Class FoldedNormalDistribution.HalfNormalDistribution
- All Implemented Interfaces:
ContinuousDistribution
- Enclosing class:
FoldedNormalDistribution
Elimination of the mu
location parameter simplifies the probability
functions and allows computation of the log density and inverse CDF/SF.
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Nested Class Summary
Nested classes/interfaces inherited from interface org.apache.commons.statistics.distribution.ContinuousDistribution
ContinuousDistribution.Sampler
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Field Summary
FieldsModifier and TypeFieldDescriptionprivate final double
The value oflog(sigma) + 0.5 * log(2*PI)
stored for faster computation.private static final double
Variance constant (1 - 2/pi).Fields inherited from class org.apache.commons.statistics.distribution.FoldedNormalDistribution
sigma, sigmaSqrt2, sigmaSqrt2pi
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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
getMu()
Gets the location parameter \( \mu \) of this distribution.double
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.double
logDensity
(double x) Returns the natural logarithm of the probability density function (PDF) of this distribution evaluated at the specified pointx
.double
probability
(double x0, double x1) For a random variableX
whose values are distributed according to this distribution, this method returnsP(x0 < X <= x1)
.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.FoldedNormalDistribution
getSigma, getSupportLowerBound, getSupportUpperBound, of
Methods inherited from class org.apache.commons.statistics.distribution.AbstractContinuousDistribution
getMedian, isSupportConnected
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Field Details
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VAR
private static final double VARVariance constant (1 - 2/pi). Computed using Matlab's VPA to 30 digits.- See Also:
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logSigmaPlusHalfLog2Pi
private final double logSigmaPlusHalfLog2PiThe value oflog(sigma) + 0.5 * log(2*PI)
stored for faster computation.
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Constructor Details
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HalfNormalDistribution
HalfNormalDistribution(double sigma) - Parameters:
sigma
- Scale parameter.
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Method Details
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getMu
public double getMu()Description copied from class:FoldedNormalDistribution
Gets the location parameter \( \mu \) of this distribution.- Specified by:
getMu
in classFoldedNormalDistribution
- Returns:
- the mu parameter.
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density
public double density(double x) Description copied from interface:ContinuousDistribution
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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probability
public double probability(double x0, double x1) Description copied from class:AbstractContinuousDistribution
For a random variableX
whose values are distributed according to this distribution, this method returnsP(x0 < X <= x1)
. The default implementation uses the identityP(x0 < X <= x1) = P(X <= x1) - P(X <= x0)
- Specified by:
probability
in interfaceContinuousDistribution
- Overrides:
probability
in classAbstractContinuousDistribution
- Parameters:
x0
- Lower bound (exclusive).x1
- Upper bound (inclusive).- Returns:
- the probability that a random variable with this distribution
takes a value between
x0
andx1
, excluding the lower and including the upper endpoint.
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logDensity
public double logDensity(double x) Description copied from interface:ContinuousDistribution
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
.
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cumulativeProbability
public double cumulativeProbability(double x) Description copied from interface:ContinuousDistribution
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) Description copied from interface:ContinuousDistribution
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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inverseCumulativeProbability
public double inverseCumulativeProbability(double p) Description copied from class:AbstractContinuousDistribution
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.
- 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.
- 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()Description copied from class:FoldedNormalDistribution
Gets the mean of this distribution.For location parameter \( \mu \) and scale parameter \( \sigma \), the mean is:
\[ \sigma \sqrt{ \frac 2 \pi } \exp \left( \frac{-\mu^2}{2\sigma^2} \right) + \mu \operatorname{erf} \left( \frac \mu {\sqrt{2\sigma^2}} \right) \]
where \( \operatorname{erf} \) is the error function.
- Specified by:
getMean
in interfaceContinuousDistribution
- Specified by:
getMean
in classFoldedNormalDistribution
- Returns:
- the mean.
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getVariance
public double getVariance()Description copied from class:FoldedNormalDistribution
Gets the variance of this distribution.For location parameter \( \mu \), scale parameter \( \sigma \) and a distribution mean \( \mu_Y \), the variance is:
\[ \mu^2 + \sigma^2 - \mu_{Y}^2 \]
- Specified by:
getVariance
in interfaceContinuousDistribution
- Specified by:
getVariance
in classFoldedNormalDistribution
- Returns:
- the variance.
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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.
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