Crime Mapping and Spatial Data Analysis using R

So far we have shown ways to visualise crime along a network, but what if we want to actually quantify the extent to which crime concentrates in these micro-places. In crime and place literature we explore the concentration of crime at micro-places. "Perhaps the first and most important empirical observation in the criminology of place is that crime concentrates at very small units of geography" ([Weisburd, 2015], p. 135). A significant proportion of crime concentrates around a small proportion of micro places.

One way to measure inequality in the distribution of a quantitative variable is to use the Lorenz curve, and associated Gini coefficient. The Lorenz curve is a probability plot (P–P plot) comparing the distribution of a variable against a hypothetical uniform distribution of that variable. It can usually be represented by a function \(L(F)\text{,}\) where \(F\text{,}\) the cumulative portion of the population, is represented by the horizontal axis, and \(L\text{,}\) the cumulative portion of the variable of interest (e.g., crime), is represented by the vertical axis. While Lorenz curves are used typically to graph inequality of distribution of wealth, they can be applied in this case to explore unequal distribution of crimes between micro-places. A perfectly equal distribution would be depicted by the straight line \(y = x\) ([Lorenz, 1905]; [Zeileis et al., 2012]). The corresponding Gini coefficient represents the ratio of the area between the line of perfect equality and the observed Lorenz curve to the area between the line of perfect equality and the line of perfect inequality ([Gastwirth, 1972]). The closer the coefficient is to 1, the more unequal the distribution is ([Zeileis et al., 2012]).

In R we can implement these tests using the functions in the ineq package. To obtain a Lorenz curve, we can use the Lc() function. Lc() computes the (empirical) ordinary Lorenz curve of a vector \(x\) (in this case, our crimes_per_m variable). The function also computes a generalised Lorenz curve (\(= \text{ordinary Lorenz curve} \times mean(x)\)). The result can be interpreted like this: \(p \times 100\)% account for \(L(p) \times 100\)% of \(x\text{.}\)

Let’s illustrate this again with the segments of Telegraph Road.

tr_lorenz <- Lc(tr_sections$summer_rate)

Our resulting tr_lorenz object has three elements. First, the p represents the cumulative percent of crimes (per metre) for each line segment. Then, L contains the values of the ordinary Lorenz curve, while the L.general element the values for the generalised Lorenz curve. We can plot the Lorenz curve with the plot() function from base R.

plot(tr_lorenz)

Upon seeing this, we can consider that many of the segments of the Telegraph Road contribute very little to overall crimes, and it is instead the top few (less than the top 20%) which contribute to all the shoplifting incidents. From a visual inspection, it appears that the Telegraph Road very precisely fits (and exceeds) the Pareto Principle, whereby 20% of the segments seem to account for 80% of the crimes (per metre length of the segment). We can quantify this further using the Gini coefficient.

ineq(tr_sections$summer_rate, type="Gini")

## [1] 0.9

This score is quite high. The Gini Index is calculated from the Lorenz curve, by taking the area between the line of equality and the Lorenz curve, and dividing this by the total area under the line of equality. This number is bounded between 0 (perfect equality where the Lorenz curve sits right on top of the line of equality) and 1 (perfect inequality, where the Lorenz curve sits right on top of the x-axis and bends at right angle); so the closer we get to 1, the higher the inequality in the distribution in our value of interest, in this case crimes per metre. Clearly we see crime concentrate in certain segments of Telegraph Road.