Package org.apache.commons.statistics.distribution

Class ZipfDistribution

  • All Implemented Interfaces:
    DiscreteDistribution
    public final class ZipfDistribution
extends Object
    Implementation of the Zipf distribution.

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

    \[ f(k; N, s) = \frac{1/k^s}{H_{N,s}} \]

    for \( N \in \{1, 2, 3, \dots\} \) the number of elements, \( s \gt 0 \) the exponent characterizing the distribution, \( k \in \{1, 2, \dots, N\} \) the element rank, and \( H_{N,s} \) is the normalizing constant which corresponds to the generalized harmonic number of order N of s.

    See Also:
    Zipf distribution (Wikipedia)

    • Method Detail

      • of

        public static ZipfDistribution of​(int numberOfElements,
                                  double exponent)
        Creates a Zipf distribution.
        Parameters:
        numberOfElements - Number of elements.
        exponent - Exponent.
        Returns:
        the distribution
        Throws:
        IllegalArgumentException - if numberOfElements <= 0 or exponent <= 0.

      • getNumberOfElements

        public int getNumberOfElements()
        Gets the number of elements parameter of this distribution.
        Returns:
        the number of elements.

      • getExponent

        public double getExponent()
        Gets the exponent parameter of this 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 - 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.
        Parameters:
        x - 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 - 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​(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 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.

      • getMean

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

        For number of elements \( N \) and exponent \( s \), the mean is:

        \[ \frac{H_{N,s-1}}{H_{N,s}} \]

        where \( H_{N,k} \) is the generalized harmonic number of order \( N \) of \( k \).

        Returns:
        the mean.

      • getVariance

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

        For number of elements \( N \) and exponent \( s \), the variance is:

        \[ \frac{H_{N,s-2}}{H_{N,s}} - \frac{H_{N,s-1}^2}{H_{N,s}^2} \]

        where \( H_{N,k} \) is the generalized harmonic number of order \( N \) of \( k \).

        Returns:
        the variance.

      • getSupportLowerBound

        public int getSupportLowerBound()
        Gets the lower bound of the support. This method must return the same value as inverseCumulativeProbability(0), i.e. \( \inf \{ x \in \mathbb Z : P(X \le x) \gt 0 \} \). By convention, Integer.MIN_VALUE should be substituted for negative infinity.

        The lower bound of the support is always 1.

        Returns:
        1.

      • getSupportUpperBound

        public int getSupportUpperBound()
        Gets the upper bound of the support. This method must return the same value as inverseCumulativeProbability(1), i.e. \( \inf \{ x \in \mathbb Z : P(X \le x) = 1 \} \). By convention, Integer.MAX_VALUE should be substituted for positive infinity.

        The upper bound of the support is the number of elements.

        Returns:
        number of elements.

      • probability

        public double probability​(int x0,
                          int x1)
        For a random variable X whose values are distributed according to this distribution, this method returns P(x0 < X <= x1). The default implementation uses the identity P(x0 < X <= x1) = P(X <= x1) - P(X <= x0)

        Special cases:

        • returns 0.0 if x0 == x1;
        • returns probability(x1) if x0 + 1 == x1;
        Specified by:
        probability in interface DiscreteDistribution
        Parameters:
        x0 - Lower bound (exclusive).
        x1 - Upper bound (inclusive).
        Returns:
        the probability that a random variable with this distribution takes a value between x0 and x1, excluding the lower and including the upper endpoint.

      • inverseSurvivalProbability

        public int 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:

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

        If the result exceeds the range of the data type int, then Integer.MIN_VALUE or Integer.MAX_VALUE is returned. In this case the result of survivalProbability(x) called using the returned (1-p)-quantile may not compute the original p.

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

        The default implementation returns:

        Specified by:
        inverseSurvivalProbability in interface DiscreteDistribution
        Parameters:
        p - Cumulative probability.
        Returns:
        the smallest (1-p)-quantile of this distribution (largest 0-quantile for p = 1).
        Throws:
        IllegalArgumentException - if p < 0 or p > 1