org.apache.commons.math3.distribution

Class LogNormalDistribution

java.lang.Object

org.apache.commons.math3.distribution.AbstractRealDistribution

org.apache.commons.math3.distribution.LogNormalDistribution

All Implemented Interfaces:

java.io.Serializable, RealDistribution

public class LogNormalDistribution
extends AbstractRealDistribution

Implementation of the log-normal (gaussian) distribution.

Parameters: X is log-normally distributed if its natural logarithm log(X) is normally distributed. The probability distribution function of X is given by (for x > 0)

exp(-0.5 * ((ln(x) - m) / s)^2) / (s * sqrt(2 * pi) * x)

• m is the scale parameter: this is the mean of the normally distributed natural logarithm of this distribution,
• s is the shape parameter: this is the standard deviation of the normally distributed natural logarithm of this distribution.

Since:

3.0

See Also:

Log-normal distribution (Wikipedia), Log Normal distribution (MathWorld), Serialized Form

Field Summary

Fields

Modifier and Type	Field and Description
static double	DEFAULT_INVERSE_ABSOLUTE_ACCURACY Default inverse cumulative probability accuracy.
private double	logShapePlusHalfLog2Pi The value of log(shape) + 0.5 * log(2*PI) stored for faster computation.
private double	scale The scale parameter of this distribution.
private static long	serialVersionUID Serializable version identifier.
private double	shape The shape parameter of this distribution.
private double	solverAbsoluteAccuracy Inverse cumulative probability accuracy.
private static double	SQRT2 √(2)
private static double	SQRT2PI √(2 π)

Fields inherited from class org.apache.commons.math3.distribution.AbstractRealDistribution

random, randomData, SOLVER_DEFAULT_ABSOLUTE_ACCURACY

Constructor Summary

Constructors

Constructor and Description
LogNormalDistribution() Create a log-normal distribution, where the mean and standard deviation of the normally distributed natural logarithm of the log-normal distribution are equal to zero and one respectively.
LogNormalDistribution(double scale, double shape) Create a log-normal distribution using the specified scale and shape.
LogNormalDistribution(double scale, double shape, double inverseCumAccuracy) Create a log-normal distribution using the specified scale, shape and inverse cumulative distribution accuracy.
LogNormalDistribution(RandomGenerator rng, double scale, double shape) Creates a log-normal distribution.
LogNormalDistribution(RandomGenerator rng, double scale, double shape, double inverseCumAccuracy) Creates a log-normal distribution.

Method Summary

Methods

Modifier and Type	Method and Description
double	cumulativeProbability(double x) For a random variable X whose values are distributed according to this distribution, this method returns P(X <= x).
double	cumulativeProbability(double x0, double x1) Deprecated. See RealDistribution.cumulativeProbability(double,double)
double	density(double x) Returns the probability density function (PDF) of this distribution evaluated at the specified point x.
double	getNumericalMean() Use this method to get the numerical value of the mean of this distribution.
double	getNumericalVariance() Use this method to get the numerical value of the variance of this distribution.
double	getScale() Returns the scale parameter of this distribution.
double	getShape() Returns the shape parameter of this distribution.
protected double	getSolverAbsoluteAccuracy() Returns the solver absolute accuracy for inverse cumulative computation.
double	getSupportLowerBound() Access the lower bound of the support.
double	getSupportUpperBound() Access the upper bound of the support.
boolean	isSupportConnected() Use this method to get information about whether the support is connected, i.e.
boolean	isSupportLowerBoundInclusive() Whether or not the lower bound of support is in the domain of the density function.
boolean	isSupportUpperBoundInclusive() Whether or not the upper bound of support is in the domain of the density function.
double	logDensity(double x) Returns the natural logarithm of the probability density function (PDF) of this distribution evaluated at the specified point x.
double	probability(double x0, double x1) For a random variable X whose values are distributed according to this distribution, this method returns P(x0 < X <= x1).
double	sample() Generate a random value sampled from this distribution.

Methods inherited from class org.apache.commons.math3.distribution.AbstractRealDistribution

inverseCumulativeProbability, probability, reseedRandomGenerator, sample

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Detail

DEFAULT_INVERSE_ABSOLUTE_ACCURACY

public static final double DEFAULT_INVERSE_ABSOLUTE_ACCURACY

Default inverse cumulative probability accuracy.

See Also:

Constant Field Values

serialVersionUID

private static final long serialVersionUID

Serializable version identifier.

See Also:

Constant Field Values

SQRT2PI

private static final double SQRT2PI

√(2 π)

SQRT2

private static final double SQRT2

√(2)

scale

private final double scale

The scale parameter of this distribution.

shape

private final double shape

The shape parameter of this distribution.

logShapePlusHalfLog2Pi

private final double logShapePlusHalfLog2Pi

The value of log(shape) + 0.5 * log(2*PI) stored for faster computation.

solverAbsoluteAccuracy

private final double solverAbsoluteAccuracy

Inverse cumulative probability accuracy.

Constructor Detail

LogNormalDistribution

public LogNormalDistribution()

Create a log-normal distribution, where the mean and standard deviation of the normally distributed natural logarithm of the log-normal distribution are equal to zero and one respectively. In other words, the scale of the returned distribution is 0, while its shape is 1.

Note: this constructor will implicitly create an instance of Well19937c as random generator to be used for sampling only (see sample() and AbstractRealDistribution.sample(int)). In case no sampling is needed for the created distribution, it is advised to pass null as random generator via the appropriate constructors to avoid the additional initialisation overhead.

LogNormalDistribution

public LogNormalDistribution(double scale,
                     double shape)
                      throws NotStrictlyPositiveException

Create a log-normal distribution using the specified scale and shape.

Note: this constructor will implicitly create an instance of Well19937c as random generator to be used for sampling only (see sample() and AbstractRealDistribution.sample(int)). In case no sampling is needed for the created distribution, it is advised to pass null as random generator via the appropriate constructors to avoid the additional initialisation overhead.

Parameters:

scale - the scale parameter of this distribution

shape - the shape parameter of this distribution

Throws:

NotStrictlyPositiveException - if shape <= 0.

LogNormalDistribution

public LogNormalDistribution(double scale,
                     double shape,
                     double inverseCumAccuracy)
                      throws NotStrictlyPositiveException

Create a log-normal distribution using the specified scale, shape and inverse cumulative distribution accuracy.

Note: this constructor will implicitly create an instance of Well19937c as random generator to be used for sampling only (see sample() and AbstractRealDistribution.sample(int)). In case no sampling is needed for the created distribution, it is advised to pass null as random generator via the appropriate constructors to avoid the additional initialisation overhead.

Parameters:

scale - the scale parameter of this distribution

shape - the shape parameter of this distribution

inverseCumAccuracy - Inverse cumulative probability accuracy.

Throws:

NotStrictlyPositiveException - if shape <= 0.

LogNormalDistribution

public LogNormalDistribution(RandomGenerator rng,
                     double scale,
                     double shape)
                      throws NotStrictlyPositiveException

Creates a log-normal distribution.

Parameters:

rng - Random number generator.

scale - Scale parameter of this distribution.

shape - Shape parameter of this distribution.

Throws:

NotStrictlyPositiveException - if shape <= 0.

Since:

3.3

LogNormalDistribution

public LogNormalDistribution(RandomGenerator rng,
                     double scale,
                     double shape,
                     double inverseCumAccuracy)
                      throws NotStrictlyPositiveException

Creates a log-normal distribution.

Parameters:

rng - Random number generator.

scale - Scale parameter of this distribution.

shape - Shape parameter of this distribution.

inverseCumAccuracy - Inverse cumulative probability accuracy.

Throws:

NotStrictlyPositiveException - if shape <= 0.

Since:

3.1

Method Detail

getScale

public double getScale()

Returns the scale parameter of this distribution.

Returns:

the scale parameter

getShape

public double getShape()

Returns the shape parameter of this distribution.

Returns:

the shape parameter

density

public double density(double x)

Returns the probability density function (PDF) of this distribution evaluated at the specified point x. In general, the PDF is the derivative of the CDF. If the derivative does not exist at x, 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 scale m, and shape s of this distribution, the PDF is given by

• 0 if x <= 0,
• exp(-0.5 * ((ln(x) - m) / s)^2) / (s * sqrt(2 * pi) * x) otherwise.

Parameters:

x - the point at which the PDF is evaluated

Returns:

the value of the probability density function at point x

logDensity

public double logDensity(double x)

Returns the natural logarithm of the probability density function (PDF) of this distribution evaluated at the specified point x. In general, the PDF is the derivative of the CDF. If the derivative does not exist at x, 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. Note that due to the floating point precision and under/overflow issues, this method will for some distributions be more precise and faster than computing the logarithm of RealDistribution.density(double). The default implementation simply computes the logarithm of density(x). See documentation of density(double) for computation details.

Overrides:

logDensity in class AbstractRealDistribution

Parameters:

x - the point at which the PDF is evaluated

Returns:

the logarithm of the value of the probability density function at point x

cumulativeProbability

public double cumulativeProbability(double x)

For a random variable X whose values are distributed according to this distribution, this method returns P(X <= x). In other words, this method represents the (cumulative) distribution function (CDF) for this distribution. For scale m, and shape s of this distribution, the CDF is given by

• 0 if x <= 0,
• 0 if ln(x) - m < 0 and m - ln(x) > 40 * s, as in these cases the actual value is within Double.MIN_VALUE of 0,
• 1 if ln(x) - m >= 0 and ln(x) - m > 40 * s, as in these cases the actual value is within Double.MIN_VALUE of 1,
• 0.5 + 0.5 * erf((ln(x) - m) / (s * sqrt(2)) otherwise.

Parameters:

x - the 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

cumulativeProbability

@Deprecated
public double cumulativeProbability(double x0,
                                      double x1)
                             throws NumberIsTooLargeException

Deprecated. See RealDistribution.cumulativeProbability(double,double)

For a random variable X whose values are distributed according to this distribution, this method returns P(x0 < X <= x1). The default implementation uses the identity

P(x0 < X <= x1) = P(X <= x1) - P(X <= x0)

Specified by:

cumulativeProbability in interface RealDistribution

Overrides:

cumulativeProbability in class AbstractRealDistribution

Parameters:

x0 - the exclusive lower bound

x1 - the inclusive upper bound

Returns:

the probability that a random variable with this distribution takes a value between x0 and x1, excluding the lower and including the upper endpoint

Throws:

NumberIsTooLargeException - if x0 > x1

probability

public double probability(double x0,
                 double x1)
                   throws NumberIsTooLargeException

For a random variable X whose values are distributed according to this distribution, this method returns P(x0 < X <= x1).

Overrides:

probability in class AbstractRealDistribution

Parameters:

x0 - Lower bound (excluded).

x1 - Upper bound (included).

Returns:

the probability that a random variable with this distribution takes a value between x0 and x1, excluding the lower and including the upper endpoint.

Throws:

NumberIsTooLargeException - if x0 > x1. The default implementation uses the identity P(x0 < X <= x1) = P(X <= x1) - P(X <= x0)

getSolverAbsoluteAccuracy

protected double getSolverAbsoluteAccuracy()

Returns the solver absolute accuracy for inverse cumulative computation. You can override this method in order to use a Brent solver with an absolute accuracy different from the default.

Overrides:

getSolverAbsoluteAccuracy in class AbstractRealDistribution

Returns:

the maximum absolute error in inverse cumulative probability estimates

getNumericalMean

public double getNumericalMean()

Use this method to get the numerical value of the mean of this distribution. For scale m and shape s, the mean is exp(m + s^2 / 2).

Returns:

the mean or Double.NaN if it is not defined

getNumericalVariance

public double getNumericalVariance()

Use this method to get the numerical value of the variance of this distribution. For scale m and shape s, the variance is (exp(s^2) - 1) * exp(2 * m + s^2).

Returns:

the variance (possibly Double.POSITIVE_INFINITY as for certain cases in TDistribution) or Double.NaN if it is not defined

getSupportLowerBound

public double getSupportLowerBound()

Access the lower bound of the support. This method must return the same value as inverseCumulativeProbability(0). In other words, this method must return

inf {x in R | P(X <= x) > 0}.

The lower bound of the support is always 0 no matter the parameters.

Returns:

lower bound of the support (always 0)

getSupportUpperBound

public double getSupportUpperBound()

Access the upper bound of the support. This method must return the same value as inverseCumulativeProbability(1). In other words, this method must return

inf {x in R | P(X <= x) = 1}.

The upper bound of the support is always positive infinity no matter the parameters.

Returns:

upper bound of the support (always Double.POSITIVE_INFINITY)

isSupportLowerBoundInclusive

public boolean isSupportLowerBoundInclusive()

Whether or not the lower bound of support is in the domain of the density function. Returns true iff getSupporLowerBound() is finite and density(getSupportLowerBound()) returns a non-NaN, non-infinite value.

Returns:

true if the lower bound of support is finite and the density function returns a non-NaN, non-infinite value there

isSupportUpperBoundInclusive

public boolean isSupportUpperBoundInclusive()

Whether or not the upper bound of support is in the domain of the density function. Returns true iff getSupportUpperBound() is finite and density(getSupportUpperBound()) returns a non-NaN, non-infinite value.

Returns:

true if the upper bound of support is finite and the density function returns a non-NaN, non-infinite value there

isSupportConnected

public boolean isSupportConnected()

Use this method to get information about whether the support is connected, i.e. whether all values between the lower and upper bound of the support are included in the support. The support of this distribution is connected.

Returns:

true

sample

public double sample()

Generate a random value sampled from this distribution. The default implementation uses the inversion method.

Specified by:

sample in interface RealDistribution

Overrides:

sample in class AbstractRealDistribution

Returns:

a random value.