GaussianDistribution (Smile Math 1.0.3 API)

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

All Implemented Interfaces:

Distribution, ExponentialFamily
```
public class GaussianDistribution
extends AbstractDistribution
implements ExponentialFamily
```
The normal distribution or Gaussian distribution is a continuous probability distribution that describes data that clusters around a mean. The graph of the associated probability density function is bell-shaped, with a peak at the mean, and is known as the Gaussian function or bell curve. The normal distribution can be used to describe any variable that tends to cluster around the mean.
The family of normal distributions is closed under linear transformations. That is, if X is normally distributed, then a linear transform aX + b (for some real numbers a ≠ 0 and b) is also normally distributed. If X₁, X₂ are two independent normal random variables, then their linear combination will also be normally distributed. The converse is also true: if X₁ and X₂ are independent and their sum X₁ + X₂ is distributed normally, then both X₁ and X₂ must also be normal, which is known as the Cramer's theorem. Of all probability distributions over the reals with mean μ and variance σ², the normal distribution N(μ, σ²) is the one with the maximum entropy.
The central limit theorem states that under certain, fairly common conditions, the sum of a large number of random variables will have approximately normal distribution. For example if X₁, …, X_n is a sequence of iid random variables, each having mean μ and variance σ² but otherwise distributions of X_i's can be arbitrary, then the central limit theorem states that
√n (1⁄n Σ X_i - μ) → N(0, σ²).
The theorem will hold even if the summands X_i are not iid, although some constraints on the degree of dependence and the growth rate of moments still have to be imposed.
Therefore, certain other distributions can be approximated by the normal distribution, for example:
- The binomial distribution B(n, p) is approximately normal N(np, np(1-p)) for large n and for p not too close to zero or one.
- The Poisson(λ) distribution is approximately normal N(λ, λ) for large values of λ.
- The chi-squared distribution Χ²(k) is approximately normal N(k, 2k) for large k.
- The Student's t-distribution t(ν) is approximately normal N(0, 1) when ν is large.
Author:

Haifeng Li

Constructor Summary

Constructors
Constructor and Description

GaussianDistribution(double[] data)
Constructor.

GaussianDistribution(double mu, double sigma)
Constructor

Constructors
Constructor and Description
`GaussianDistribution(double[] data)` Constructor.
`GaussianDistribution(double mu, double sigma)` Constructor

Method Summary

Methods
Modifier and Type	Method and Description
`double`	`cdf(double x)` Cumulative distribution function.
`double`	`entropy()` Shannon entropy of the distribution.
`static GaussianDistribution`	`getInstance()`
`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(p) is p.
`double`	`rand()` Uses the Box-Muller algorithm to transform Random.random()'s into Gaussian deviates.
`double`	`randInverseCDF()` Uses Inverse CDF method to generate a Gaussian deviate.
`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
  - GaussianDistribution
```
public GaussianDistribution(double mu,
                    double sigma)
```
    Constructor
    
    Parameters:
    mu - mean.
    sigma - standard deviation.
  - GaussianDistribution
```
public GaussianDistribution(double[] data)
```
    Constructor. Mean and standard deviation will be estimated from the data by MLE.
- Method Detail
  - getInstance
```
public static GaussianDistribution getInstance()
```
  - 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()
```
    Uses the Box-Muller algorithm to transform Random.random()'s into Gaussian deviates.
    
    Specified by:
    
    rand in interface Distribution
  - randInverseCDF
```
public double randInverseCDF()
```
    Uses Inverse CDF method to generate a Gaussian deviate.
  - 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)
```
    The quantile, the probability to the left of quantile(p) is p. This is actually the inverse of cdf. Original algorythm and Perl implementation can be found at: http://www.math.uio.no/~jacklam/notes/invnorm/index.html
    
    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 GaussianDistribution

Constructor Summary

Method Summary

Methods inherited from class smile.stat.distribution.AbstractDistribution

Methods inherited from class java.lang.Object

Constructor Detail

GaussianDistribution

GaussianDistribution

Method Detail

getInstance

npara

mean

var

sd

entropy

toString

rand

randInverseCDF

p

logp

cdf

quantile

M