Package org.apache.commons.statistics.distribution

Class TruncatedNormalDistribution

java.lang.Object
org.apache.commons.statistics.distribution.TruncatedNormalDistribution
All Implemented Interfaces:
ContinuousDistribution
public final class TruncatedNormalDistribution extends Object
Implementation of the truncated normal distribution.

The probability density function of \( X \) is:

\[ f(x;\mu,\sigma,a,b) = \frac{1}{\sigma}\,\frac{\phi(\frac{x - \mu}{\sigma})}{\Phi(\frac{b - \mu}{\sigma}) - \Phi(\frac{a - \mu}{\sigma}) } \]

for \( \mu \) mean of the parent normal distribution, \( \sigma \) standard deviation of the parent normal distribution, \( -\infty \le a \lt b \le \infty \) the truncation interval, and \( x \in [a, b] \), where \( \phi \) is the probability density function of the standard normal distribution and \( \Phi \) is its cumulative distribution function.

See Also:

  • Method Details

    • of

      public static TruncatedNormalDistribution of(double mean, double sd, double lower, double upper)
      Creates a truncated normal distribution.

      Note that the mean and sd is of the parent normal distribution, and not the true mean and standard deviation of the truncated normal distribution. The lower and upper bounds define the truncation of the parent normal distribution.

      Parameters:
      mean - Mean for the parent distribution.
      sd - Standard deviation for the parent distribution.
      lower - Lower bound (inclusive) of the distribution, can be Double.NEGATIVE_INFINITY.
      upper - Upper bound (inclusive) of the distribution, can be Double.POSITIVE_INFINITY.
      Returns:
      the distribution
      Throws:
      IllegalArgumentException - if sd <= 0; if lower >= upper; or if the truncation covers no probability range in the parent distribution.

    • 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.
      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
      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.
      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.
      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
      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
      Parameters:
      p - Survival probability.
      Returns:
      the smallest (1-p)-quantile of this distribution (largest 0-quantile for p = 1).

    • 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.

    • getMean

      public double getMean()
      Gets the mean of this distribution.

      Represents the true mean of the truncated normal distribution rather than the parent normal distribution mean.

      For \( \mu \) mean of the parent normal distribution, \( \sigma \) standard deviation of the parent normal distribution, and \( a \lt b \) the truncation interval of the parent normal distribution, the mean is:

      \[ \mu + \frac{\phi(a)-\phi(b)}{\Phi(b) - \Phi(a)}\sigma \]

      where \( \phi \) is the probability density function of the standard normal distribution and \( \Phi \) is its cumulative distribution function.

      Returns:
      the mean.

    • getVariance

      public double getVariance()
      Gets the variance of this distribution.

      Represents the true variance of the truncated normal distribution rather than the parent normal distribution variance.

      For \( \mu \) mean of the parent normal distribution, \( \sigma \) standard deviation of the parent normal distribution, and \( a \lt b \) the truncation interval of the parent normal distribution, the variance is:

      \[ \sigma^2 \left[1 + \frac{a\phi(a)-b\phi(b)}{\Phi(b) - \Phi(a)} - \left( \frac{\phi(a)-\phi(b)}{\Phi(b) - \Phi(a)} \right)^2 \right] \]

      where \( \phi \) is the probability density function of the standard normal distribution and \( \Phi \) is its cumulative distribution function.

      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 equal to the lower bound parameter of the distribution.

      Returns:
      the lower bound of the support.

    • 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 equal to the upper bound parameter of the distribution.

      Returns:
      the upper bound of the support.