Class Skewness

  • All Implemented Interfaces:
    java.util.function.DoubleConsumer, java.util.function.DoubleSupplier, java.util.function.IntSupplier, java.util.function.LongSupplier, DoubleStatistic, StatisticAccumulator<Skewness>, StatisticResult

    public final class Skewness
    extends java.lang.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 int LENGTH_THREE
      3, the length limit where the unbiased skewness is undefined.
      private static int LENGTH_TWO
      2, the length limit where the biased skewness is undefined.
      private 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

      All Methods Static Methods Instance Methods Concrete Methods 
      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.
      Skewness 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
    • Field Detail

      • 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
      • biased

        private boolean biased
        Flag to control if the statistic is biased, or should use a bias correction.
    • Constructor Detail

      • 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 Detail

      • 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 java.util.function.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 java.util.function.DoubleSupplier
        Returns:
        skewness of all values.
      • 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