Class TrapezoidalDistribution.RegularTrapezoidalDistribution
- All Implemented Interfaces:
ContinuousDistribution
- Enclosing class:
TrapezoidalDistribution
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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
Cached value (b - a).private final double
Cumulative probability at b.private final double
Cumulative probability at c.private final double
Cached value (d + c - a - b).private final double
Cached value (d - c).private final double
Survival probability at b.private final double
Survival probability at c.Fields inherited from class org.apache.commons.statistics.distribution.TrapezoidalDistribution
a, b, c, d
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Constructor Summary
Constructors -
Method Summary
Modifier and TypeMethodDescriptiondouble
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
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.private static double
nonCentralMoment
(int k, double b, double c) Compute thek
-th non-central moment of the standardized trapezoidal 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.TrapezoidalDistribution
getB, getC, getSupportLowerBound, getSupportUpperBound, of
Methods inherited from class org.apache.commons.statistics.distribution.AbstractContinuousDistribution
createSampler, getMedian, isSupportConnected, probability
Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
Methods inherited from interface org.apache.commons.statistics.distribution.ContinuousDistribution
logDensity
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Field Details
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divisor
private final double divisorCached value (d + c - a - b). -
bma
private final double bmaCached value (b - a). -
dmc
private final double dmcCached value (d - c). -
cdfB
private final double cdfBCumulative probability at b. -
cdfC
private final double cdfCCumulative probability at c. -
sfB
private final double sfBSurvival probability at b. -
sfC
private final double sfCSurvival probability at c.
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Constructor Details
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RegularTrapezoidalDistribution
RegularTrapezoidalDistribution(double a, double b, double c, double d) - Parameters:
a
- Lower limit of this distribution (inclusive).b
- Start of the trapezoid constant density.c
- End of the trapezoid constant density.d
- Upper limit of this distribution (inclusive).
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Method Details
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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 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
.
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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) Description copied from class:AbstractContinuousDistribution
Computes the inverse survival probability function of this distribution. For a random variableX
distributed according to this distribution, the returned value is:\[ x = \begin{cases} \inf \{ x \in \mathbb R : P(X \ge x) \le p\} & \text{for } 0 \le p \lt 1 \\ \inf \{ x \in \mathbb R : P(X \ge x) \lt 1 \} & \text{for } p = 1 \end{cases} \]
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:TrapezoidalDistribution
Gets the mean of this distribution.For lower limit \( a \), start of the density constant region \( b \), end of the density constant region \( c \) and upper limit \( d \), the mean is:
\[ \frac{1}{3(d+c-b-a)}\left(\frac{d^3-c^3}{d-c}-\frac{b^3-a^3}{b-a}\right) \]
- Specified by:
getMean
in interfaceContinuousDistribution
- Specified by:
getMean
in classTrapezoidalDistribution
- Returns:
- the mean.
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getVariance
public double getVariance()Description copied from class:TrapezoidalDistribution
Gets the variance of this distribution.For lower limit \( a \), start of the density constant region \( b \), end of the density constant region \( c \) and upper limit \( d \), the variance is:
\[ \frac{1}{6(d+c-b-a)}\left(\frac{d^4-c^4}{d-c}-\frac{b^4-a^4}{b-a}\right) - \mu^2 \]
where \( \mu \) is the mean.
- Specified by:
getVariance
in interfaceContinuousDistribution
- Specified by:
getVariance
in classTrapezoidalDistribution
- Returns:
- the variance.
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nonCentralMoment
private static double nonCentralMoment(int k, double b, double c) Compute thek
-th non-central moment of the standardized trapezoidal distribution.Shifting the distribution by scale
(d - a)
and locationa
creates a standardized trapezoidal distribution. This has a simplified non-central moment asa = 0, d = 1, 0 <= b < c <= 1
.2 1 ( 1 - c^(k+2) ) E[X^k] = ----------- -------------- ( ----------- - b^(k+1) ) (1 + c - b) (k + 1)(k + 2) ( 1 - c )
Simplification eliminates issues computing the moments when
a == b
orc == d
in the original (non-standardized) distribution.- Parameters:
k
- Moment to computeb
- Start of the trapezoid constant density (shape parameter in [0, 1]).c
- End of the trapezoid constant density (shape parameter in [0, 1]).- Returns:
- the moment
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