Class NakagamiDistribution
- All Implemented Interfaces:
ContinuousDistribution
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:
-
Nested Class Summary
Nested classes/interfaces inherited from interface org.apache.commons.statistics.distribution.ContinuousDistribution
ContinuousDistribution.Sampler
-
Field Summary
FieldsModifier and TypeFieldDescriptionprivate final double
Density prefactor.private static final double
Natural logarithm of 2.private final double
Log density prefactor.private final double
Cached value for inverse probability function.private final double
The shape parameter.private final double
The scale parameter.private static final double
Support upper bound.private static final double
Support lower bound.private final double
Cached value for inverse probability function. -
Constructor Summary
Constructors -
Method Summary
Modifier and TypeMethodDescriptioncreateSampler
(org.apache.commons.rng.UniformRandomProvider rng) Creates a sampler.double
cumulativeProbability
(double x) For a random variableX
whose values are distributed according to this distribution, this method returnsP(X <= x)
.double
density
(double x) Returns the probability density function (PDF) of this distribution evaluated at the specified pointx
.double
getMean()
Gets the mean of this distribution.double
getScale()
Gets the scale parameter of this distribution.double
getShape()
Gets the shape parameter of this distribution.double
Gets the lower bound of the support.double
Gets the upper bound of the support.double
Gets the variance of this distribution.double
logDensity
(double x) Returns the natural logarithm of the probability density function (PDF) of this distribution evaluated at the specified pointx
.static NakagamiDistribution
of
(double mu, double omega) Creates a Nakagami distribution.double
survivalProbability
(double x) For a random variableX
whose values are distributed according to this distribution, this method returnsP(X > x)
.Methods inherited from class org.apache.commons.statistics.distribution.AbstractContinuousDistribution
getMedian, inverseCumulativeProbability, inverseSurvivalProbability, isSupportConnected, probability
-
Field Details
-
SUPPORT_LO
private static final double SUPPORT_LOSupport lower bound.- See Also:
-
SUPPORT_HI
private static final double SUPPORT_HISupport upper bound.- See Also:
-
LN_2
private static final double LN_2Natural logarithm of 2.- See Also:
-
mu
private final double muThe shape parameter. -
omega
private final double omegaThe scale parameter. -
densityPrefactor
private final double densityPrefactorDensity prefactor. -
logDensityPrefactor
private final double logDensityPrefactorLog density prefactor. -
mean
private final double meanCached value for inverse probability function. -
variance
private final double varianceCached 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
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
- ifmu <= 0
or ifomega <= 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 pointx
. In general, the PDF is the derivative of theCDF
. If the derivative does not exist atx
, 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 pointx
.- 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 variableX
whose values are distributed according to this distribution, this method returnsP(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 variableX
whose values are distributed according to this distribution, this method returnsP(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 asinverseCumulativeProbability(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 asinverseCumulativeProbability(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 interfaceContinuousDistribution
- Overrides:
createSampler
in classAbstractContinuousDistribution
- Parameters:
rng
- Generator of uniformly distributed numbers.- Returns:
- a sampler that produces random numbers according this distribution.
-