Alex Herbert created STATISTICS-85:
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Summary: Quantile implementation
Key: STATISTICS-85
URL: https://issues.apache.org/jira/browse/STATISTICS-85
Project: Commons Statistics
Issue Type: New Feature
Components: descriptive
Reporter: Alex Herbert
Add a quantile implementation. This will interpolate the value of a sorted
array of data for probability p in [0, 1].
Replace the legacy API from Commons Math Percentile with an updated API. The
new API should:
* Decouple estimation of quantile positions inside data of length n; and the
selection of correctly-ordered indices in array data.
* Support multiple data types.
* Support pre-sorted data.
* Avoid performance issues observed in the CM Percentile implementation.
h2. Proposed API
{code:java}
org.apache.commons.statistics.descriptive
public final class Quantile {
// overwrite=true; EstimationMethod.HF8; NaNPolicy.ERROR
public static Quantile withDefaults();
public Quantile withOverwrite(boolean);
public Quantile with(EstimationMethod);
// Could support NaN handling ... see below for NaNPolicy
public Quantile with(NaNPolicy);
// Create n uniform probabilities in range [p1, p2]
public static double[] probabilities(int n);
public static double[] probabilities(int n, double p1, double p2);
// Quantiles on sorted data a of size n
public double evaluate(int n, java.util.function.IntToDoubleFunction a,
double p);
public double[] evaluate(int n, java.util.function.IntToDoubleFunction a,
double... p);
// Quantiles on the primitive types that cannot be easily sorted
public double evaluate(double[] a, double p);
public double[] evaluate(double[] a, double... p);
public double evaluate(int[] a, double p);
public double[] evaluate(int[] a, double... p);
public double evaluate(long[] a, double p);
public double[] evaluate(long[] a, double... p);
public double evaluate(float[] a, double p);
public double[] evaluate(float[] a, double... p);
// Provide the 9 methods in Hyndman and Fan (1996)
// Sample Quantiles in Statistical Packages.
// The American Statistician, 50, 361-365.
public abstract class Quantile$EstimationMethod extends
java.lang.Enum<Quantile$EstimationMethod> {
public static final Quantile$EstimationMethod HF1;
public static final Quantile$EstimationMethod HF2;
public static final Quantile$EstimationMethod HF3;
public static final Quantile$EstimationMethod HF4;
public static final Quantile$EstimationMethod HF5;
public static final Quantile$EstimationMethod HF6;
public static final Quantile$EstimationMethod HF7;
public static final Quantile$EstimationMethod HF8;
public static final Quantile$EstimationMethod HF9;
}
}
Note: The CM API used the 9 methods from Hyndman and Fann but labelled them as
R1-9; this may be derived from the same names used in the R language. I propose
to rename as HF1-9 to reflect the origin.
{code}
h2. NaNPolicy
There are multiple options here. For reference R and Python's numpy only
provide the option to exclude NaN:
* R: quantile errors if NaN is present. median returns NaN. They is an option
to exclude NaN.
* numpy: two methods are provided: median/nanmedian + quantile/nanquantile
(the non-nan versions will return NaN if any NaNs are present)
Commons Math provides a remapping. Note the Statistics ranking module has the
same NaNStrategy as that in CM:
* MINIMAL: map to -infinity
* MAXIMAL: map to +infinity
* REMOVED: ignore from the data
* FIXED: leave in place. This makes no sense for quantiles. It is done by
moving to the end following the order imposed by Double.compare.
* FAILED: raise an exception
I favour the simpler option of: treating NaN so they are above/below all other
values; removing them from the data; or raising an exception. I do not see the
requirement to remap NaN to infinity. This can be done by the user.
The API can be simplified by using:
{code:java}
public final class NaNPolicy extends java.lang.Enum<NaNPolicy> {
public static final NaNPolicy LAST; // Move to end of data
public static final NaNPolicy FIRST; // Move to start of data
public static final NaNPolicy REMOVE; // Remove from data
public static final NaNPolicy ERROR; // Raise exception
}
{code}
Note that the FIRST option is not strictly required. But if there is an option
to order the NaNs (i.e. LAST) then both orders should be supported.
Which option to use as the default is not clear. As a drop in substitute for
Arrays.sort the default would be handle NaN with a move to the end (LAST). As
an API to signal to the user that the quantiles are possibly corrupted then
ERROR would be the default. A user can then decide what to do if they receive
errors during their analysis. Note that a separation of the Quantile API from
the partitioning API (see below) may result in doing NaN processing twice
introducing a performance overhead. If this will occur by default it should be
documented so a user can override the NaN behaviour if they do not expect NaNs.
Commons Math places the Percentile class in the rank package. I propose to move
this implementation to place it in the descriptive package where it will sit
beside other descriptive statistics for data such as mean and standard
deviation.
h2. Examples
{code:java}
double[] data = ...
double q = Quantile.withDefaults().evaluate(data, 0.25);
double[] quantiles = Quantile.withDefaults().evaluate(data, 0.25, 0.5, 0.75);
// Use cases:
// 1. Median / other quantile estimation
// 2. Box plot of data
// 3. Interquartile range analysis
double[] p = Quantile.probabilities(100, 0.01, 0.99);
double[] quantiles = Quantile.withDefaults().evaluate(data, p);
// Use cases:
// 1. plot p vs quantiles
// 2. use p with an expected (inverse) probability distribution to create a QQ
plot
// Sorted input / unsupported datatype
short[] data = ...
Arrays.sort(data);
double[] quantiles = Quantile.withDefaults().evaluate(data.length, i ->
data[i], p);
{code}
h2. Implementation
The Quantile API and the underlying implementation can be separated. Performing
quantile estimation requires knowing the value of elements (i, i+1) in a sorted
array a. Note that i+1 is used for interpolation:
{noformat}
value = a[i] + alpha * a[i+1] ; alpha in [0, 1)
{noformat}
Note: The alpha factor and the index i are chosen from the percentile p using
the EstimationMethod.
The array does not have to be fully sorted. It can be partitioned so that each
required element i is equal to the value in a fully sorted array. A
partitioning API can be placed in Commons Numbers:
{code:java}
org.apache.commons.numbers.arrays
public final class Selection {
// Operate in-place and support a range as per Arrays.sort
public void select(double[] a, int k);
public void select(double[] a, int... k);
public void select(int from, int to, double[] a, int k);
public void select(int from, int to, double[] a, int... k);
// Extend to other primitive types where sorting is slow
// Sorting of 16-bit data is fast using a histogram so selection
// has no major speed gain above a small length:
// byte[] does counting sort at length 64 (temurin JDK 11)
// short[]/char[] at length 1750 (temurin JDK 11)
}
{code}
h3. Examples
{code:java}
// Find the bottom 100 (with 0-based indexing)
Selection.select(data, 99);
double[] bottom100 = Arrays.copyOf(data, 100);
// Find the bottom and top 10
Selection.select(data, 9, data.length - 10);
double[] bottom10 = Arrays.copyOf(data, 10);
double[] top10 = Arrays.copyOfRange(data, data.length - 10, data.length);
{code}
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