so then, discounting making time itself a causal variable, it seems like using methods that rely on stationary distributions still treat the data, after pre-processing, as i.i.d, rather than predicting values from their correlated history.
I'm interested in methods that don't "subtract" simple "trends" and "seasonality" from the data (which may work for bog-standard templates such as sales data but not what I'm interested in), and rather responds to sequential relationships in the data itself, that exploits exactly the correlations you describe directly.
> I'm interested in methods that don't "subtract" simple "trends" and "seasonality"
a 2nd order difference equation can model a single harmonic frequency - that is, if your data is a pure sine and sampled at regular intervals, then
x_n =~ a x_n-1 + b x_n-2
can model any frequency with the proper a and b values (machine precision limits apply in real world scenarios, of course); That is, if your data looks like a sine wave with a yearly period, you still need no more than one sample per minute and 2nd order model to filter it out.
It's likely not a perfect sinewave, so you'd need a lot more - but if you are incredibly lucky and your periodic underlying admits a (relatively) sparse harmonic decomposition, and the signal riding on it has (very) low amplitude compared to the periodic signal, you can model very long periods implicitly by just having enough recent samples.
The name itself states this. Autocorrelation and autoregression. It regress on it's past values hence the "auto". We're interested on past value so each other value is dependent on the past.
An example of this is taking blood pressure. You would assume that taking blood pressure two consecutive days means that the previous day is highly correlated to the present day test.
Where as if you compare your blood test a year ago compare to today blood test it won't be as correlated. This is why ARIMA is dealing with correlation with time lags/lengths.
> it seems like using methods that rely on stationary distributions still treat the data, after pre-processing, as i.i.d, rather than predicting values from their correlated history.
Not sure what you mean exactly. Stationarity in time series is not i.i.d. The whole point of ARIMA modelling is to model, after transforming to stationarity, the remaining temporal correlation. ARMA is just a limited-parameter fit of the autocorrelation function.
I'm interested in methods that don't "subtract" simple "trends" and "seasonality" from the data (which may work for bog-standard templates such as sales data but not what I'm interested in), and rather responds to sequential relationships in the data itself, that exploits exactly the correlations you describe directly.