Uni-variate hourly time series anomaly detection by TBATS I have hourly and +4 years length of air pollution data (PM10). It has 24 (daily), 168 (weekly) and 8766 (yearly) seasonality. Also distribution is right skewed and has very long tail. I want to make anomaly detection on residuals of current data by fitting TBATS model but residuals still have seasonality (in terms of variance) at 24th hour (By the help of ACF and PACF plots). I know TBATS is very slow on long time series but my main goal is to be able to identify seasonally adjusted anomalies. For instance, 22C temperature can be an anomaly in winter but it is unlikely in summer but 50C is another anomaly in summer, too.


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*Is this a good approach to determine seasonally adjusted
anomalies? If not, why?

*If not, can you advice another method for a multiple seasonal and non-gaussian data?

*If yes, I expect a completely random residuals but variance of my residuals change by time i.e. high variance in winter time and low variance in summer time. Is it good interpretation to say that model can not handle seasonality well in my time series? or having seasonality in residuals is a common situation?


Thanks in advance
 A: stats.stackexchange.com/questions/437053/… discusses how to form a useful model i.e. extract latent features present in hourly data by considering possible improvements 1-9 . You can safely ignore the fact that it was explicitly a causal model as the same lessons apply when one doesn't pre-specify possible supporting series . Perhaps you should augment your BATS approach to develop/identify/add these possible model features. 
Your post is somewhat confusing , do you have 24 values per day for 365/366 days ...this would be about 8766 .  Do you have data for 4 years and lots of missing values ? Please clarify ..
See the cited residual plot from a deficient model suggesting changing error model variance through the year . This would need to be treated to identify anomalies as you had suggested.
You asked for suggestions ... I can suggest the SARIMAX Model presented here https://autobox.com/pdfs/SARMAX.pdf as your objective. It is also referred to as Dynamic Regression or Transfer Function Model OR a PDL model or an ADL model in different literatures.
