[R] confidence / prediction ellipse

Rolf Turner rolf.turner at xtra.co.nz
Tue Jan 29 02:04:03 CET 2013

I believe that the value of "radius" that you are using is incorrect. 
If you have a data
matrix X whose columns  are jointly distributed N(mu,Sigma) then a 
ellipse for mu is determined by

     n * (x - Xbar)' S^{-1}(x - Xbar) ~ T^2

where Xbar is the mean vector for X and S is the sample covariance matrix,
and where "T^2" means Hotelling's T-squared distribution, which is equal to

     (n-1)*2/(n-2) * F_{2,n-2}

the latter representing the F distribution on 2 and n-2 degrees of freedom.

Thus (I think) your radius should be

     radius <- sqrt(2 * (npts-1) * qf(0.95, 2, npts-2)/(npts*(npts-2)))

where npts <- length(a).  Note that it is qf(0.95,2,npts-2) and *NOT*

To get the corresponding *prediction* ellipse simply multiply the foregoing
radius by sqrt(npts+1).  By "prediction ellipse" I mean an ellipse such that
the probability that a new independent observation from the same population
will fall in that ellipse is the given probability (e.g. 0.95). Note 
that this does
not mean that 95% of the data will fall in the calculated ellipse 
(basically because
of the *dependence* between S and the individual observations).

These confidence and prediction ellipses are (I'm pretty sure) valid under
the assumption that the data are (two dimensional, independent) Gaussian,
and that you use the sample covariance and sample mean as "shape" and
"centre".  I don't know what impact your robustification procedure of using
cov.trob() will/would have on the properties of these ellipses.

A script which does the ellipses for your toy data, using the sample 
and sample mean (rather than output from cov.trob()) is as follows:

# Script scr.amatulli

a <- c(12,12,4,5,63,63,23)
b <- c(13,15,7,10,73,83,43)
npts   <- length(a)
shape  <- var(cbind(a, b))
center <- c(mean(a),mean(b))
rconf  <- sqrt(2 * (npts-1) * qf(0.95, 2, npts-2)/(npts*(npts-2)))
rpred  <- sqrt(npts+1)*rconf

conf.elip <- ellipse(center, shape, rconf,draw = FALSE)
pred.elip <- ellipse(center, shape, rpred,draw = FALSE)
plot(pred.elip, type='l')


         Rolf Turner

On 01/27/2013 10:12 AM, Giuseppe Amatulli wrote:
> Hi,
> I'm using the R library(car) to draw confidence/prediction ellipses in a
> scatterplot.
> >From what i understood  the ellipse() function return an ellipse based
> parameters:  shape, center,  radius .
> If i read  dataEllipse() function i can see how these parameters are
> calculated for a confidence ellipse.
> ibrary(car)
> a=c(12,12,4,5,63,63,23)
> b=c(13,15,7,10,73,83,43)
> v <- cov.trob(cbind(a, b))
> shape <- v$cov
> center <- v$center
> radius <- sqrt(2 * qf(0.95, 2, length(a) - 1))   # radius <- sqrt(dfn *
> qf(level, dfn, dfd))
> conf.elip = ellipse(center, shape, radius,draw = F)
> plot(conf.elip, type='l')
> points(a,b)
> My question is how I can calculate shape, center and radius  to obtain a
> prediction ellipses rather than a confidence ellipse?
> Thanks in Advance
> Giuseppe

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