Mixture of Two Poisson Distributions
Estimates the three parameters of a mixture of two Poisson distributions by maximum likelihood estimation.
mix2poisson(lphi = "logitlink", llambda = "loglink",
iphi = 0.5, il1 = NULL, il2 = NULL,
qmu = c(0.2, 0.8), nsimEIM = 100, zero = "phi")lphi, llambda |
Link functions for the parameter phi and
lambda.
See |
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 |
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 |
nsimEIM, zero |
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.
An object of class "vglmff" (see vglmff-class).
The object is used by modelling functions such as vglm
and vgam.
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.
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.
T. W. Yee
## 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)), digits = 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), data = ltdata2)
coef(Mfit, matrix = TRUE)
Coef(Mfit)
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