Class Skewness

java.lang.Object

org.apache.commons.statistics.descriptive.Skewness

All Implemented Interfaces:

DoubleConsumer, DoubleSupplier, IntSupplier, LongSupplier, DoubleStatistic, StatisticAccumulator<Skewness>, StatisticResult

public final class Skewness
extends Object
implements DoubleStatistic, StatisticAccumulator<Skewness>

Computes the skewness of the available values. The skewness is defined as:

\[ \gamma_1 = \operatorname{E}\left[ \left(\frac{X-\mu}{\sigma}\right)^3 \right] = \frac{\mu_3}{\sigma^3} \]

where \( \mu \) is the mean of \( X \), \( \sigma \) is the standard deviation of \( X \), \( \operatorname{E} \) represents the expectation operator, and \( \mu_3 \) is the third central moment.

The default implementation uses the following definition of the sample skewness:

\[ G_1 = \frac{k_3}{k_2^{3/2}} = \frac{\sqrt{n(n-1)}}{n-2}\; g_1 = \frac{n^2}{(n-1)(n-2)}\; \frac{\tfrac{1}{n} \sum_{i=1}^n (x_i-\overline{x})^3} {\left[\tfrac{1}{n-1} \sum_{i=1}^n (x_i-\overline{x})^2 \right]^{3/2}} \]

where \( k_3 \) is the unique symmetric unbiased estimator of the third cumulant, \( k_2 \) is the symmetric unbiased estimator of the second cumulant (i.e. the sample variance), \( g_1 \) is a method of moments estimator (see below), \( \overline{x} \) is the sample mean, and \( n \) is the number of samples.

• The result is NaN if less than 3 values are added.
• The result is NaN if any of the values is NaN or infinite.
• The result is NaN if the sum of the cubed deviations from the mean is infinite.

The default computation is for the adjusted Fisher–Pearson standardized moment coefficient \( G_1 \). If the biased option is enabled the following equation applies:

\[ g_1 = \frac{m_3}{m_2^{3/2}} = \frac{\tfrac{1}{n} \sum_{i=1}^n (x_i-\overline{x})^3} {\left[\tfrac{1}{n} \sum_{i=1}^n (x_i-\overline{x})^2 \right]^{3/2}} \]

where \( g_2 \) is a method of moments estimator, \( m_3 \) is the (biased) sample third central moment and \( m_2^{3/2} \) is the (biased) sample second central moment.

In this case the computation only requires 2 values are added (i.e. the result is NaN if less than 2 values are added).

Note that the computation requires division by the second central moment \( m_2 \). If this is effectively zero then the result is NaN. This occurs when the value \( m_2 \) approaches the machine precision of the mean: \( m_2 \le (m_1 \times 10^{-15})^2 \).

The accept(double) method uses a recursive updating algorithm.

The of(double...) method uses a two-pass algorithm, starting with computation of the mean, and then computing the sum of deviations in a second pass.

Note that adding values using accept and then executing getAsDouble will sometimes give a different result than executing of with the full array of values. The former approach should only be used when the full array of values is not available.

Supports up to 2⁶³ (exclusive) observations. This implementation does not check for overflow of the count.

This class is designed to work with (though does not require) streams.

Note that this instance is not synchronized. If multiple threads access an instance of this class concurrently, and at least one of the threads invokes the accept or combine method, it must be synchronized externally.

However, it is safe to use accept and combine as accumulator and combiner functions of Collector on a parallel stream, because the parallel instance of Stream.collect() provides the necessary partitioning, isolation, and merging of results for safe and efficient parallel execution.

Since:

1.1

See Also:

• Skewness (Wikipedia)

Field Summary

Fields

Modifier and Type	Field	Description
private boolean	biased	Flag to control if the statistic is biased, or should use a bias correction.
private static final int	LENGTH_THREE	3, the length limit where the unbiased skewness is undefined.
private static final int	LENGTH_TWO	2, the length limit where the biased skewness is undefined.
private final SumOfCubedDeviations	sc	An instance of SumOfCubedDeviations, which is used to compute the skewness.

Constructor Summary

Constructors

Modifier	Constructor	Description
private	Skewness()	Create an instance.
(package private)	Skewness(SumOfCubedDeviations sc)	Creates an instance with the sum of cubed deviations from the mean.

Method Summary

Modifier and Type	Method	Description
void	accept(double value)	Updates the state of the statistic to reflect the addition of value.
Skewness	combine(Skewness other)	Combines the state of the other statistic into this one.
static Skewness	create()	Creates an instance.
double	getAsDouble()	Gets the skewness of all input values.
static Skewness	of(double... values)	Returns an instance populated using the input values.
static Skewness	of(int... values)	Returns an instance populated using the input values.
static Skewness	of(long... values)	Returns an instance populated using the input values.
static Skewness	ofRange(double[] values, int from, int to)	Returns an instance populated using the specified range of values.
static Skewness	ofRange(int[] values, int from, int to)	Returns an instance populated using the specified range of values.
static Skewness	ofRange(long[] values, int from, int to)	Returns an instance populated using the specified range of values.
Skewness	setBiased(boolean v)	Sets the value of the biased flag.

Methods inherited from class Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Methods inherited from interface DoubleConsumer

andThen

Methods inherited from interface StatisticResult

getAsBigInteger, getAsInt, getAsLong

Field Details

LENGTH_TWO

private static final int LENGTH_TWO

2, the length limit where the biased skewness is undefined. This limit effectively imposes the result m3 / m2^1.5 = 0 / 0 = NaN when 1 value has been added. Note that when more samples are added and the variance approaches zero the result is also returned as NaN.

See Also:

• Constant Field Values

LENGTH_THREE

private static final int LENGTH_THREE

3, the length limit where the unbiased skewness is undefined.

See Also:

• Constant Field Values

sc

private final SumOfCubedDeviations sc

An instance of SumOfCubedDeviations, which is used to compute the skewness.

biased

private boolean biased

Flag to control if the statistic is biased, or should use a bias correction.

Constructor Details

Skewness

private Skewness()

Create an instance.

Skewness

Skewness(SumOfCubedDeviations sc)

Creates an instance with the sum of cubed deviations from the mean.

Parameters:

sc - Sum of cubed deviations.

Method Details

create

public static Skewness create()

Creates an instance.

The initial result is NaN.

Returns:

Skewness instance.

of

public static Skewness of(double... values)

Returns an instance populated using the input values.

Note: Skewness computed using accept may be different from this instance.

Parameters:

values - Values.

Returns:

Skewness instance.

ofRange

public static Skewness ofRange(double[] values,
 int from,
 int to)

Returns an instance populated using the specified range of values.

Note: Skewness computed using accept may be different from this instance.

Parameters:

values - Values.

from - Inclusive start of the range.

to - Exclusive end of the range.

Returns:

Skewness instance.

Throws:

IndexOutOfBoundsException - if the sub-range is out of bounds

Since:

1.2

of

public static Skewness of(int... values)

Returns an instance populated using the input values.

Note: Skewness computed using accept may be different from this instance.

Parameters:

values - Values.

Returns:

Skewness instance.

ofRange

public static Skewness ofRange(int[] values,
 int from,
 int to)

Returns an instance populated using the specified range of values.

Note: Skewness computed using accept may be different from this instance.

Parameters:

values - Values.

from - Inclusive start of the range.

to - Exclusive end of the range.

Returns:

Skewness instance.

Throws:

IndexOutOfBoundsException - if the sub-range is out of bounds

Since:

1.2

of

public static Skewness of(long... values)

Returns an instance populated using the input values.

Note: Skewness computed using accept may be different from this instance.

Parameters:

values - Values.

Returns:

Skewness instance.

ofRange

public static Skewness ofRange(long[] values,
 int from,
 int to)

Returns an instance populated using the specified range of values.

Note: Skewness computed using accept may be different from this instance.

Parameters:

values - Values.

from - Inclusive start of the range.

to - Exclusive end of the range.

Returns:

Skewness instance.

Throws:

IndexOutOfBoundsException - if the sub-range is out of bounds

Since:

1.2

accept

public void accept(double value)

Updates the state of the statistic to reflect the addition of value.

Specified by:

accept in interface DoubleConsumer

Parameters:

value - Value.

getAsDouble

public double getAsDouble()

Gets the skewness of all input values.

When fewer than 3 values have been added, the result is NaN.

Specified by:

getAsDouble in interface DoubleSupplier

Returns:

skewness of all values.

combine

public Skewness combine(Skewness other)

Description copied from interface: StatisticAccumulator

Combines the state of the other statistic into this one.

Specified by:

combine in interface StatisticAccumulator<Skewness>

Parameters:

other - Another statistic to be combined.

Returns:

this instance after combining other.

setBiased

public Skewness setBiased(boolean v)

Sets the value of the biased flag. The default value is false. See Skewness for details on the computing algorithm.

This flag only controls the final computation of the statistic. The value of this flag will not affect compatibility between instances during a combine operation.

Parameters:

v - Value.

Returns:

this instance