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133 lines
5.2 KiB
Plaintext
133 lines
5.2 KiB
Plaintext
[section:weibull_dist Weibull Distribution]
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``#include <boost/math/distributions/weibull.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 weibull_distribution;
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typedef weibull_distribution<> weibull;
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template <class RealType, class ``__Policy``>
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class weibull_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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// Construct:
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weibull_distribution(RealType shape, RealType scale = 1)
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// Accessors:
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RealType shape()const;
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RealType scale()const;
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};
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}} // namespaces
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The [@http://en.wikipedia.org/wiki/Weibull_distribution Weibull distribution]
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is a continuous distribution
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with the
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[@http://en.wikipedia.org/wiki/Probability_density_function probability density function]:
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f(x; [alpha], [beta]) = ([alpha]\/[beta]) * (x \/ [beta])[super [alpha] - 1] * e[super -(x\/[beta])[super [alpha]]]
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For shape parameter [alpha][space] > 0, and scale parameter [beta][space] > 0, and x > 0.
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The Weibull distribution is often used in the field of failure analysis;
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in particular it can mimic distributions where the failure rate varies over time.
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If the failure rate is:
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* constant over time, then [alpha][space] = 1, suggests that items are failing from random events.
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* decreases over time, then [alpha][space] < 1, suggesting "infant mortality".
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* increases over time, then [alpha][space] > 1, suggesting "wear out" - more likely to fail as time goes by.
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The following graph illustrates how the PDF varies with the shape parameter [alpha]:
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[graph weibull_pdf1]
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While this graph illustrates how the PDF varies with the scale parameter [beta]:
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[graph weibull_pdf2]
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[h4 Related distributions]
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When [alpha][space] = 3, the
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[@http://en.wikipedia.org/wiki/Weibull_distribution Weibull distribution] appears similar to the
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[@http://en.wikipedia.org/wiki/Normal_distribution normal distribution].
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When [alpha][space] = 1, the Weibull distribution reduces to the
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[@http://en.wikipedia.org/wiki/Exponential_distribution exponential distribution].
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The relationship of the types of extreme value distributions, of which the Weibull is but one, is
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discussed by
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[@http://www.worldscibooks.com/mathematics/p191.html Extreme Value Distributions, Theory and Applications
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Samuel Kotz & Saralees Nadarajah].
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[h4 Member Functions]
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weibull_distribution(RealType shape, RealType scale = 1);
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Constructs a [@http://en.wikipedia.org/wiki/Weibull_distribution
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Weibull distribution] with shape /shape/ and scale /scale/.
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Requires that the /shape/ and /scale/ parameters are both greater than zero,
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otherwise calls __domain_error.
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RealType shape()const;
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Returns the /shape/ parameter of this distribution.
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RealType scale()const;
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Returns the /scale/ parameter of this distribution.
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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 \[0, [infin]\].
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[h4 Accuracy]
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The Weibull distribution is implemented in terms of the
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standard library `log` and `exp` functions plus __expm1 and __log1p
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and as such should have very low error rates.
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[h4 Implementation]
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In the following table [alpha][space] is the shape parameter of the distribution,
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[beta][space] is its scale parameter, /x/ is the random variate, /p/ is the probability
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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 = [alpha][beta][super -[alpha] ]x[super [alpha] - 1] e[super -(x/beta)[super alpha]] ]]
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[[cdf][Using the relation: p = -__expm1(-(x\/[beta])[super [alpha]]) ]]
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[[cdf complement][Using the relation: q = e[super -(x\/[beta])[super [alpha]]] ]]
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[[quantile][Using the relation: x = [beta] * (-__log1p(-p))[super 1\/[alpha]] ]]
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[[quantile from the complement][Using the relation: x = [beta] * (-log(q))[super 1\/[alpha]] ]]
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[[mean][[beta] * [Gamma](1 + 1\/[alpha]) ]]
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[[variance][[beta][super 2]([Gamma](1 + 2\/[alpha]) - [Gamma][super 2](1 + 1\/[alpha])) ]]
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[[mode][[beta](([alpha] - 1) \/ [alpha])[super 1\/[alpha]] ]]
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[[skewness][Refer to [@http://mathworld.wolfram.com/WeibullDistribution.html Weisstein, Eric W. "Weibull Distribution." From MathWorld--A Wolfram Web Resource.] ]]
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[[kurtosis][Refer to [@http://mathworld.wolfram.com/WeibullDistribution.html Weisstein, Eric W. "Weibull Distribution." From MathWorld--A Wolfram Web Resource.] ]]
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[[kurtosis excess][Refer to [@http://mathworld.wolfram.com/WeibullDistribution.html Weisstein, Eric W. "Weibull Distribution." From MathWorld--A Wolfram Web Resource.] ]]
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]
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[h4 References]
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* [@http://en.wikipedia.org/wiki/Weibull_distribution ]
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* [@http://mathworld.wolfram.com/WeibullDistribution.html Weisstein, Eric W. "Weibull Distribution." From MathWorld--A Wolfram Web Resource.]
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* [@http://www.itl.nist.gov/div898/handbook/eda/section3/eda3668.htm Weibull in NIST Exploratory Data Analysis]
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[endsect][/section:weibull Weibull]
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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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