Using time series analysis to analyze/predict violent behavior This is a bit of a flippant question, but I have a serious interest in the answer. I work in a psychiatric hospital and I have three years' of data, collected every day across each ward regarding the level of violence on that ward.
Clearly the model which fits these data is a time series model. I had to difference the scores in order to make them more normal. I fit an ARMA model with the differenced data, and the best fit I think was a model with one degree of differencing and first order auto-correlation at lag 2.
My question is, what on earth can I use this model for? Time series always seems so useful in the textbooks when it's about hare populations and oil prices, but now I've done my own the result seems so abstract as to be completely opaque. The differenced scores correlate with each other at lag two, but I can't really advise everyone to be on high alert two days after a serious incident in all seriousness.
Or can I?
 A: The model that fits the data doesn't have to be a time series model; I would advise thinking outside the box a little.  
If you have multiple variables (e.g. age, gender, diet, ethnicity, illness, medication) you can use these for a different model.  Maybe having certain patients in the same room is an important predictor?  Or perhaps it has to do with the attending staff?  Or consider using a multi-variate time series model (e.g. VECM) if you have other variables that you can use.  Look at the relationships between violence across patients: do certain patients act out together?  
The time series model is useful if time has some important role in the behavior.  For instance, there might be a clustering of violence.  Look at the volatility clustering literature.  As @Jonas suggests, with a lag order of 2, you may need to be on higher alert on the day following a burst in violence.  But that doesn't help you prevent the first day: there may be other information that you can link into the analysis to actually understand the cause of the violence, rather than simply forcasting it in a time series fashion.
Lastly, as a technical suggestion: if you're using R for the analysis, you might have a look at the forecast package from Rob Hyndman (the creator of this site).  This has many very nice features; see the paper "Automatic Time Series Forecasting: The forecast Package for R" in the Journal of Statistical Software.
A: You fitted the model to the differences, which means that you're describing the change in levels of violence. You get a lag of 2 days. A lag is indicative of the memory of the process. In other words, the change in levels of violence today has some dependency on the change in levels of violence in the last two days. For longer time-scales, the contribution of random influences becomes strong enough so that there is no clear link anymore.
Is the auto-correlation positive? Then a change of levels of violence today suggests a similar change in levels of violence in two days. Is it negative? Then violence might stay higher for two days.
Of course, you may want to have to control for confounding effects. For example, after a serious incident, people may be more likely to report minor incidents, but this "sensitization" would be going away after two days.
