Hi, Thanks for your message. I tried to prove that the R-squared of the averaged model is always greater than or equals to the average R-squared of individual models (supposed m=2), Please see the attached r2.pdf. I hope this can be generalized to general case (m > 2).
Any comment would be very appreciated! Kan Cambridge University, UK --- "Liaw, Andy" <[EMAIL PROTECTED]> wrote: > The Y1, Y2, etc. that Kan mentioned are predicted > values of a test set data > from models that supposedly were fitted to the same > (or similar) data. It's > hard for me to imagine the outcome would be as > `severe' as Y1 = -Y2. > > That said, I do not think that the R-squared (or > q-squared as some call it) > of the aggregate model is necessarily larger or > equal to the average > R-squared of the component models. It obviously > depends on how the > component models are generated. As a hypothetical > example (because I > haven't acutally tried it, just speculating): > Suppose the data are > generated from a step function, the sort that would > be perfect for > regression trees. If one grows several well-pruned > trees, I'd guess that > the average R-squared of the individual trees has a > chance of being larger > than the R-squared of the averaged model. > > Best, > Andy > > > From: Gabor Grothendieck > > > > Suppose m=2, Y1=Y and Y2= -Y. Then (b) is zero so > (a) must be > > greater or equal to (b). Thus (b) is not > necessarily greater > > than (a). > > > > > > kan Liu <kan_liu1 <at> yahoo.com> writes: > > > > : > > : Hi, > > : > > : We got a question about interpretating > R-suqared. > > : > > : The actual outputs for a test dataset is > X=(x1,x2, ..., xn). > > : model 1 predicted the outputs as > Y1=(y11,y12,..., y1n) > > : model n predicted the outputs as > Y2=(y21,y22,..., y2n) > > : > > : ... > > : model m predicted the outputs as > Ym=(ym1,ym2,..., ymn) > > : > > : Now we have two ways to calculate R squared to > evaluate the average > > performance of committee model. > > : > > : (a) Calculate R squared between (X, Y1), (X, > Y2), ..., > > (X,Ym), and then > > averaging the R squared > > : (b) Calculate average Y=(Y1+Y2, + ... Ym)/m, and > then > > calculate the R > > squared between (X, Y). > > : > > : We found it seemed that R squared calculated in > (b) is > > 'always' higher than > > that in (a). > > : > > : Does this result depends on the test dataset or > this > > happened by chance?Can > > you advise me any reference for > > : this issue? > > : > > : Many thanks in advance! > > : > > : Kan > > : > > : > > : > > : --------------------------------- > > : > > : [[alternative HTML version deleted]] > > : > > : ______________________________________________ > > : R-help <at> stat.math.ethz.ch mailing list > > : > https://www.stat.math.ethz.ch/mailman/listinfo/r-help > > : PLEASE do read the posting guide! > > http://www.R-project.org/posting-guide.html > > : > > : > > > > > > ______________________________________________ > > [EMAIL PROTECTED] mailing list > > > https://www.stat.math.ethz.ch/mailman/listinfo/r-help > > PLEASE do read the posting guide! > > http://www.R-project.org/posting-guide.html > > > > > > > ------------------------------------------------------------------------------ > Notice: This e-mail message, together with any > attachments, contains information of Merck & Co., > Inc. (One Merck Drive, Whitehouse Station, New > Jersey, USA 08889), and/or its affiliates (which may > be known outside the United States as Merck Frosst, > Merck Sharp & Dohme or MSD and in Japan, as Banyu) > that may be confidential, proprietary copyrighted > and/or legally privileged. It is intended solely for > the use of the individual or entity named on this > message. If you are not the intended recipient, and > have received this message in error, please notify > us immediately by reply e-mail and then delete it > from your system. > ------------------------------------------------------------------------------ __________________________________
r2.pdf
Description: r2.pdf
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