Example code showing how transformations to the raw data affect estimation of covariance matrices, and hence ellipse size and shape.

z_score_covariance_matrices

rm(list=ls())
graphics.off()
library(mnormt)
set.seed(1)

# start of notes on re-scaling
#if x and y are two correlated variables with covariance matrix S then
#let x.z and y.z be their independently z-score transformed values.

# some multivariate normal random data
n <- 50

rr <- rmnorm(n, c(10, 20), matrix(c(1, 2, 2, 5), 2, 2))

# extract and label them x and y
x <- rr[,1]
y <- rr[,2]

# some summary statistics on our raw data
mu.x <- mean(x)
mu.y <- mean(y)

var.x <- var(x)
var.y <- var(y)

sd.x <- sd(x)
sd.y <- sd(y)

# and the sample covariance matrix
S <- cov(data.frame(x,y))

# our z-score transformed data
x.z <- (x - mu.x) / sd.x
y.z <- (y - mu.y) / sd.y

# ---------------------------------------------------------------------
# Note that their correlation is preserved in this transform
dev.new()
par(mfrow = c(1,2))
plot(x,y, 
     main = paste("raw data, cor = ", round(cor(x,y), digits = 2), sep=""))
plot(x.z, y.z, 
     main =  paste("z-score data, cor = ", round(cor(x,y), digits = 2), sep=""))

# ---------------------------------------------------------------------
# now estimate the covariance matrix on the z-scored data and 
# back-transform to the original scale. Although the 
# correlation coefficients are unchanged by the z-score transform,
# the covariance matrices are not the same, and hence an ellipse
# fitted to these will be different. We can back-transform though.

# compare S with S.z
S.z <- cov(data.frame(x.z,y.z))

print(S)
print(S.z)

# back-transform
S.back <- matrix(0, 2, 2)
S.back[1,1] <- S.z[1,1] * var.x
S.back[2,2] <- S.z[2,2] * var.y
S.back[1,2] <- S.z[1,2] * sd.x * sd.y
S.back[2,1] <- S.back[1,2]

# and now our back-transformed covariance matrix is the same
# as the covariance matrix calculated on the raw data.
# Hence, the ellipses fit to both these will be identical.
print(S.back)