confidence / prediction ellipse

Hi Rolf,
sorry for this late answer and thanks for your kind explanation and
relative R code. I really appreciate.
In reality the concept that I'm trying to address is a bit more complex.
I'm fitting a model y vs 6 predictors with MARS / RandomForest /
Multiple Linear Regression Models having 140 observations.
I have the prediction of each model and would like to delineate the
prediction ellipses for 3 models, for the 95% probability, and
plotting them together with the observation vs prediction.
I think that the prediction-ellipses code that you provide to me is
valid also for predictions derived by not-linear model (such as MARS
and RF).
Is it correct? or should i use an alternative solution ?

Moreover, I was expecting that the  abline (lm(b,a)) would be
correspond to the main axis of the prediction ellipse, but is not this
the case.
why?

Thanks in advance
Giuseppe

On 28 January 2013 19:04, Rolf Turner <rolf.turner at xtra.co.nz> wrote:

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
confidence
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*
qf(0.95,2,npts-1).

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
covariance
and sample mean (rather than output from cov.trob()) is as follows:

#
# Script scr.amatulli
#

require(car)
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')
points(a,b)
lines(conf.elip,col="red")

     cheers,

         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

--
Giuseppe Amatulli
Web: www.spatial-ecology.net