Class WeibullDistribution

java.lang.Object
org.apache.commons.statistics.distribution.AbstractContinuousDistribution
org.apache.commons.statistics.distribution.WeibullDistribution
All Implemented Interfaces:
ContinuousDistribution
public final class WeibullDistribution extends AbstractContinuousDistribution
Implementation of the Weibull distribution.

The probability density function of \( X \) is:

\[ f(x;k,\lambda) = \frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1}e^{-(x/\lambda)^{k}} \]

for \( k > 0 \) the shape, \( \lambda > 0 \) the scale, and \( x \in (0, \infty) \).

Note the special cases:

  • \( k = 1 \) is the exponential distribution
  • \( k = 2 \) is the Rayleigh distribution with scale \( \sigma = \frac {\lambda}{\sqrt{2}} \)
See Also:

  • Field Details

    • SUPPORT_LO

      private static final double SUPPORT_LO
      Support lower bound.
      See Also:

    • SUPPORT_HI

      private static final double SUPPORT_HI
      Support upper bound.
      See Also:

    • shape

      private final double shape
      The shape parameter.

    • scale

      private final double scale
      The scale parameter.

    • shapeOverScale

      private final double shapeOverScale
      shape / scale.

    • logShapeOverScale

      private final double logShapeOverScale
      log(shape / scale).

  • Constructor Details

    • WeibullDistribution

      private WeibullDistribution(double shape, double scale)
      Parameters:
      shape - Shape parameter.
      scale - Scale parameter.

  • Method Details

    • of

      public static WeibullDistribution of(double shape, double scale)
      Creates a Weibull distribution.
      Parameters:
      shape - Shape parameter.
      scale - Scale parameter.
      Returns:
      the distribution
      Throws:
      IllegalArgumentException - if shape <= 0 or scale <= 0.

    • getShape

      public double getShape()
      Gets the shape parameter of this distribution.
      Returns:
      the shape parameter.

    • getScale

      public double getScale()
      Gets the scale parameter of this distribution.
      Returns:
      the scale parameter.

    • density

      public double density(double x)
      Returns the probability density function (PDF) of this distribution evaluated at the specified point x. In general, the PDF is the derivative of the CDF. If the derivative does not exist at x, then an appropriate replacement should be returned, e.g. Double.POSITIVE_INFINITY, Double.NaN, or the limit inferior or limit superior of the difference quotient.

      Returns the limit when x = 0:

      • shape < 1: Infinity
      • shape == 1: 1 / scale
      • shape > 1: 0
      Parameters:
      x - Point at which the PDF is evaluated.
      Returns:
      the value of the probability density function at x.

    • logDensity

      public double logDensity(double x)
      Returns the natural logarithm of the probability density function (PDF) of this distribution evaluated at the specified point x.

      Returns the limit when x = 0:

      • shape < 1: Infinity
      • shape == 1: log(1 / scale)
      • shape > 1: -Infinity
      Parameters:
      x - Point at which the PDF is evaluated.
      Returns:
      the logarithm of the value of the probability density function at x.

    • cumulativeProbability

      public double cumulativeProbability(double x)
      For a random variable X whose values are distributed according to this distribution, this method returns P(X <= x). In other words, this method represents the (cumulative) distribution function (CDF) for this distribution.
      Parameters:
      x - Point at which the CDF is evaluated.
      Returns:
      the probability that a random variable with this distribution takes a value less than or equal to x.

    • survivalProbability

      public double survivalProbability(double x)
      For a random variable X whose values are distributed according to this distribution, this method returns P(X > x). In other words, this method represents the complementary cumulative distribution function.

      By default, this is defined as 1 - cumulativeProbability(x), but the specific implementation may be more accurate.

      Parameters:
      x - Point at which the survival function is evaluated.
      Returns:
      the probability that a random variable with this distribution takes a value greater than x.

    • inverseCumulativeProbability

      public double inverseCumulativeProbability(double p)
      Computes the quantile function of this distribution. For a random variable X distributed according to this distribution, the returned value is:

      \[ x = \begin{cases} \inf \{ x \in \mathbb R : P(X \le x) \ge p\} & \text{for } 0 \lt p \le 1 \\ \inf \{ x \in \mathbb R : P(X \le x) \gt 0 \} & \text{for } p = 0 \end{cases} \]

      The default implementation returns:

      Returns 0 when p == 0 and Double.POSITIVE_INFINITY when p == 1.

      Specified by:
      inverseCumulativeProbability in interface ContinuousDistribution
      Overrides:
      inverseCumulativeProbability in class AbstractContinuousDistribution
      Parameters:
      p - Cumulative probability.
      Returns:
      the smallest p-quantile of this distribution (largest 0-quantile for p = 0).

    • inverseSurvivalProbability

      public double inverseSurvivalProbability(double p)
      Computes the inverse survival probability function of this distribution. For a random variable X distributed according to this distribution, the returned value is:

      \[ x = \begin{cases} \inf \{ x \in \mathbb R : P(X \gt x) \le p\} & \text{for } 0 \le p \lt 1 \\ \inf \{ x \in \mathbb R : P(X \gt x) \lt 1 \} & \text{for } p = 1 \end{cases} \]

      By default, this is defined as inverseCumulativeProbability(1 - p), but the specific implementation may be more accurate.

      The default implementation returns:

      Returns 0 when p == 1 and Double.POSITIVE_INFINITY when p == 0.

      Specified by:
      inverseSurvivalProbability in interface ContinuousDistribution
      Overrides:
      inverseSurvivalProbability in class AbstractContinuousDistribution
      Parameters:
      p - Survival probability.
      Returns:
      the smallest (1-p)-quantile of this distribution (largest 0-quantile for p = 1).

    • getMean

      public double getMean()
      Gets the mean of this distribution.

      For shape parameter \( k \) and scale parameter \( \lambda \), the mean is:

      \[ \lambda \, \Gamma(1+\frac{1}{k}) \]

      where \( \Gamma \) is the Gamma-function.

      Returns:
      the mean.

    • getVariance

      public double getVariance()
      Gets the variance of this distribution.

      For shape parameter \( k \) and scale parameter \( \lambda \), the variance is:

      \[ \lambda^2 \left[ \Gamma\left(1+\frac{2}{k}\right) - \left(\Gamma\left(1+\frac{1}{k}\right)\right)^2 \right] \]

      where \( \Gamma \) is the Gamma-function.

      Returns:
      the variance.

    • getSupportLowerBound

      public double getSupportLowerBound()
      Gets the lower bound of the support. It must return the same value as inverseCumulativeProbability(0), i.e. \( \inf \{ x \in \mathbb R : P(X \le x) \gt 0 \} \).

      The lower bound of the support is always 0.

      Returns:
      0.

    • getSupportUpperBound

      public double getSupportUpperBound()
      Gets the upper bound of the support. It must return the same value as inverseCumulativeProbability(1), i.e. \( \inf \{ x \in \mathbb R : P(X \le x) = 1 \} \).

      The upper bound of the support is always positive infinity.

      Returns:
      positive infinity.

    • createSampler

      public ContinuousDistribution.Sampler createSampler(org.apache.commons.rng.UniformRandomProvider rng)
      Creates a sampler.
      Specified by:
      createSampler in interface ContinuousDistribution
      Overrides:
      createSampler in class AbstractContinuousDistribution
      Parameters:
      rng - Generator of uniformly distributed numbers.
      Returns:
      a sampler that produces random numbers according this distribution.