Compound Poisson Generalized Linear Models

Description

This function fits compound Poisson generalized linear models.

Usage

1
2
3
4
cpglm(formula, link = "log", data, weights, offset, 
          subset, na.action = NULL, contrasts = NULL, 
          control = list(), chunksize = 0, 
          optimizer = "nlminb", ...)                      

Arguments

formula

an object of class formula. See also in glm.

link

a specification for the model link function. This can be either a literal character string or a numeric number. If it is a character string, it must be one of "log", "identity", "sqrt" or "inverse". If it is numeric, it is the same as the link.power argument in the tweedie function. The default is link = "log".

data

an optional data frame, list or environment (or object coercible by as.data.frame to a data frame) containing the variables in the model.

weights

an optional vector of weights. Should be either NULL or a numeric vector. When it is numeric, it must be positive. Zero weights are not allowed in cpglm.

subset

an optional vector specifying a subset of observations to be used in the fitting process.

na.action

a function which indicates what should happen when the data contain NAs. The default is set by the na.action setting of options, and is na.fail if that is unset. Another possible value is NULL, no action. Value na.exclude can be useful.

offset

this can be used to specify an a priori known component to be included in the linear predictor during fitting. This should be either NULL or a numeric vector of length equal to the number of cases. One or more offset terms can be included in the formula instead or as well, and if more than one is specified their sum is used.

contrasts

an optional list. See contrasts.arg.

control

a list of parameters for controling the fitting process. See 'Details' below.

chunksize

an integer that indicates the size of chunks for processing the data frame as used in bigglm. The value of this argument also determines how the model is estimated. When it is 0 (the default), regular Fisher's scoring algorithms are used, which may run into memory issues when handling large data sets. In contrast, a value greater than 0 indicates that the bigglm is employed to fit the GLMs. The function bigglm relies on the bounded memory regression technique, and thus is well suited to large data GLMs.

optimizer

a character string that determines which optimization routine is to be used in estimating the index and the dispersion parameters. Possible choices are "nlminb" (the default, see nlminb), "bobyqa" (bobyqa) and "L-BFGS-B" (optim).

...

additional arguments to be passed to bigglm. Not used when chunksize = 0. The maxit argument defaults to 50 in cpglm if not specified.

Details

This function implements the profile likelihood approach in Tweedie compound Poisson generalized linear models. First, the index and the dispersion parameters are estimated by maximizing (numerically) the profile likelihood (profile out the mean parameters as they are determined for a given value of the index parameter). Then the mean parameters are estimated using a GLM with the above-estimated index parameter. To compute the profile likelihood, one must resort to numerical methods provided in the tweedie package for approximating the density of the compound Poisson distribution. Indeed, the function tweedie.profile in that package makes available the profile likelihood approach. The cpglm function differs from tweedie.profile in two aspects. First, the user does not need to specify the grid of possible values the index parameter can take. Rather, the optimization of the profile likelihood is automated. Second, big data sets can be handled where the bigglm function from the biglm package is used in fitting GLMs. The bigglm is invoked when the argument chunksize is greater than 0. It is also to be noted that only MLE estimate for the dispersion parameter is included here, while tweedie.profile provides several other possibilities.

The package used to implement a second approach using the Monte Carlo EM algorithm, but it is now removed because it does not offer obvious advantages over the profile likelihood approach for this model.

The control argument is a list that can supply various controlling elements used in the optimization process, and it has the following components:

bound.p

a vector of lower and upper bounds for the index parameter p used in the optimization. The default is c(1.01, 1.99).

trace

if greater than 0, tracing information on the progress of the fitting is produced. For optimizer = "nlminb" or optimizer = "L-BFGS-B", this is the same as the trace control parameter, and for optimizer = "bobyqa", this is the same as the iprint control parameter. See the corresponding documentation for details.

max.iter

maximum number of iterations allowed in the optimization. The default is 300.

max.fun

maximum number of function evaluations allowed in the optimizer. The default is 2000.

Value

cpglm returns an object of class "cpglm". See cpglm-class for details of the return values as well as various methods available for this class.

Author(s)

Yanwei (Wayne) Zhang actuary_zhang@hotmail.com

References

Dunn, P.K. and Smyth, G.K. (2005). Series evaluation of Tweedie exponential dispersion models densities. Statistics and Computing, 15, 267-280.

See Also

The users are recommended to see the documentation for cpglm-class, glm, tweedie, and tweedie.profile for related information.

Examples

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
fit1 <- cpglm(RLD ~ factor(Zone) * factor(Stock),
  data = FineRoot)
     
# residual and qq plot
parold <- par(mfrow = c(2, 2), mar = c(5, 5, 2, 1))
# 1. regular plot
r1 <- resid(fit1) / sqrt(fit1$phi)
plot(r1 ~ fitted(fit1), cex = 0.5)
qqnorm(r1, cex = 0.5)
# 2. quantile residual plot to avoid overlapping
u <- tweedie::ptweedie(fit1$y, fit1$p, fitted(fit1), fit1$phi)
u[fit1$y == 0] <- runif(sum(fit1$y == 0), 0, u[fit1$y == 0])
r2 <- qnorm(u)
plot(r2 ~ fitted(fit1), cex = 0.5)
qqnorm(r2, cex = 0.5)
par(parold)

# use bigglm 
fit2 <- cpglm(RLD ~ factor(Zone), 
  data = FineRoot, chunksize = 250)