Package smile.stat.distribution

Class ChiSquareDistribution

java.lang.Object
smile.stat.distribution.ChiSquareDistribution
All Implemented Interfaces:
Serializable, Distribution, ExponentialFamily
public class ChiSquareDistribution extends Object implements ExponentialFamily
Chi-square (or chi-squared) distribution with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. It's mean and variance are k and 2k, respectively. The chi-square distribution is a special case of the gamma distribution. It follows from the definition of the chi-square distribution that the sum of independent chi-square variables is also chi-square distributed. Specifically, if Xi are independent chi-square variables with ki degrees of freedom, respectively, then Y = Σ Xi is chi-square distributed with Σ ki degrees of freedom.

The chi-square distribution has numerous applications in inferential statistics, for instance in chi-square tests and in estimating variances. Many other statistical tests also lead to a use of this distribution, like Friedman's analysis of variance by ranks.

See Also:

  • Field Summary

    Fields
    Modifier and Type
    Field
    Description
    final int
    nu
    The degrees of freedom.

  • Constructor Summary

    Constructors
    Constructor
    Description
    ChiSquareDistribution(int nu)
    Constructor.

  • Method Summary

    Modifier and Type
    Method
    Description
    double
    cdf(double x)
    Cumulative distribution function.
    double
    entropy()
    Returns Shannon entropy of the distribution.
    int
    length()
    Returns the number of parameters of the distribution.
    double
    logp(double x)
    The density at x in log scale, which may prevents the underflow problem.
    Mixture.Component
    M(double[] x, double[] posteriori)
    The M step in the EM algorithm, which depends on the specific distribution.
    double
    mean()
    Returns the mean of distribution.
    double
    p(double x)
    The probability density function for continuous distribution or probability mass function for discrete distribution at x.
    double
    quantile(double p)
    The quantile, the probability to the left of quantile is p.
    double
    rand()
    Generates a random number following this distribution.
    double
    sd()
    Returns the standard deviation of distribution.
    String
    toString()
     
    double
    variance()
    Returns the variance of distribution.

    Methods inherited from class java.lang.Object

    clone, equals, finalize, getClass, hashCode, notify, notifyAll, wait, wait, wait

    Methods inherited from interface smile.stat.distribution.Distribution

    inverseTransformSampling, likelihood, logLikelihood, quantile, quantile, rand, rejectionSampling

  • Field Details

    • nu

      public final int nu
      The degrees of freedom.

  • Constructor Details

    • ChiSquareDistribution

      public ChiSquareDistribution(int nu)
      Constructor.
      Parameters:
      nu - the degree of freedom.

  • Method Details

    • length

      public int length()
      Description copied from interface: Distribution
      Returns the number of parameters of the distribution. The "length" is in the sense of the minimum description length principle.
      Specified by:
      length in interface Distribution
      Returns:
      The number of parameters.

    • mean

      public double mean()
      Description copied from interface: Distribution
      Returns the mean of distribution.
      Specified by:
      mean in interface Distribution
      Returns:
      The mean.

    • variance

      public double variance()
      Description copied from interface: Distribution
      Returns the variance of distribution.
      Specified by:
      variance in interface Distribution
      Returns:
      The variance.

    • sd

      public double sd()
      Description copied from interface: Distribution
      Returns the standard deviation of distribution.
      Specified by:
      sd in interface Distribution
      Returns:
      The standard deviation.

    • entropy

      public double entropy()
      Description copied from interface: Distribution
      Returns Shannon entropy of the distribution.
      Specified by:
      entropy in interface Distribution
      Returns:
      Shannon entropy.

    • toString

      public String toString()
      Overrides:
      toString in class Object

    • rand

      public double rand()
      Description copied from interface: Distribution
      Generates a random number following this distribution.
      Specified by:
      rand in interface Distribution
      Returns:
      a random number.

    • p

      public double p(double x)
      Description copied from interface: Distribution
      The probability density function for continuous distribution or probability mass function for discrete distribution at x.
      Specified by:
      p in interface Distribution
      Parameters:
      x - a real number.
      Returns:
      the density.

    • logp

      public double logp(double x)
      Description copied from interface: Distribution
      The density at x in log scale, which may prevents the underflow problem.
      Specified by:
      logp in interface Distribution
      Parameters:
      x - a real number.
      Returns:
      the log density.

    • cdf

      public double cdf(double x)
      Description copied from interface: Distribution
      Cumulative distribution function. That is the probability to the left of x.
      Specified by:
      cdf in interface Distribution
      Parameters:
      x - a real number.
      Returns:
      the probability.

    • quantile

      public double quantile(double p)
      Description copied from interface: Distribution
      The quantile, the probability to the left of quantile is p. It is actually the inverse of cdf.
      Specified by:
      quantile in interface Distribution
      Parameters:
      p - the probability.
      Returns:
      the quantile.

    • M

      public Mixture.Component M(double[] x, double[] posteriori)
      Description copied from interface: ExponentialFamily
      The M step in the EM algorithm, which depends on the specific distribution.
      Specified by:
      M in interface ExponentialFamily
      Parameters:
      x - the input data for estimation
      posteriori - the posteriori probability.
      Returns:
      the (unnormalized) weight of this distribution in the mixture.