Class Kurtosis
- All Implemented Interfaces:
DoubleConsumer
,DoubleSupplier
,IntSupplier
,LongSupplier
,DoubleStatistic
,StatisticAccumulator<Kurtosis>
,StatisticResult
\[ \operatorname{Kurt} = \operatorname{E}\left[ \left(\frac{X-\mu}{\sigma}\right)^4 \right] = \frac{\mu_4}{\sigma^4} \]
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 isNaN
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:
-
Field Summary
FieldsModifier and TypeFieldDescriptionprivate boolean
Flag to control if the statistic is biased, or should use a bias correction.private static final int
4, the length limit where the kurtosis is undefined.private static final int
2, the length limit where the biased skewness is undefined.private final SumOfFourthDeviations
An instance ofSumOfFourthDeviations
, which is used to compute the kurtosis. -
Constructor Summary
ConstructorsModifierConstructorDescriptionprivate
Kurtosis()
Create an instance.(package private)
Creates an instance with the sum of fourth deviations from the mean. -
Method Summary
Modifier and TypeMethodDescriptionvoid
accept
(double value) Updates the state of the statistic to reflect the addition ofvalue
.Combines the state of theother
statistic into this one.static Kurtosis
create()
Creates an instance.double
Gets the kurtosis of all input values.static Kurtosis
of
(double... values) Returns an instance populated using the inputvalues
.static Kurtosis
of
(int... values) Returns an instance populated using the inputvalues
.static Kurtosis
of
(long... values) Returns an instance populated using the inputvalues
.setBiased
(boolean v) Sets the value of the biased flag.Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
Methods inherited from interface java.util.function.DoubleConsumer
andThen
Methods inherited from interface org.apache.commons.statistics.descriptive.StatisticResult
getAsBigInteger, getAsInt, getAsLong
-
Field Details
-
LENGTH_TWO
private static final int LENGTH_TWO2, 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:
-
LENGTH_FOUR
private static final int LENGTH_FOUR4, the length limit where the kurtosis is undefined.- See Also:
-
sq
An instance ofSumOfFourthDeviations
, which is used to compute the kurtosis. -
biased
private boolean biasedFlag 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
Creates an instance.The initial result is
NaN
.- Returns:
Kurtosis
instance.
-
of
Returns an instance populated using the inputvalues
.Note:
Kurtosis
computed usingaccept
may be different from this instance.- Parameters:
values
- Values.- Returns:
Kurtosis
instance.
-
of
Returns an instance populated using the inputvalues
.Note:
Kurtosis
computed usingaccept
may be different from this instance.- Parameters:
values
- Values.- Returns:
Kurtosis
instance.
-
of
Returns an instance populated using the inputvalues
.Note:
Kurtosis
computed usingaccept
may be different from this instance.- Parameters:
values
- Values.- Returns:
Kurtosis
instance.
-
accept
public void accept(double value) Updates the state of the statistic to reflect the addition ofvalue
.- Specified by:
accept
in interfaceDoubleConsumer
- 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 interfaceDoubleSupplier
- Returns:
- kurtosis of all values.
-
combine
Description copied from interface:StatisticAccumulator
Combines the state of theother
statistic into this one.- Specified by:
combine
in interfaceStatisticAccumulator<Kurtosis>
- Parameters:
other
- Another statistic to be combined.- Returns:
this
instance after combiningother
.
-
setBiased
Sets the value of the biased flag. The default value isfalse
. SeeKurtosis
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
-