Package org.apache.commons.statistics.distribution

Class ZipfDistribution

java.lang.Object

org.apache.commons.statistics.distribution.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)

Nested Class Summary

Nested classes/interfaces inherited from interface org.apache.commons.statistics.distribution.DiscreteDistribution

DiscreteDistribution.Sampler

Method Summary

Modifier and Type	Method	Description
DiscreteDistribution.Sampler	createSampler(org.apache.commons.rng.UniformRandomProvider rng)	Creates a sampler.
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()	Gets the exponent parameter of this distribution.
double	getMean()	Gets the mean of this distribution.
int	getNumberOfElements()	Gets the number of elements parameter of this distribution.
int	getSupportLowerBound()	Gets the lower bound of the support.
int	getSupportUpperBound()	Gets the upper bound of the support.
double	getVariance()	Gets the variance of this distribution.
int	inverseCumulativeProbability(double p)	Computes the quantile function of this distribution.
int	inverseSurvivalProbability(double p)	Computes the inverse survival probability function of this distribution.
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.
static ZipfDistribution	of(int numberOfElements, double exponent)	Creates a Zipf distribution.
double	probability(int x)	For a random variable X whose values are distributed according to this distribution, this method returns P(X = x).
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).
double	survivalProbability(int x)	For a random variable X whose values are distributed according to this distribution, this method returns P(X > x).

Methods inherited from class java.lang.Object

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

Method Details

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.

createSampler

public DiscreteDistribution.Sampler createSampler(org.apache.commons.rng.UniformRandomProvider rng)

Creates a sampler.

Specified by:

createSampler in interface DiscreteDistribution

Parameters:

rng - Generator of uniformly distributed numbers.

Returns:

a sampler that produces random numbers according this distribution.

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.

inverseCumulativeProbability

public int inverseCumulativeProbability(double p)

Computes the quantile function of this distribution. For a random variable X distributed according to this distribution, the returned value is:

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

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 cumulativeProbability(x) called using the returned p-quantile may not compute the original p.

The default implementation returns:

• DiscreteDistribution.getSupportLowerBound() for p = 0,
• DiscreteDistribution.getSupportUpperBound() for p = 1, or
• the result of a binary search between the lower and upper bound using cumulativeProbability(x). The bounds may be bracketed for efficiency.

Specified by:

inverseCumulativeProbability in interface DiscreteDistribution

Parameters:

p - Cumulative probability.

Returns:

the smallest p-quantile of this distribution (largest 0-quantile for p = 0).

Throws:

IllegalArgumentException - if p < 0 or p > 1

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:

• DiscreteDistribution.getSupportLowerBound() for p = 1,
• DiscreteDistribution.getSupportUpperBound() for p = 0, or
• the result of a binary search between the lower and upper bound using survivalProbability(x). The bounds may be bracketed for efficiency.

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