribution;
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typedef triangular_distribution<> triangular;
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template <class RealType, class ``__Policy``>
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class triangular_distribution
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{
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public:
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typedef RealType value_type;
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typedef Policy policy_type;
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triangular_distribution(RealType lower = -1, RealType mode = 0) RealType upper = 1); // Constructor.
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: m_lower(lower), m_mode(mode), m_upper(upper) // Default is -1, 0, +1 symmetric triangular distribution.
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// Accessor functions.
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RealType lower()const;
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RealType mode()const;
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RealType upper()const;
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}; // class triangular_distribution
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}} // namespaces
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The [@http://en.wikipedia.org/wiki/Triangular_distribution triangular distribution]
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is a [@http://en.wikipedia.org/wiki/Continuous_distribution continuous]
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[@http://en.wikipedia.org/wiki/Probability_distribution probability distribution]
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with a lower limit a,
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[@http://en.wikipedia.org/wiki/Mode_%28statistics%29 mode c],
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and upper limit b.
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The triangular distribution is often used where the distribution is only vaguely known,
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but, like the [@http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29 uniform distribution],
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upper and limits are 'known', but a 'best guess', the mode or center point, is also added.
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It has been recommended as a
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[@http://www.worldscibooks.com/mathematics/etextbook/5720/5720_chap1.pdf proxy for the beta distribution.]
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The distribution is used in business decision making and project planning.
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The [@http://en.wikipedia.org/wiki/Triangular_distribution triangular distribution]
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is a distribution with the
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[@http://en.wikipedia.org/wiki/Probability_density_function probability density function]:
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__spaces f(x) =
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* 2(x-a)/(b-a) (c-a) for a <= x <= c
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* 2(b-x)/(b-a)(b-c) for c < x <= b
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Parameter ['a] (lower) can be any finite value.
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Parameter ['b] (upper) can be any finite value > a (lower).
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Parameter ['c] (mode) a <= c <= b. This is the most probable value.
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The [@http://en.wikipedia.org/wiki/Random_variate random variate] x must also be finite, and is supported lower <= x <= upper.
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The triangular distribution may be appropriate when an assumption of a normal distribution
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is unjustified because uncertainty is caused by rounding and quantization from analog to digital conversion.
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Upper and lower limits are known, and the most probable value lies midway.
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The distribution simplifies when the 'best guess' is either the lower or upper limit - a 90 degree angle triangle.
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The 001 triangular distribution which expresses an estimate that the lowest value is the most likely;
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for example, you believe that the next-day quoted delivery date is most likely
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(knowing that a quicker delivery is impossible - the postman only comes once a day),
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and that longer delays are decreasingly likely,
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and delivery is assumed to never take more than your upper limit.
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The following graph illustrates how the
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[@http://en.wikipedia.org/wiki/Probability_density_function probability density function PDF]
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varies with the various parameters:
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[graph triangular_pdf]
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and cumulative distribution function
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[graph triangular_cdf]
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[h4 Member Functions]
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triangular_distribution(RealType lower = 0, RealType mode = 0 RealType upper = 1);
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Constructs a [@http://en.wikipedia.org/wiki/triangular_distribution triangular distribution]
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with lower /lower/ (a) and upper /upper/ (b).
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Requires that the /lower/, /mode/ and /upper/ parameters are all finite,
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otherwise calls __domain_error.
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[warning These constructors are slightly different from the analogs provided by __Mathworld
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[@http://reference.wolfram.com/language/ref/TriangularDistribution.html Triangular distribution],
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where
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[^TriangularDistribution\[{min, max}\]] represents a [*symmetric] triangular statistical distribution giving values between min and max.[br]
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[^TriangularDistribution\[\]] represents a [*symmetric] triangular statistical distribution giving values between 0 and 1.[br]
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[^TriangularDistribution\[{min, max}, c\]] represents a triangular distribution with mode at c (usually [*asymmetric]).[br]
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So, for example, to compute a variance using __WolframAlpha, use
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[^N\[variance\[TriangularDistribution{1, +2}\], 50\]]
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]
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The parameters of a distribution can be obtained using these member functions:
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RealType lower()const;
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Returns the ['lower] parameter of this distribution (default -1).
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RealType mode()const;
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Returns the ['mode] parameter of this distribution (default 0).
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RealType upper()const;
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Returns the ['upper] parameter of this distribution (default+1).
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[h4 Non-member Accessors]
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All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions] that are generic to all
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distributions are supported: __usual_accessors.
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The domain of the random variable is \lower\ to \upper\,
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and the supported range is lower <= x <= upper.
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[h4 Accuracy]
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The triangular distribution is implemented with simple arithmetic operators and so should have errors within an epsilon or two,
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except quantiles with arguments nearing the extremes of zero and unity.
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[h4 Implementation]
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In the following table, a is the /lower/ parameter of the distribution,
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c is the /mode/ parameter,
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b is the /upper/ parameter,
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/x/ is the random variate, /p/ is the probability and /q = 1-p/.
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[table
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[[Function][Implementation Notes]]
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[[pdf][Using the relation: pdf = 0 for x < mode, 2(x-a)\/(b-a)(c-a) else 2*(b-x)\/((b-a)(b-c))]]
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[[cdf][Using the relation: cdf = 0 for x < mode (x-a)[super 2]\/((b-a)(c-a)) else 1 - (b-x)[super 2]\/((b-a)(b-c))]]
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[[cdf complement][Using the relation: q = 1 - p ]]
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[[quantile][let p0 = (c-a)\/(b-a) the point of inflection on the cdf,
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then given probability p and q = 1-p:
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x = sqrt((b-a)(c-a)p) + a ; for p < p0
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x = c ; for p == p0
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x = b - sqrt((b-a)(b-c)q) ; for p > p0
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(See [@../../../../boost/math/distributions/triangular.hpp /boost/math/distributions/triangular.hpp] for details.)]]
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[[quantile from the complement][As quantile (See [@../../../../boost/math/distributions/triangular.hpp /boost/math/distributions/triangular.hpp] for details.)]]
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[[mean][(a + b + 3) \/ 3 ]]
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[[variance][(a[super 2]+b[super 2]+c[super 2] - ab - ac - bc)\/18]]
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[[mode][c]]
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[[skewness][(See [@../../../../boost/math/distributions/triangular.hpp /boost/math/distributions/triangular.hpp] for details). ]]
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[[kurtosis][12\/5]]
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[[kurtosis excess][-3\/5]]
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]
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Some 'known good' test values were obtained using __WolframAlpha.
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[h4 References]
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* [@http://en.wikipedia.org/wiki/Triangular_distribution Wikpedia triangular distribution]
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* [@http://mathworld.wolfram.com/TriangularDistribution.html Weisstein, Eric W. "Triangular Distribution." From MathWorld--A Wolfram Web Resource.]
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* Evans, M.; Hastings, N.; and Peacock, B. "Triangular Distribution." Ch. 40 in Statistical Distributions, 3rd ed. New York: Wiley, pp. 187-188, 2000, ISBN - 0471371246.
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* [@http://www.measurement.sk/2002/S1/Wimmer2.pdf Gejza Wimmer, Viktor Witkovsky and Tomas Duby,
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Measurement Science Review, Volume 2, Section 1, 2002, Proper Rounding Of The Measurement Results Under The Assumption Of Triangular Distribution.]
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[endsect][/section:triangular_dist triangular]
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[/
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Copyright 2006 John Maddock and Paul A. Bristow.
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Distributed under the Boost Software License, Version 1.0.
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(See accompanying file LICENSE_1_0.txt or copy at
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http://www.boost.org/LICENSE_1_0.txt).
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]
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