mix2normal: Mixture of Two Univariate Normal Distributions

Description

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

Usage

mix2normal(lphi = "logit", lmu = "identitylink", lsd = "loge",
           iphi = 0.5, imu1 = NULL, imu2 = NULL, isd1 = NULL, isd2 = NULL,
           qmu = c(0.2, 0.8), eq.sd = TRUE, nsimEIM = 100, zero = "phi")

Arguments

lphi,lmu,lsd

Link functions for the parameters $\phi$, $\mu$, and $\sigma$. See Links for more choices.

iphi

Initial value for $\phi$, whose value must lie between 0 and 1.

imu1, imu2

Optional initial value for $\mu_1$ and $\mu_2$. The default is to compute initial values internally using the argument qmu.

isd1, isd2

Optional initial value for $\sigma_1$ and $\sigma_2$. The default is to compute initial values internally based on the argument qmu. Currently these are not great, therefore using these arguments where practical is a good idea.

qmu

Vector with two values giving the probabilities relating to the sample quantiles for obtaining initial values for $\mu_1$ and $\mu_2$. The two values are fed in as the probs argument into quantile.

eq.sd

Logical indicating whether the two standard deviations should be constrained to be equal. If TRUE then the appropriate constraint matrices will be used.

nsimEIM

See CommonVGAMffArguments.

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 $\{1,2,\ldots,5\}$. 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. See CommonVGAMffArguments for more information.

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 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.

Details

The probability density function can be loosely written as $$f(y) = \phi \, N(\mu_1,\sigma_1) + (1-\phi) \, N(\mu_2, \sigma_2)$$ where $\phi$ is the probability an observation belongs to the first group. The parameters $\mu_1$ and $\mu_2$ are the means, and $\sigma_1$ and $\sigma_2$ are the standard deviations. The parameter $\phi$ satisfies $0 < \phi < 1$. The mean of $Y$ is $\phi \mu_1 + (1-\phi) \mu_2$ and this is returned as the fitted values. By default, the five linear/additive predictors are $(logit(\phi), \mu_1, \log(\sigma_1), \mu_2, \log(\sigma_2))^T$. If eq.sd = TRUE then $\sigma_1 = \sigma_2$ is enforced.

References

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.

Examples

Run this code

# NOT RUN {
 mu1 <-  99; mu2 <- 150; nn <- 1000
sd1 <- sd2 <- exp(3)
(phi <- logit(-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(logit(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 <- 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 = "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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