Telling the future with mathematics: Probability and crime prediction

Summer School of Science, Požega, Croatia, 2019-08-04.

Telling the future with mathematics: probability and crime prediction

If you've ever watched late-night TV, you've probably seen fortune tellers claiming they can tell the future if you call them – usually using some magic instruments. While the aforementioned is far-fetched, mathematicians can actually use probability calculations to predict the certainty of an outcome. Furthermore, we can use our knowledge of probability to help us understand when and where crime will (potentially) happen so police departments can work pre-emptively. Using mathematical models combined with knowledge from social media platforms such as Twitter, mathematics can help police departments battle crime before it even happens. In this swapshop, we'll discuss how this application of mathematics works and we'll try to build some simplified models for prediction.

Errata: There have been some concerns regarding predictive policing, voicing discrimination concerns or opinions regarding its role in combatting discrimination, including some voiced by the Human Rights Data Analysis Group. While working on this swapshop, I was not aware of such issues nor was my intent to promote an inherently biased tool: I was working on the (wrong) assumption that the data collected is perfectly unbiased. However, I did aim to show in task handouts how humans are inherently biased when intuitively interpreting data. I am sorry if this came across wrong and obviously I should have put more thought into the human rights aspect of this topic.

Transcript

1. Telling the future with mathematics Probability and crime prediction Mario Borna Mjertan, Faculty of Science, University of Zagreb marimjer@student.math.hr
2. A bit about me
3. Quick question... How many of you are from Croatia? (or nearby) Nick Savchenko, Wikimedia Commons, CC-BY-SA
4. Because if you are, you probably remember this guy.
5. This is much closer to what we’ll be doing.
6. Here’s what we’ll be doing. Basic probability (or: how can maths predict anything?) Text mining (or: what can your DMs tell us?) Spacial-temporal generalized additive models (or: how can we put this all together to tell the future?) Building our own small, localised models (or: DIY N3MBERS)
7. Let’s talk about simple probability.
8. = $ % wanted outcomes all possible outcomes
9. For instance... • There are 52 cards in a deck. There are 13 red cards. • We draw one card. What’s the probability that it’s red? • nw = 13 • np = 52 • p = 0.25 = 25% = ¼
10. For instance... • There are 52 cards in a deck. There are 13 red cards. • We draw four cards. What’s the probability that they’re all red? • nw = 13*12*11*10 • np = 52*51*50*49 • p = 0.00264 = 0.26%
11. Probability differs wildly depending on scale.
12. Try it out yourselves. • A phone number is made out of: • a three digit network code • a six digit user number How many nine digit phone numbers can be made if the network code is made up out of the digits 0, 7, 8 or 9, while the user number is made out of the digits 0 through 9? • A railroad network consists of 30 stations. How many different one-way tickets can that network sell? • We throw a six-sided game dice three times. What are the chances that we get a sum of 13?
13. Why are we doing this?
14. We can’t be sure about the future.
15. How to predict crime?
16. Two things to take into consideration Space Time
17. We need a way to predict where and when crime will happen
18. Where is easy
19. Where is easy-ish
20. Hotspot maps
21. What do we need to take into account? Just plotting crime on a map won’t help us predict where crime will happen.
22. What do we need to take into account? • Usually, we draw layers on top of a map: • The base layer • Land use • Demographics • Crime initatives • Transport networks • Regenration areas • Crime incidents • Also, we usually take into account the motives behind crime (like police officers do)
23. We want to identify hotspots.
24. What is a hotspot?
26. A hotspot is a geographical area of higher than average crime. It is an area of crime concentration, relative to the distribution of crime across the whole region of interest (e.g. a city centre, census ward or tract, municipal district, county or state). Hotspots are often clusters of crime that can exist at different scales of interest.
27. Let’s try something. We’ll try to identify crime clusters: • In a 2D coordinate system • On a map
28. Try to plot this and see if you can spot crime clusters.
32. Now let’s look at a map.
33. Murder Fight Drug abuse Train accident Speeding
34. Murder Fight Drug abuse Train accident Speeding Murder – around the river Fights – near nightclubs, in the city centre Drug abuse – parks, low population area Train accidents – near the railroad Speeding – long roads in a good state
35. All of the data here is purely hypothetical.
36. Let’s imagine we’re the police. How does this help us?
37. This helps us: • Know where crime hotspots for certain crime types are • Make decent arguments about why crime happens where • for instance, drug abuse happens in low population areas and parks because there’s less chance of getting reported to the police • fights happen around nightclubs because people get drunk and emotional etc. • Know where to efficiently have patrol cars
38. But that’s not enough. We need to have confidence in the fact that an area is a crime hotspot.
39. We also need: • A formal, mathematical way of finding out where the hotspots are • A confidence measurement • A way to do everything fast
40. There are methods to do so.
41. Continuous surface smoothing methods • We can visualise the distribution of crime and identify hotspots by creating a smooth continuous surface to represent the density or volume of crimes distributed across the study area. èInterpolation techniques, include inverse distance weighting and kriging. • They use a population or intensity value – with crime data, we usually don’t have those and we’re not trying to estimate a number of crimes. • What’s a more suitable method?
42. Quartic kernel density estimation • The quartic kernel density estimation creates a smooth surface of the variation in the density of point events across an area. • The method is explained in these steps: • A fine grid is generated over the point distribution • A moving 3D function of a specified radius visits each cell and calculates weights for each point within the kernel’s radius • Points closer to the centre will receieve a higher weight, and therefore contribute more to the cell’s total density value • Final grid cell values are calculated by summing the values of all kernel estimates for each location • We get a result which tells us something about the density or clustering of crime points at all locations in our study area.
43. K-means clustering • A general method for finding clusters of data • This can be useful here as well - for instance, in our 2D mapping • However, actually implementing the algorithm here can be challenging • NP-hard à slow • It’s a method of vector quantization • Given an initial set of k means m1, ..., mk, the algorithm proceeds by alternating between two steps • Asign each observation to the cluster whose mean has the least squared Euclidean distance • Calculate the new means of the observation in the new clusters • The algorithm has converged when the assignments no longer changed, but doesn’t guarantee to find the optimum
44. Is data about previous events enough to prevent crime in the future?
45. maybe
46. But probably not.
47. How can we gather more data about people’s intentions?
48. Text mining
49. What can your DMs tell us?
50. Let’s compare these two chats.
51. Let’s compare these two chats. A friendly chat between two gossipy friends No signs of aggression
52. Let’s compare these two chats. Clear signs of aggression Loudly voicing his/her displeasure (this one is fake)
53. We could... Collect and analyse text messages Assign a score of “harmfulness” to them Assign a category of potential crime Alert services
54. What’s the best way to do this?
55. AI & Machine Learning
56. How does machine learning work? • A lot of complicated maths • Linear algebra, tensors etc. are really useful here! • There are some general tools that can help us • For example, Tensorflow is a great framework for building machine learning services • However, it comes at a cost • Expensive • Takes a lot of existing data to draw a meaningful conclusion • Takes a lot of time and computer resources
57. There’s a different way as well
58. Spacial-temporal additive models
59. I know you’re going what the... • Let’s break it down. • Spacial • We take into account the place that the crime could potentially occur • Temporal • We take into account the time that the crime could potentially occur • Additive • A method in statistics • Nonparametric regression method • It uses a 1D smoother to build a restricted class of nonparametric regression models • In translation, it takes some data, leaves out noise and random incidents and builds out ‘different classes’ of crime • Model • Basically, a way to use maths to predict real life
60. However... • This is complicated • We’d really need a strong statistics background • We’ll be going for an intuitive understanding of how this works • I don’t want to torture you with university-level maths here
61. So let’s try to put together everything we’ve learned.
62. You’ll get a few papers with some data and tasks.
63. Questions?
64. We’re done, I guess. FEEL FREE TO CONTACT ME FOR ANY MATHS- RELATED QUESTIONS (ESPECIALLY IF YOU’D LIKE TO STUDY MATHS) EMAIL ME: MARIMJER@STUDENT.MATH.HR DM ME: @MARIOBORNAMJERTAN ON MESSENGER @MBMJERTAN ON INSTAGRAM 00385958455325 ON WHATSAPP