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