Class FDistribution
- java.lang.Object
-
- org.apache.commons.statistics.distribution.AbstractContinuousDistribution
-
- org.apache.commons.statistics.distribution.FDistribution
-
- All Implemented Interfaces:
ContinuousDistribution
public final class FDistribution extends AbstractContinuousDistribution
Implementation of the F-distribution.The probability density function of \( X \) is:
\[ \begin{aligned} f(x; n, m) &= \frac{1}{\operatorname{B}\left(\frac{n}{2},\frac{m}{2}\right)} \left(\frac{n}{m}\right)^{n/2} x^{n/2 - 1} \left(1+\frac{n}{m} \, x \right)^{-(n+m)/2} \\ &= \frac{n^{\frac n 2} m^{\frac m 2} x^{\frac{n}{2}-1} }{ (nx+m)^{\frac{(n+m)}{2}} \operatorname{B}\left(\frac{n}{2},\frac{m}{2}\right)} \end{aligned} \]
for \( n, m > 0 \) the degrees of freedom, \( \operatorname{B}(a, b) \) is the beta function, and \( x \in [0, \infty) \).
-
-
Nested Class Summary
-
Nested classes/interfaces inherited from interface org.apache.commons.statistics.distribution.ContinuousDistribution
ContinuousDistribution.Sampler
-
-
Field Summary
Fields Modifier and Type Field Description private double
denominatorDegreesOfFreedom
The denominator degrees of freedom.private double
logBetaNhalfMhalf
LogBeta(n/2, n/2) with n = numerator DF.private double
mean
Cached value for inverse probability function.private static double
MIN_DENOMINATOR_DF_FOR_MEAN
The minimum degrees of freedom for the denominator when computing the mean.private static double
MIN_DENOMINATOR_DF_FOR_VARIANCE
The minimum degrees of freedom for the denominator when computing the variance.private double
nHalfLogNmHalfLogM
n/2 * log(n) + m/2 * log(m) with n = numerator DF and m = denominator DF.private double
numeratorDegreesOfFreedom
The numerator degrees of freedom.private static double
SUPPORT_HI
Support upper bound.private static double
SUPPORT_LO
Support lower bound.private double
variance
Cached value for inverse probability function.
-
Constructor Summary
Constructors Modifier Constructor Description private
FDistribution(double numeratorDegreesOfFreedom, double denominatorDegreesOfFreedom)
-
Method Summary
All Methods Static Methods Instance Methods Concrete Methods Modifier and Type Method Description private double
computeDensity(double x, boolean log)
Compute the density at point x.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
getDenominatorDegreesOfFreedom()
Gets the denominator degrees of freedom parameter of this distribution.double
getMean()
Gets the mean of this distribution.double
getNumeratorDegreesOfFreedom()
Gets the numerator degrees of freedom 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
logDensity(double x)
Returns the natural logarithm of the probability density function (PDF) of this distribution evaluated at the specified pointx
.static FDistribution
of(double numeratorDegreesOfFreedom, double denominatorDegreesOfFreedom)
Creates an F-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
createSampler, getMedian, inverseCumulativeProbability, inverseSurvivalProbability, isSupportConnected, probability
-
-
-
-
Field Detail
-
SUPPORT_LO
private static final double SUPPORT_LO
Support lower bound.- See Also:
- Constant Field Values
-
SUPPORT_HI
private static final double SUPPORT_HI
Support upper bound.- See Also:
- Constant Field Values
-
MIN_DENOMINATOR_DF_FOR_MEAN
private static final double MIN_DENOMINATOR_DF_FOR_MEAN
The minimum degrees of freedom for the denominator when computing the mean.- See Also:
- Constant Field Values
-
MIN_DENOMINATOR_DF_FOR_VARIANCE
private static final double MIN_DENOMINATOR_DF_FOR_VARIANCE
The minimum degrees of freedom for the denominator when computing the variance.- See Also:
- Constant Field Values
-
numeratorDegreesOfFreedom
private final double numeratorDegreesOfFreedom
The numerator degrees of freedom.
-
denominatorDegreesOfFreedom
private final double denominatorDegreesOfFreedom
The denominator degrees of freedom.
-
nHalfLogNmHalfLogM
private final double nHalfLogNmHalfLogM
n/2 * log(n) + m/2 * log(m) with n = numerator DF and m = denominator DF.
-
logBetaNhalfMhalf
private final double logBetaNhalfMhalf
LogBeta(n/2, n/2) with n = numerator DF.
-
mean
private final double mean
Cached value for inverse probability function.
-
variance
private final double variance
Cached value for inverse probability function.
-
-
Method Detail
-
of
public static FDistribution of(double numeratorDegreesOfFreedom, double denominatorDegreesOfFreedom)
Creates an F-distribution.- Parameters:
numeratorDegreesOfFreedom
- Numerator degrees of freedom.denominatorDegreesOfFreedom
- Denominator degrees of freedom.- Returns:
- the distribution
- Throws:
java.lang.IllegalArgumentException
- ifnumeratorDegreesOfFreedom <= 0
ordenominatorDegreesOfFreedom <= 0
.
-
getNumeratorDegreesOfFreedom
public double getNumeratorDegreesOfFreedom()
Gets the numerator degrees of freedom parameter of this distribution.- Returns:
- the numerator degrees of freedom.
-
getDenominatorDegreesOfFreedom
public double getDenominatorDegreesOfFreedom()
Gets the denominator degrees of freedom parameter of this distribution.- Returns:
- the denominator degrees of freedom.
-
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
:df1 < 2
: Infinitydf1 == 2
: 1df1 > 2
: 0
Where
df1
is the numerator degrees of freedom.- 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
.Returns the limit when
x = 0
:df1 < 2
: Infinitydf1 == 2
: 0df1 > 2
: -Infinity
Where
df1
is the numerator degrees of freedom.- Parameters:
x
- Point at which the PDF is evaluated.- Returns:
- the logarithm of the value of the probability density function
at
x
.
-
computeDensity
private double computeDensity(double x, boolean log)
Compute the density at point x. Assumes x is within the support bound.- Parameters:
x
- Valuelog
- true to compute the log density- Returns:
- pdf(x) or logpdf(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 denominator degrees of freedom parameter \( m \), the mean is:
\[ \mathbb{E}[X] = \begin{cases} \frac{m}{m-2} & \text{for } m \gt 2 \\ \text{undefined} & \text{otherwise} \end{cases} \]
- Returns:
- the mean, or
NaN
if it is not defined.
-
getVariance
public double getVariance()
Gets the variance of this distribution.For numerator degrees of freedom parameter \( n \) and denominator degrees of freedom parameter \( m \), the variance is:
\[ \operatorname{var}[X] = \begin{cases} \frac{2m^2 (n+m-2)}{n (m-2)^2 (m-4)} & \text{for } m \gt 4 \\ \text{undefined} & \text{otherwise} \end{cases} \]
- Returns:
- the variance, or
NaN
if it is not defined.
-
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.
-
-