GammaDistribution (Smile Math 1.0.4 API)

java.lang.Object
- smile.stat.distribution.AbstractDistribution
- - smile.stat.distribution.GammaDistribution

All Implemented Interfaces:

Distribution, ExponentialFamily
```
public class GammaDistribution
extends AbstractDistribution
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) = x^k-1e^-x/θ / (θ^kΓ(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.
Author:

Haifeng Li

Constructor Summary

Constructors
Constructor and Description

GammaDistribution(double[] data)
Constructor.

GammaDistribution(double shape, double scale)
Constructor.

Constructors
Constructor and Description
`GammaDistribution(double[] data)` Constructor.
`GammaDistribution(double shape, double scale)` Constructor.

Method Summary

All Methods Instance Methods Concrete Methods
Modifier and Type	Method and Description
`double`	`cdf(double x)` Cumulative distribution function.
`double`	`entropy()` Shannon entropy of the distribution.
`double`	`getScale()` Returns the scale parameter.
`double`	`getShape()` Returns the shape parameter.
`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 the specific distribution.
`double`	`mean()` The mean of distribution.
`int`	`npara()` The number of parameters of the 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()` The standard deviation of distribution.
`String`	`toString()`
`double`	`var()` The variance of distribution.

Methods inherited from class smile.stat.distribution.AbstractDistribution
inverseTransformSampling, likelihood, logLikelihood, quantile, quantile, rejection

Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, wait, wait, wait

- Constructor Detail
  - GammaDistribution
```
public GammaDistribution(double shape,
                         double scale)
```
    Constructor.
    
    Parameters:
    
    shape - the shape parameter.
    
    scale - the scale parameter.
  - GammaDistribution
```
public GammaDistribution(double[] data)
```
    Constructor. Parameter will be estimated from the data by (approximate) MLE.
- Method Detail
  - getScale
```
public double getScale()
```
    Returns the scale parameter.
  - getShape
```
public double getShape()
```
    Returns the shape parameter.
  - npara
```
public int npara()
```
    Description copied from interface: Distribution
    
    The number of parameters of the distribution.
    
    Specified by:
    
    npara in interface Distribution
  - mean
```
public double mean()
```
    Description copied from interface: Distribution
    
    The mean of distribution.
    
    Specified by:
    
    mean in interface Distribution
  - var
```
public double var()
```
    Description copied from interface: Distribution
    
    The variance of distribution.
    
    Specified by:
    
    var in interface Distribution
  - sd
```
public double sd()
```
    Description copied from interface: Distribution
    
    The standard deviation of distribution.
    
    Specified by:
    
    sd in interface Distribution
  - entropy
```
public double entropy()
```
    Description copied from interface: Distribution
    
    Shannon entropy of the distribution.
    
    Specified by:
    
    entropy in interface Distribution
  - 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
  - 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
  - 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
  - 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
  - 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
  - M
```
public Mixture.Component M(double[] x,
                           double[] posteriori)
```
    Description copied from interface: ExponentialFamily
    
    The M step in the EM algorithm, which depends 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.

Class GammaDistribution

Constructor Summary

Method Summary

Methods inherited from class smile.stat.distribution.AbstractDistribution

Methods inherited from class java.lang.Object

Constructor Detail

GammaDistribution

GammaDistribution

Method Detail

getScale

getShape

npara

mean

var

sd

entropy

toString

rand

p

logp

cdf

quantile

M