# Autocorrelated residuals in GAM even with lagged variables or AR process

I'm using GAMs to model ozone as a predictor, and using time , temperature and another pollutant (poll) as covariates. I have 20 years hourly data (large dataset), but I'm using 2 periods of 10 years, since I aim to see how the interaction term temperature-poll changes, as well as the differences between both periods. My main interest is to model ozone as a function of temperature-pollutant and be able to interpret how the interaction is changing (and how it affects ozone). I'm only using summer months (3months), so I am not including seasonal terms , only year to account long-term trends.

So far, I tried to fit different models, but I have serious problems with the residuals correlations, even when using lagged variables. I also tried to use AR1 process within the model, which seem not to improve it, but I'm probably missing something here, so I'd really appreciate any suggestion about it. I have read similar posts, but I'm still struggling to find the proper way to model this.

These are the models I tried:

m0 <- <- gam(o3 ~  s(year) + te(temp,poll), data=mydata)

I also fitted some previous models to test the significance of the interaction temp, poll, which is always very significant as expected.

The plots I got from this:

par(mfrow=c(1,2))
acf(residuals(m0))
pacf(residuals(m0))

Model with lag (previous hour of O3)

m1l <-gam(o3 ~  s(year) + lag + te(temp, poll), data=mydata)

which gives me:

still, autocorrelation problems.

Introducing AR1 process (since usually the concentrations depends also on the previous (time) concentrations.

m1.cor <- gamm(o3 ~  s(year)  + te(temp, poll), data=mydata,
correlation = corARMA(form = ~ 1|H, p = 1))

But, still:

When plotting normalized residuals as:

acf(resid(m1.cor$$lme, type = "normalized")) pacf(resid(m1.cor$$lme, type = "normalized"))

I got the following patterns:

Then, I included also the lag in:

m1l.cor <- gamm(o3 ~  s(year)  + lag + te(temp, poll), data=mydata,
correlation = corARMA(form = ~ 1|H, p = 1))

With normalized residuals:

I still see some problems, and I don't know how to improve the models.

I'm probably missing something here, and I might use another term or the autocorrelation should be handled in a different way. Then, I'd appreciate any suggestion or comment about it.

For the gamm() models, you need to pass acf() and pacf() the normalized residuals to have them take into account the correlation structure of the model.

Use

resid(m1.cor, type = "normalized")

and

resid(m1l.cor, type = "normalized")

to get these normalized residuals for each of the two models.

• Thanks a lot for your comment @Gavin. I just edited my question, adding the plots using normalised residuals. Unfortunately I didn't see too much improvement, so I'm still a bit lost here. Hope to get some feedback about it. Thanks again! Oct 31, 2019 at 19:49