org.hipparchus.stat.descriptive.rank

Enum Percentile.EstimationType

java.lang.Object

java.lang.Enum<Percentile.EstimationType>

org.hipparchus.stat.descriptive.rank.Percentile.EstimationType

All Implemented Interfaces:

Serializable, Comparable<Percentile.EstimationType>

Enclosing class:

Percentile

public static enum Percentile.EstimationType
extends Enum<Percentile.EstimationType>

An enum for various estimation strategies of a percentile referred in wikipedia on quantile with the names of enum matching those of types mentioned in wikipedia.

Each enum corresponding to the specific type of estimation in wikipedia implements the respective formulae that specializes in the below aspects

• An index method to calculate approximate index of the estimate
• An estimate method to estimate a value found at the earlier computed index
• A minLimit on the quantile for which first element of sorted input is returned as an estimate
• A maxLimit on the quantile for which last element of sorted input is returned as an estimate

Users can now create Percentile by explicitly passing this enum; such as by invoking Percentile.withEstimationType(EstimationType)

References:

1. Wikipedia on quantile
2. Hyndman, R. J. and Fan, Y. (1996) Sample quantiles in statistical packages, American Statistician 50, 361–365
3. R-Manual

Enum Constant Summary

Enum Constants

Enum Constant and Description
LEGACY This is the default type used in the Percentile.This method has the following formulae for index and estimates \( \begin{align} &index = (N+1)p\ \\ &estimate = x_{\lceil h\,-\,1/2 \rceil} \\ &minLimit = 0 \\ &maxLimit = 1 \\ \end{align}\)
R_1 The method R_1 has the following formulae for index and estimates \( \begin{align} &index= Np + 1/2\, \\ &estimate= x_{\lceil h\,-\,1/2 \rceil} \\ &minLimit = 0 \\ \end{align}\)
R_2 The method R_2 has the following formulae for index and estimates \( \begin{align} &index= Np + 1/2\, \\ &estimate=\frac{x_{\lceil h\,-\,1/2 \rceil} + x_{\lfloor h\,+\,1/2 \rfloor}}{2} \\ &minLimit = 0 \\ &maxLimit = 1 \\ \end{align}\)
R_3 The method R_3 has the following formulae for index and estimates \( \begin{align} &index= Np \\ &estimate= x_{\lfloor h \rceil}\, \\ &minLimit = 0.5/N \\ \end{align}\)
R_4 The method R_4 has the following formulae for index and estimates \( \begin{align} &index= Np\, \\ &estimate= x_{\lfloor h \rfloor} + (h - \lfloor h \rfloor) (x_{\lfloor h \rfloor + 1} - x_{\lfloor h \rfloor}) \\ &minLimit = 1/N \\ &maxLimit = 1 \\ \end{align}\)
R_5 The method R_5 has the following formulae for index and estimates \( \begin{align} &index= Np + 1/2\\ &estimate= x_{\lfloor h \rfloor} + (h - \lfloor h \rfloor) (x_{\lfloor h \rfloor + 1} - x_{\lfloor h \rfloor}) \\ &minLimit = 0.5/N \\ &maxLimit = (N-0.5)/N \end{align}\)
R_6 The method R_6 has the following formulae for index and estimates \( \begin{align} &index= (N + 1)p \\ &estimate= x_{\lfloor h \rfloor} + (h - \lfloor h \rfloor) (x_{\lfloor h \rfloor + 1} - x_{\lfloor h \rfloor}) \\ &minLimit = 1/(N+1) \\ &maxLimit = N/(N+1) \\ \end{align}\)
R_7 The method R_7 implements Microsoft Excel style computation has the following formulae for index and estimates. \( \begin{align} &index = (N-1)p + 1 \\ &estimate = x_{\lfloor h \rfloor} + (h - \lfloor h \rfloor) (x_{\lfloor h \rfloor + 1} - x_{\lfloor h \rfloor}) \\ &minLimit = 0 \\ &maxLimit = 1 \\ \end{align}\)
R_8 The method R_8 has the following formulae for index and estimates \( \begin{align} &index = (N + 1/3)p + 1/3 \\ &estimate = x_{\lfloor h \rfloor} + (h - \lfloor h \rfloor) (x_{\lfloor h \rfloor + 1} - x_{\lfloor h \rfloor}) \\ &minLimit = (2/3)/(N+1/3) \\ &maxLimit = (N-1/3)/(N+1/3) \\ \end{align}\)
R_9 The method R_9 has the following formulae for index and estimates \( \begin{align} &index = (N + 1/4)p + 3/8\\ &estimate = x_{\lfloor h \rfloor} + (h - \lfloor h \rfloor) (x_{\lfloor h \rfloor + 1} - x_{\lfloor h \rfloor}) \\ &minLimit = (5/8)/(N+1/4) \\ &maxLimit = (N-3/8)/(N+1/4) \\ \end{align}\)

Method Summary

Modifier and Type	Method and Description
protected double	estimate(double[] work, int[] pivotsHeap, double pos, int length, KthSelector selector) Estimation based on Kth selection.
double	evaluate(double[] work, double p, KthSelector selector) Evaluate method to compute the percentile for a given bounded array.
protected double	evaluate(double[] work, int[] pivotsHeap, double p, KthSelector selector) Evaluate method to compute the percentile for a given bounded array using earlier computed pivots heap. This basically calls the index and then estimate functions to return the estimated percentile value.
protected abstract double	index(double p, int length) Finds the index of array that can be used as starting index to estimate percentile.
static Percentile.EstimationType	valueOf(String name) Returns the enum constant of this type with the specified name.
static Percentile.EstimationType[]	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 Detail

LEGACY

public static final Percentile.EstimationType LEGACY

This is the default type used in the Percentile.This method has the following formulae for index and estimates
\( \begin{align} &index = (N+1)p\ \\ &estimate = x_{\lceil h\,-\,1/2 \rceil} \\ &minLimit = 0 \\ &maxLimit = 1 \\ \end{align}\)

R_1

public static final Percentile.EstimationType R_1

The method R_1 has the following formulae for index and estimates
\( \begin{align} &index= Np + 1/2\, \\ &estimate= x_{\lceil h\,-\,1/2 \rceil} \\ &minLimit = 0 \\ \end{align}\)

R_2

public static final Percentile.EstimationType R_2

The method R_2 has the following formulae for index and estimates
\( \begin{align} &index= Np + 1/2\, \\ &estimate=\frac{x_{\lceil h\,-\,1/2 \rceil} + x_{\lfloor h\,+\,1/2 \rfloor}}{2} \\ &minLimit = 0 \\ &maxLimit = 1 \\ \end{align}\)

R_3

public static final Percentile.EstimationType R_3

The method R_3 has the following formulae for index and estimates
\( \begin{align} &index= Np \\ &estimate= x_{\lfloor h \rceil}\, \\ &minLimit = 0.5/N \\ \end{align}\)

R_4

public static final Percentile.EstimationType R_4

The method R_4 has the following formulae for index and estimates
\( \begin{align} &index= Np\, \\ &estimate= x_{\lfloor h \rfloor} + (h - \lfloor h \rfloor) (x_{\lfloor h \rfloor + 1} - x_{\lfloor h \rfloor}) \\ &minLimit = 1/N \\ &maxLimit = 1 \\ \end{align}\)

R_5

public static final Percentile.EstimationType R_5

The method R_5 has the following formulae for index and estimates
\( \begin{align} &index= Np + 1/2\\ &estimate= x_{\lfloor h \rfloor} + (h - \lfloor h \rfloor) (x_{\lfloor h \rfloor + 1} - x_{\lfloor h \rfloor}) \\ &minLimit = 0.5/N \\ &maxLimit = (N-0.5)/N \end{align}\)

R_6

public static final Percentile.EstimationType R_6

The method R_6 has the following formulae for index and estimates
\( \begin{align} &index= (N + 1)p \\ &estimate= x_{\lfloor h \rfloor} + (h - \lfloor h \rfloor) (x_{\lfloor h \rfloor + 1} - x_{\lfloor h \rfloor}) \\ &minLimit = 1/(N+1) \\ &maxLimit = N/(N+1) \\ \end{align}\)

Note: This method computes the index in a manner very close to the default Hipparchus Percentile existing implementation. However the difference to be noted is in picking up the limits with which first element (p<1(N+1)) and last elements (p>N/(N+1))are done. While in default case; these are done with p=0 and p=1 respectively.

R_7

public static final Percentile.EstimationType R_7

The method R_7 implements Microsoft Excel style computation has the following formulae for index and estimates.
\( \begin{align} &index = (N-1)p + 1 \\ &estimate = x_{\lfloor h \rfloor} + (h - \lfloor h \rfloor) (x_{\lfloor h \rfloor + 1} - x_{\lfloor h \rfloor}) \\ &minLimit = 0 \\ &maxLimit = 1 \\ \end{align}\)

R_8

public static final Percentile.EstimationType R_8

The method R_8 has the following formulae for index and estimates
\( \begin{align} &index = (N + 1/3)p + 1/3 \\ &estimate = x_{\lfloor h \rfloor} + (h - \lfloor h \rfloor) (x_{\lfloor h \rfloor + 1} - x_{\lfloor h \rfloor}) \\ &minLimit = (2/3)/(N+1/3) \\ &maxLimit = (N-1/3)/(N+1/3) \\ \end{align}\)

As per Ref [2,3] this approach is most recommended as it provides an approximate median-unbiased estimate regardless of distribution.

R_9

public static final Percentile.EstimationType R_9

The method R_9 has the following formulae for index and estimates
\( \begin{align} &index = (N + 1/4)p + 3/8\\ &estimate = x_{\lfloor h \rfloor} + (h - \lfloor h \rfloor) (x_{\lfloor h \rfloor + 1} - x_{\lfloor h \rfloor}) \\ &minLimit = (5/8)/(N+1/4) \\ &maxLimit = (N-3/8)/(N+1/4) \\ \end{align}\)

Method Detail

values

public static Percentile.EstimationType[] 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 (Percentile.EstimationType c : Percentile.EstimationType.values())
    System.out.println(c);

Returns:

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

valueOf

public static Percentile.EstimationType 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

index

protected abstract double index(double p,
                                int length)

Finds the index of array that can be used as starting index to estimate percentile. The calculation of index calculation is specific to each Percentile.EstimationType.

Parameters:

p - the p^th quantile

length - the total number of array elements in the work array

Returns:

a computed real valued index as explained in the wikipedia

estimate

protected double estimate(double[] work,
                          int[] pivotsHeap,
                          double pos,
                          int length,
                          KthSelector selector)

Estimation based on K^th selection. This may be overridden in specific enums to compute slightly different estimations.

Parameters:

work - array of numbers to be used for finding the percentile

pos - indicated positional index prior computed from calling index(double, int)

pivotsHeap - an earlier populated cache if exists; will be used

length - size of array considered

selector - a KthSelector used for pivoting during search

Returns:

estimated percentile

evaluate

protected double evaluate(double[] work,
                          int[] pivotsHeap,
                          double p,
                          KthSelector selector)

Evaluate method to compute the percentile for a given bounded array using earlier computed pivots heap.
This basically calls the index and then estimate functions to return the estimated percentile value.

Parameters:

work - array of numbers to be used for finding the percentile

pivotsHeap - a prior cached heap which can speed up estimation

p - the p^th quantile to be computed

selector - a KthSelector used for pivoting during search

Returns:

estimated percentile

Throws:

MathIllegalArgumentException - if p is out of range

NullArgumentException - if work array is null

evaluate

public double evaluate(double[] work,
                       double p,
                       KthSelector selector)

Evaluate method to compute the percentile for a given bounded array. This basically calls the index and then estimate functions to return the estimated percentile value. Please note that this method does not make use of cached pivots.

Parameters:

work - array of numbers to be used for finding the percentile

p - the p^th quantile to be computed

selector - a KthSelector used for pivoting during search

Returns:

estimated percentile

Throws:

MathIllegalArgumentException - if length or p is out of range

NullArgumentException - if work array is null