org.apache.commons.math3.distribution

## Class WeibullDistribution

- java.lang.Object
- - org.apache.commons.math3.distribution.AbstractRealDistribution
  - - org.apache.commons.math3.distribution.WeibullDistribution

- All Implemented Interfaces:
    Serializable, RealDistribution

      public class WeibullDistribution
      extends AbstractRealDistribution

  Implementation of the Weibull distribution. This implementation uses the two parameter form of the distribution defined by Weibull Distribution, equations (1) and (2).
  Since:
    1.1 (changed to concrete class in 3.0)
  See Also:
    Weibull distribution (Wikipedia), Weibull distribution (MathWorld), Serialized Form

- - ### Field Summary

    Fields
    Modifier and Type | Field and Description
    --- | ---
    `static double` | `DEFAULT_INVERSE_ABSOLUTE_ACCURACY` Default inverse cumulative probability accuracy.

    - ### Fields inherited from class org.apache.commons.math3.distribution.AbstractRealDistribution

      `random, randomData, SOLVER_DEFAULT_ABSOLUTE_ACCURACY`

  - ### Constructor Summary

    Constructors
    Constructor and Description
    `WeibullDistribution(double alpha, double beta)` Create a Weibull distribution with the given shape and scale and a location equal to zero.
    `WeibullDistribution(double alpha, double beta, double inverseCumAccuracy)` Create a Weibull distribution with the given shape, scale and inverse cumulative probability accuracy and a location equal to zero.
    `WeibullDistribution(RandomGenerator rng, double alpha, double beta)` Creates a Weibull distribution.
    `WeibullDistribution(RandomGenerator rng, double alpha, double beta, double inverseCumAccuracy)` Creates a Weibull distribution.

  - ### Method Summary

    Methods
    Modifier and Type | Method and Description
    --- | ---
    `protected double` | `calculateNumericalMean()` used by `getNumericalMean()`
    `protected double` | `calculateNumericalVariance()` used by `getNumericalVariance()`
    `double` | `cumulativeProbability(double x)` For a random variable `X` whose values are distributed according to this distribution, this method returns `P(X <= x)`.
    `double` | `density(double x)` Returns the probability density function (PDF) of this distribution evaluated at the specified point `x`.
    `double` | `getNumericalMean()` Use this method to get the numerical value of the mean of this distribution.
    `double` | `getNumericalVariance()` Use this method to get the numerical value of the variance of this distribution.
    `double` | `getScale()` Access the scale parameter, `beta`.
    `double` | `getShape()` Access the shape parameter, `alpha`.
    `protected double` | `getSolverAbsoluteAccuracy()` Return the absolute accuracy setting of the solver used to estimate inverse cumulative probabilities.
    `double` | `getSupportLowerBound()` Access the lower bound of the support.
    `double` | `getSupportUpperBound()` Access the upper bound of the support.
    `double` | `inverseCumulativeProbability(double p)` Computes the quantile function of this distribution.
    `boolean` | `isSupportConnected()` Use this method to get information about whether the support is connected, i.e.
    `boolean` | `isSupportLowerBoundInclusive()` Whether or not the lower bound of support is in the domain of the density function.
    `boolean` | `isSupportUpperBoundInclusive()` Whether or not the upper bound of support is in the domain of the density function.
    `double` | `logDensity(double x)` Returns the natural logarithm of the probability density function (PDF) of this distribution evaluated at the specified point `x`.

    - ### Methods inherited from class org.apache.commons.math3.distribution.AbstractRealDistribution

      `cumulativeProbability, probability, probability, reseedRandomGenerator, sample, sample`

    - ### Methods inherited from class java.lang.Object

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

- - ### Field Detail

    - #### DEFAULT_INVERSE_ABSOLUTE_ACCURACY

          public static final double DEFAULT_INVERSE_ABSOLUTE_ACCURACY

      Default inverse cumulative probability accuracy.
      Since:
        2.1
      See Also:
        Constant Field Values

  - ### Constructor Detail

    - #### WeibullDistribution

          public WeibullDistribution(double alpha,
                             double beta)
                              throws NotStrictlyPositiveException

      Create a Weibull distribution with the given shape and scale and a location equal to zero.

      Note: this constructor will implicitly create an instance of `Well19937c` as random generator to be used for sampling only (see `AbstractRealDistribution.sample()` and `AbstractRealDistribution.sample(int)`). In case no sampling is needed for the created distribution, it is advised to pass `null` as random generator via the appropriate constructors to avoid the additional initialisation overhead.

      Parameters:
        `alpha` - Shape parameter.
        `beta` - Scale parameter.
      Throws:
        `NotStrictlyPositiveException` - if `alpha <= 0` or `beta <= 0`.

    - #### WeibullDistribution

          public WeibullDistribution(double alpha,
                             double beta,
                             double inverseCumAccuracy)

      Create a Weibull distribution with the given shape, scale and inverse cumulative probability accuracy and a location equal to zero.

      Note: this constructor will implicitly create an instance of `Well19937c` as random generator to be used for sampling only (see `AbstractRealDistribution.sample()` and `AbstractRealDistribution.sample(int)`). In case no sampling is needed for the created distribution, it is advised to pass `null` as random generator via the appropriate constructors to avoid the additional initialisation overhead.

      Parameters:
        `alpha` - Shape parameter.
        `beta` - Scale parameter.
        `inverseCumAccuracy` - Maximum absolute error in inverse cumulative probability estimates (defaults to `DEFAULT_INVERSE_ABSOLUTE_ACCURACY`).
      Throws:
        `NotStrictlyPositiveException` - if `alpha <= 0` or `beta <= 0`.
      Since:
        2.1

    - #### WeibullDistribution

          public WeibullDistribution(RandomGenerator rng,
                             double alpha,
                             double beta)
                              throws NotStrictlyPositiveException

      Creates a Weibull distribution.
      Parameters:
        `rng` - Random number generator.
        `alpha` - Shape parameter.
        `beta` - Scale parameter.
      Throws:
        `NotStrictlyPositiveException` - if `alpha <= 0` or `beta <= 0`.
      Since:
        3.3

    - #### WeibullDistribution

          public WeibullDistribution(RandomGenerator rng,
                             double alpha,
                             double beta,
                             double inverseCumAccuracy)
                              throws NotStrictlyPositiveException

      Creates a Weibull distribution.
      Parameters:
        `rng` - Random number generator.
        `alpha` - Shape parameter.
        `beta` - Scale parameter.
        `inverseCumAccuracy` - Maximum absolute error in inverse cumulative probability estimates (defaults to `DEFAULT_INVERSE_ABSOLUTE_ACCURACY`).
      Throws:
        `NotStrictlyPositiveException` - if `alpha <= 0` or `beta <= 0`.
      Since:
        3.1

  - ### Method Detail

    - #### getShape

          public double getShape()

      Access the shape parameter, `alpha`.
      Returns:
        the shape parameter, `alpha`.

    - #### getScale

          public double getScale()

      Access the scale parameter, `beta`.
      Returns:
        the scale parameter, `beta`.

    - #### 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.
      Parameters:
        `x` - the point at which the PDF is evaluated
      Returns:
        the value of the probability density function at point `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`. 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. Note that due to the floating point precision and under/overflow issues, this method will for some distributions be more precise and faster than computing the logarithm of `RealDistribution.density(double)`. The default implementation simply computes the logarithm of `density(x)`.
      Overrides:
        `logDensity` in class `AbstractRealDistribution`
      Parameters:
        `x` - the point at which the PDF is evaluated
      Returns:
        the logarithm of the value of the probability density function at point `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` - the 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`

    - #### 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
      - `inf{x in R | P(X<=x) >= p}` for `0 < p <= 1`,
      - `inf{x in R | P(X<=x) > 0}` for `p = 0`.
      The default implementation returns
      - `RealDistribution.getSupportLowerBound()` for `p = 0`,
      - `RealDistribution.getSupportUpperBound()` for `p = 1`.
      Returns `0` when `p == 0` and `Double.POSITIVE_INFINITY` when `p == 1`.
      Specified by:
        `inverseCumulativeProbability` in interface `RealDistribution`
      Overrides:
        `inverseCumulativeProbability` in class `AbstractRealDistribution`
      Parameters:
        `p` - the cumulative probability
      Returns:
        the smallest `p`-quantile of this distribution (largest 0-quantile for `p = 0`)

    - #### getSolverAbsoluteAccuracy

          protected double getSolverAbsoluteAccuracy()

      Return the absolute accuracy setting of the solver used to estimate inverse cumulative probabilities.
      Overrides:
        `getSolverAbsoluteAccuracy` in class `AbstractRealDistribution`
      Returns:
        the solver absolute accuracy.
      Since:
        2.1

    - #### getNumericalMean

          public double getNumericalMean()

      Use this method to get the numerical value of the mean of this distribution. The mean is `scale * Gamma(1 + (1 / shape))`, where `Gamma()` is the Gamma-function.
      Returns:
        the mean or `Double.NaN` if it is not defined

    - #### calculateNumericalMean

          protected double calculateNumericalMean()

      used by `getNumericalMean()`
      Returns:
        the mean of this distribution

    - #### getNumericalVariance

          public double getNumericalVariance()

      Use this method to get the numerical value of the variance of this distribution. The variance is `scale^2 * Gamma(1 + (2 / shape)) - mean^2` where `Gamma()` is the Gamma-function.
      Returns:
        the variance (possibly `Double.POSITIVE_INFINITY` as for certain cases in `TDistribution`) or `Double.NaN` if it is not defined

    - #### calculateNumericalVariance

          protected double calculateNumericalVariance()

      used by `getNumericalVariance()`
      Returns:
        the variance of this distribution

    - #### getSupportLowerBound

          public double getSupportLowerBound()

      Access the lower bound of the support. This method must return the same value as `inverseCumulativeProbability(0)`. In other words, this method must return

      `inf {x in R | P(X <= x) > 0}`.

      The lower bound of the support is always 0 no matter the parameters.
      Returns:
        lower bound of the support (always 0)

    - #### getSupportUpperBound

          public double getSupportUpperBound()

      Access the upper bound of the support. This method must return the same value as `inverseCumulativeProbability(1)`. In other words, this method must return

      `inf {x in R | P(X <= x) = 1}`.

      The upper bound of the support is always positive infinity no matter the parameters.
      Returns:
        upper bound of the support (always `Double.POSITIVE_INFINITY`)

    - #### isSupportLowerBoundInclusive

          public boolean isSupportLowerBoundInclusive()

      Whether or not the lower bound of support is in the domain of the density function. Returns true iff `getSupporLowerBound()` is finite and `density(getSupportLowerBound())` returns a non-NaN, non-infinite value.
      Returns:
        true if the lower bound of support is finite and the density function returns a non-NaN, non-infinite value there

    - #### isSupportUpperBoundInclusive

          public boolean isSupportUpperBoundInclusive()

      Whether or not the upper bound of support is in the domain of the density function. Returns true iff `getSupportUpperBound()` is finite and `density(getSupportUpperBound())` returns a non-NaN, non-infinite value.
      Returns:
        true if the upper bound of support is finite and the density function returns a non-NaN, non-infinite value there

    - #### isSupportConnected

          public boolean isSupportConnected()

      Use this method to get information about whether the support is connected, i.e. whether all values between the lower and upper bound of the support are included in the support. The support of this distribution is connected.
      Returns:
        `true`

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