# mix2normal: Mixture of Two Univariate Normal Distributions In VGAM: Vector Generalized Linear and Additive Models

## Description

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

## Usage

 ```1 2 3``` ```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(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.

## 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.

## 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.

T. W. Yee

## 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.

`uninormal`, `Normal`, `mix2poisson`.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27``` ```## 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") ## End(Not run) ```