Talk:Trend-stationary process

Surely the definition shouldn't be so general?

The page says that a process {Y} is said to be trend stationary if $Y_{t}=f(t)+e_{t}$, where f is any function mapping from the reals to the reals, and {e} is a stationary process. But if any f is allowed, doesn't that mean that every process is trend stationary? Choose your favourite stationary process {e}, and then just define f(t) to equal $Y_{t}-e_{t}$. Surely the definition should place some restriction on the type of functions f that are allowed? Or have I misunderstood something here? Jowa fan (talk) 03:43, 28 June 2012 (UTC)

With the definition given, {e} is not aribtrary but has to be stationary. Thus the process {Y} is such that an additive trend can be removed leaving a stationary process, as opposed to one where removing a trend leaves a process that may be nonstationary in variance (or more general spread) for example. In practice, for analyzing real data it may be necesary to try various transformations of the data before a plausible model with an additive trend might be acceptable, but it is not guaranteeed that one can be found. The next layer of modelling might be to introduce a variable scaling function so that both location and scale can be adjusted, possibly leaving stationary residuals. Even if only second-order stationarity is being considered, this location-scale detrending still leaves the possibility of leaving a non-stationary autocorrelation function. Melcombe (talk) 08:02, 28 June 2012 (UTC)

Thanks for the response. I understand that {e} isn't arbitrary. What bothers me is the idea that f can be arbitrary. (You used the word "plausible" in your reply. If arbitrary f are allowed, then some very implausible models come into consideration.) Can you give a concrete example of a process that is not trend-stationary according to this definition? Jowa fan (talk) 01:50, 29 June 2012 (UTC)

Real world examples ... almost any real seasonal process such as temperature, rainfall etc.. Consider monthly total river flows ... these show variations in both location and scale with calendar month, and the correlation between adjacent months also varies seasonally. Theoretical examples ... take any non-trivial trend-stationary process and create a new one by taking the exponential of the first. But you may be looking at this the wrong way ... the idea isn't really to define a class of interesting processes to be studied for their mathematical properties ... it is to describe the assumptions sometimes made in data analysis, that subtracting a trend leaves a process that can be treated as stationary. As to f ... it has to a be a deterministic function (otherwise the article would have said that it was a random function). Melcombe (talk) 10:38, 29 June 2012 (UTC)

Thanks, I think I get it now. My intuition about Stationary processes was a bit off. Variations in location are not the issue here: you can fix that by choosing the appropriate function f. But subtracting a function won't affect variations of scale, and won't remove correlations. Jowa fan (talk) 14:30, 29 June 2012 (UTC)