Interface GLMExponentialDistribution
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- All Known Subinterfaces:
QuasiDistribution
- All Known Implementing Classes:
GLMBinomial
,GLMGamma
,GLMGaussian
,GLMInverseGaussian
,GLMPoisson
,QuasiBinomial
,QuasiGamma
,QuasiGaussian
,QuasiInverseGaussian
,QuasiPoisson
public interface GLMExponentialDistribution
This interface represents a probability distribution from the exponential family. \[ f_{Y}(y;\theta,\phi)=\exp[\frac{y * \theta - b(\theta)}{a(\phi)}+c(y,\phi)] \] where the parameter θ is called the canonical parameter, b(θ) the cumulant function, and Φ the dispersion parameter.
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Method Summary
All Methods Instance Methods Abstract Methods Modifier and Type Method Description double
AIC(Vector y, Vector mu, Vector weight, double preLogLike, double deviance, int nFactors)
AIC = 2 * #param - 2 * log-likelihooddouble
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.
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Method Detail
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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
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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
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cumulant
double cumulant(double theta)
The cumulant function of the exponential distribution.- Parameters:
theta
- θ- Returns:
- the value of the cumulant function at θ
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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 observationmu
- the distribution mean, μnFactors
- the number of factors- Returns:
- the dispersion
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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.
X^2 estimates a(Φ) = Φ, the dispersion parameter (assuming w = 1). For,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
GLMGamma
,GLMGaussian
,GLMInverseGaussian
, over-dispersion is the same as dispersion.- Parameters:
y
- an observationmu
- the distribution mean, μnFactors
- the number of factors- Returns:
- the dispersion
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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 to2 * [(y * θ(y) - b(θ(y))) - (y * θ(μ^) - b(θ(μ^)]
where b is the cumulant function of the distribution.- Parameters:
y
- an observationmu
- the estimated mean, μ^- Returns:
- the deviance
- See Also:
- P. J. MacCullagh and J. A. Nelder, "Section 2.3, pp.34, Measuring the goodness-of-fit," Generalized Linear Models, 2nd ed.
- Wikipedia: Deviance
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AIC
double AIC(Vector y, Vector mu, Vector weight, double preLogLike, double deviance, int nFactors)
AIC = 2 * #param - 2 * log-likelihood- Parameters:
y
- an observationmu
- the distribution mean, μweight
- the weights assigned to the observationspreLogLike
- sum of (yi * θi - b(θi))deviance
- the deviancenFactors
- the number of factors- Returns:
- the AIC the Akaike information criterion
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