mix2poisson {VGAM}R Documentation

Mixture of Two Poisson Distributions

Description

Estimates the three parameters of a mixture of two Poisson distributions by maximum likelihood estimation.

Usage

mix2poisson(lphi = "logit", llambda = "loge",
            ephi=list(), el1=list(), el2=list(),
            iphi = 0.5, il1 = NULL, il2 = NULL,
            qmu = c(0.2, 0.8), nsimEIM=100, zero = 1)

Arguments

lphi Link function for the parameter phi. See Links for more choices.
llambda Link function applied to each lambda parameter. See Links for more choices.
ephi, el1, el2 List. Extra argument for each of the links. See earg in Links for general information.
iphi Initial value for phi, whose value must lie between 0 and 1.
il1, il2 Optional initial value for lambda1 and lambda2. These values must be positive. The default is to compute initial values internally using the argument qmu.
qmu Vector with two values giving the probabilities relating to the sample quantiles for obtaining initial values for lambda1 and lambda2. The two values are fed in as the probs argument into quantile.
nsimEIM See CommonVGAMffArguments.
zero An integer specifying which linear/additive predictor is modelled as intercepts only. If given, the value must be either 1 and/or 2 and/or 3, and the default is the first one only, meaning phi is a single parameter even when there are explanatory variables. Set zero=NULL to model all linear/additive predictors as functions of the explanatory variables. See CommonVGAMffArguments for more information.

Details

The probability function can be loosely written as

P(Y=y) = phi * Poisson(lambda1) + (1-phi) * Poisson(lambda2)

where phi is the probability an observation belongs to the first group, and y=0,1,2,.... The parameter phi satisfies 0 < phi < 1. The mean of Y is phi*lambda1 + (1-phi)*lambda2 and this is returned as the fitted values. By default, the three linear/additive predictors are (logit(phi), log(lambda1), log(lambda2))^T.

Value

An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm and vgam.

Warning

This VGAM family function requires care for a successful application. In particular, good initial values are required because of the presence of local solutions. Therefore running this function with several different combinations of arguments such as iphi, il1, il2, qmu is highly recommended. Graphical methods such as hist can be used as an aid.

With grouped data (i.e., using the weights argument) one has to use a large value of nsimEIM; see the example below.

Note

The response must be integer-valued since dpois is invoked.

Fitting this model successfully to data can be difficult due to local solutions and ill-conditioned data. It pays to fit the model several times with different initial values, and check that the best fit looks reasonable. Plotting the results is recommended. This function works better as lambda1 and lambda2 become more different. The default control argument trace=TRUE is to encourage monitoring convergence.

Author(s)

T. W. Yee

See Also

rpois, poissonff, mix2normal1.

Examples

# Example 1: simulated data
n = 1000
mu1 = exp(2.5) # also known as lambda1
mu2 = exp(3)
(phi = logit(-0.5, inverse=TRUE))
y = ifelse(runif(n) < phi, rpois(n, mu1), rpois(n, mu2))
fit = vglm(y ~ 1, mix2poisson)
coef(fit, matrix=TRUE)

# Compare the results with the truth
round(rbind('Estimated'=Coef(fit), 'Truth'=c(phi, mu1, mu2)), dig=2)

## Not run: 
# Plot the results
ty = table(y)
plot(names(ty), ty, type="h", main="Red=estimate, blue=truth")
abline(v=Coef(fit)[-1], lty=2, col="red", lwd=2)
abline(v=c(mu1, mu2), lty=2, col="blue", lwd=2)
## End(Not run)

# Example 2: London Times data (Lange, 1997, p.31)
deaths = 0:9
freq = c(162, 267, 271, 185, 111, 61, 27, 8, 3, 1)
y = rep(deaths, freq)

# Usually this does not work well unless nsimEIM is large
fit = vglm(deaths ~ 1, weight = freq,
           mix2poisson(iphi=0.3, il1=1, il2=2.5, nsimEIM=5000))

# This works better in general
fit = vglm(y ~ 1, mix2poisson(iphi=0.3, il1=1, il2=2.5))

coef(fit, matrix=TRUE)
Coef(fit)

[Package VGAM version 0.7-7 Index]