Package org.apache.commons.statistics.distribution

Class TrapezoidalDistribution

java.lang.Object
org.apache.commons.statistics.distribution.TrapezoidalDistribution
All Implemented Interfaces:
ContinuousDistribution
public abstract class TrapezoidalDistribution extends Object
Implementation of the trapezoidal distribution.

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

\[ f(x; a, b, c, d) = \begin{cases} \frac{2}{d+c-a-b}\frac{x-a}{b-a} & \text{for } a\le x \lt b \\ \frac{2}{d+c-a-b} & \text{for } b\le x \lt c \\ \frac{2}{d+c-a-b}\frac{d-x}{d-c} & \text{for } c\le x \le d \end{cases} \]

for \( -\infty \lt a \le b \le c \le d \lt \infty \) and \( x \in [a, d] \).

Note the special cases:

  • \( b = c \) is the triangular distribution
  • \( a = b \) and \( c = d \) is the uniform distribution
See Also:

  • Field Details

    • a

      protected final double a
      Lower limit of this distribution (inclusive).

    • b

      protected final double b
      Start of the trapezoid constant density.

    • c

      protected final double c
      End of the trapezoid constant density.

    • d

      protected final double d
      Upper limit of this distribution (inclusive).

  • Method Details

    • of

      public static TrapezoidalDistribution of(double a, double b, double c, double d)
      Creates a trapezoidal distribution.

      The distribution density is represented as an up sloping line from a to b, constant from b to c, and then a down sloping line from c to d.

      Parameters:
      a - Lower limit of this distribution (inclusive).
      b - Start of the trapezoid constant density (first shape parameter).
      c - End of the trapezoid constant density (second shape parameter).
      d - Upper limit of this distribution (inclusive).
      Returns:
      the distribution
      Throws:
      IllegalArgumentException - if a >= d, if b < a, if c < b or if c > d.

    • getMean

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

      For lower limit \( a \), start of the density constant region \( b \), end of the density constant region \( c \) and upper limit \( d \), the mean is:

      \[ \frac{1}{3(d+c-b-a)}\left(\frac{d^3-c^3}{d-c}-\frac{b^3-a^3}{b-a}\right) \]

      Returns:
      the mean.

    • getVariance

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

      For lower limit \( a \), start of the density constant region \( b \), end of the density constant region \( c \) and upper limit \( d \), the variance is:

      \[ \frac{1}{6(d+c-b-a)}\left(\frac{d^4-c^4}{d-c}-\frac{b^4-a^4}{b-a}\right) - \mu^2 \]

      where \( \mu \) is the mean.

      Returns:
      the variance.

    • getB

      public double getB()
      Gets the start of the constant region of the density function.

      This is the first shape parameter b of the distribution.

      Returns:
      the first shape parameter b

    • getC

      public double getC()
      Gets the end of the constant region of the density function.

      This is the second shape parameter c of the distribution.

      Returns:
      the second shape parameter c

    • 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 limit parameter a 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 limit parameter d of the distribution.

      Returns:
      the upper bound of the support.

    • 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.

    • 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).
      Throws:
      IllegalArgumentException - if p < 0 or p > 1

    • 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).
      Throws:
      IllegalArgumentException - if p < 0 or 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.