Package org.apache.commons.statistics.descriptive

Enum Quantile.EstimationMethod

  • All Implemented Interfaces:
    Serializable, Comparable<Quantile.EstimationMethod>
    Enclosing class:
    Quantile
    public static enum Quantile.EstimationMethod
extends Enum<Quantile.EstimationMethod>
    Estimation methods for a quantile. Provides the nine quantile algorithms defined in Hyndman and Fan (1996)[1] as HF1 - HF9.

    Samples quantiles are defined by:

    \[ Q(p) = (1 - \gamma) x_j + \gamma x_{j+1} \]

    where \( \frac{j-m}{n} \leq p \le \frac{j-m+1}{n} \), \( x_j \) is the \( j \)th order statistic, \( n \) is the sample size, the value of \( \gamma \) is a function of \( j = \lfloor np+m \rfloor \) and \( g = np + m - j \), and \( m \) is a constant determined by the sample quantile type.

    Note that the real-valued position \( np + m \) is a 1-based index and \( j \in [1, n] \). If the real valued position is computed as beyond the lowest or highest values in the sample, this implementation will return the minimum or maximum observation respectively.

    Types 1, 2, and 3 are discontinuous functions of \( p \); types 4 to 9 are continuous functions of \( p \).

    For the continuous functions, the probability \( p_k \) is provided for the \( k \)-th order statistic in size \( n \). Samples quantiles are equivalently obtained to \( Q(p) \) by linear interpolation between points \( (p_k, x_k) \) and \( (p_{k+1}, x_{k+1}) \) for any \( p_k \leq p \leq p_{k+1} \).

    1. Hyndman and Fan (1996) Sample Quantiles in Statistical Packages. The American Statistician, 50, 361-365. doi.org/10.2307/2684934
    2. Quantile (Wikipedia)

    • Enum Constant Summary

      Enum Constants 
      Enum Constant Description
      HF1
      Inverse of the empirical distribution function.
      HF2
      Similar to HF1 with averaging at discontinuities.
      HF3
      The observation closest to \( np \).
      HF4
      Linear interpolation of the inverse of the empirical CDF.
      HF5
      A piecewise linear function where the knots are the values midway through the steps of the empirical CDF.
      HF6
      Linear interpolation of the expectations for the order statistics for the uniform distribution on [0,1].
      HF7
      Linear interpolation of the modes for the order statistics for the uniform distribution on [0,1].
      HF8
      Linear interpolation of the approximate medians for order statistics.
      HF9
      Quantile estimates are approximately unbiased for the expected order statistics if \( x \) is normally distributed.

    • Enum Constant Detail

      • HF1

        public static final Quantile.EstimationMethod HF1
        Inverse of the empirical distribution function.

        \( m = 0 \). \( \gamma = 0 \) if \( g = 0 \), and 1 otherwise.

      • HF2

        public static final Quantile.EstimationMethod HF2
        Similar to HF1 with averaging at discontinuities.

        \( m = 0 \). \( \gamma = 0.5 \) if \( g = 0 \), and 1 otherwise.

      • HF3

        public static final Quantile.EstimationMethod HF3
        The observation closest to \( np \). Ties are resolved to the nearest even order statistic.

        \( m = -1/2 \). \( \gamma = 0 \) if \( g = 0 \) and \( j \) is even, and 1 otherwise.

      • HF4

        public static final Quantile.EstimationMethod HF4
        Linear interpolation of the inverse of the empirical CDF.

        \( m = 0 \). \( p_k = \frac{k}{n} \).

      • HF5

        public static final Quantile.EstimationMethod HF5
        A piecewise linear function where the knots are the values midway through the steps of the empirical CDF. Proposed by Hazen (1914) and popular amongst hydrologists.

        \( m = 1/2 \). \( p_k = \frac{k - 1/2}{n} \).

      • HF6

        public static final Quantile.EstimationMethod HF6
        Linear interpolation of the expectations for the order statistics for the uniform distribution on [0,1]. Proposed by Weibull (1939).

        \( m = p \). \( p_k = \frac{k}{n + 1} \).

        This method computes the quantile as per the Apache Commons Math Percentile legacy implementation.

      • HF7

        public static final Quantile.EstimationMethod HF7
        Linear interpolation of the modes for the order statistics for the uniform distribution on [0,1]. Proposed by Gumbull (1939).

        \( m = 1 - p \). \( p_k = \frac{k - 1}{n - 1} \).

      • HF8

        public static final Quantile.EstimationMethod HF8
        Linear interpolation of the approximate medians for order statistics.

        \( m = (p + 1)/3 \). \( p_k = \frac{k - 1/3}{n + 1/3} \).

        As per Hyndman and Fan (1996) this approach is most recommended as it provides an approximate median-unbiased estimate regardless of distribution.

      • HF9

        public static final Quantile.EstimationMethod HF9
        Quantile estimates are approximately unbiased for the expected order statistics if \( x \) is normally distributed.

        \( m = p/4 + 3/8 \). \( p_k = \frac{k - 3/8}{n + 1/4} \).

    • Method Detail

      • values

        public static Quantile.EstimationMethod[] values()
        Returns an array containing the constants of this enum type, in the order they are declared. This method may be used to iterate over the constants as follows: 
        for (Quantile.EstimationMethod c : Quantile.EstimationMethod.values())
    System.out.println(c);
        Returns:
        an array containing the constants of this enum type, in the order they are declared

      • valueOf

        public static Quantile.EstimationMethod valueOf​(String name)
        Returns the enum constant of this type with the specified name. The string must match exactly an identifier used to declare an enum constant in this type. (Extraneous whitespace characters are not permitted.)
        Parameters:
        name - the name of the enum constant to be returned.
        Returns:
        the enum constant with the specified name
        Throws:
        IllegalArgumentException - if this enum type has no constant with the specified name
        NullPointerException - if the argument is null