Centroids, Clusters and Crime: Anchoring the Geographic Profiles of ...

Team # 7507 Page 13
committed.
A logical next step is to ask how much a model outperforms another, which can simply
be done by taking the ratio of the amount of resources allocated at the crime point under the
first model to the amount allocated under the second. When this ratio is greater than 1, the
first metric outperforms the second, and vice-versa. Assuming that the police force is twice as
effective at a location when they have twice the resources allocated, we can think of this ratio
as an effectiveness multiplier, representing how many times more prepared to stop the criminal
the police will be under the first model compared to the second. In this paper, effectiveness
ratios will be denoted by the symbol κ:
κ ≡
resource allocation at crime point, model 1
resource allocation at crime point, model 2
Observe that if we normalize the integral of each likelihood plot to the total number of
resources available to the department in question, the equation above is simply a ratio of the
percentage of department resources allocated to the point, and since the total resources available
are the same under both models, we can evaluate κ simply as
κ = Z 1(CrimeP oint)
Z 2 (CrimeP oint)
where Z n is the likelihood function of model n and CrimeP oint is the location of the actual
next crime. We note that though this information is obviously not available in most cases, we
use our metric by truncating a data sequence and using that set to predict the truncated crime.
Finally, note that since a randomly guessing algorithm, like an unconvincing psychic, will
have no reason to privilege any one point over another, averaging over a large number of
attempts to predict a crime location, their resource allocation appears the same as a model
which produces a perfectly flat likelihood plot. This allows us to compare our model to a
random guess by computing a “standard effectiveness multiplier,” in which model 2 is assumed
to be a uniform, flat distribution
κ s =
Z 1(CrimeP oint)
Z flat (CrimeP oint)
This standard effectiveness multiplier κ s is the metric we will use to assess how well our
model performs. A value of exactly 1 indicated it is no better (on average) than a random guess,
and a fractional value would indicate we are actively misleading the police. The upper bound
of this number is simply the magnitude of the maximum potential striking area considered by
the model, which is achieved only if the model predicts the next crime point exactly and assigns
it 100% likelihood 1 .
1 Although much of the formalism developed around anchor point finding algorithms is applicable, this metric