Is there a mathematical model that distinguishes between volatility and trend? Say we are studying Twitter hashtags over time.  We monitor how popular they are day to day.  Some hashtags may be volatile (i.e. "lunch", "Celtics", "Friday").  Their popularity rises and falls frequently. Some hashtags may be in the process of becoming unpopular (i.e. "Gulf oil spill", "Transformers 2", "Christine O'Donnell").
Is there a mathematical model that can distinguish between a hashtag that has temporarily fallen in popularity but is likely to go up in popularity later and a hashtag that is sinking and likely to stay sunk in popularity?
thanks
 A: Maybe I'm missing something here, but if you plot the number of times these hashtags are mentioned over time, shouldn't that tell you something?
Of course, maybe you need automated processing. In that case fit splines to these series, and take the derivative. (They are easy : just look up what the Function Data Analysts do.) Sharply trending topics will have high derivatives. How high, will come from your data.
Do tell if this worked or not?
A: As far as I can tell from my reading, a common method for determining a time series' trend is to smooth the series, perhaps in an iterated fashion, as in:
A Pakistan SBP paper. In the Seasonal Adjustment Methodology section, it describes how X-12 ARIMA does it, though they also use a seasonal factor which perhaps you could also use or perhaps you could simply ignore.
Other links might include A Bank of England web page and A US Census Bureau paper (pages 8-12).
A: A simple start using historical tweets data: Create a weekly variable called popularity change based on week to week changes in tweets for a tag for the past 25 weeks from the current time.
Calculate these 2 measures:


*

*Trend: Mean of popularity change

*Volatility: Standard Deviation (square root of
variance) of popularity change.


Meaning: A change from the number of tweets in your current week from the past week that is greater than 2 standard deviations can be considered a big change, and a change of 3 standard deviations would be a very big change (a very rare occurrence assuming the distribution of popularity changes for the time span looks like a normal distribution). 
