Class LogNormalDistribution

java.lang.Object
org.apache.commons.statistics.distribution.AbstractContinuousDistribution
org.apache.commons.statistics.distribution.LogNormalDistribution
All Implemented Interfaces:
ContinuousDistribution
public final class LogNormalDistribution extends AbstractContinuousDistribution
Implementation of the log-normal distribution.

\( X \) 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:

  • Nested Class Summary

    Nested classes/interfaces inherited from interface ContinuousDistribution

    ContinuousDistribution.Sampler

  • Field Summary

    Fields
    Modifier and Type
    Field
    Description
    private final double
    logSigmaPlusHalfLog2Pi
    The value of log(sigma) + 0.5 * log(2*PI) stored for faster computation.
    private final double
    mu
    The mu parameter of this distribution.
    private final double
    sigma
    The sigma parameter of this distribution.
    private final double
    sigmaSqrt2
    Sigma multiplied by sqrt(2).
    private final double
    sigmaSqrt2Pi
    Sigma multiplied by sqrt(2 * pi).
    private static final double
    SQRT2PI
    √(2 π).

  • Constructor Summary

    Constructors
    Modifier
    Constructor
    Description
    private
    LogNormalDistribution(double mu, double sigma)
     

  • Method Summary

    Modifier and Type
    Method
    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
    getMu()
    Gets the mu parameter of this distribution.
    double
    getSigma()
    Gets the sigma 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.
    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 point x.
    static LogNormalDistribution
    of(double mu, double sigma)
    Creates a log-normal distribution.
    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
    survivalProbability(double x)
    For a random variable X whose values are distributed according to this distribution, this method returns P(X > x).

    Methods inherited from class AbstractContinuousDistribution

    getMedian, isSupportConnected

    Methods inherited from class Object

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

  • Field Details

    • SQRT2PI

      private static final double SQRT2PI
      √(2 π).

    • mu

      private final double mu
      The mu parameter of this distribution.

    • sigma

      private final double sigma
      The sigma parameter of this distribution.

    • logSigmaPlusHalfLog2Pi

      private final double logSigmaPlusHalfLog2Pi
      The value of log(sigma) + 0.5 * log(2*PI) stored for faster computation.

    • sigmaSqrt2

      private final double sigmaSqrt2
      Sigma multiplied by sqrt(2).

    • sigmaSqrt2Pi

      private final double sigmaSqrt2Pi
      Sigma multiplied by sqrt(2 * pi).

  • Constructor Details

    • 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.

  • Method Details

    • of

      public static LogNormalDistribution of(double mu, double sigma)
      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 - if sigma <= 0.

    • getMu

      public double getMu()
      Gets the mu 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.

    • getSigma

      public double getSigma()
      Gets the sigma 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.

    • 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 mu, and sigma s of this distribution, the PDF is given by

      • 0 if x <= 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.

    • 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)
      Specified by:
      probability in interface ContinuousDistribution
      Overrides:
      probability in class AbstractContinuousDistribution
      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.

    • 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 mu, and sigma s of this distribution, the CDF is given by

      • 0 if x <= 0,
      • 0 if ln(x) - mu < 0 and mu - ln(x) > 40 * s, as in these cases the actual value is within Double.MIN_VALUE of 0,
      • 1 if ln(x) - mu >= 0 and ln(x) - mu > 40 * s, as in these cases the actual value is within Double.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.

    • survivalProbability

      public double survivalProbability(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 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.

    • 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:

      \[ x = \begin{cases} \inf \{ x \in \mathbb R : P(X \le x) \ge p\} & \text{for } 0 \lt p \le 1 \\ \inf \{ x \in \mathbb R : P(X \le x) \gt 0 \} & \text{for } p = 0 \end{cases} \]

      The default implementation returns:

      Specified by:
      inverseCumulativeProbability in interface ContinuousDistribution
      Overrides:
      inverseCumulativeProbability in class AbstractContinuousDistribution
      Parameters:
      p - Cumulative probability.
      Returns:
      the smallest p-quantile of this distribution (largest 0-quantile for p = 0).

    • inverseSurvivalProbability

      public double inverseSurvivalProbability(double p)
      Computes the inverse survival probability function of this distribution. For a random variable X distributed according to this distribution, the returned value is:

      \[ x = \begin{cases} \inf \{ x \in \mathbb R : P(X \gt x) \le p\} & \text{for } 0 \le p \lt 1 \\ \inf \{ x \in \mathbb R : P(X \gt x) \lt 1 \} & \text{for } p = 1 \end{cases} \]

      By default, this is defined as inverseCumulativeProbability(1 - p), but the specific implementation may be more accurate.

      The default implementation returns:

      Specified by:
      inverseSurvivalProbability in interface ContinuousDistribution
      Overrides:
      inverseSurvivalProbability in class AbstractContinuousDistribution
      Parameters:
      p - Survival probability.
      Returns:
      the smallest (1-p)-quantile of this distribution (largest 0-quantile for p = 1).

    • 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.

    • 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.

    • getSupportLowerBound

      public double getSupportLowerBound()
      Gets the lower bound of the support. It must return the same value as inverseCumulativeProbability(0), i.e. \( \inf \{ x \in \mathbb R : P(X \le x) \gt 0 \} \).

      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 as inverseCumulativeProbability(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.

    • createSampler

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