mixAK (version 5.8)

MVNmixture: Mixture of (multivariate) normal distributions

Description

Density and random generation for the mixture of the \(p\)-variate normal distributions with means given by mean, precision matrix given by Q (or covariance matrices given bySigma).

Usage

dMVNmixture(x, weight, mean, Q, Sigma, log=FALSE)
dMVNmixture2(x, weight, mean, Q, Sigma, log=FALSE)
rMVNmixture(n, weight, mean, Q, Sigma)
rMVNmixture2(n, weight, mean, Q, Sigma)

Value

Some objects.

Arguments

weight

vector of length \(K\) with the mixture weights or values which are proportional to the weights.

mean

vector or matrix of mixture means.

For \(p=1\) this should be a vector of length \(K\), for \(p>1\) this should be a \(K\times p\) matrix with mixture means in rows.

Q

precision matrices of the multivariate normal distribution. Ignored if Sigma is given.

For \(p=1\) this should be a vector of length \(K\), for \(p > 1\) this should be a list of length \(K\) with the mixture precision matrices as components of the list.

Sigma

covariance matrix of the multivariate normal distribution. If Sigma is supplied, precisions are computed from \(\Sigma\) as \(Q = \Sigma^{-1}\).

For \(p=1\) this should be a vector of length \(K\), for \(p > 1\) this should be a list of length \(K\) with the mixture covariance matrices as components of the list.

n

number of observations to be sampled.

x

vector or matrix of the points where the density should be evaluated.

log

logical; if TRUE, log-density is computed

Value for dMVNmixture

A vector with evaluated values of the (log-)density.

Value for dMVNmixture2

A vector with evaluated values of the (log-)density.

Value for rMVNmixture

A vector (for n=1 or for univariate mixture) or matrix with sampled values (in rows of the matrix).

Value for rMVNmixture2

A list with components named x which is a vector (for n=1 or for univariate mixture) or matrix with sampled values (in rows of the matrix) and dens which are the values of the density evaluated in x.

Author

Arnošt Komárek

Details

Functions dMVNmixture and dMVNmixture2 differ only internally in the way they compute the mixture density. In dMVNmixture, only multivariate normal densities are evaluated in compiled C++ code and mixing is done directly in R. In dMVNmixture2, everything is evaluated in compiled C++ code. Normally, both dMVNmixture and dMVNmixture2 should return the same results.

Similarly for rMVNmixture and rMVNmixture2. Another difference is that rMVNmixture returns only random generated points and rMVNmixture2 also values of the density evaluated in the generated points.

See Also

dnorm, MVN, Mvnorm.

Examples

set.seed(1977)

##### Univariate normal mixture
##### =========================
mu <- c(-1, 1)
Sigma <- c(0.25^2, 0.4^2)
Q <- 1/Sigma
w <- c(0.3, 0.7)

xx <- seq(-2, 2.5, length=100)
yyA <- dMVNmixture(xx, weight=w, mean=mu, Sigma=Sigma)
yyB <- dMVNmixture(xx, weight=w, mean=mu, Q=Q)
yyC <- dMVNmixture2(xx, weight=w, mean=mu, Sigma=Sigma)
yyD <- dMVNmixture2(xx, weight=w, mean=mu, Q=Q)

xxSample <- rMVNmixture(1000, weight=w, mean=mu, Sigma=Sigma)
xxSample2 <- rMVNmixture2(1000, weight=w, mean=mu, Sigma=Sigma)

sum(abs(xxSample2$dens - dMVNmixture(xxSample2$x, weight=w, mean=mu, Sigma=Sigma)) > 1e-15)
xxSample2 <- xxSample2$x

par(mfrow=c(2, 2), bty="n")
plot(xx, yyA, type="l", col="red", xlab="x", ylab="f(x)")
points(xx, yyB, col="darkblue")
hist(xxSample, col="lightblue", prob=TRUE, xlab="x", xlim=range(xx), ylim=c(0, max(yyA)),
     main="Sampled values")
lines(xx, yyA, col="red")
plot(xx, yyC, type="l", col="red", xlab="x", ylab="f(x)")
points(xx, yyD, col="darkblue")
hist(xxSample2, col="sandybrown", prob=TRUE, xlab="x", xlim=range(xx), ylim=c(0, max(yyA)),
     main="Sampled values")
lines(xx, yyC, col="red")


##### Bivariate normal mixture
##### ========================
### Choice 1
sd11 <- sd12 <- 1.1
sd21 <- 0.5
sd22 <- 1.5
rho2 <- 0.7
Xlim <- c(-3, 4)
Ylim <- c(-6, 4)

### Choice 2
sd11 <- sd12 <- 0.3
sd21 <- 0.5
sd22 <- 0.3
rho2 <- 0.8
Xlim <- c(-3, 2.5)
Ylim <- c(-2.5, 2.5)

mu <- matrix(c(1,1, -1,-1), nrow=2, byrow=TRUE)
Sigma <- list(diag(c(sd11^2, sd12^2)),
              matrix(c(sd21^2, rho2*sd21*sd22, rho2*sd21*sd22, sd22^2), nrow=2))
Q <- list(chol2inv(chol(Sigma[[1]])), chol2inv(chol(Sigma[[2]])))
w <- c(0.3, 0.7)

xx1 <- seq(mu[2,1]-3*sd21, mu[1,1]+3*sd11, length=100)
xx2 <- seq(mu[2,2]-3*sd22, mu[1,2]+3*sd12, length=90)
XX <- cbind(rep(xx1, length(xx2)), rep(xx2, each=length(xx1)))
yyA <- matrix(dMVNmixture(XX, weight=w, mean=mu, Sigma=Sigma), nrow=length(xx1), ncol=length(xx2))
yyB <- matrix(dMVNmixture(XX, weight=w, mean=mu, Q=Q), nrow=length(xx1), ncol=length(xx2))
yyC <- matrix(dMVNmixture2(XX, weight=w, mean=mu, Sigma=Sigma), nrow=length(xx1), ncol=length(xx2))
yyD <- matrix(dMVNmixture2(XX, weight=w, mean=mu, Q=Q), nrow=length(xx1), ncol=length(xx2))

#xxSample <- rMVNmixture(1000, weight=w, mean=mu, Sigma=Sigma)
### Starting from version 3.6, the above command led to SegFault
### on CRAN r-patched-solaris-sparc check.
### Commented here on 20140806 (version 3.6-1).
xxSample2 <- rMVNmixture2(1000, weight=w, mean=mu, Sigma=Sigma)

sum(abs(xxSample2$dens - dMVNmixture(xxSample2$x, weight=w, mean=mu, Sigma=Sigma)) > 1e-15)
xxSample2 <- xxSample2$x

par(mfrow=c(1, 2), bty="n")
plot(xxSample, col="darkblue", xlab="x1", ylab="x2", xlim=Xlim, ylim=Ylim)
contour(xx1, xx2, yyA, col="red", add=TRUE)
plot(xxSample2, col="darkblue", xlab="x1", ylab="x2", xlim=Xlim, ylim=Ylim)
contour(xx1, xx2, yyC, col="red", add=TRUE)

par(mfrow=c(2, 2), bty="n")
contour(xx1, xx2, yyA, col="red", xlab="x1", ylab="x2")
points(mu[,1], mu[,2], col="darkgreen")
persp(xx1, xx2, yyA, col="lightblue", xlab="x1", ylab="x2", zlab="f(x1, x2)")
contour(xx1, xx2, yyB, col="red", xlab="x1", ylab="x2")
points(mu[,1], mu[,2], col="darkgreen")
persp(xx1, xx2, yyB, col="lightblue", xlab="x1", ylab="x2", zlab="f(x1, x2)", phi=30, theta=30)

par(mfrow=c(2, 2), bty="n")
contour(xx1, xx2, yyC, col="red", xlab="x1", ylab="x2")
points(mu[,1], mu[,2], col="darkgreen")
persp(xx1, xx2, yyC, col="lightblue", xlab="x1", ylab="x2", zlab="f(x1, x2)")
contour(xx1, xx2, yyD, col="red", xlab="x1", ylab="x2")
points(mu[,1], mu[,2], col="darkgreen")
persp(xx1, xx2, yyD, col="lightblue", xlab="x1", ylab="x2", zlab="f(x1, x2)", phi=30, theta=30)