Class TrapezoidalDistribution.RegularTrapezoidalDistribution

java.lang.Object
org.apache.commons.statistics.distribution.AbstractContinuousDistribution
org.apache.commons.statistics.distribution.TrapezoidalDistribution
org.apache.commons.statistics.distribution.TrapezoidalDistribution.RegularTrapezoidalDistribution
All Implemented Interfaces:
ContinuousDistribution
Enclosing class:
TrapezoidalDistribution
private static class TrapezoidalDistribution.RegularTrapezoidalDistribution extends TrapezoidalDistribution
Regular implementation of the trapezoidal distribution.

  • Field Details

    • divisor

      private final double divisor
      Cached value (d + c - a - b).

    • bma

      private final double bma
      Cached value (b - a).

    • dmc

      private final double dmc
      Cached value (d - c).

    • cdfB

      private final double cdfB
      Cumulative probability at b.

    • cdfC

      private final double cdfC
      Cumulative probability at c.

    • sfB

      private final double sfB
      Survival probability at b.

    • sfC

      private final double sfC
      Survival probability at c.

  • Constructor Details

    • RegularTrapezoidalDistribution

      RegularTrapezoidalDistribution(double a, double b, double c, double d)
      Parameters:
      a - Lower limit of this distribution (inclusive).
      b - Start of the trapezoid constant density.
      c - End of the trapezoid constant density.
      d - Upper limit of this distribution (inclusive).

  • Method Details

    • 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.

    • 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)
      Description copied from class: AbstractContinuousDistribution
      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: TrapezoidalDistribution
      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:

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

      Specified by:
      getMean in interface ContinuousDistribution
      Specified by:
      getMean in class TrapezoidalDistribution
      Returns:
      the mean.

    • getVariance

      public double getVariance()
      Description copied from class: TrapezoidalDistribution
      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.

      Specified by:
      getVariance in interface ContinuousDistribution
      Specified by:
      getVariance in class TrapezoidalDistribution
      Returns:
      the variance.

    • nonCentralMoment

      private static double nonCentralMoment(int k, double b, double c)
      Compute the k-th non-central moment of the standardized trapezoidal distribution.

      Shifting the distribution by scale (d - a) and location a creates a standardized trapezoidal distribution. This has a simplified non-central moment as a = 0, d = 1, 0 <= b < c <= 1. 

                    2             1       ( 1 - c^(k+2)           )
E[X^k] = ----------- -------------- ( ----------- - b^(k+1) )
         (1 + c - b) (k + 1)(k + 2) (    1 - c              )

      Simplification eliminates issues computing the moments when a == b or c == d in the original (non-standardized) distribution.

      Parameters:
      k - Moment to compute
      b - Start of the trapezoid constant density (shape parameter in [0, 1]).
      c - End of the trapezoid constant density (shape parameter in [0, 1]).
      Returns:
      the moment