Class LogNormalDistribution
- All Implemented Interfaces:
ContinuousDistribution
\( X \) is log-normally distributed if its natural logarithm \( \ln(x) \) is normally distributed. The probability density function of \( X \) is:
\[ f(x; \mu, \sigma) = \frac 1 {x\sigma\sqrt{2\pi\,}} e^{-{\frac 1 2}\left( \frac{\ln x-\mu}{\sigma} \right)^2 } \]
for \( \mu \) the mean of the normally distributed natural logarithm of this distribution, \( \sigma > 0 \) the standard deviation of the normally distributed natural logarithm of this distribution, and \( x \in (0, \infty) \).
- See Also:
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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 static final double
0.5 * ln(2 * pi).private final double
The value oflog(sigma) + 0.5 * log(2*PI)
stored for faster computation.private final double
The mu parameter of this distribution.private final double
The sigma parameter of this distribution.private final double
Sigma multiplied by sqrt(2).private final double
Sigma multiplied by sqrt(2 * pi).private static final double
√(2 π). -
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 themu
parameter of this distribution.double
getSigma()
Gets thesigma
parameter of this distribution.double
Gets the lower bound of the support.double
Gets the upper bound of the support.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
.static LogNormalDistribution
of
(double mu, double sigma) Creates a log-normal distribution.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.AbstractContinuousDistribution
getMedian, isSupportConnected
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Field Details
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HALF_LOG_TWO_PI
private static final double HALF_LOG_TWO_PI0.5 * ln(2 * pi). Computed to 25-digits precision.- See Also:
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SQRT2PI
private static final double SQRT2PI√(2 π). -
mu
private final double muThe mu parameter of this distribution. -
sigma
private final double sigmaThe sigma parameter of this distribution. -
logSigmaPlusHalfLog2Pi
private final double logSigmaPlusHalfLog2PiThe value oflog(sigma) + 0.5 * log(2*PI)
stored for faster computation. -
sigmaSqrt2
private final double sigmaSqrt2Sigma multiplied by sqrt(2). -
sigmaSqrt2Pi
private final double sigmaSqrt2PiSigma multiplied by sqrt(2 * pi).
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Constructor Details
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LogNormalDistribution
private LogNormalDistribution(double mu, double sigma) - Parameters:
mu
- Mean of the natural logarithm of the distribution values.sigma
- Standard deviation of the natural logarithm of the distribution values.
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Method Details
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of
Creates a log-normal distribution.- Parameters:
mu
- Mean of the natural logarithm of the distribution values.sigma
- Standard deviation of the natural logarithm of the distribution values.- Returns:
- the distribution
- Throws:
IllegalArgumentException
- ifsigma <= 0
.
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getMu
public double getMu()Gets themu
parameter of this distribution. This is the mean of the natural logarithm of the distribution values, not the mean of distribution.- Returns:
- the mu parameter.
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getSigma
public double getSigma()Gets thesigma
parameter of this distribution. This is the standard deviation of the natural logarithm of the distribution values, not the standard deviation of distribution.- Returns:
- the sigma 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 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.For
mu
, and sigmas
of this distribution, the PDF is given by0
ifx <= 0
,exp(-0.5 * ((ln(x) - mu) / s)^2) / (s * sqrt(2 * pi) * x)
otherwise.
- 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) 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) Returns the natural logarithm of the probability density function (PDF) of this distribution evaluated at the specified pointx
.See documentation of
density(double)
for computation details.- 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.For
mu
, and sigmas
of this distribution, the CDF is given by0
ifx <= 0
,0
ifln(x) - mu < 0
andmu - ln(x) > 40 * s
, as in these cases the actual value is withinDouble.MIN_VALUE
of 0,1
ifln(x) - mu >= 0
andln(x) - mu > 40 * s
, as in these cases the actual value is withinDouble.MIN_VALUE
of 1,0.5 + 0.5 * erf((ln(x) - mu) / (s * sqrt(2))
otherwise.
- 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.
- 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:\[ x = \begin{cases} \inf \{ x \in \mathbb R : P(X \ge x) \le p\} & \text{for } 0 \le p \lt 1 \\ \inf \{ x \in \mathbb R : P(X \ge x) \lt 1 \} & \text{for } p = 1 \end{cases} \]
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()Gets the mean of this distribution.For \( \mu \) the mean of the normally distributed natural logarithm of this distribution, \( \sigma > 0 \) the standard deviation of the normally distributed natural logarithm of this distribution, the mean is:
\[ \exp(\mu + \frac{\sigma^2}{2}) \]
- Returns:
- the mean.
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getVariance
public double getVariance()Gets the variance of this distribution.For \( \mu \) the mean of the normally distributed natural logarithm of this distribution, \( \sigma > 0 \) the standard deviation of the normally distributed natural logarithm of this distribution, the variance is:
\[ [\exp(\sigma^2) - 1)] \exp(2 \mu + \sigma^2) \]
- 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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