Package org.apache.commons.statistics.descriptive

Enum Quantile.EstimationMethod

java.lang.Object

java.lang.Enum<Quantile.EstimationMethod>

org.apache.commons.statistics.descriptive.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.

Method Summary

Modifier and Type	Method	Description
static Quantile.EstimationMethod	valueOf(String name)	Returns the enum constant of this type with the specified name.
static Quantile.EstimationMethod[]	values()	Returns an array containing the constants of this enum type, in the order they are declared.

Methods inherited from class java.lang.Enum

clone, compareTo, equals, finalize, getDeclaringClass, hashCode, name, ordinal, toString, valueOf

Methods inherited from class java.lang.Object

getClass, notify, notifyAll, wait, wait, wait

Enum Constant Details

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 Details

values

public static Quantile.EstimationMethod[] values()

Returns an array containing the constants of this enum type, in the order they are declared.

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