# Mixture of Two Univariate Normal Distributions Estimates the five parameters of a mixture of two univariate normal distributions by maximum likelihood estimation. ```r 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 $sd$. See `Links` for more choices. - `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`. 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 $mu1$ and $mu2$. 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 $1,2,...,5$. 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 $$ f(y) = \phi \, N(\mu_1,\sigma_1) +(1-\phi) \, N(\mu_2, \sigma_2)%f(y) = phi * N(mu1, sd1) + (1-phi) * N(mu2, sd2) $$ 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 c("$\n$", "$ (logit(phi), mu1, log(sd1), mu2, log(sd2))^T$"). If `eq.sd = TRUE` then $sd1=sd2$ is enforced. ## Returns 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(s) 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 $mu1$ 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. ## See Also `uninormal`, `Normal`, `mix2poisson`. ## Examples ```r ## Not run: mu1 <- 99; mu2 <- 150; nn <- 1000 sd1 <- sd2 <- exp(3) (phi <- logitlink(-1, inverse = TRUE)) rrn <- runif(nn) mdata <- data.frame(y = ifelse(rrn < 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 = "red = 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 = "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) ```

mix2normal function