Package org.apache.commons.statistics.distribution

Class NakagamiDistribution

  • All Implemented Interfaces:
    ContinuousDistribution
    public final class NakagamiDistribution
extends AbstractContinuousDistribution
    Implementation of the Nakagami distribution.

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

    \[ f(x; \mu, \Omega) = \frac{2\mu^\mu}{\Gamma(\mu)\Omega^\mu}x^{2\mu-1}\exp\left(-\frac{\mu}{\Omega}x^2\right) \]

    for \( \mu > 0 \) the shape, \( \Omega > 0 \) the scale, and \( x \in (0, \infty) \).

    See Also:
    Nakagami distribution (Wikipedia)

    • Field Detail

      • SUPPORT_LO

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

      • SUPPORT_HI

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

      • mu

        private final double mu
        The shape parameter.

      • omega

        private final double omega
        The scale parameter.

      • densityPrefactor

        private final double densityPrefactor
        Density prefactor.

      • logDensityPrefactor

        private final double logDensityPrefactor
        Log density prefactor.

      • mean

        private final double mean
        Cached value for inverse probability function.

      • variance

        private final double variance
        Cached value for inverse probability function.

    • Constructor Detail

      • NakagamiDistribution

        private NakagamiDistribution​(double mu,
                             double omega)
        Parameters:
        mu - Shape parameter (must be positive).
        omega - Scale parameter (must be positive). Controls the spread of the distribution.

    • Method Detail

      • of

        public static NakagamiDistribution of​(double mu,
                                      double omega)
        Creates a Nakagami distribution.
        Parameters:
        mu - Shape parameter (must be positive).
        omega - Scale parameter (must be positive). Controls the spread of the distribution.
        Returns:
        the distribution
        Throws:
        java.lang.IllegalArgumentException - if mu <= 0 or if omega <= 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.
        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.
        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.

      • getMean

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

        For shape parameter \( \mu \) and scale parameter \( \Omega \), the mean is:

        \[ \frac{\Gamma(m+\frac{1}{2})}{\Gamma(m)}\left(\frac{\Omega}{m}\right)^{1/2} \]

        Returns:
        the mean.

      • getVariance

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

        For shape parameter \( \mu \) and scale parameter \( \Omega \), the variance is:

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

        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.