Class FoldedNormalDistribution.HalfNormalDistribution

All Implemented Interfaces:
ContinuousDistribution
Enclosing class:
FoldedNormalDistribution

private static class FoldedNormalDistribution.HalfNormalDistribution extends FoldedNormalDistribution
Specialisation for the half-normal distribution.

Elimination of the mu location parameter simplifies the probability functions and allows computation of the log density and inverse CDF/SF.

  • Field Details

    • VAR

      private static final double VAR
      Variance constant (1 - 2/pi). Computed using Matlab's VPA to 30 digits.
      See Also:
    • logSigmaPlusHalfLog2Pi

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

    • HalfNormalDistribution

      HalfNormalDistribution(double sigma)
      Parameters:
      sigma - Scale parameter.
  • Method Details

    • getMu

      public double getMu()
      Description copied from class: FoldedNormalDistribution
      Gets the location parameter \( \mu \) of this distribution.
      Specified by:
      getMu in class FoldedNormalDistribution
      Returns:
      the mu parameter.
    • density

      public double density(double x)
      Description copied from interface: ContinuousDistribution
      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.
      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)
      Description copied from class: AbstractContinuousDistribution
      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)
      Description copied from interface: ContinuousDistribution
      Returns the natural logarithm of the probability density function (PDF) of this distribution evaluated at the specified point x.
      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)
      Description copied from interface: ContinuousDistribution
      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.
      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)
      Description copied from interface: ContinuousDistribution
      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)
      Description copied from class: AbstractContinuousDistribution
      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()
      Description copied from class: FoldedNormalDistribution
      Gets the mean of this distribution.

      For location parameter \( \mu \) and scale parameter \( \sigma \), the mean is:

      \[ \sigma \sqrt{ \frac 2 \pi } \exp \left( \frac{-\mu^2}{2\sigma^2} \right) + \mu \operatorname{erf} \left( \frac \mu {\sqrt{2\sigma^2}} \right) \]

      where \( \operatorname{erf} \) is the error function.

      Specified by:
      getMean in interface ContinuousDistribution
      Specified by:
      getMean in class FoldedNormalDistribution
      Returns:
      the mean.
    • getVariance

      public double getVariance()
      Description copied from class: FoldedNormalDistribution
      Gets the variance of this distribution.

      For location parameter \( \mu \), scale parameter \( \sigma \) and a distribution mean \( \mu_Y \), the variance is:

      \[ \mu^2 + \sigma^2 - \mu_{Y}^2 \]

      Specified by:
      getVariance in interface ContinuousDistribution
      Specified by:
      getVariance in class FoldedNormalDistribution
      Returns:
      the variance.
    • createSampler

      public ContinuousDistribution.Sampler createSampler(org.apache.commons.rng.UniformRandomProvider rng)
      Description copied from class: AbstractContinuousDistribution
      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.