confidence / prediction ellipse
On Feb 7, 2013, at 8:20 AM, Giuseppe Amatulli wrote:
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 ?
Well, if a method provides a probability estimate or even if it only provides a rankable order, could always use color change to highlight the the "most predicted" 95%. I say "most predicted" rather than "most probable", since it's not clear that these are probability or credibility estimates.
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?
Presumably you meant lm(b~a)? You might have also expected ( ... also incorrectly) that the line for lm(a~b) would be along the major axis. Perhaps reading some material on orthogonal regression (AKA total least squares regression, AKA Deming regression) would be of interest. The major axis should "split the difference" between those two estimates.
David
>
> 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
>
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David Winsemius
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