Mixture of Two Univariate Normal Distributions
Estimates the five parameters of a mixture of two univariate normal distributions by maximum likelihood estimation.
mix2normal(lphi = "logitlink", lmu = "identitylink", lsd = "loglink",
iphi = 0.5, imu1 = NULL, imu2 = NULL, isd1 = NULL, isd2 = NULL,
qmu = c(0.2, 0.8), eq.sd = TRUE, nsimEIM = 100, zero = "phi")lphi,lmu,lsd |
Link functions for the parameters phi,
mu, and
sd.
See |
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 |
isd1, isd2 |
Optional initial value for sd1 and sd2.
The default is to compute initial values internally based on
the argument |
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 |
eq.sd |
Logical indicating whether the two standard deviations should be
constrained to be equal. If |
nsimEIM |
|
zero |
May be an integer vector
specifying which linear/additive predictors are modelled as
intercept-only. If given, the value or values can 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 |
The probability density function can be loosely written as
f(y) = phi * N(mu1, sd1) + (1-phi) * N(mu2, sd2)
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 eq.sd = TRUE then sd1=sd2
is enforced.
An object of class "vglmff" (see vglmff-class).
The object is used by modelling functions such as vglm,
and vgam.
Numerical problems can occur and
half-stepping is not uncommon.
If failure to converge occurs, try inputting better initial values,
e.g., by using iphi,
qmu,
imu1,
imu2,
isd1,
isd2,
etc.
This VGAM family function is experimental and should be used with care.
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 can be slow, especially when the two component
distributions are not well separated.
The default control argument trace = TRUE is to encourage
monitoring convergence.
Having eq.sd = TRUE often makes the overall optimization problem
easier.
T. W. Yee
McLachlan, G. J. and Peel, D. (2000). Finite Mixture Models. New York: Wiley.
Everitt, B. S. and Hand, D. J. (1981). Finite Mixture Distributions. London: Chapman & Hall.
## Not run: mu1 <- 99; mu2 <- 150; nn <- 1000
sd1 <- sd2 <- exp(3)
(phi <- logitlink(-1, inverse = TRUE))
mdata <- data.frame(y = ifelse(runif(nn) < phi, rnorm(nn, mu1, sd1),
rnorm(nn, mu2, sd2)))
fit <- vglm(y ~ 1, mix2normal(eq.sd = TRUE), data = mdata)
# Compare the results
cfit <- coef(fit)
round(rbind('Estimated' = c(logitlink(cfit[1], inverse = TRUE),
cfit[2], exp(cfit[3]), cfit[4]),
'Truth' = c(phi, mu1, sd1, mu2)), digits = 2)
# Plot the results
xx <- with(mdata, seq(min(y), max(y), len = 200))
plot(xx, (1-phi) * dnorm(xx, mu2, sd2), type = "l", xlab = "y",
main = "Orange = estimate, blue = truth",
col = "blue", ylab = "Density")
phi.est <- logitlink(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 = "orange")
lines(xx, (1-phi.est) * dnorm(xx, Coef(fit)[4], sd.est), col = "orange")
abline(v = Coef(fit)[c(2,4)], lty = 2, col = "orange")
abline(v = c(mu1, mu2), lty = 2, col = "blue")
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