I have datasets that can form several different curvy patterns between the dependent and independent variables. The 'true' relationship likely depends on a large number of factors that aren't easily measured and so it remains unknown. One method of describing this relationship has been through LOESS.

However, I am trying to optimize the LOESS fit by changing the tuning parameter (span or f-value) using k-fold cross-validation (with the loess.wrapper function in the bisoreg package in R). I mostly understand how the optimal span is chosen by minimizing the estimated predictive error via the CV. I've been studying the bias-variance tradeoff topic in sources such as Elements of Statistical Learning and I'm still left with a difficult question. Is it possible to calculate (or estimate) the bias and variance of a LOESS fit for different span values? I'd like to be able to compare different LOESS fits on the same data with these statistics but I think the issue lies in the fact that I don't know the 'true' relationship of the data.

Any guidance or recommended readings are appreciated.


1 Answer 1


I'm not an expert in this area at all, but I'm also interested in the question, so I did some digging.

First, It definitely is, possible to calculate the variance of a LOESS model, because such variance estimates are available from some packages, e.g. R's loess (see this post for a usage example).

The citation for that loess function states[1] that:

We can specify properties of the variances of the in one of two ways. The first is simply that they are a constant, $\sigma^2$. The second is that $a_i\epsilon_i$ has constant variance $\sigma^2$, where the a priori weights, $a_i\epsilon_i$, are positive and known.Howe

Meaning, I think, that you either have to assume that the error variance is globally constant, or that you know your weights a priori. However, the chapter doesn't actually describe the process for calculating the error variance in either situation, and I haven't yet found another reference that does...


  1. W. S. Cleveland, E. Grosse and W. M. Shyu (1992) Local regression models. Chapter 8 of Statistical Models in S eds J.M. Chambers and T.J. Hastie, Wadsworth & Brooks/Cole.

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.