I have a time series of monthly temperature data for ~ 100 years. My aim is to turn it into a time series which has neither a trend nor autocorrelation. I want to make sure this is true for the whole time series as well as for the series of each of the months seperately (e.g. looking at a time series of August temperature from 1900-2000). How do I go about that? Especially taking into account that different months might have different trends?
While I don't quite understand what your goal would be, but some general remarks that might be helpful for you:
Removing Trend (DetrendedSignal = Signal - Trend): Judging from the picture you posted, there seems to be a slight positive, linear trend overall. So you might fit a linear model (temp as a function of time) and subtract its estimation from your original data. Of course, you can also try non-linear models..Depending on which software you are using, there surely exist a package/method that does this automatically.
Removing Autocorrelation: One way would be to fit an ARIMA-Model on your data and keep only the residuals (the part of your data that can't be explained via auto-correlation). There are many softwarepackages for this too-
whole time-series vs. separate month: hard to say anything concrete without knowing your application, but wouldn't it be possible to make separate analyses for the whole data and for specific months? If not, you would have to rely on a more complex trend-model, maybe a multiplicative one or seasonal trend decomposition. I found these two tutorials helpful : some general information & seasonal Trend decomposition in R
Hope this was somewhat helpful.