mix2normal1 {VGAM}R Documentation

Mixture of Two Univariate Normal Distributions

Description

Estimates the five parameters of a mixture of two univariate normal distributions by maximum likelihood estimation.

Usage

mix2normal1(lphi="logit", lmu="identity", lsd="loge",
            ephi=list(), emu1=list(), emu2=list(), esd1=list(), esd2=list(),
            iphi=0.5, imu1=NULL, imu2=NULL, isd1=NULL, isd2=NULL,
            qmu=c(0.2, 0.8), esd=FALSE, zero=1)

Arguments

lphi Link function for the parameter phi. See below for more details. See Links for more choices.
lmu Link function applied to each mu parameter. See Links for more choices.
lsd Link function applied to each sd parameter. See Links for more choices.
ephi, emu1, emu2, esd1, esd2 List. Extra argument for each of the links. See earg in Links for general information. If esd=TRUE then esd1 is used and not esd2.
iphi Initial value for phi, whose value must lie between 0 and 1.
imu1, imu2 Optional initial value for mu1 and mu2. The default is to compute initial values internally using the argument qmu.
isd1, isd2 Optional initial value for sd1 and sd2. The default is to compute initial values internally based on the argument qmu.
qmu Vector with two values giving the probabilities relating to the sample quantiles for obtaining initial values for mu1 and mu2. The two values are fed in as the probs argument into quantile.
esd Logical indicating whether the two standard deviations should be constrained to be equal. If set TRUE, the appropriate constraint matrices will be used.
zero An integer specifying which linear/additive predictor is modelled as intercepts only. If given, the value or values must be from the set 1,2,...,5. 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.

Details

The probability function can be loosely written as

f(y) = phi * N(mu1, sd1^2) + (1-phi) * N(mu2, sd2^2)

where phi is the probability an observation belongs to the first group. The parameters mu1 and mu2 are the means, and sd1 and sd2 are the standard deviations. The parameter phi satisfies 0 < phi < 1. The mean of Y is phi*mu1 + (1-phi)*mu2 and this is returned as the fitted values. By default, the five linear/additive predictors are (logit(phi), mu1, log(sd1), mu2, log(sd2))^T. If esd=TRUE then sd1=sd2 is enforced.

Value

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

Warning

Numerical problems can occur. Half-stepping is not uncommon. If failure to converge occurs, try obtaining better initial values, e.g., by using iphi and qmu etc.

This function uses a quasi-Newton update for the working weight matrices (BFGS variant). It builds up approximations to the weight matrices, and currently the code is not fully tested. In particular, results based on the weight matrices (e.g., from vcov and summary) may be quite incorrect, especially when the arguments weights is used to input prior weights.

This VGAM family function should be used with caution.

Note

Fitting this model successfully to data can be difficult due to numerical problems 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 mu1 and mu2 become more different.

Convergence is often slow, especially when the two component distributions are not well separated. The control argument maxit should be set to a higher value, e.g., 200, and use trace=TRUE to monitor convergence. If appropriate in the first place, setting esd=TRUE often makes the optimization problem much easier in general.

Author(s)

T. W. Yee

References

Everitt, B. S. and Hand, D. J. (1981) Finite Mixture Distributions. London: Chapman & Hall.

See Also

normal1, Normal, mix2poisson.

Examples

n = 1000
mu1 =  99   # Mean IQ of geography professors
mu2 = 150   # Mean IQ of mathematics professors
sd1 = sd2 = 16
phi = 0.3
y = ifelse(runif(n) < phi, rnorm(n, mu1, sd1), rnorm(n, mu2, sd2))

# Good idea to have trace=TRUE:
fit = vglm(y ~ 1, mix2normal1(esd=TRUE), maxit=200)
coef(fit, matrix=TRUE) # the estimates
c(phi, mu1, sd1, mu2, sd2) # the truth

## Not run: 
# Plot the results
xx = seq(min(y), max(y), len=200)
plot(xx, (1-phi)*dnorm(xx, mu2, sd2), type="l", xlab="IQ",
     main="Red=estimate, blue=truth", col="blue")
phi.est = logit(coef(fit)[1], inverse=TRUE)
sd.est = exp(coef(fit)[3])
lines(xx, phi*dnorm(xx, mu1, sd1), col="blue")
lines(xx, phi.est * dnorm(xx, Coef(fit)[2], sd.est), col="red")
lines(xx, (1-phi.est) * dnorm(xx, Coef(fit)[4], sd.est), col="red")
abline(v=Coef(fit)[c(2,4)], lty=2, col="red")
abline(v=c(mu1, mu2), lty=2, col="blue")
## End(Not run)

[Package VGAM version 0.7-4 Index]