Density of a multivariate normal distribution

Multivariate Probability Distributions in R

Surajit Ray

Professor, University of Glasgow

Why calculate the density of a distribution?

Univariate normal functions dnorm() normal

Probability density of a bivariate normal

Standard bivariate normal with $$ \mu = \begin{pmatrix} 0 \\ 0 \end{pmatrix} , \Sigma = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} $$ animation

dmvnorm() function

Density heights calculated at several locations (xy coordinates)

density

Density using dmvnorm

library(mvtnorm)
dmvnorm(x, mean, sigma)

x can be a row vector or a matrix

mu1 <- c(1, 2)
sigma1 <- matrix(c(1, .5, .5, 2), 2)
dmvnorm(x = c(0, 0), mean = mu1, sigma = sigma1)

0.0384

Density at multiple points using dmvnorm

x <- rbind(c(0, 0), c(1, 1), c(0, 1)); x

[1,]    0    0
[2,]    1    1
[3,]    0    1

dmvnorm(x = x, mean = mu, sigma = sigma)

[1] 0.0384 0.0904 0.0679

Plotting bivariate densities with perspective plot

Steps:

Create grid of $x$ and $y$ coordinates
Calculate density on grid

Plotting bivariate densities with perspective plot

Steps:

Create grid of $x$ and $y$ coordinates
Calculate density on grid
Convert densities into a matrix
Create perspective plot using persp() function

Code for plotting bivariate densities

# Create grid
d <- expand.grid(seq(-3, 6, length.out = 50 ), seq(-3,  6, length.out = 50))                   

# Calculate density on grid
dens1 <- dmvnorm(as.matrix(d), mean=c(1,2), sigma=matrix(c(1, .5, .5, 2), 2))

# Convert to matrix 
dens1 <-  matrix(dens1, nrow = 50 )

# Use perspective plot
persp(dens1, theta = 80, phi = 30, expand = 0.6, shade = 0.2, col = "lightblue", xlab = "x", ylab = "y", zlab = "dens")

Changing viewing angle in perspective plot

persp() with theta = 30, phi = 30

persp() with theta = 80, phi = 10

Let's practice!

Multivariate Probability Distributions in R