com.numericalmethod.suanshu.stats.regression.linear.glm.distribution

Class Poisson

java.lang.Object

com.numericalmethod.suanshu.stats.regression.linear.glm.distribution.Family

com.numericalmethod.suanshu.stats.regression.linear.glm.distribution.Poisson

All Implemented Interfaces:

ExponentialDistribution, java.io.Serializable

Direct Known Subclasses:

Poisson

public class Poisson
extends Family

The Poisson distribution for the error distribution in a GLM model.

The R equivalent function is poisson.

See Also:

Serialized Form

Constructor Summary

Constructors

Constructor and Description
Poisson() Construct an instance of Poisson.
Poisson(LinkFunction link) Construct an instance of Poisson with an overriding link function.

Method Summary

Modifier and Type	Method and Description
double	AIC(Vector y, Vector mu, Vector weight, double preLogLike, double deviance, int nFactors) AIC = 2 * #param - 2 * log-likelihood
double	cumulant(double theta) The cumulant function of the exponential distribution.
double	deviance(double y, double mu) Deviance D(y;μ^) measures the goodness-of-fit of a model, which is defined as the difference between the maximum log likelihood achievable and that achieved by the model.
double	dispersion(Vector y, Vector mu, int nFactors) Different distribution models have different ways to compute dispersion, φ.
double	overdispersion(Vector y, Vector mu, int nFactors) Overdispersion is the presence of greater variability (statistical dispersion) in a data set than would be expected based on the nominal variance of a given simple statistical model.
double	theta(double mu) The canonical parameter of the distribution in terms of the mean μ.
double	variance(double mu) The variance function of the distribution in terms of the mean μ.

Methods inherited from class com.numericalmethod.suanshu.stats.regression.linear.glm.distribution.Family

link

Methods inherited from class java.lang.Object

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

Constructor Detail

Poisson

public Poisson()

Construct an instance of Poisson. The canonical link is Log.

See Also:

"pp.32. Section 2.2.4. Measuring the goodness-of-fit. Generalized Linear Models. 2nd ed. P. J. MacCullagh and J. A. Nelder."

Poisson

public Poisson(LinkFunction link)

Construct an instance of Poisson with an overriding link function.

Parameters:

link - the overriding link function

Method Detail

variance

public double variance(double mu)

Description copied from interface: ExponentialDistribution

The variance function of the distribution in terms of the mean μ.

Parameters:

mu - the distribution mean, μ

Returns:

the value of variance function at μ

See Also:

"P. J. MacCullagh and J. A. Nelder, Generalized Linear Models, 2nd ed. Chapter 2. Table 2.1. pp.30"

theta

public double theta(double mu)

Description copied from interface: ExponentialDistribution

The canonical parameter of the distribution in terms of the mean μ.

Parameters:

mu - the distribution mean, μ

Returns:

the value of canonical parameter θ at μ

See Also:

"P. J. MacCullagh and J. A. Nelder, Generalized Linear Models, 2nd ed. Chapter 2. Table 2.1. pp.30."

cumulant

public double cumulant(double theta)

Description copied from interface: ExponentialDistribution

The cumulant function of the exponential distribution.

Parameters:

theta - the input argument of the cumulant function

Returns:

the value of the cumulant function at (@code θ}

See Also:

"P. J. MacCullagh and J. A. Nelder, Generalized Linear Models, 2nd ed. Chapter 2. Table 2.1. pp.30."

deviance

public double deviance(double y,
                       double mu)

Description copied from class: Family

Deviance D(y;μ^) measures the goodness-of-fit of a model, which is defined as the difference between the maximum log likelihood achievable and that achieved by the model.

D(y;μ^) = 2 * [l(y;y) - l(μ^;y)]

where l is the log-likelihood.

For an exponential family distribution, this is equivalent to

2 * [(y * θ(y) - b(θ(y))) - (y * θ(μ^) - b(θ(μ^)]

where b() is the cumulant function of the distribution.

The definition above is the default implementation of this function, a subclass of ExponentialDistribution may override this function to implement a simplified expression for efficiency or handle special values.

Specified by:

deviance in interface ExponentialDistribution

Overrides:

deviance in class Family

Parameters:

y - the observed value

mu - the estimated mean, μ^

Returns:

the deviance

See Also:

• P. J. MacCullagh and J. A. Nelder, "Measuring the goodness-of-fit," Generalized Linear Models, 2nd ed. Section 2.3. pp.34.
• Wikipedia: Deviance

overdispersion

public double overdispersion(Vector y,
                             Vector mu,
                             int nFactors)

Description copied from interface: ExponentialDistribution

Overdispersion is the presence of greater variability (statistical dispersion) in a data set than would be expected based on the nominal variance of a given simple statistical model. σ^2 = X^2/(n-p), 4.23 X^2 = sum{(y-μ)^2}/V(μ), p.34 = sum{(y-μ)^2}/b''(θ), p.29 X^2 estimates a(φ) = φ, the dispersion parameter (assuming w = 1).

For, Gamma, Gaussian, InverseGaussian, over-dispersion is the same as dispersion.

mu - μ

Returns:

the dispersion

See Also:

"P. J. MacCullagh and J. A. Nelder, Generalized Linear Models, 2nd ed. Section 4.5. Equation 4.23."

dispersion

public double dispersion(Vector y,
                         Vector mu,
                         int nFactors)

Description copied from interface: ExponentialDistribution

Different distribution models have different ways to compute dispersion, φ.

Note that in R's output, this is called "over-dispersion".

mu - μ

Returns:

the dispersion

See Also:

"P. J. MacCullagh and J. A. Nelder, Generalized Linear Models, 2nd ed. Section 2.2.2. Table 2.1."

AIC

public double AIC(Vector y,
                  Vector mu,
                  Vector weight,
                  double preLogLike,
                  double deviance,
                  int nFactors)

Description copied from interface: ExponentialDistribution

AIC = 2 * #param - 2 * log-likelihood

mu - μ

preLogLike - sum of y * θ_i - b(θ_i)

Returns:

the AIC