smile.stat.distribution

Class PoissonDistribution

java.lang.Object

smile.stat.distribution.AbstractDistribution

smile.stat.distribution.DiscreteDistribution

smile.stat.distribution.PoissonDistribution

All Implemented Interfaces:

java.io.Serializable, DiscreteExponentialFamily, Distribution

public class PoissonDistribution
extends DiscreteDistribution
implements DiscreteExponentialFamily

Poisson distribution expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate and independently of the time since the last event. The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume. If the expected number of occurrences in this interval is λ, then the probability that there are exactly n occurrences (n = 0, 1, 2, ...) is equal to

             λn e-λ
 f(n; λ) = ---------
               n!
 

For sufficiently large values of λ, (say λ > 1000), the normal distribution with mean λ and variance λ, is an excellent approximation to the Poisson distribution. If λ is greater than about 10, then the normal distribution is a good approximation if an appropriate continuity correction is performed, i.e., P(X ≤ x), where (lower-case) x is a non-negative integer, is replaced by P(X ≤ x + 0.5).

When a variable is Poisson distributed, its square root is approximately normally distributed with expected value of about λ^1/2 and variance of about 1/4.

See Also:

Serialized Form

Field Summary

Fields

Modifier and Type	Field and Description
double	lambda The average number of events per interval.

Constructor Summary

Constructors

Constructor and Description
PoissonDistribution(double lambda) Constructor.

Method Summary

Modifier and Type	Method and Description
double	cdf(double k) Cumulative distribution function.
double	entropy() Shannon entropy of the distribution.
static PoissonDistribution	fit(int[] data) Estimates the distribution parameters by MLE.
int	length() The number of parameters of the distribution.
double	logp(int k) The probability mass function in log scale.
DiscreteMixture.Component	M(int[] x, double[] posteriori) The M step in the EM algorithm, which depends the specific distribution.
double	mean() The mean of distribution.
double	p(int k) The probability mass function.
double	quantile(double p) The quantile, the probability to the left of quantile is p.
double	rand() This function generates a random variate with the poisson distribution.
double	sd() The standard deviation of distribution.
java.lang.String	toString()
double	variance() The variance of distribution.

Methods inherited from class smile.stat.distribution.DiscreteDistribution

likelihood, logLikelihood, logp, p, quantile, randi, randi

Methods inherited from class smile.stat.distribution.AbstractDistribution

inverseTransformSampling, quantile, quantile, rejection

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

likelihood, logLikelihood, rand

Field Detail

lambda

public final double lambda

The average number of events per interval.

Constructor Detail

PoissonDistribution

public PoissonDistribution(double lambda)

Constructor.

Parameters:

lambda - the average number of events per interval.

Method Detail

fit

public static PoissonDistribution fit(int[] data)

Estimates the distribution parameters by MLE.

length

public int length()

Description copied from interface: Distribution

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

mean

public double mean()

Description copied from interface: Distribution

The mean of distribution.

Specified by:

mean in interface Distribution

variance

public double variance()

Description copied from interface: Distribution

The variance of distribution.

Specified by:

variance 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 java.lang.String toString()

Overrides:

toString in class java.lang.Object

p

public double p(int k)

Description copied from class: DiscreteDistribution

The probability mass function.

Specified by:

p in class DiscreteDistribution

logp

public double logp(int k)

Description copied from class: DiscreteDistribution

The probability mass function in log scale.

Specified by:

logp in class DiscreteDistribution

cdf

public double cdf(double k)

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 DiscreteMixture.Component M(int[] x,
                                   double[] posteriori)

Description copied from interface: DiscreteExponentialFamily

The M step in the EM algorithm, which depends the specific distribution.

Specified by:

M in interface DiscreteExponentialFamily

Parameters:

x - the input data for estimation

posteriori - the posteriori probability.

Returns:

the (unnormalized) weight of this distribution in the mixture.

rand

public double rand()

This function generates a random variate with the poisson distribution.

Uses down/up search from the mode by chop-down technique for λ < 20, and patchwork rejection method for λ ≥ 20.

For λ < 1.E-6 numerical inaccuracy is avoided by direct calculation.

Specified by:

rand in interface Distribution