Package smile.stat.distribution

Class WeibullDistribution

java.lang.Object
smile.stat.distribution.WeibullDistribution
All Implemented Interfaces:
Serializable, Distribution
public class WeibullDistribution extends Object implements Distribution
The Weibull distribution is one of the most widely used lifetime distributions in reliability engineering. It is a versatile distribution that can take on the characteristics of other types of distributions, based on the value of the shape parameter. The distribution has two parameters: k > 0 is the shape parameter and λ > 0 is the scale parameter of the distribution. The probability density function is f(x;λ,k) = k/λ (x/λ)k-1e-(x/λ)^k for x >= 0.

The Weibull distribution is often used in the field of life data analysis due to its flexibility - it can mimic the behavior of other statistical distributions such as the normal and the exponential. If the failure rate decreases over time, then k < 1. If the failure rate is constant over time, then k = 1. If the failure rate increases over time, then k > 1.

An understanding of the failure rate may provide insight as to what is causing the failures:

  • A decreasing failure rate would suggest "infant mortality". That is, defective items fail early and the failure rate decreases over time as they fall out of the population.
  • A constant failure rate suggests that items are failing from random events.
  • An increasing failure rate suggests "wear out" - parts are more likely to fail as time goes on.
Under certain parameterizations, the Weibull distribution reduces to several other familiar distributions:
  • When k = 1, it is the exponential distribution.
  • When k = 2, it becomes equivalent to the Rayleigh distribution, which models the modulus of a two-dimensional uncorrelated bivariate normal vector.
  • When k = 3.4, it appears similar to the normal distribution.
  • As k goes to infinity, the Weibull distribution asymptotically approaches the Dirac delta function.
See Also:

  • Field Summary

    Fields
    Modifier and Type
    Field
    Description
    final double
    k
    The shape parameter.
    final double
    lambda
    The scale parameter.

  • Constructor Summary

    Constructors
    Constructor
    Description
    WeibullDistribution(double k)
    Constructor.
    WeibullDistribution(double k, double lambda)
    Constructor.

  • Method Summary

    Modifier and Type
    Method
    Description
    double
    cdf(double x)
    Cumulative distribution function.
    double
    entropy()
    Returns Shannon entropy of the distribution.
    int
    length()
    Returns the number of parameters of the distribution.
    double
    logp(double x)
    The density at x in log scale, which may prevents the underflow problem.
    double
    mean()
    Returns the mean of distribution.
    double
    p(double x)
    The probability density function for continuous distribution or probability mass function for discrete distribution at x.
    double
    quantile(double p)
    The quantile, the probability to the left of quantile is p.
    double
    rand()
    Generates a random number following this distribution.
    String
    toString()
     
    double
    variance()
    Returns the variance of distribution.

    Methods inherited from class java.lang.Object

    clone, equals, finalize, getClass, hashCode, notify, notifyAll, wait, wait, wait

    Methods inherited from interface smile.stat.distribution.Distribution

    inverseTransformSampling, likelihood, logLikelihood, quantile, quantile, rand, rejectionSampling, sd

  • Field Details

    • k

      public final double k
      The shape parameter.

    • lambda

      public final double lambda
      The scale parameter.

  • Constructor Details

    • WeibullDistribution

      public WeibullDistribution(double k)
      Constructor. The default scale parameter is 1.0.
      Parameters:
      k - the shape parameter.

    • WeibullDistribution

      public WeibullDistribution(double k, double lambda)
      Constructor.
      Parameters:
      k - the shape parameter.
      lambda - the scale parameter.

  • Method Details

    • length

      public int length()
      Description copied from interface: Distribution
      Returns the number of parameters of the distribution. The "length" is in the sense of the minimum description length principle.
      Specified by:
      length in interface Distribution
      Returns:
      The number of parameters.

    • mean

      public double mean()
      Description copied from interface: Distribution
      Returns the mean of distribution.
      Specified by:
      mean in interface Distribution
      Returns:
      The mean.

    • variance

      public double variance()
      Description copied from interface: Distribution
      Returns the variance of distribution.
      Specified by:
      variance in interface Distribution
      Returns:
      The variance.

    • entropy

      public double entropy()
      Description copied from interface: Distribution
      Returns Shannon entropy of the distribution.
      Specified by:
      entropy in interface Distribution
      Returns:
      Shannon entropy.

    • toString

      public String toString()
      Overrides:
      toString in class Object

    • rand

      public double rand()
      Description copied from interface: Distribution
      Generates a random number following this distribution.
      Specified by:
      rand in interface Distribution
      Returns:
      a random number.

    • p

      public double p(double x)
      Description copied from interface: Distribution
      The probability density function for continuous distribution or probability mass function for discrete distribution at x.
      Specified by:
      p in interface Distribution
      Parameters:
      x - a real number.
      Returns:
      the density.

    • logp

      public double logp(double x)
      Description copied from interface: Distribution
      The density at x in log scale, which may prevents the underflow problem.
      Specified by:
      logp in interface Distribution
      Parameters:
      x - a real number.
      Returns:
      the log density.

    • cdf

      public double cdf(double x)
      Description copied from interface: Distribution
      Cumulative distribution function. That is the probability to the left of x.
      Specified by:
      cdf in interface Distribution
      Parameters:
      x - a real number.
      Returns:
      the probability.

    • quantile

      public double quantile(double p)
      Description copied from interface: Distribution
      The quantile, the probability to the left of quantile is p. It is actually the inverse of cdf.
      Specified by:
      quantile in interface Distribution
      Parameters:
      p - the probability.
      Returns:
      the quantile.