mix2poisson: Mixture of Two Poisson Distributions

In VGAM: Vector Generalized Linear and Additive Models

Mixture of Two Poisson Distributions

Description

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

Usage

mix2poisson(lphi = "logitlink", llambda = "loglink",
            iphi = 0.5, il1 = NULL, il2 = NULL,
            qmu = c(0.2, 0.8), nsimEIM = 100, zero = "phi")

Arguments

lphi, llambda	Link functions for the parameter \phi and \lambda. See Links for more choices.
iphi	Initial value for \phi, whose value must lie between 0 and 1.
il1, il2	Optional initial value for \lambda_1 and \lambda_2. 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 \lambda_1 and \lambda_2. The two values are fed in as the probs argument into quantile.
nsimEIM, zero	See CommonVGAMffArguments.

Details

The probability function can be loosely written as

P(Y=y) = \phi \, Poisson(\lambda_1) + (1-\phi) \, Poisson(\lambda_2)

where \phi is the probability an observation belongs to the first group, and y=0,1,2,\ldots. The parameter \phi satisfies 0 < \phi < 1. The mean of Y is \phi\lambda_1+(1-\phi)\lambda_2 and this is returned as the fitted values. By default, the three linear/additive predictors are (logit(\phi), \log(\lambda_1), \log(\lambda_2))^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.

This VGAM family function is experimental and should be used with care.

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 \lambda_1 and \lambda_2 become more different. The default control argument trace = TRUE is to encourage monitoring convergence.

Author(s)

T. W. Yee

See Also

rpois, poissonff, mix2normal.

Examples

## Not run:  # Example 1: simulated data
nn <- 1000
mu1 <- exp(2.5)  # Also known as lambda1
mu2 <- exp(3)
(phi <- logitlink(-0.5, inverse = TRUE))
mdata <- data.frame(y = rpois(nn, ifelse(runif(nn) < phi, mu1, mu2)))
mfit <- vglm(y ~ 1, mix2poisson, data = mdata)
coef(mfit, matrix = TRUE)

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

ty <- with(mdata, table(y))
plot(names(ty), ty, type = "h", main = "Orange=estimate, blue=truth",
     ylab = "Frequency", xlab = "y")
abline(v = Coef(mfit)[-1], lty = 2, col = "orange", lwd = 2)
abline(v = c(mu1, mu2), lty = 2, col = "blue", lwd = 2)

# Example 2: London Times data (Lange, 1997, p.31)
ltdata1 <- data.frame(deaths = 0:9,
                      freq = c(162,267,271, 185,111,61,27,8,3,1))
ltdata2 <- data.frame(y = with(ltdata1, rep(deaths, freq)))

# Usually this does not work well unless nsimEIM is large
Mfit <- vglm(deaths ~ 1, weight = freq, data = ltdata1,
        mix2poisson(iphi=0.3, il1=1, il2=2.5, nsimEIM=5000))

# This works better in general
Mfit = vglm(y ~ 1, mix2poisson(iphi=0.3, il1=1, il2=2.5), ltdata2)
coef(Mfit, matrix = TRUE)
Coef(Mfit)

## End(Not run)

VGAM documentation built on Sept. 18, 2024, 9:09 a.m.