Class WeibullDistribution
- java.lang.Object
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- org.apache.commons.statistics.distribution.AbstractContinuousDistribution
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- org.apache.commons.statistics.distribution.WeibullDistribution
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- All Implemented Interfaces:
ContinuousDistribution
public final class WeibullDistribution extends AbstractContinuousDistribution
Implementation of the Weibull distribution.The probability density function of \( X \) is:
\[ f(x;k,\lambda) = \frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1}e^{-(x/\lambda)^{k}} \]
for \( k > 0 \) the shape, \( \lambda > 0 \) the scale, and \( x \in (0, \infty) \).
Note the special cases:
- \( k = 1 \) is the exponential distribution
- \( k = 2 \) is the Rayleigh distribution with scale \( \sigma = \frac {\lambda}{\sqrt{2}} \)
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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
logShapeOverScale
log(shape / scale).private double
scale
The scale parameter.private double
shape
The shape parameter.private double
shapeOverScale
shape / scale.private static double
SUPPORT_HI
Support upper bound.private static double
SUPPORT_LO
Support lower bound.
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Constructor Summary
Constructors Modifier Constructor Description private
WeibullDistribution(double shape, double scale)
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Method Summary
All Methods Static Methods Instance Methods Concrete Methods Modifier and Type Method Description 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
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
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.double
logDensity(double x)
Returns the natural logarithm of the probability density function (PDF) of this distribution evaluated at the specified pointx
.static WeibullDistribution
of(double shape, double scale)
Creates a Weibull 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
getMedian, isSupportConnected, probability
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Field Detail
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SUPPORT_LO
private static final double SUPPORT_LO
Support lower bound.- See Also:
- Constant Field Values
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SUPPORT_HI
private static final double SUPPORT_HI
Support upper bound.- See Also:
- Constant Field Values
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shape
private final double shape
The shape parameter.
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scale
private final double scale
The scale parameter.
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shapeOverScale
private final double shapeOverScale
shape / scale.
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logShapeOverScale
private final double logShapeOverScale
log(shape / scale).
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Method Detail
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of
public static WeibullDistribution of(double shape, double scale)
Creates a Weibull distribution.- Parameters:
shape
- Shape parameter.scale
- Scale parameter.- Returns:
- the distribution
- Throws:
java.lang.IllegalArgumentException
- ifshape <= 0
orscale <= 0
.
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getShape
public double getShape()
Gets the shape parameter of this distribution.- Returns:
- the shape 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.Returns the limit when
x = 0
:shape < 1
: Infinityshape == 1
: 1 / scaleshape > 1
: 0
- Parameters:
x
- Point at which the PDF is evaluated.- Returns:
- the value of the probability density function at
x
.
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logDensity
public double logDensity(double x)
Returns the natural logarithm of the probability density function (PDF) of this distribution evaluated at the specified pointx
.Returns the limit when
x = 0
:shape < 1
: Infinityshape == 1
: log(1 / scale)shape > 1
: -Infinity
- 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)
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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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
0
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
0
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.For shape parameter \( k \) and scale parameter \( \lambda \), the mean is:
\[ \lambda \, \Gamma(1+\frac{1}{k}) \]
where \( \Gamma \) is the Gamma-function.
- Returns:
- the mean.
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getVariance
public double getVariance()
Gets the variance of this distribution.For shape parameter \( k \) and scale parameter \( \lambda \), the variance is:
\[ \lambda^2 \left[ \Gamma\left(1+\frac{2}{k}\right) - \left(\Gamma\left(1+\frac{1}{k}\right)\right)^2 \right] \]
where \( \Gamma \) is the Gamma-function.
- Returns:
- the variance.
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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 0.
- Returns:
- 0.
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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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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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