# SuanShu, a Java numerical and statistical library

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

## Interface GLMExponentialDistribution

• ### Method Summary

All Methods
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)
Over-dispersion 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.
• ### Method Detail

• #### variance

double variance(double mu)
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
"P. J. MacCullagh and J. A. Nelder, "Chapter 2, Table 2.1, pp.30," Generalized Linear Models, 2nd ed."
• #### theta

double theta(double mu)
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
"P. J. MacCullagh and J. A. Nelder, "Chapter 2, Table 2.1, pp.30," Generalized Linear Models, 2nd ed."
• #### cumulant

double cumulant(double theta)
The cumulant function of the exponential distribution.
Parameters:
theta - θ
Returns:
the value of the cumulant function at θ
"P. J. MacCullagh and J. A. Nelder, "Chapter 2, Table 2.1, pp.30," Generalized Linear Models, 2nd ed."
• #### dispersion

double dispersion(Vector y,
Vector mu,
int nFactors)
Different distribution models have different ways to compute dispersion, Φ.

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

Parameters:
y - an observation
mu - the distribution mean, μ
nFactors - the number of factors
Returns:
the dispersion
"P. J. MacCullagh and J. A. Nelder, "Section 2.2.2, Table 2.1," Generalized Linear Models, 2nd ed."
• #### overdispersion

double overdispersion(Vector y,
Vector mu,
int nFactors)
Over-dispersion 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), eq. 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, GLMGamma, GLMGaussian, GLMInverseGaussian, over-dispersion is the same as dispersion.

Parameters:
y - an observation
mu - the distribution mean, μ
nFactors - the number of factors
Returns:
the dispersion
"P. J. MacCullagh and J. A. Nelder, "Section 4.5, Equation 4.23," Generalized Linear Models, 2nd ed."
• #### deviance

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.
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.
Parameters:
y - an observation
mu - the estimated mean, μ^
Returns:
the deviance
• P. J. MacCullagh and J. A. Nelder, "Section 2.3, pp.34, Measuring the goodness-of-fit," Generalized Linear Models, 2nd ed.
• Wikipedia: Deviance
• #### AIC

double AIC(Vector y,
Vector mu,
Vector weight,
double preLogLike,
double deviance,
int nFactors)
AIC = 2 * #param - 2 * log-likelihood
Parameters:
y - an observation
mu - the distribution mean, μ
weight - the weights assigned to the observations
preLogLike - sum of (yi * θi - b(θi))
deviance - the deviance
nFactors - the number of factors
Returns:
the AIC the Akaike information criterion