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")

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.

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.

Value

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

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.

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.

Author

T. W. Yee

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 $\mu_1$ and $\mu_2$ 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.

Examples

if (FALSE)  mu1 <-  99; mu2 <- 150; nn <- 1000
sd1 <- sd2 <- exp(3)
(phi <- logitlink(-1, inverse = TRUE))
#> [1] 0.2689414
rrn <- runif(nn)
mdata <- data.frame(y = ifelse(rrn < phi, rnorm(nn, mu1, sd1),
                                          rnorm(nn, mu2, sd2)))
#> Error: object 'mu1' not found
fit <- vglm(y ~ 1, mix2normal(eq.sd = TRUE), data = mdata)
#> Error in eval(mf, parent.frame()): object 'mdata' not found

# Compare the results
cfit <- coef(fit)
#> Error in h(simpleError(msg, call)): error in evaluating the argument 'object' in selecting a method for function 'coef': object 'fit' not found
round(rbind('Estimated' = c(logitlink(cfit[1], inverse = TRUE),
            cfit[2], exp(cfit[3]), cfit[4]),
            'Truth' = c(phi, mu1, sd1, mu2)), digits = 2)
#> Error: object 'cfit' not found

# Plot the results
xx <- with(mdata, seq(min(y), max(y), len = 200))
#> Error: object 'mdata' not found
plot(xx, (1-phi) * dnorm(xx, mu2, sd2), type = "l", xlab = "y",
     main = "red = estimate, blue = truth",
     col = "blue", ylab = "Density")
#> Error in h(simpleError(msg, call)): error in evaluating the argument 'x' in selecting a method for function 'plot': object 'xx' not found
phi.est <- logitlink(coef(fit)[1], inverse = TRUE)
#> Error in h(simpleError(msg, call)): error in evaluating the argument 'object' in selecting a method for function 'coef': object 'fit' not found
sd.est <- exp(coef(fit)[3])
#> Error in h(simpleError(msg, call)): error in evaluating the argument 'object' in selecting a method for function 'coef': object 'fit' not found
lines(xx, phi*dnorm(xx, mu1, sd1), col = "blue")
#> Error: object 'xx' not found
lines(xx, phi.est * dnorm(xx, Coef(fit)[2], sd.est), col = "red")
#> Error: object 'xx' not found
lines(xx, (1-phi.est)*dnorm(xx, Coef(fit)[4], sd.est), col="red")
#> Error: object 'xx' not found
abline(v = Coef(fit)[c(2,4)], lty = 2, col = "red")
#> Error in h(simpleError(msg, call)): error in evaluating the argument 'object' in selecting a method for function 'Coef': object 'fit' not found
abline(v = c(mu1, mu2), lty = 2, col = "blue")
#> Error: object 'mu1' not found
 # \dontrun{}