Question

I am searching for a function/package-name in R which allows one to separate two superimposed normal distributions. The distribution looks something like this:

x<-c(3.95, 3.99, 4.0, 4.04, 4.1, 10.9, 11.5, 11.9, 11.7, 12.3)

Answer 1

I had good results in the past using vector generalized linear models. The VGAM package is useful for that.

The mix2normal1 function allows to estimate the parameters of a mix of two univariate normal distributions.

Little example

require(VGAM)
set.seed(12345)

# Create a binormal distribution with means 10 and 20
data <- c(rnorm(100, 10, 1.5), rnorm(200, 20, 3))

# Initial parameters for minimization algorithm
# You may want to create some logic to estimate this a priori... not always easy but possible
# m, m2: Means - s, s2: SDs - w: relative weight of the first distribution (the second is 1-w)
init.params <- list(m=5, m2=8, s=1, s2=1, w=0.5)

fit <<- vglm(data ~ 1, mix2normal1(equalsd=FALSE), 
                iphi=init.params$w, imu=init.params$m, imu2=init.params$m2, 
                isd1=init.params$s, isd2=init.params$s2)

# Calculated parameters
pars = as.vector(coef(fit))
w = logit(pars[1], inverse=TRUE)
m1 = pars[2]
sd1 = exp(pars[3])
m2 = pars[4]
sd2 = exp(pars[5])

# Plot an histogram of the data
hist(data, 30, col="black", freq=F)
# Superimpose the fitted distribution
x <- seq(0, 30, 0.1)
points(x, w*dnorm(x, m1, sd1)+(1-w)*dnorm(x,m2,sd2), "l", col="red", lwd=2)

This correctly gives (“true” parameters – 10, 20, 1.5, 3)

> m1
[1] 10.49236
> m2
[1] 20.06296
> sd1
[1] 1.792519
> sd2
[1] 2.877999