Non-constant standard deviation in residuals

Question

I am fitting a model in the frequency domain, and my fit looks as follows:

As you can see, the model function does not fit the data perfectly, especially in the higher frequencies. So, I examined the residuals and found that a polynomial of the first degree is the missing term in the model function.

What concerns me more is that variance of the data decreases with frequency. What implications does that have on my analysis?

(It seems that looking at the autocorrelation of the residuals also doesn't prove very useful.)

Any ideas would be very helpful.

Answer 1

The spread looks approximately proportional to mean.

This suggests a couple of possibilities:

i) Generalized linear models, which have an explicit model for variance as a function of the mean. The Gamma glm has variance proportional to mean squared (spread proportional to mean). You could fit a linear, exponential or inverse mean-function relatively easily.

ii) taking logs and modelling on the log-scale, since logs will approximately stabilize the variance. Depending on the function you want to fit, you may end up with nonlinear least squares on the log scale rather than linear least squares. There's also the issue of transformation bias - the expectation of the log is not the log of the expectation, so if you were after the mean on the untransformed scale you need to account for that effect (it's reasonably easy).

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However, one thing that does concern me somewhat is the dependence structure between nearby frequencies in the frequency domain. I am not sure that the usual independence assumptions on which the standard errors rely will be appropriate.