3

In keeping with the times, i was reading a news article describing the rapid uptake of predictive policing analytics software by public city police departments in the US. (eg. See vice.com "Dozens of Cities Have Secretly Experimented With Predictive Policing Software", 2019).

In the public interest, FOIA requests turned up a powerpoint presentation given by the subject company PredPol Inc to prospective clients titled "PREDICTIVE POLICING TACOMA OVERVIEW DECK (2012 JULY)". (If you search, it is avail on web).

If you look through the presentation, besides assuaging public fear of prejudicial surveillance, the company says they merely rely on the "what, where, and when" of past crime reports in their predictive algorithm to target police patrols on as fine as 500'x500' grids of cities. Then on page 8 of the presentation, they present a "formula" allegedly used:

$$\frac{\partial{B}}{\partial{t}} = \frac{\eta*D}{4}\nabla^2B -\omega{B}+\theta\omega\delta$$

Q: has anyone seen this equation? Is it a diffusion equation? Heat flow equation? I tried a google search but google is not good for math entry.

Kiers
  • 141
  • 1
    Just to seed the discussion, i would hazard they are attempting to use vector field theory for "badness" (B) over area grids? So the time rate of increase of "badness" is proportional to the Laplacian, which tends to decays on its own proportional to the extent of "badness"; and perhaps there's a background rate of increase related to exogenous population density, traffic flow etc etc. Just taking a stab at this. – Kiers May 16 '23 at 18:57
  • Interesting to think, by analogy with heat conduction, about $-(\eta D/4)\nabla B$ being a flux density. – A rural reader May 16 '23 at 19:51
  • 1
    @rural...so the badness has "sources" and a "flux" it would seem and the source of badness in one location flows to all points neighbors as well...? – Kiers May 16 '23 at 20:02
  • 2
    A longer review paper with focus on PredPol can be found here: "The mathematics of policing", by Tian An Wong. In particular the PDE of interest shows up, with only slightly different notation, as equation (2) on page 5. The article contextualizes it as an example of a continuous reaction-diffusion system. – Semiclassical May 16 '23 at 20:14
  • @Semiclassical this is TOPS. Thank you! I will dig into this paper and see if it makes sense to apply a diffusion model combined with historical crime metadata. You should have put your "answer" as a full block answer rather than comment! – Kiers May 17 '23 at 15:36
  • Would anyone care to comment on the "validity" of this approach? I realize analytical s/w corporations like to add "fancy looking equations" in their marketing to bowl over prospective clients, but isn't PredPol guilty of the same? Surely, crime does not spread like an ink blot on tissue paper? And even so, it would not be helpful to police departments to react to this model if it were the case? – Kiers May 29 '23 at 00:43

0 Answers0