Class GLMPoisson

    • Constructor Summary

      Constructors 
      Constructor Description
      GLMPoisson()  
    • Method Summary

      All Methods Instance Methods Concrete 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-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.
    • Constructor Detail

      • GLMPoisson

        public GLMPoisson()
    • Method Detail

      • variance

        public double variance​(double mu)
        Description copied from interface: GLMExponentialDistribution
        The variance function of the distribution in terms of the mean μ.
        Specified by:
        variance in interface GLMExponentialDistribution
        Parameters:
        mu - the distribution mean, μ
        Returns:
        the value of variance function at μ
      • theta

        public double theta​(double mu)
        Description copied from interface: GLMExponentialDistribution
        The canonical parameter of the distribution in terms of the mean μ.
        Specified by:
        theta in interface GLMExponentialDistribution
        Parameters:
        mu - the distribution mean, μ
        Returns:
        the value of canonical parameter θ at μ
      • cumulant

        public double cumulant​(double theta)
        Description copied from interface: GLMExponentialDistribution
        The cumulant function of the exponential distribution.
        Specified by:
        cumulant in interface GLMExponentialDistribution
        Parameters:
        theta - θ
        Returns:
        the value of the cumulant function at θ
      • deviance

        public double deviance​(double y,
                               double mu)
        Description copied from interface: GLMExponentialDistribution
        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.
        Specified by:
        deviance in interface GLMExponentialDistribution
        Parameters:
        y - an observation
        mu - 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
      • overdispersion

        public double overdispersion​(Vector y,
                                     Vector mu,
                                     int nFactors)
        Description copied from interface: GLMExponentialDistribution
        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.

        Specified by:
        overdispersion in interface GLMExponentialDistribution
        Parameters:
        y - an observation
        mu - the distribution mean, μ
        nFactors - the number of factors
        Returns:
        the dispersion
      • dispersion

        public double dispersion​(Vector y,
                                 Vector mu,
                                 int nFactors)
        Description copied from interface: GLMExponentialDistribution
        Different distribution models have different ways to compute dispersion, Φ.

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

        Specified by:
        dispersion in interface GLMExponentialDistribution
        Parameters:
        y - an observation
        mu - the distribution mean, μ
        nFactors - the number of factors
        Returns:
        the dispersion
      • AIC

        public double AIC​(Vector y,
                          Vector mu,
                          Vector weight,
                          double preLogLike,
                          double deviance,
                          int nFactors)
        Description copied from interface: GLMExponentialDistribution
        AIC = 2 * #param - 2 * log-likelihood
        Specified by:
        AIC in interface GLMExponentialDistribution
        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