Understanding Uncertainty – crime counts and the Poisson distribution

A regular occurrence for me when I was a crime analyst went along the lines of, "There was a noteworthy crime event in the media, can you provide some related analysis". Most of the time this followed one or multiple noteworthy crimes that caught the public’s attention, which could range from a series of thefts from vehicles over a month, the same gas station being robbed on consecutive days, or a single murder.

Any single violent crime is awful, and this is not meant to deny that. But often single noteworthy events are often misconstrued as crime waves, or general notions that the neighborhood is in decline or the city is a more dangerous place now than it ever was. The media is intentionally hyperbole, so it is not an effective gauge whether or not crime is really increasing or decreasing. Here I will show an example of using the Poisson distribution to show whether or not a recent spree of crimes is more than you would expect by chance.

So lets say that over the course of 20 years, the mean number of homicides in a jurisdiction is 2. Lets also say that in the year so far, we have 5 homicides. Ignoring that the year has not concluded, what is the probability of observing 5 or more homicides? Assuming the number of homicides is a Poisson distributed random variable (often not too unreasonable for low counts over long time periods) the probability is 5.7%. Small, but not totally improbable. To calculate this probability it is just 1 minus the cumulative distribution function for a Poisson distribution with the given mean. This can be calculated easily in R by using ppois, i.e. 1 - ppois(5-1,2) (just replace 5 with your observed count and 2 with your mean). Note that I subtract 1 from 5, otherwise it would be testing the probability of over 5 instead of 5 or more. For those analysts using Excel, the formula is =1 - POISSON.DIST(5-1,2,TRUE). For SPSS it would be COMPUTE Prob = 1 - CDF.POISSON(5-1,2).

The reason for making these calculations is specifically to understand chance variations given the numbers historically. For the crime analyst it is necessary to avoid chasing noise. For the public it is necessary to understand the context of the current events in light of historical data. For another example, say that the average number of robberies in a month is 15, what is the probability of observing 21 or more robberies? It is 8%, so basically you would expect this high of a number to happen at least one time every year (i.e. 0.08*12~1). Without any other information, there is little reason for a crime analyst to spend extra time examining 21 robberies in a month based on the total number of events alone. Ditto for the public there is little reason to be alarmed by that many robberies in a month given the historical data. (The analyst may want to examine the robberies for other reasons, but there is no reason to be fooled into thinking there is an unexpected increase.)

Here I’ve ignored some complications for the sake of simplicity in the analysis. One is that crime may not be Poisson distributed, but may be under or over-dispersed. In the case of over-dispersion (which seems to happen more often with crime data) the series likely has a higher number of 0’s and then high bursts of activity. In this case you would expect the higher bursts more often than you would with the Poisson distribution. For under-dispersed Poisson data, the variance is smaller than the mean, and so higher bursts of activity are less likely. These are fairly simple to check (at least to see if they are grossly violated), either see if the mean approximately equals the variance, or draw a histogram and superimpose a density estimate for the Poisson distribution. This also ignores seasonal fluctuations in crime (e.g. more burglaries occur in the summer than in the winter).

Even if you do not like making the Poisson assumption a very simple analysis to conduct is to plot the time series of the event over a long period. The rarer the crime the larger aggregation and time series you might need, but this is pretty straightforward to conduct with a SQL query and whatever program you use to conduct analysis. If it is for UCR crime counts, you can try going onto the UCR data tool to see if your jurisdiction has historical annual data going back to 1985. My experience is the vast majority of crime waves depicted in the media are simply chance fluctuations, clearly visible as such just by inspecting the time series plot. Similarly such a plot will show if there is an increase or a decrease compared to historical numbers. Another simple analysis is to take the current numbers and rank them against, say the prior 50 to 100 values in the series. If it is abnormal it should be the the highest or very near the highest in those prior values.

Another complication I have ignored is that of multiple testing. When one is constantly observing a series, even a rare event is likely to happen over a long period of observation. So lets say that in your jurisdiction on average there are 3 domestic assaults in a week, and one week you observe 9. The probability of observing 9 or more is 0.003, but over a whole year the probability of this happening at least once is around 18% (i.e. 1 - ppois(8,3)^52 in R code). Over the course of 10 years (around 520 weeks) we would expect around 2 weeks to have 9 or more domestic assaults (i.e. 520*(1-ppois(8,3))). (This probability goes up higher if we consider sliding windows, e.g. 9 or more domestic assaults in any 7 day span, instead of just over different weeks.) These statistics make the assumption that events are independent (likely not true in practice) but I rather make that false assumption to get a sense of the probability then rely on gut feelings or opinions based on the notoriety of the recent crime(s).

The title for this blog post is taken from David Spiegelhalter’s site Understanding Uncertainty, and that link provides a synonymous example with the recent cluster of plane crashes.

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3 Comments

  1. Jon Peck

     /  August 25, 2014

    You might like trying the STATS ZEROINFL extension command on such data. It fits zero-inflated Poisson or negative binomial models.

    Reply
  2. David Spiegelhalter has another example based on murders in London per day that is pretty similar to what I note here, http://understandinguncertainty.org/node/200

    Reply

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