Class CauchyDistribution
- java.lang.Object
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- org.apache.commons.statistics.distribution.AbstractContinuousDistribution
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- org.apache.commons.statistics.distribution.CauchyDistribution
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- All Implemented Interfaces:
ContinuousDistribution
public final class CauchyDistribution extends AbstractContinuousDistribution
Implementation of the Cauchy distribution.The probability density function of \( X \) is:
\[ f(x; x_0, \gamma) = { 1 \over \pi \gamma } \left[ { \gamma^2 \over (x - x_0)^2 + \gamma^2 } \right] \]
for \( x_0 \) the location, \( \gamma > 0 \) the scale, and \( x \in (-\infty, \infty) \).
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Nested Class Summary
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Nested classes/interfaces inherited from interface org.apache.commons.statistics.distribution.ContinuousDistribution
ContinuousDistribution.Sampler
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Field Summary
Fields Modifier and Type Field Description private double
location
The location of this distribution.private double
scale
The scale of this distribution.private double
scale2
Density factor (scale^2).private double
scaleOverPi
Density factor (scale / pi).
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Constructor Summary
Constructors Modifier Constructor Description private
CauchyDistribution(double location, double scale)
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Method Summary
All Methods Static Methods Instance Methods Concrete Methods Modifier and Type Method Description private static double
cdf(double x)
Compute the CDF of the Cauchy distribution with location 0 and scale 1.ContinuousDistribution.Sampler
createSampler(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
getLocation()
Gets the location parameter of this distribution.double
getMean()
Gets the mean of this distribution.(package private) double
getMedian()
Gets the median.double
getScale()
Gets the scale parameter of this distribution.double
getSupportLowerBound()
Gets the lower bound of the support.double
getSupportUpperBound()
Gets the upper bound of the support.double
getVariance()
Gets the variance of this distribution.double
inverseCumulativeProbability(double p)
Computes the quantile function of this distribution.double
inverseSurvivalProbability(double p)
Computes the inverse survival probability function of this distribution.static CauchyDistribution
of(double location, double scale)
Creates a Cauchy distribution.double
survivalProbability(double x)
For a random variableX
whose values are distributed according to this distribution, this method returnsP(X > x)
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Methods inherited from class org.apache.commons.statistics.distribution.AbstractContinuousDistribution
isSupportConnected, probability
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Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
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Methods inherited from interface org.apache.commons.statistics.distribution.ContinuousDistribution
logDensity
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Method Detail
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of
public static CauchyDistribution of(double location, double scale)
Creates a Cauchy distribution.- Parameters:
location
- Location parameter.scale
- Scale parameter.- Returns:
- the distribution
- Throws:
java.lang.IllegalArgumentException
- ifscale <= 0
.
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getLocation
public double getLocation()
Gets the location parameter of this distribution.- Returns:
- the location parameter.
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getScale
public double getScale()
Gets the scale parameter of this distribution.- Returns:
- the scale parameter.
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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 the CDF. 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
.
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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
.
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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
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cdf
private static double cdf(double x)
Compute the CDF of the Cauchy distribution with location 0 and scale 1.- Parameters:
x
- Point at which the CDF is evaluated- Returns:
- CDF(x)
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inverseCumulativeProbability
public double inverseCumulativeProbability(double p)
Computes the quantile function of this distribution. For a random variableX
distributed according to this distribution, the returned value is:\[ x = \begin{cases} \inf \{ x \in \mathbb R : P(X \le x) \ge p\} & \text{for } 0 \lt p \le 1 \\ \inf \{ x \in \mathbb R : P(X \le x) \gt 0 \} & \text{for } p = 0 \end{cases} \]
The default implementation returns:
ContinuousDistribution.getSupportLowerBound()
forp = 0
,ContinuousDistribution.getSupportUpperBound()
forp = 1
, or- the result of a search for a root between the lower and upper bound using
cumulativeProbability(x) - p
. The bounds may be bracketed for efficiency.
Returns
Double.NEGATIVE_INFINITY
whenp == 0
andDouble.POSITIVE_INFINITY
whenp == 1
.- Specified by:
inverseCumulativeProbability
in interfaceContinuousDistribution
- Overrides:
inverseCumulativeProbability
in classAbstractContinuousDistribution
- Parameters:
p
- Cumulative probability.- Returns:
- the smallest
p
-quantile of this distribution (largest 0-quantile forp = 0
).
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inverseSurvivalProbability
public double inverseSurvivalProbability(double p)
Computes the inverse survival probability function of this distribution. For a random variableX
distributed according to this distribution, the returned value is:\[ x = \begin{cases} \inf \{ x \in \mathbb R : P(X \gt x) \le p\} & \text{for } 0 \le p \lt 1 \\ \inf \{ x \in \mathbb R : P(X \gt x) \lt 1 \} & \text{for } p = 1 \end{cases} \]
By default, this is defined as
inverseCumulativeProbability(1 - p)
, but the specific implementation may be more accurate.The default implementation returns:
ContinuousDistribution.getSupportLowerBound()
forp = 1
,ContinuousDistribution.getSupportUpperBound()
forp = 0
, or- the result of a search for a root between the lower and upper bound using
survivalProbability(x) - p
. The bounds may be bracketed for efficiency.
Returns
Double.NEGATIVE_INFINITY
whenp == 1
andDouble.POSITIVE_INFINITY
whenp == 0
.- Specified by:
inverseSurvivalProbability
in interfaceContinuousDistribution
- Overrides:
inverseSurvivalProbability
in classAbstractContinuousDistribution
- Parameters:
p
- Survival probability.- Returns:
- the smallest
(1-p)
-quantile of this distribution (largest 0-quantile forp = 1
).
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getMean
public double getMean()
Gets the mean of this distribution.The mean is always undefined.
- Returns:
NaN
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getVariance
public double getVariance()
Gets the variance of this distribution.The variance is always undefined.
- Returns:
NaN
.
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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 negative infinity.
- Returns:
- negative infinity.
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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.
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getMedian
double getMedian()
Gets the median. This is used to determine if the arguments to theAbstractContinuousDistribution.probability(double, double)
function are in the upper or lower domain.The default implementation calls
AbstractContinuousDistribution.inverseCumulativeProbability(double)
with a value of 0.5.- Overrides:
getMedian
in classAbstractContinuousDistribution
- Returns:
- the median
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createSampler
public ContinuousDistribution.Sampler createSampler(org.apache.commons.rng.UniformRandomProvider rng)
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.
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