org.apache.commons.statistics.distribution

## Class LogNormalDistribution

• java.lang.Object
• org.apache.commons.statistics.distribution.LogNormalDistribution
• All Implemented Interfaces:
ContinuousDistribution

public class LogNormalDistribution
extends Object
Implementation of the log-normal 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.

• ### Nested classes/interfaces inherited from interface org.apache.commons.statistics.distribution.ContinuousDistribution

ContinuousDistribution.Sampler
• ### Constructor Summary

Constructors
Constructor and Description
LogNormalDistribution(double scale, double shape)
Creates a log-normal distribution.
• ### Method Summary

All Methods
Modifier and Type Method and Description
ContinuousDistribution.Sampler createSampler(org.apache.commons.rng.UniformRandomProvider rng)
Creates a sampler.
double cumulativeProbability(double x)
For a random variable X whose values are distributed according to this distribution, this method returns P(X <= x).
double density(double x)
Returns the probability density function (PDF) of this distribution evaluated at the specified point x.
double getMean()
Gets the mean of this distribution.
double getScale()
Returns the scale parameter of this distribution.
double getShape()
Returns 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.
boolean isSupportConnected()
Indicates whether the support is connected, i.e.
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).
static double[] sample(int n, ContinuousDistribution.Sampler sampler)
Utility function for allocating an array and filling it with n samples generated by the given sampler.
• ### 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

probability
• ### Constructor Detail

• #### LogNormalDistribution

public LogNormalDistribution(double scale,
double shape)
Creates a log-normal distribution.
Parameters:
scale - Scale parameter of this distribution.
shape - Shape parameter of this distribution.
Throws:
IllegalArgumentException - if shape <= 0.
• ### 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 - 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 point x. 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.
• #### 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 - 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.
• #### probability

public 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). The default implementation uses the identity P(x0 < X <= x1) = P(X <= x1) - P(X <= x0)
Parameters:
x0 - Lower bound (exclusive).
x1 - Upper bound (inclusive).
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.
• #### getMean

public double getMean()
Gets 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.
• #### getVariance

public double getVariance()
Gets 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, or Double.NaN if it is not defined.
• #### getSupportLowerBound

public double getSupportLowerBound()
Gets the lower bound of the support. It must return the same value as inverseCumulativeProbability(0), i.e. 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()
Gets the upper bound of the support. It must return the same value as inverseCumulativeProbability(1), i.e. 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)
• #### isSupportConnected

public boolean isSupportConnected()
Indicates 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
• #### createSampler

public ContinuousDistribution.Sampler createSampler(org.apache.commons.rng.UniformRandomProvider rng)
Creates a sampler.
Specified by:
createSampler in interface ContinuousDistribution
Parameters:
rng - Generator of uniformly distributed numbers.
Returns:
a sampler that produces random numbers according this distribution.
• #### inverseCumulativeProbability

public double inverseCumulativeProbability(double p)
Computes the quantile function of this distribution. For a random variable X distributed according to this distribution, the returned value is
• inf{x in R | P(X<=x) >= p} for 0 < p <= 1,
• inf{x in R | P(X<=x) > 0} for p = 0.
The default implementation returns
Specified by:
inverseCumulativeProbability in interface ContinuousDistribution
Parameters:
p - Cumulative probability.
Returns:
the smallest p-quantile of this distribution (largest 0-quantile for p = 0).
• #### sample

public static double[] sample(int n,
ContinuousDistribution.Sampler sampler)
Utility function for allocating an array and filling it with n samples generated by the given sampler.
Parameters:
n - Number of samples.
sampler - Sampler.
Returns:
an array of size n.