I have time series data for two signals, $X$ and $Y$. At $\text{time}=0$, the both $X$ and $Y$ are roughly 50. As time progresses, $X$ increases to settle at 100 and $Y$ decreases to settle at 0. Hence, the time-series go from (50,50) to (0,100).
My question is, if I plot this data, clearly $X$ any $Y$ will appear negatively correlated. Is there a way to control for the correlation that develops just by virtue of the time? Or am I thinking about this all wrong?
You're thinking about it quite right. There certainly are ways to deal with that. Transfer function models are among the simpler if you're already familiar with ARIMA. Basically you 'pre-whiten' the series to remove autocorrelations before examining the correlations between them. The answers here might help & the book by Box, Jenkins, & Reisel I'd recommend.
You want the partial correlation of $X$ and $Y$, controlling for time. The general notion is that you do separate regressions of $X$ and $Y$ on time, then correlate the residuals. If the regressions of $X$ and $Y$ on time were linear and homoscedastic then it would be a simple textbook problem, but the fact that the general levels of $X$ and $Y$ are asymptoting says that the regressions are not linear, so you need to know (or assume) the forms of those regressions. Then you fit the regressions and get the residuals $x - \hat{x}$ and $y - \hat{y}$. Look at their scatterplot, get the simplest form of correlation that is consistent with the plot.
