How to evaluate deterministic vs stochastic components of a time series? Suppose we are working with a real data set , say in R.
What are the approaches/methods for determining underlying structure.
That is,
How can we separate stochastic components vs deterministic?
Can we use state space modelling, and if so, how to best implement in R?
Thank you all for any advice.
 A: Thank you for your question! I must reflect that this is a profound question as it is never treated in either texts or the classroom.
There is an R version of AUTOBOX (a commercial piece of software I helped to write) which identifies both the stochastic components (ARIMA & possible variance stabilizing transforms) while sorting out/identifying possible deterministic components. This is accomplished by evaluating alternative threads or approaches. The deterministic components are found primarily through proprietary scripted search procedures. For example a particular day-of-the month or week-of-the-month may be important or a change in daily-effects over time which would be detected by evaluating alternative hypothesis/data structures. Pulses,level shifts,local time trends,seasonal pulses are also discovered/detected by search procedures. Long-weekend effects, days-before and after each holiday can also be identified in a similar manner.
The interaction between ARIMA and possible deterministic structure is an added dimension/complication/opportunity that needs to be woven into a self-checking iterative procedure to ensure parsimony and estimability. All of the above needs to be done while accounting for the timely response and anticipation of user specified variables. 
I am unaware of any state-space modelling that conducts this level of signal /noise deconstruction . If you wanted to post a real world data set I would be glad to demonstrate the results of AUTOBOX to routinely/automatically perform this task of forming a minimally sufficient model containing both stochastic and deterministic components.
