# detecting change point in the time series on a 2 dimensional space

I have a 2 dimensional geographic space. There are crime events occuring at different regions in the space over time. I am looking particularly at property crimes like burglary. If I look at the time series of burglary events over the entire 2-d space, I can see an obvious periodicity based upon the frequency spectrum of the Fourier Transform.

I believe that different parts of the space exhibit different periodicities, i.e., different frequency spectrums.

So I have events per location indexed by date and time. So I can say a burglary happened on Wed May 3, 2017 at 12:00noon on the block of Venice blvd and Robertson in Los Angeles. I have tens of thousands of events indexed this way, with data from 2010 till present. You are correct that there is a seasonal trend and also a weekly trend, and a daily trend. However there tend to be differences in two ways. First, there is some variation in the number of events in a day--so the height of the frequency. There might also be the introduction of some new frequencies--though small--in intermittent ranges between the normal daily, weekly, and seasonal trends. These new frequencies are relatively hard to detect unless I have a good window on the period.

My goal was to find a way to detect the boundaries of regions in which the spectrum seems equivalent, so that I can thereby identify the regions of distinct periodicities.

I have been looking at material on change point detection and and such, but have not found any articles relating to my question. I just might be looking under the wrong names or something.

In looking at the Signal Processing literature, seems like I need to do a sliding window. But I am not sure how to detect the change point in the frequency spectrum as the window grows in size.

Once I have the change detection locked down, then I can basically just growth the sliding window by randomly starting at different points in the space. If I ran the simulation hundreds of times, I could try and find the regions where 90% of the simulations overlap, etc.

Does anyone know of a technique to detect the change points that I am trying to detect?

• What is your data? An image time series? A set of $(x,y,t)$ event-occurrence points? What are you taking a Fourier transform of? You need to provide some more information. (There is a lot of work on periodic change-detection in univariate time-series, e.g. pitch detection.) – GeoMatt22 May 4 '17 at 16:51
• @GeoMatt22 I am looking at crime data. Sorry, I can update the OP. – krishnab May 4 '17 at 16:54
• OK, but it is still not clear to me. For example, if you had no spatial component, then the burglary events might be something like a point process in time. But a Fourier transform would typically be applied to a time-series e.g. $n[t]$ = # events/month. So do you have something like events per day per police precinct? What exactly is the data? (Also, for stochastic data the better term may be seasonality.) – GeoMatt22 May 4 '17 at 17:08
• I can add this to the OP. – krishnab May 4 '17 at 17:27
• One other note: change-point detection would normally be in time. For variations in space (e.g. for periodicity = diurnal for x<0, annual for x>0), probably segmentation is the more general term (or perhaps edge detection, in an image context). – GeoMatt22 May 5 '17 at 3:07