org.apache.commons.math3.distribution

## Class ZipfDistribution

- java.lang.Object
- - org.apache.commons.math3.distribution.AbstractIntegerDistribution
  - - org.apache.commons.math3.distribution.ZipfDistribution

- All Implemented Interfaces:
    Serializable, IntegerDistribution

      public class ZipfDistribution
      extends AbstractIntegerDistribution

  Implementation of the Zipf distribution.

  Parameters: For a random variable `X` whose values are distributed according to this distribution, the probability mass function is given by

         P(X = k) = H(N,s) * 1 / k^s    for k = 1,2,...,N.

  `H(N,s)` is the normalizing constant which corresponds to the generalized harmonic number of order N of s.

  - `N` is the number of elements
  - `s` is the exponent
  See Also:
    Zipf's law (Wikipedia), Generalized harmonic numbers, Serialized Form

- - ### Field Summary

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

      `random, randomData`

  - ### Constructor Summary

    Constructors
    Constructor and Description
    `ZipfDistribution(int numberOfElements, double exponent)` Create a new Zipf distribution with the given number of elements and exponent.
    `ZipfDistribution(RandomGenerator rng, int numberOfElements, double exponent)` Creates a Zipf distribution.

  - ### Method Summary

    Methods
    Modifier and Type | Method and Description
    --- | ---
    `protected double` | `calculateNumericalMean()` Used by `getNumericalMean()`.
    `protected double` | `calculateNumericalVariance()` Used by `getNumericalVariance()`.
    `double` | `cumulativeProbability(int x)` For a random variable `X` whose values are distributed according to this distribution, this method returns `P(X <= x)`.
    `double` | `getExponent()` Get the exponent characterizing the distribution.
    `int` | `getNumberOfElements()` Get the number of elements (e.g.
    `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.
    `int` | `getSupportLowerBound()` Access the lower bound of the support.
    `int` | `getSupportUpperBound()` Access the upper bound of the support.
    `boolean` | `isSupportConnected()` Use this method to get information about whether the support is connected, i.e.
    `double` | `logProbability(int x)` For a random variable `X` whose values are distributed according to this distribution, this method returns `log(P(X = x))`, where `log` is the natural logarithm.
    `double` | `probability(int x)` For a random variable `X` whose values are distributed according to this distribution, this method returns `P(X = x)`.
    `int` | `sample()` Generate a random value sampled from this distribution.

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

      `cumulativeProbability, inverseCumulativeProbability, reseedRandomGenerator, sample, solveInverseCumulativeProbability`

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

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

- - ### Constructor Detail

    - #### ZipfDistribution

          public ZipfDistribution(int numberOfElements,
                          double exponent)

      Create a new Zipf distribution with the given number of elements and exponent.

      Note: this constructor will implicitly create an instance of `Well19937c` as random generator to be used for sampling only (see `sample()` and `AbstractIntegerDistribution.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:
        `numberOfElements` - Number of elements.
        `exponent` - Exponent.
      Throws:
        `NotStrictlyPositiveException` - if `numberOfElements <= 0` or `exponent <= 0`.

    - #### ZipfDistribution

          public ZipfDistribution(RandomGenerator rng,
                          int numberOfElements,
                          double exponent)
                           throws NotStrictlyPositiveException

      Creates a Zipf distribution.
      Parameters:
        `rng` - Random number generator.
        `numberOfElements` - Number of elements.
        `exponent` - Exponent.
      Throws:
        `NotStrictlyPositiveException` - if `numberOfElements <= 0` or `exponent <= 0`.
      Since:
        3.1

  - ### Method Detail

    - #### getNumberOfElements

          public int getNumberOfElements()

      Get the number of elements (e.g. corpus size) for the distribution.
      Returns:
        the number of elements

    - #### getExponent

          public double getExponent()

      Get the exponent characterizing the distribution.
      Returns:
        the exponent

    - #### probability

          public double probability(int 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 probability mass function (PMF) for the distribution.
      Parameters:
        `x` - the point at which the PMF is evaluated
      Returns:
        the value of the probability mass function at `x`

    - #### logProbability

          public double logProbability(int x)

      For a random variable `X` whose values are distributed according to this distribution, this method returns `log(P(X = x))`, where `log` is the natural logarithm. In other words, this method represents the logarithm of the probability mass function (PMF) for the distribution. 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 `IntegerDistribution.probability(int)`.

      The default implementation simply computes the logarithm of `probability(x)`.

      Overrides:
        `logProbability` in class `AbstractIntegerDistribution`
      Parameters:
        `x` - the point at which the PMF is evaluated
      Returns:
        the logarithm of the value of the probability mass function at `x`

    - #### cumulativeProbability

          public double cumulativeProbability(int 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`

    - #### getNumericalMean

          public double getNumericalMean()

      Use this method to get the numerical value of the mean of this distribution. For number of elements `N` and exponent `s`, the mean is `Hs1 / Hs`, where
      - `Hs1 = generalizedHarmonic(N, s - 1)`,
      - `Hs = generalizedHarmonic(N, s)`.
      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. For number of elements `N` and exponent `s`, the mean is `(Hs2 / Hs) - (Hs1^2 / Hs^2)`, where
      - `Hs2 = generalizedHarmonic(N, s - 2)`,
      - `Hs1 = generalizedHarmonic(N, s - 1)`,
      - `Hs = generalizedHarmonic(N, s)`.
      Returns:
        the variance (possibly `Double.POSITIVE_INFINITY` or `Double.NaN` if it is not defined)

    - #### calculateNumericalVariance

          protected double calculateNumericalVariance()

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

    - #### getSupportLowerBound

          public int 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 Z | P(X <= x) > 0}`.

      The lower bound of the support is always 1 no matter the parameters.
      Returns:
        lower bound of the support (always 1)

    - #### getSupportUpperBound

          public int 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 the number of elements.
      Returns:
        upper bound of the support

    - #### isSupportConnected

          public boolean isSupportConnected()

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

    - #### sample

          public int sample()

      Generate a random value sampled from this distribution. The default implementation uses the inversion method.
      Specified by:
        `sample` in interface `IntegerDistribution`
      Overrides:
        `sample` in class `AbstractIntegerDistribution`
      Returns:
        a random value

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