Class NakagamiDistribution

java.lang.Object
org.apache.commons.statistics.distribution.AbstractContinuousDistribution
org.apache.commons.statistics.distribution.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:
  • Field Details

    • SUPPORT_LO

      private static final double SUPPORT_LO
      Support lower bound.
      See Also:
    • SUPPORT_HI

      private static final double SUPPORT_HI
      Support upper bound.
      See Also:
    • LN_2

      private static final double LN_2
      Natural logarithm of 2.
      See Also:
    • 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 Details

    • 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 Details

    • 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:
      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.
    • createSampler

      public ContinuousDistribution.Sampler createSampler(org.apache.commons.rng.UniformRandomProvider rng)
      Description copied from class: AbstractContinuousDistribution
      Creates a sampler.
      Specified by:
      createSampler in interface ContinuousDistribution
      Overrides:
      createSampler in class AbstractContinuousDistribution
      Parameters:
      rng - Generator of uniformly distributed numbers.
      Returns:
      a sampler that produces random numbers according this distribution.