Original User: whuber
Thank you very much, Eric.
I am trying to connect that document to what the software produces, but am failing to do so. My understanding of the "Prediction Standard Error Map" is that it contains the square root of the estimated residual variance: that is, it's the entire square root term in the "Formula for quantile map." (Cressie's notation for this term is \sigma_k(s_0).) When no transformation occurs (as in my test cases), the formula thereby looks like it should reduce to what is in Cressie: namely, the quantile map (\widehat{Z_q}) looks as if it's supposed to be the prediction map (\hat(Y)) plus a constant times the prediction SE map. *But it is not.*
The only difference I can see between what is in the document and what I have been expecting concerns the sign of x^T\mu within the square root: on one line of your document it appears with a negative sign but in the formula for the quantile map it appears with a positive sign. Isn't that negative sign a typographical error? Even if not, the discrepancies I am seeing cannot be explained by an additional term of that nature.
Allow me to explain how I know there are problems. Your formula implies W = (t(\widehat{Z_q}) - \hat{Y})^2 must be a multiple of the terms found under the square root sign. We can find that multiple by subtracting the prediction map \hat{Y} from the (transformed) quantile map t(\widehat{Z_q}) and squaring the result to produce W, and then regressing that against the squared prediction error map. I would hope to achieve a fit that is accurate at least to single precision floating point error. I haven't managed to do that--in some cases it's close but in others they are wildly different. I haven't ruled out errors on my part, but failing to find any I'm looking for all the information I can obtain concerning what the software is actually doing.
Please note that I am unconcerned with bias, accuracy, or uncertainty here: I am only trying to understand how the three GA outputs--prediction map, prediction SE map, and quantile map--are related mathematically.
