Package smile.stat.distribution

Class GammaDistribution

java.lang.Object
smile.stat.distribution.GammaDistribution
All Implemented Interfaces:
Serializable, Distribution, ExponentialFamily
public class GammaDistribution extends Object implements ExponentialFamily
The Gamma distribution is a continuous probability distributions with a scale parameter θ and a shape parameter k. If k is an integer then the distribution represents the sum of k independent exponentially distributed random variables, each of which has a rate parameter of θ). The gamma distribution is frequently a probability model for waiting times; for instance, the waiting time until death in life testing. The probability density function is f(x; k,θ) = xk-1e-x/θ / (θkΓ(k)) for x > 0 and k, θ > 0.
  • If X ∼ Γ(k=1, θ=1/λ), then X has an exponential distribution with rate parameter λ.
  • If X ∼ Γ(k=ν/2, θ=2), then X is identical to Χ2(ν), the chi-square distribution with ν degrees of freedom. Conversely, if Q ∼ Χ2(ν), and c is a positive constant, then c ⋅ Q ∼ Γ(k=ν/2, θ=2c).
  • If k is an integer, the gamma distribution is an Erlang distribution and is the probability distribution of the waiting time until the k-th "arrival" in a one-dimensional Poisson process with intensity 1/θ.
  • If X2 ∼ Γ(3/2, 2a2), then X has a Maxwell-Boltzmann distribution with parameter a.
In Bayesian inference, the gamma distribution is the conjugate prior to many likelihood distributions: the Poisson, exponential, normal (with known mean), Pareto, gamma with known shape, and inverse gamma with known shape parameter.
See Also:

  • Field Summary

    Fields
    Modifier and Type
    Field
    Description
    final double
    k
    The shape parameter.
    final double
    theta
    The scale parameter.

  • Constructor Summary

    Constructors
    Constructor
    Description
    GammaDistribution(double shape, double scale)
    Constructor.

  • Method Summary

    Modifier and Type
    Method
    Description
    double
    cdf(double x)
    Cumulative distribution function.
    double
    entropy()
    Returns Shannon entropy of the distribution.
    static GammaDistribution
    fit(double[] data)
    Estimates the distribution parameters by (approximate) MLE.
    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()
    Only support shape parameter k of integer.
    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

    • theta

      public final double theta
      The scale parameter.

    • k

      public final double k
      The shape parameter.

  • Constructor Details

    • GammaDistribution

      public GammaDistribution(double shape, double scale)
      Constructor.
      Parameters:
      shape - the shape parameter.
      scale - the scale parameter.

  • Method Details

    • fit

      public static GammaDistribution fit(double[] data)
      Estimates the distribution parameters by (approximate) MLE.
      Parameters:
      data - the training data.
      Returns:
      the distribution.

    • 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()
      Only support shape parameter k of integer.
      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.