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:
  • Field Summary

    Fields
    Modifier and Type
    Field
    Description
    private 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 of SumOfFourthDeviations, which is used to compute the kurtosis.
  • Constructor Summary

    Constructors
    Modifier
    Constructor
    Description
    private
    Create an instance.
    (package private)
    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.
    Combines the state of the other statistic into this one.
    static Kurtosis
    Creates an instance.
    double
    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.
    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_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:
    • LENGTH_FOUR

      private static final int LENGTH_FOUR
      4, the length limit where the kurtosis is undefined.
      See Also:
    • 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.
    • 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.
    • 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.
    • 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