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