# Multivariate normal distribution Density and random generation for the multivariate normal distribution with mean equal to `mean`, precision matrix equal to `Q` (or covariance matrix equal to `Sigma`). Function `rcMVN` samples from the multivariate normal distribution with a canonical mean $b$, i.e., the mean is $mu = Q^{-1} * b.$ ```r dMVN(x, mean=0, Q=1, Sigma, log=FALSE) rMVN(n, mean=0, Q=1, Sigma) rcMVN(n, b=0, Q=1, Sigma) ``` ## Arguments - `mean`: vector of mean. - `b`: vector of a canonical mean. - `Q`: precision matrix of the multivariate normal distribution. Ignored if `Sigma` is given. - `Sigma`: covariance matrix of the multivariate normal distribution. If `Sigma` is supplied, precision is computed from $Sigma$ as $Q = Sigma^{-1}$. - `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 ## Returns Some objects. ## Value for dMVN A vector with evaluated values of the (log-)density ## Value for rMVN A list with the components: - **x**: vector or matrix with sampled values - **log.dens**: vector with the values of the log-density evaluated in the sampled values ## Value for rcMVN A list with the components: - **x**: vector or matrix with sampled values - **mean**: vector or the mean of the normal distribution - **log.dens**: vector with the values of the log-density evaluated in the sampled values ## References Rue, H. and Held, L. (2005). **Gaussian Markov Random Fields: Theory and Applications**. Boca Raton: Chapman and Hall/CRC. ## See Also `dnorm`, `Mvnorm`. ## Author(s) Arnošt Komárek arnost.komarek@mff.cuni.cz ## Examples ```r set.seed(1977) ### Univariate normal distribution ### ============================== c(dMVN(0), dnorm(0)) c(dMVN(0, log=TRUE), dnorm(0, log=TRUE)) rbind(dMVN(c(-1, 0, 1)), dnorm(c(-1, 0, 1))) rbind(dMVN(c(-1, 0, 1), log=TRUE), dnorm(c(-1, 0, 1), log=TRUE)) c(dMVN(1, mean=1.2, Q=0.5), dnorm(1, mean=1.2, sd=sqrt(2))) c(dMVN(1, mean=1.2, Q=0.5, log=TRUE), dnorm(1, mean=1.2, sd=sqrt(2), log=TRUE)) rbind(dMVN(0:2, mean=1.2, Q=0.5), dnorm(0:2, mean=1.2, sd=sqrt(2))) rbind(dMVN(0:2, mean=1.2, Q=0.5, log=TRUE), dnorm(0:2, mean=1.2, sd=sqrt(2), log=TRUE)) ## ### ================================ mu <- c(0, 6, 8) L <- matrix(1:9, nrow=3) L[upper.tri(L, diag=FALSE)] <- 0 Sigma <- L %*% t(L) Q <- chol2inv(chol(Sigma)) b <- solve(Sigma, mu) dMVN(mu, mean=mu, Q=Q) dMVN(mu, mean=mu, Sigma=Sigma) dMVN(mu, mean=mu, Q=Q, log=TRUE) dMVN(mu, mean=mu, Sigma=Sigma, log=TRUE) xx <- matrix(c(0,6,8, 1,5,7, -0.5,5.5,8.5, 0.5,6.5,7.5), ncol=3, byrow=TRUE) dMVN(xx, mean=mu, Q=Q) dMVN(xx, mean=mu, Sigma=Sigma) dMVN(xx, mean=mu, Q=Q, log=TRUE) dMVN(xx, mean=mu, Sigma=Sigma, log=TRUE) zz <- rMVN(1000, mean=mu, Sigma=Sigma) rbind(apply(zz$x, 2, mean), mu) var(zz$x) Sigma cbind(dMVN(zz$x, mean=mu, Sigma=Sigma, log=TRUE), zz$log.dens)[1:10,] zz <- rcMVN(1000, b=b, Sigma=Sigma) rbind(apply(zz$x, 2, mean), mu) var(zz$x) Sigma cbind(dMVN(zz$x, mean=mu, Sigma=Sigma, log=TRUE), zz$log.dens)[1:10,] zz <- rMVN(1000, mean=rep(0, 3), Sigma=Sigma) rbind(apply(zz$x, 2, mean), rep(0, 3)) var(zz$x) Sigma cbind(dMVN(zz$x, mean=rep(0, 3), Sigma=Sigma, log=TRUE), zz$log.dens)[1:10,] ### The same using the package mvtnorm ### ================================== # require(mvtnorm) # c(dMVN(mu, mean=mu, Sigma=Sigma), dmvnorm(mu, mean=mu, sigma=Sigma)) # c(dMVN(mu, mean=mu, Sigma=Sigma, log=TRUE), dmvnorm(mu, mean=mu, sigma=Sigma, log=TRUE)) # # rbind(dMVN(xx, mean=mu, Sigma=Sigma), dmvnorm(xx, mean=mu, sigma=Sigma)) # rbind(dMVN(xx, mean=mu, Sigma=Sigma, log=TRUE), dmvnorm(xx, mean=mu, sigma=Sigma, log=TRUE)) ```

MVN function