Class Kurtosis

java.lang.Object

org.apache.commons.statistics.descriptive.Kurtosis

All Implemented Interfaces:

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

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

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

\[ \operatorname{Kurt} = \operatorname{E}\left[ \left(\frac{X-\mu}{\sigma}\right)^4 \right] = \frac{\mu_4}{\sigma^4} \]

where \( \mu \) is the mean of \( X \), \( \sigma \) is the standard deviation of \( X \), \( \operatorname{E} \) represents the expectation operator, and \( \mu_4 \) is the fourth central moment.

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

\[ G_2 = \frac{k_4}{k_2^2} = \; \frac{n-1}{(n-2)\,(n-3)} \left[(n+1)\,\frac{m_4}{m_{2}^2} - 3\,(n-1) \right] \]

where \( k_4 \) is the unique symmetric unbiased estimator of the fourth cumulant, \( k_2 \) is the symmetric unbiased estimator of the second cumulant (i.e. the sample variance), \( m_4 \) is the fourth sample moment about the mean, \( m_2 \) is the second sample moment about the mean, \( \overline{x} \) is the sample mean, and \( n \) is the number of samples.

• The result is NaN if less than 4 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 fourth deviations from the mean is infinite.

The default computation is for the adjusted Fisher–Pearson standardized moment coefficient \( G_2 \). If the biased option is enabled the following equation applies:

\[ g_2 = \frac{m_4}{m_2^2} - 3 = \frac{\tfrac{1}{n} \sum_{i=1}^n (x_i-\overline{x})^4} {\left[\tfrac{1}{n} \sum_{i=1}^n (x_i-\overline{x})^2 \right]^2} - 3 \]

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:

• Kurtosis (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_FOUR	4, the length limit where the kurtosis is undefined.
private static final int	LENGTH_TWO	2, the length limit where the biased skewness is undefined.
private final SumOfFourthDeviations	sq	An instance of SumOfFourthDeviations, which is used to compute the kurtosis.

Constructor Summary

Constructors

Modifier	Constructor	Description
private	Kurtosis()	Create an instance.
(package private)	Kurtosis(SumOfFourthDeviations sq)	Creates an instance with the sum of fourth 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.
Kurtosis	combine(Kurtosis other)	Combines the state of the other statistic into this one.
static Kurtosis	create()	Creates an instance.
double	getAsDouble()	Gets the kurtosis of all input values.
static Kurtosis	of(double... values)	Returns an instance populated using the input values.
static Kurtosis	of(int... values)	Returns an instance populated using the input values.
static Kurtosis	of(long... values)	Returns an instance populated using the input values.
static Kurtosis	ofRange(double[] values, int from, int to)	Returns an instance populated using the specified range of values.
static Kurtosis	ofRange(int[] values, int from, int to)	Returns an instance populated using the specified range of values.
static Kurtosis	ofRange(long[] values, int from, int to)	Returns an instance populated using the specified range of values.
Kurtosis	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 m4 / m2^2 = 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_FOUR

private static final int LENGTH_FOUR

4, the length limit where the kurtosis is undefined.

See Also:

• Constant Field Values

sq

private final SumOfFourthDeviations sq

An instance of SumOfFourthDeviations, which is used to compute the kurtosis.

biased

private boolean biased

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

Constructor Details

Kurtosis

private Kurtosis()

Create an instance.

Kurtosis

Kurtosis(SumOfFourthDeviations sq)

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

Parameters:

sq - Sum of fourth deviations.

Method Details

create

public static Kurtosis create()

Creates an instance.

The initial result is NaN.

Returns:

Kurtosis instance.

of

public static Kurtosis of(double... values)

Returns an instance populated using the input values.

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

Parameters:

values - Values.

Returns:

Kurtosis instance.

ofRange

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

Returns an instance populated using the specified range of values.

Note: Kurtosis 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:

Kurtosis instance.

Throws:

IndexOutOfBoundsException - if the sub-range is out of bounds

Since:

1.2

of

public static Kurtosis of(int... values)

Returns an instance populated using the input values.

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

Parameters:

values - Values.

Returns:

Kurtosis instance.

ofRange

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

Returns an instance populated using the specified range of values.

Note: Kurtosis 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:

Kurtosis instance.

Throws:

IndexOutOfBoundsException - if the sub-range is out of bounds

Since:

1.2

of

public static Kurtosis of(long... values)

Returns an instance populated using the input values.

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

Parameters:

values - Values.

Returns:

Kurtosis instance.

ofRange

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

Returns an instance populated using the specified range of values.

Note: Kurtosis 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:

Kurtosis 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 kurtosis of all input values.

When fewer than 4 values have been added, the result is NaN.

Specified by:

getAsDouble in interface DoubleSupplier

Returns:

kurtosis of all values.

combine

public Kurtosis combine(Kurtosis other)

Description copied from interface: StatisticAccumulator

Combines the state of the other statistic into this one.

Specified by:

combine in interface StatisticAccumulator<Kurtosis>

Parameters:

other - Another statistic to be combined.

Returns:

this instance after combining other.

setBiased

public Kurtosis setBiased(boolean v)

Sets the value of the biased flag. The default value is false. See Kurtosis 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