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:
  • 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
    3, the length limit where the unbiased skewness is undefined.
    private static final int
    2, the length limit where the biased skewness is undefined.
    private final SumOfCubedDeviations
    An instance of SumOfCubedDeviations, which is used to compute the skewness.
  • Constructor Summary

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

      private static final int LENGTH_THREE
      3, the length limit where the unbiased skewness is undefined.
      See Also:
    • 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.
    • 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.
    • 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.
    • 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