relationship between RMSE and R^2

First of all this is more like a theoretical question than a methodological one. I made a script to fit some time series data. I fitted many data series and calculated their goodness of fit statistics. When I analyzed the resulting data I found an inverse relationship between RMSE and R^2.

I´ve look around the web and my statistics books looking for a possible explanation but with no luck.

Is there anyone here who can give me some ideas ?

Answer 1

[Tom Lane]

R^2 = 1 - SSE/SST = 1 - DFE*RMSE^2/SST

Here SSE is the error sum of squares, SST is the total sum of squares, and DFE is the degrees of freedom for error. So you would expect R^2 to go down as RMSE goes up. Is that what you meant by an inverse relationship?

Answer 2

[Samuel Fonseca]

Sorry I now realize that my previous post was not clear, what I mean is that I'm getting a relationship inverse from what I expected to get...

Lets say in series 1 i'm geting R^2 =0.8 and RMSE= 1.5 while in series 2 I get R^2= 0.65 and RMSE=0.75

After looking at data in some more detail I found that point 2 has a biger range (max(x)-min(x)) so having a biger range and maybe standar deviation could be related?

[Tom Lane]

R^2 is sensitive to the x range. That's what some people dislike about it. RMSE should not be sensitive if the model is correct. However, usually a bigger range leads to large R^2 and no change in RMSE. You seem to be saying R^2 is smaller and RMSE is smaller. That is unexpected.

[Samuel Fonseca]

yup that's my point... R^2 values are telling me that fit works good (R^2=0.8) but RMSE id big and then for other series I get the opposite result... unexpectedin deed.. i'll get back and re-check code but i'm pretty sure i'm just using RMSE and R^2 values generated by the fit statistics