Friday, February 19, 2010

Rossmo 's formula-find serial killers using math!

Geographic profiling19th feb(less than 50 days b4 JEE!)

Dr. Rossmo had been interested in mathematics since his early youth, and believed it could be used in his professional work involved with fighting crime. In fact, sometime around 1991, as the story goes, he was travelling in the dining room of the "bullet train" in Japan when a formula for locating criminals occurred to him. In true mathematician-style --- can't you imagine Charlie doing this? --- he wrote it down on the only available piece of paper: a napkin. The formula gives a method for finding the likelihood that a criminal might live in a certain quadrant of a neighborhood where apparently serial crimes had been committed.

The neighborhood in question is divided into a grid of squares or quadrants, denoted Xr s, with the "r" denoting the rth row and the "s" the sth column. What seem to be related crimes have been committed in the quadrants in blue (see arrows in the diagram below). Note that distances between quadrants are computed by adding up horizontal and vertical distances as shown. These are sometimes called "taxicab" distances since they are the distances covered in an urban street grid by a taxi going from one quadrant to another. Here's a diagram:



In studying many crimes, Rossmo came to two realizations:

(1) criminals tended not to go too far into strange neighborhoods -- the likelihood of them committing their misdeeds at a certain place decreases the further its distance from their residence and

(2) criminals also tended not to commit crimes too near their own homes, lest they leave clues as to where they live; thus, they leave a crime-free "buffer zone" around where they actually live (or work).

Instead of trying to predict where a criminal might strike again, Rossmo tried to find in which quadrant he most likely lived. Rossmo therefore tried to find a formula which gave an estimate, based on the crimes committed, of the probability that the perpetrator lived in some particular quadrant. This formula should give lower probabilities far from the "perp's" home and also lower probabilites within the buffer zone. Once outside this buffer zone, the likelihoods should decrease with distance from home, but increase near where actual crimes were committed. In mulling this over, Rossmo created his first basic formula:





Here, on the left, P i j represents the probability or likelihood that the perp lives in the quadrant on the ith row, jth column. On the right is a sum, designated by the capital Greek sigma (Σ), of some terms, one for each of the crime quadrants centered at (xk, yk) (k = 1, 2, ...up to N = # of crimes). The numbers xi and yj are the coordinates of the center of the quadrant in row i, column j; in practice they could be just i and j themselves. Each term has two parts. Part A says that the likelihood of the residence being in row i column j gets smaller the further the residence is from the crime scene. Part B says that the probability is small when the residence is within the buffer zone, but gets big just outside the buffer zone (B = the "width" of the buffer zone). Thus, parts A and B tend to fight against each other: the criminal doesn't want to go too far away into unfamiliar neighborhoods, but doesn't want to call attention to his home by committing crimes nearby.

So how do we balance these tendencies? Note that there are several other numbers here: the Greek letter φ and the powers f and g are ways of weighting or giving different emphases to the the two parts A and B. They are assigned values based on studies of past crimes, and can be fiddled with in order to fine-tune the formula.

This formula is used to produce a "geographical profile", which is a map of the neighborhood in which the different quadrants are assigned colors based on the probabilities that Rossmo's formula assigns to them; note that in this particular plot there are two red areas of very high probability: hot zones.



Rossmo's computer software produced a similar geo profile for the Lafayette LA case. Based on the profile, police investigated residences in the hot zone. Unfortunately, none of these investigations panned out. Then, following up a tip, they looked into the background of a new suspect, and found that he had previously lived exactly in the hot zone, but had recently moved. He was subsequently investigated, arrested, tried, and convicted.

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