编程实现EM算法，并用以下数据和初始值估计一个two-component GMM。使用contour plot展示估计的正态分布 library(MASS) set.seed(123) n <- 1000 mu1 <- c(0,4) mu2 <- c(-2,0) Sigma1 <- matrix(c(3,0,0,0.5),nr=2,nc=2) Sigma2 <- matrix(c(1,0,0,2),nr=2,nc=2) phi <- c(0.6,0.4) X <- matrix(0,nr=2,nc=n) for (i in 1:n) { if (runif(1)<=phi[1]) { X[,i] <- mvrnorm(1,mu=mu1,Sigma=Sigma1) }else{ X[,i] <- mvrnorm(1,mu=mu2,Sigma=Sigma2) } } ##initial guess for parameters mu10 <- runif(2) mu20 <- runif(2) Sigma10 <- diag(2) Sigma20 <- diag(2) phi0 <- runif(2) phi0 <- phi0/sum(phi0)

EM算法是一种常用的参数估计方法，用于估计数据生成模型中的参数。下面给出了用R语言编写的EM算法实现代码，用来估计two-component GMM的参数。

library(MASS)
set.seed(123)
n <- 1000
mu1 <- c(0,4)
mu2 <- c(-2,0)
Sigma1 <- matrix(c(3,0,0,0.5),nr=2,nc=2)
Sigma2 <- matrix(c(1,0,0,2),nr=2,nc=2)
phi <- c(0.6,0.4)
X <- matrix(0,nr=2,nc=n)
for (i in 1:n) {
if (runif(1)<=phi[1]) {
X[,i] <- mvrnorm(1,mu=mu1,Sigma=Sigma1)
}else{
X[,i] <- mvrnorm(1,mu=mu2,Sigma=Sigma2)
}
}

# 定义EM算法函数
EM_GMM <- function(X, k){
# 初始化参数
n <- ncol(X)
d <- nrow(X)
w <- rep(1/k, k)
mu <- matrix(rnorm(k*d, mean(X), sd(X)), nrow=k, ncol=d)
sigma <- array(aperm(array(rnorm(k*d*d), dim=c(k,d,d)), c(2,3,1)), dim=c(d,d,k))
R <- numeric(k*n)

# EM算法迭代
for (iter in 1:100){
# E步
for (i in 1:k){
R[(i-1)*n+1:i*n] <- w[i] * dnorm(X, mean=mu[i,], sd=sigma[,,i])
}
R <- matrix(R, nrow=n, byrow=TRUE)
R <- R / rowSums(R)

# M步
Nk <- colSums(R) # 每个分量的权重
w <- Nk / n # 均值
for (i in 1:k){
mu[i,] <- colSums(R[,i] * X) / Nk[i] # 均值
sigma[,,i] <- (t(X) %*% (R[,i] * X)) / Nk[i] - mu[i,] %*% t(mu[i,]) # 协方差矩阵
}
}

# 返回估计的参数
list(w=w, mu=mu, sigma=sigma)
}

# 估计two-component GMM
result <- EM_GMM(X, 2)

# 绘制contour plot展示估计的正态分布
xgrid <- seq(min(X[1,]), max(X[1,]), length.out=100)
ygrid <- seq(min(X[2,]), max(X[2,]), length.out=100)
z <- outer(xgrid, ygrid,
function(x,y) {
z <- numeric(length(x))
for (i in 1:nrow(result$mu)){
z <- z + result$w[i] * dnorm(c(x, y), mean=result$mu[i,], sd=sqrt(result$sigma[1,1,i]))
}
z
})
contour(xgrid, ygrid, z, nlev=10, color.palette=heat.colors, main="Two-component GMM Contours")