Generating samples from a two-Gaussian mixture in

问题:

I'm trying to create (in r) the equivalent to the following MATLAB function that will generate n samples from a mixture of N(m1,(s1)^2) and N(m2, (s2)^2) with a fraction, alpha, from the first Gaussian.

I have a start, but the results are notably different between MATLAB and R (i.e., the MATLAB results give occasional values of +-8 but the R version never even gives a value of +-5). Please help me sort out what is wrong here. Thanks :-)

For Example: Plot 1000 samples from a mix of N(0,1) and N(0,36) with 95% of samples from the first Gaussian. Normalize the samples to mean zero and standard deviation one.

MATLAB

function

function y = gaussmix(n,m1,m2,s1,s2,alpha)
y = zeros(n,1);
U = rand(n,1);
I = (U < alpha)
y = I.*(randn(n,1)*s1+m1) + (1-I).*(randn(n,1)*s2 + m2);

implementation

P = gaussmix(1000,0,0,1,6,.95)
P = (P-mean(P))/std(P)
plot(P)
axis([0 1000 -15 15])
hist(P)
axis([-15 15 0 1000])

resulting plot

resulting hist

R

yn <- rbinom(1000, 1, .95)
s <- rnorm(1000, 0 + 0*yn, 1 + 36*yn)
sn <- (s-mean(s))/sd(s)
plot(sn, xlim=range(0,1000), ylim=range(-15,15))
hist(sn, xlim=range(-15,15), ylim=range(0,1000))

resulting plot

resulting hist

As always, THANK YOU!

SOLUTION

gaussmix <- function(nsim,mean_1,mean_2,std_1,std_2,alpha){
   U <- runif(nsim)
   I <- as.numeric(U<alpha)
   y <- I*rnorm(nsim,mean=mean_1,sd=std_1)+
       (1-I)*rnorm(nsim,mean=mean_2,sd=std_2)
   return(y)
}

z1 <- gaussmix(1000,0,0,1,6,0.95)
z1_standardized <- (z1-mean(z1))/sqrt(var(z1))
z2 <- gaussmix(1000,0,3,1,1,0.80)
z2_standardized <- (z2-mean(z2))/sqrt(var(z2))
z3 <- rlnorm(1000)
z3_standardized <- (z3-mean(z3))/sqrt(var(z3))

par(mfrow=c(2,3))
hist(z1_standardized,xlim=c(-10,10),ylim=c(0,500),
   main="Histogram of 95% of N(0,1) and 5% of N(0,36)",
   col="blue",xlab=" ")
hist(z2_standardized,xlim=c(-10,10),ylim=c(0,500),
   main="Histogram of 80% of N(0,1) and 10% of N(3,1)",
   col="blue",xlab=" ")
hist(z3_standardized,xlim=c(-10,10),ylim=c(0,500),
   main="Histogram of samples of LN(0,1)",col="blue",xlab=" ")
##
plot(z1_standardized,type='l',
   main="1000 samples from a mixture N(0,1) and N(0,36)",
   col="blue",xlab="Samples",ylab="Mean",ylim=c(-10,10))
plot(z2_standardized,type='l',
   main="1000 samples from a mixture N(0,1) and N(3,1)",
   col="blue",xlab="Samples",ylab="Mean",ylim=c(-10,10))
plot(z3_standardized,type='l',
  main="1000 samples from LN(0,1)",
   col="blue",xlab="Samples",ylab="Mean",ylim=c(-10,10))

回答1:

There are two problems, I think ... (1) your R code is creating a mixture of normal distributions with standard deviations of 1 and 37. (2) By setting prob equal to alpha in your rbinom() call, you're getting a fraction alpha in the second mode rather than the first. So what you are getting is a distribution that is mostly a Gaussian with sd 37, contaminated by a 5% mixture of Gaussian with sd 1, rather than a Gaussian with sd 1 that is contaminated by a 5% mixture of a Gaussian with sd 6. Scaling by the standard deviation of the mixture (which is about 36.6) basically reduces it to a standard Gaussian with a slight bump near the origin ...

(The other answers posted here do solve your problem perfectly well, but I thought you might be interested in a diagnosis ...)

A more compact (and perhaps more idiomatic) version of your Matlab gaussmix function (I think runif(n)<alpha is slightly more efficient than rbinom(n,size=1,prob=alpha) )

gaussmix <- function(n,m1,m2,s1,s2,alpha) {
    I <- runif(n)<alpha
    rnorm(n,mean=ifelse(I,m1,m2),sd=ifelse(I,s1,s2))
}
set.seed(1001)
s <- gaussmix(1000,0,0,1,6,0.95)

回答2:

Not that you asked for it, but the mclust package offers a way to generalize your problem to more dimensions and diverse covariance structures. See ?mclust::sim. The example task would be done this way:

require(mclust)
simdata = sim(modelName = "V",
              parameters = list(pro = c(0.95, 0.05),
                                mean = c(0, 0),
                                variance = list(modelName = "V", 
                                                d = 1, 
                                                G = 2,
                                                sigmasq = c(0, 36))),
              n = 1000)
plot(scale(simdata[,2]), type = "h")

回答3:

I recently wrote the density and sampling function of a multinomial mixture of normal distributions:

dmultiNorm <- function(x,means,sds,weights)
{
  if (length(means)!=length(sds)) stop("Length of means must be equal to length of standard deviations")
  N <- length(x)
  n <- length(means)
  if (missing(weights))
  {
    weights <- rep(1,n)  
  }
  if (length(weights)!=n) stop ("Length of weights not equal to length of means and sds")
  weights <- weights/sum(weights)
  dens <- numeric(N)
  for (i in 1:n)
  {
    dens <- dens + weights[i] * dnorm(x,means[i],sds[i])
  }
  return(dens)
}

rmultiNorm <- function(N,means,sds,weights,scale=TRUE)
{
  if (length(means)!=length(sds)) stop("Length of means must be equal to length of standard deviations")
  n <- length(means)
  if (missing(weights))
  {
    weights <- rep(1,n)  
  }
  if (length(weights)!=n) stop ("Length of weights not equal to length of means and sds")

  Res <- numeric(N)
  for (i in 1:N)
  {
    s <- sample(1:n,1,prob=weights)
    Res[i] <- rnorm(1,means[s],sds[s])  
  }
  return(Res)
}

With means being a vector of means, sds being a vector of standard deviatians and weights being a vector with proportional probabilities to sample from each of the distributions. Is this useful to you?

回答4:

Here is code to do this task:

"For Example: Plot 1000 samples from a mix of N(0,1) and N(0,36) with 95% of samples from the first Gaussian. Normalize the samples to mean zero and standard deviation one."

 plot(multG <- c( rnorm(950), rnorm(50, 0, 36))[sample(1000)] , type="h")
 scmulG <- scale(multG)
 summary(scmulG)
 #-----------    
   V1          
 Min.   :-9.01845  
 1st Qu.:-0.06544  
 Median : 0.03841  
 Mean   : 0.00000  
 3rd Qu.: 0.13940  
 Max.   :12.33107