mix2normal: Mixture of Two Univariate Normal Distributions

Description Usage Arguments Details Value Warning Note Author(s) References See Also Examples

View source: R/family.mixture.R

Description

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

Usage

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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.

Author(s)

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.

See Also

uninormal, Normal, mix2poisson.

Examples

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## 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)

VGAM documentation built on Jan. 16, 2021, 5:21 p.m.