org.apache.commons.math3.distribution

## Class NakagamiDistribution

- java.lang.Object
- - org.apache.commons.math3.distribution.AbstractRealDistribution
  - - org.apache.commons.math3.distribution.NakagamiDistribution

- All Implemented Interfaces:
    Serializable, RealDistribution

      public class NakagamiDistribution
      extends AbstractRealDistribution

  This class implements the Nakagami distribution.
  Since:
    3.4
  See Also:
    Nakagami Distribution (Wikipedia), Serialized Form

- - ### Field Summary

    Fields
    Modifier and Type | Field and Description
    --- | ---
    `static double` | `DEFAULT_INVERSE_ABSOLUTE_ACCURACY` Default inverse cumulative probability accuracy.

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

      `random, randomData, SOLVER_DEFAULT_ABSOLUTE_ACCURACY`

  - ### Constructor Summary

    Constructors
    Constructor and Description
    `NakagamiDistribution(double mu, double omega)` Build a new instance.
    `NakagamiDistribution(double mu, double omega, double inverseAbsoluteAccuracy)` Build a new instance.
    `NakagamiDistribution(RandomGenerator rng, double mu, double omega, double inverseAbsoluteAccuracy)` Build a new instance.

  - ### Method Summary

    Methods
    Modifier and Type | Method and Description
    --- | ---
    `double` | `cumulativeProbability(double x)` For a random variable `X` whose values are distributed according to this distribution, this method returns `P(X <= x)`.
    `double` | `density(double x)` Returns the probability density function (PDF) of this distribution evaluated at the specified point `x`.
    `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.
    `double` | `getScale()` Access the scale parameter, `omega`.
    `double` | `getShape()` Access the shape parameter, `mu`.
    `protected double` | `getSolverAbsoluteAccuracy()` Returns the solver absolute accuracy for inverse cumulative computation.
    `double` | `getSupportLowerBound()` Access the lower bound of the support.
    `double` | `getSupportUpperBound()` Access the upper bound of the support.
    `boolean` | `isSupportConnected()` Use this method to get information about whether the support is connected, i.e.
    `boolean` | `isSupportLowerBoundInclusive()` Whether or not the lower bound of support is in the domain of the density function.
    `boolean` | `isSupportUpperBoundInclusive()` Whether or not the upper bound of support is in the domain of the density function.

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

      `cumulativeProbability, inverseCumulativeProbability, logDensity, probability, probability, reseedRandomGenerator, sample, sample`

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

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

- - ### Field Detail

    - #### DEFAULT_INVERSE_ABSOLUTE_ACCURACY

          public static final double DEFAULT_INVERSE_ABSOLUTE_ACCURACY

      Default inverse cumulative probability accuracy.
      See Also:
        Constant Field Values

  - ### Constructor Detail

    - #### NakagamiDistribution

          public NakagamiDistribution(double mu,
                              double omega)

      Build a new instance.

      Note: this constructor will implicitly create an instance of `Well19937c` as random generator to be used for sampling only (see `AbstractRealDistribution.sample()` and `AbstractRealDistribution.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:
        `mu` - shape parameter
        `omega` - scale parameter (must be positive)
      Throws:
        `NumberIsTooSmallException` - if `mu < 0.5`
        `NotStrictlyPositiveException` - if `omega <= 0`

    - #### NakagamiDistribution

          public NakagamiDistribution(double mu,
                              double omega,
                              double inverseAbsoluteAccuracy)

      Build a new instance.

      Note: this constructor will implicitly create an instance of `Well19937c` as random generator to be used for sampling only (see `AbstractRealDistribution.sample()` and `AbstractRealDistribution.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:
        `mu` - shape parameter
        `omega` - scale parameter (must be positive)
        `inverseAbsoluteAccuracy` - the maximum absolute error in inverse cumulative probability estimates (defaults to `DEFAULT_INVERSE_ABSOLUTE_ACCURACY`).
      Throws:
        `NumberIsTooSmallException` - if `mu < 0.5`
        `NotStrictlyPositiveException` - if `omega <= 0`

    - #### NakagamiDistribution

          public NakagamiDistribution(RandomGenerator rng,
                              double mu,
                              double omega,
                              double inverseAbsoluteAccuracy)

      Build a new instance.
      Parameters:
        `rng` - Random number generator
        `mu` - shape parameter
        `omega` - scale parameter (must be positive)
        `inverseAbsoluteAccuracy` - the maximum absolute error in inverse cumulative probability estimates (defaults to `DEFAULT_INVERSE_ABSOLUTE_ACCURACY`).
      Throws:
        `NumberIsTooSmallException` - if `mu < 0.5`
        `NotStrictlyPositiveException` - if `omega <= 0`

  - ### Method Detail

    - #### getShape

          public double getShape()

      Access the shape parameter, `mu`.
      Returns:
        the shape parameter.

    - #### getScale

          public double getScale()

      Access the scale parameter, `omega`.
      Returns:
        the scale parameter.

    - #### getSolverAbsoluteAccuracy

          protected double getSolverAbsoluteAccuracy()

      Returns the solver absolute accuracy for inverse cumulative computation. You can override this method in order to use a Brent solver with an absolute accuracy different from the default.
      Overrides:
        `getSolverAbsoluteAccuracy` in class `AbstractRealDistribution`
      Returns:
        the maximum absolute error in inverse cumulative probability estimates

    - #### 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` - the point at which the PDF is evaluated
      Returns:
        the value of the probability density function at point `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` - 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.
      Returns:
        the mean or `Double.NaN` if it is not defined

    - #### getNumericalVariance

          public double getNumericalVariance()

      Use this method to get the numerical value of the variance of this distribution.
      Returns:
        the variance (possibly `Double.POSITIVE_INFINITY` as for certain cases in `TDistribution`) or `Double.NaN` if it is not defined

    - #### getSupportLowerBound

          public double 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 R | P(X <= x) > 0}`.

      Returns:
        lower bound of the support (might be `Double.NEGATIVE_INFINITY`)

    - #### getSupportUpperBound

          public double 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}`.

      Returns:
        upper bound of the support (might be `Double.POSITIVE_INFINITY`)

    - #### isSupportLowerBoundInclusive

          public boolean isSupportLowerBoundInclusive()

      Whether or not the lower bound of support is in the domain of the density function. Returns true iff `getSupporLowerBound()` is finite and `density(getSupportLowerBound())` returns a non-NaN, non-infinite value.
      Returns:
        true if the lower bound of support is finite and the density function returns a non-NaN, non-infinite value there

    - #### isSupportUpperBoundInclusive

          public boolean isSupportUpperBoundInclusive()

      Whether or not the upper bound of support is in the domain of the density function. Returns true iff `getSupportUpperBound()` is finite and `density(getSupportUpperBound())` returns a non-NaN, non-infinite value.
      Returns:
        true if the upper bound of support is finite and the density function returns a non-NaN, non-infinite value there

    - #### isSupportConnected

          public boolean isSupportConnected()

      Use this method to get information about whether the support is connected, i.e. whether all values between the lower and upper bound of the support are included in the support.
      Returns:
        whether the support is connected or not

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