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WeibullDistribution Class

Weibull distribution.
Inheritance Hierarchy
SystemObject
  Accord.Statistics.DistributionsDistributionBase
    Accord.Statistics.Distributions.UnivariateUnivariateContinuousDistribution
      Accord.Statistics.Distributions.UnivariateWeibullDistribution

Namespace:  Accord.Statistics.Distributions.Univariate
Assembly:  Accord.Statistics (in Accord.Statistics.dll) Version: 3.5.0
Syntax
[SerializableAttribute]
public class WeibullDistribution : UnivariateContinuousDistribution, 
	ISampleableDistribution<double>, IDistribution<double>, IDistribution, 
	ICloneable, IRandomNumberGenerator<double>, IFormattable
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The WeibullDistribution type exposes the following members.

Constructors
  NameDescription
Public methodWeibullDistribution
Initializes a new instance of the WeibullDistribution class.
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Properties
Methods
  NameDescription
Public methodClone
Creates a new object that is a copy of the current instance.
(Overrides DistributionBaseClone.)
Public methodComplementaryDistributionFunction
Gets the complementary cumulative distribution function (ccdf) for this distribution evaluated at point x. This function is also known as the Survival function.
(Overrides UnivariateContinuousDistributionComplementaryDistributionFunction(Double).)
Public methodCumulativeHazardFunction
Gets the cumulative hazard function for this distribution evaluated at point x.
(Overrides UnivariateContinuousDistributionCumulativeHazardFunction(Double).)
Public methodDistributionFunction(Double)
Gets the cumulative distribution function (cdf) for this distribution evaluated at point x.
(Overrides UnivariateContinuousDistributionDistributionFunction(Double).)
Public methodDistributionFunction(Double, Double)
Gets the cumulative distribution function (cdf) for this distribution in the semi-closed interval (a; b] given as P(a < X ≤ b).
(Inherited from UnivariateContinuousDistribution.)
Public methodEquals
Determines whether the specified object is equal to the current object.
(Inherited from Object.)
Protected methodFinalize
Allows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection.
(Inherited from Object.)
Public methodFit(Double)
Fits the underlying distribution to a given set of observations.
(Inherited from UnivariateContinuousDistribution.)
Public methodFit(Double, IFittingOptions)
Fits the underlying distribution to a given set of observations.
(Inherited from UnivariateContinuousDistribution.)
Public methodFit(Double, Double)
Fits the underlying distribution to a given set of observations.
(Inherited from UnivariateContinuousDistribution.)
Public methodFit(Double, Int32)
Fits the underlying distribution to a given set of observations.
(Inherited from UnivariateContinuousDistribution.)
Public methodFit(Double, Int32, IFittingOptions)
Fits the underlying distribution to a given set of observations.
(Inherited from UnivariateContinuousDistribution.)
Public methodFit(Double, Double, IFittingOptions)
Fits the underlying distribution to a given set of observations.
(Overrides UnivariateContinuousDistributionFit(Double, Double, IFittingOptions).)
Public methodGenerate
Generates a random observation from the current distribution.
(Overrides UnivariateContinuousDistributionGenerate.)
Public methodGenerate(Int32)
Generates a random vector of observations from the current distribution.
(Inherited from UnivariateContinuousDistribution.)
Public methodGenerate(Int32, Double)
Generates a random vector of observations from the current distribution.
(Overrides UnivariateContinuousDistributionGenerate(Int32, Double).)
Public methodGetHashCode
Serves as the default hash function.
(Inherited from Object.)
Public methodGetRange
Gets the distribution range within a given percentile.
(Inherited from UnivariateContinuousDistribution.)
Public methodGetType
Gets the Type of the current instance.
(Inherited from Object.)
Public methodHazardFunction
Gets the hazard function, also known as the failure rate or the conditional failure density function for this distribution evaluated at point x.
(Overrides UnivariateContinuousDistributionHazardFunction(Double).)
Public methodInverseComplementaryDistributionFunction
Gets the inverse of the ComplementaryDistributionFunction(Double). The inverse complementary distribution function is also known as the inverse survival Function.
Public methodInverseDistributionFunction
Gets the inverse of the cumulative distribution function (icdf) for this distribution evaluated at probability p. This function is also known as the Quantile function.
(Inherited from UnivariateContinuousDistribution.)
Public methodLogCumulativeHazardFunction
Gets the log of the cumulative hazard function for this distribution evaluated at point x.
(Inherited from UnivariateContinuousDistribution.)
Public methodLogProbabilityDensityFunction
Gets the log-probability density function (pdf) for this distribution evaluated at point x.
(Overrides UnivariateContinuousDistributionLogProbabilityDensityFunction(Double).)
Protected methodMemberwiseClone
Creates a shallow copy of the current Object.
(Inherited from Object.)
Public methodProbabilityDensityFunction
Gets the probability density function (pdf) for this distribution evaluated at point x.
(Overrides UnivariateContinuousDistributionProbabilityDensityFunction(Double).)
Public methodQuantileDensityFunction
Gets the first derivative of the inverse distribution function (icdf) for this distribution evaluated at probability p.
(Inherited from UnivariateContinuousDistribution.)
Public methodStatic memberRandom(Double, Double)
Generates a random observation from the Weibull distribution with the given parameters.
Public methodStatic memberRandom(Double, Double, Int32)
Generates a random vector of observations from the Weibull distribution with the given parameters.
Public methodStatic memberRandom(Double, Double, Int32, Double)
Generates a random vector of observations from the Weibull distribution with the given parameters.
Public methodToString
Returns a String that represents this instance.
(Inherited from DistributionBase.)
Public methodToString(IFormatProvider)
Returns a String that represents this instance.
(Inherited from DistributionBase.)
Public methodToString(String)
Returns a String that represents this instance.
(Inherited from DistributionBase.)
Public methodToString(String, IFormatProvider)
Returns a String that represents this instance.
(Overrides DistributionBaseToString(String, IFormatProvider).)
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Extension Methods
  NameDescription
Public Extension MethodHasMethod
Checks whether an object implements a method with the given name.
(Defined by ExtensionMethods.)
Public Extension MethodIsEqual
Compares two objects for equality, performing an elementwise comparison if the elements are vectors or matrices.
(Defined by Matrix.)
Public Extension MethodToTOverloaded.
Converts an object into another type, irrespective of whether the conversion can be done at compile time or not. This can be used to convert generic types to numeric types during runtime.
(Defined by ExtensionMethods.)
Public Extension MethodToTOverloaded.
Converts an object into another type, irrespective of whether the conversion can be done at compile time or not. This can be used to convert generic types to numeric types during runtime.
(Defined by Matrix.)
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Remarks

In probability theory and statistics, the Weibull distribution is a continuous probability distribution. It is named after Waloddi Weibull, who described it in detail in 1951, although it was first identified by Fréchet (1927) and first applied by Rosin and Rammler (1933) to describe a particle size distribution.

The Weibull distribution is related to a number of other probability distributions; in particular, it interpolates between the exponential distribution (for k = 1) and the Rayleigh distribution (when k = 2).

If the quantity x is a "time-to-failure", the Weibull distribution gives a distribution for which the failure rate is proportional to a power of time. The shape parameter, k, is that power plus one, and so this parameter can be interpreted directly as follows:

  • A value of k < 1 indicates that the failure rate decreases over time. This happens if there is significant "infant mortality", or defective items failing early and the failure rate decreasing over time as the defective items are weeded out of the population.
  • A value of k = 1 indicates that the failure rate is constant over time. This might suggest random external events are causing mortality, or failure.
  • A value of k > 1 indicates that the failure rate increases with time. This happens if there is an "aging" process, or parts that are more likely to fail as time goes on.

In the field of materials science, the shape parameter k of a distribution of strengths is known as the Weibull modulus.

References:

Examples
// Create a new Weibull distribution with λ = 0.42 and k = 1.2
var weilbull = new WeibullDistribution(scale: 0.42, shape: 1.2);

// Common measures
double mean = weilbull.Mean;     // 0.39507546046784414
double median = weilbull.Median; // 0.30945951550913292
double var = weilbull.Variance;  // 0.10932249666369542
double mode = weilbull.Mode;     // 0.094360430821809421

// Cumulative distribution functions
double cdf = weilbull.DistributionFunction(x: 1.4);           //  0.98560487188700052
double pdf = weilbull.ProbabilityDensityFunction(x: 1.4);     //  0.052326687031379278
double lpdf = weilbull.LogProbabilityDensityFunction(x: 1.4); // -2.9502487697674415

// Probability density functions
double ccdf = weilbull.ComplementaryDistributionFunction(x: 1.4); // 0.22369885565908001
double icdf = weilbull.InverseDistributionFunction(p: cdf);       // 1.400000001051205

// Hazard (failure rate) functions
double hf = weilbull.HazardFunction(x: 1.4);            // 1.1093328057258516
double chf = weilbull.CumulativeHazardFunction(x: 1.4); // 1.4974545260150962

// String representation
string str = weilbull.ToString(CultureInfo.InvariantCulture); // Weibull(x; λ = 0.42, k = 1.2)
See Also