I've written a macro to conduct aoristic analysis with SPSS. Here I will briefly describe what it is, provide alternative references and demonstrate some of its utility on example data from Arlington PD.

In short, crime event data are frequently recorded as occurring within some indefinate time frame. For example, you may park your car and go to work at 08:00, and when you come back out at 04:30 on the same day to find your car window broken and your GPS stolen. Unless there happens to be other witnesses to the crime, you don't know when the criminal event occurred besides between those two times. Where this is problematic for crime analysis is, you want to be able to look at the distribution of when events occurred, so as to give suggestions to understand why the event is occurring and how to potentially address it. Allocating patrols geographically and temporally to areas of high crime incidence has been regular practice for a long time (Wilson, 1963)! Aoristic analysis is simply a means to take into account that uncertainty of when the event occurred when we examine the overal incidence of crimes occurring across a set of times.

For a very brief illustrative example, lets say we want to know the number of crimes occuring for within the hours of 08:00, 09:00 and 10:00. If we had a criminal event that potentially occurred between 08:00 and 10:00, which is a total time span of 120 minutes (2 hours), instead of counting that event as occurring at 08:00 (the begin time), 10:00 (the end time) or 09:00 (the middle time), we spread the event out over the time frame, and only partially count it within any particular interval. So here it would count as a total of 0.50 weight in both the 08:00 and 09:00 category (60/120=0.25) and assign 0 weight in the 10:00 category (note the weights sum back to the value of 1). This just ends up being a way to estimate the incidence of some event within a given time bin knowing that the event did not necessarily occur in that time bin, so only partially counts towards the total in that bin (where partially is defined by how long the interval is and how much of that interval overlaps with the bin).

Here I illustrate the macro with some examples from the Arlington PD data downloaded on 1/11/2013. This is the only dataset I've found publicly available online that has both start and end dates for events (I first looked at NIBRS, and was slightly surprised that they did not have this information). I have the macro code, with examples therein of fake data and the same Arlington PD data, and compare them to this online calculator.

Note, my results will be slightly different than most other programs (including the online app I pointed to) because of one arbitrary (but what I feel is reasonable) coding decision. When an event is over the time period evaluated in the particular estimate (either days or week for my functions), it simply returns the event coded as having equal weight across the time period. Other's don't do this as far as I'm aware. So say an event takes place between 08:00 on 1/2/2013 and 10:00 on 1/3/2013. For my functions that only evaluate times over the day, I would return equal probability within every time slot, although some calculators would say there is a higher probability of occurring for times between 08:00 and 10:00 (because of the wrap-around). I believe this practice is a bit of a stretch, and any uncertainty over a day is essentially saying it is totally useless information to determine when during the time of day at all (although my week functions would be equivalent in that example). In those cases the begin and end times say more about when people check there cars, wake up in the morning, get home from work, come back from vacation etc. than they do about when the actual crime occurred.

Some examples

If you want to follow along right within SPSS, I suggest going to the google code site I've posted the code and data, but otherwise you can just take my word for it and see how the macro works in practice. I provide several seperate functions to either estimate the frequency of crimes occurring during 1 hour bins over the day, 15 minute bins over the day, days over the week, 1 hour bins over the week, or 15 minute bins over the week.

Here is an example call of the macro and the output from the 1 hour bins over the day with all crimes for the Arlington data.

!aoristic_day1hour begin_date = Date1 begin_time = Time1 end_date = Date2 end_time = Time2.

Date1 and Date2 are the begin and end dates respectively, Time1 and Time2 are the begin and end times respectively. Below is (close to) the automated graph the macro produces, which is just a line chart super-imposing the both the aoristic estimate and what the estimate would be if using the begin, end or middle time. The only differences are I post-hoc edited the aoristoc estimate line to be thicker and in the front (so it is more prominent) plus my personal chart template. Paramaterizing GGRAPH charts to work in macros is quite annoying, and python is a preferable solution (I'm personally happy with just a helper function to return the data in a nice format for me to generate the approprate GGRAPH code to generate the chart myself, there is more power in knowing the grammar than being complacent with the default chart).

So you can see here that overall, the aoristoc number does not make much of a difference. Here is the same info for the 15 minute bins across the week.

!aoristic_day15min begin_date = Date1 begin_time = Time1 end_date = Date2 end_time = Time2.

You can see here the aoristic estimate smooths out the data quite a bit more (which is nice above and beyond just worrying about whether one approach is correct or not. With the smaller time bins you can also see patterns to over-report incidents at natural times of hour and half-hour intervals. You can also see midnight, noon and 08:00 are aberently popular times to report either beginning or end times of incidents. You can spot a few other ones as well that differ between begin and end times, for instance it appears 07:00 is a popular end time but not so popular a begin time. The obverse is true for events in the middle afternoon, late evening and early night (e.g. the big spikes in the green begin time line for hours between 17:00 till midnight). Also note that it is near universal that crime dips to its lowest around 4~5 am, and you can see using either the aoristic estimate or the middle point of the event brings the number of events up during this period (as expected).

Also in the set of functions I have the capability to specify an arbitrary category to split the results by, and here is an example splitting the day of week aoristic estimate by the beat variable provided with the Arlington data. Again this isn't the direct code, but a subsequent GGRAPH command to produce a nicer chart (the original is ok if you make it much bigger, but with so many categories facet wrapping is appropriate to save space).

!aoristic_week begin_date = Date1 begin_time = Time1 end_date = Date2 end_time = Time2 split = Beat.

The main thing that draws attention in this graph is the difference in levels of calls and different trends between beats (there were no obvious differences in the aoristic estimates versus the naive estimates, which is unsurprising since most incidents don't have uncertainty of over a day). I know nothing about Arlington, and I don't know where these beats are, so I can't say anything about why these differences may potentially occur. In SPSS days start at Sunday (so Sunday = 1 and Saturday = 7). It isn't that strange to expect slightly more crime on Fridays and Saturdays (people out and about doing things that make them more vulnerable), but for most of the beats showing a flat profile this is not strange either. But it is interesting to see Beat 260 have an atypical pattern of obviously more crimes during the week, and if I had to hazard a guess I would assume there is a middle or high school in Beat 260.

Although you could argue aoristic analysis is called for based purely on theoretical grounds, all these examples show events that it doesn't make much of a difference whether you use it or simpler methods. Where it is likely to make the most difference though are events which have the longest unknown time intervals. Property crimes tend to be committed when the victim is not around, and so here I compare the aoristic estimate for 15 minute intervals over the day for burglaries, which will show an example where the aoristic estimate makes an actual difference!

One can see the property crimes have a larger difference for the aoristic estimate across the day, and it is largely flat compared to begin and end times. The end and begin times are likely biased to report when people discovered that the victimization occurred or when they last left their home vulnerable. There is some slight trend for more burglaries to occur during the daytime, and somewhat higher periods during the night (with lulls around 08:00 and 18:00. These are near the exact opposite conclusions you would make with utilizing the begin and end times as to when most burglaries occurred! Middle times results in some weird differences as well, with a high spike in the 01:00 to 04:00 range.

Some closing

This project kicked my butt alittle, and took much longer than I expected. Certainly the code could use improvement and re-factoring, but I'm glad it is done (and seemingly working). You will see there is certainly alot of redundancy between the functions, some temporary variables are computed multiple times, and the week long functions take a while to compute.

In the SPSS macro what I do in a nutshell is make a variable for every time bin, calculate the weight each case has for that time bin, then reshape the dataset wide to long, then at last aggregate the total weight within each time bin. Note this results in many more cases than the original data. For example, the Arlington data I will display later in the post has slightly over 49,000 incidents. For the 15 minute intervals per day (96), this results in over 4.7 million cases (49,000*96). For the 1 hour bins across the week, this results in n times 168 more cases, for the 15 minute bins across the week in n times 672! Subsequently those latter two take an appreciable amount of time to compute for larger datasets (if you don't run out of local memory on your computer entirely, which I'm guessing could easily happen for some of the older systems and when you have upwards of probably 60,000 cases).

For those interested, the bottle-neck is obviously the VARSTOCASES procedure. But, I have a substantive reason for going through that step, and that is if one wants to use the original data weighted, for say kernel density maps sliced by time of day having the data in that format (long with a field identifying the factor) is more convienant than in the wide format. Thinking about it I could generate NULL data for 0 weight categories and then drop those cases during the VARSTOCASES, but it remains to be seen if that will have much of an appreciable effect on real world datasets. Hopefully in the near future I will get the time to provide examples of that (probably in R using facetting and small multiple maps). If anyone has improvements to the code feel free to send them to me (or just shoot me an email).

In the future I plan on talking about some more visualization techniques to explore crime data with intervals like this. In particular I have a plot manipulating the grammar of graphics a bit to produce a visualization of individual incidents, but it still needs some work and writing up into a nice function. Here is an example though.

Citations

Wilson, O.W. 1963. Police Administration. McGraw-Hill.

Ratcliffe, Jerry H. 2002. Aoristic signatures and the spatio-temporal analysis of high volume crime patterns. Journal of Quantitative Criminology 18(1): 23-43. PDF Here.