DISCLAIMER: This is a blog post from my graduate school years. The blog is no longer maintained and the material might include typos and errors.

An interesting measure that I’ve encountered in my research on polarization is the coefficient of overlapping defined as

\[OVL(f,g) = \int_X \min\{f(x),g(x)\}dx,\]

where $ f $ and $ g $ are two density functions defined over the same region $ X $. The measure is quite intuitive in that it measures the overlapping region of two densities. If the densities are defined over disjoint subsets of $ X $, say $ X_f $ and $ X_g $, $ X_f \cap X_g = \varnothing $, then $ OVL(f,g)=0 $. Also, as the area under each of the densities has to integrate to one, $ OVL(f,g)=1 $ if and only if the two functions are equal almost everywhere. For the discrete case, we might write

\[OVL(f,g) = \sum_{x\in X} \min\{f(x),g(x)\},\]

with $ f,g $ being PMFs.

Another interesting measure, encountered in the literature on neighborhood segregation, is the dissimilarity index. Consider two populations $ A $ and $ B $, which are distributed over the same area. Suppose the area is divided into cells (e.g., census tracks) which we denote by $ x\in X $. The dissimilarity index is defined as

\[D(f,g) = \frac{1}{2}\sum_{x\in X} \left|f(x)-g(x)\right|,\]

where $ f(x)=n(x)/N $ and $ g(x)=m(x)/M $ with $ N $ and $ M $ being, respectively, the total number of individuals in group $ A $ and $ B $, and $ n(x) $ and $ m(x) $ are the number of individuals from group $ A $ and $ B $ which are found in cell $ x $.

Now, let us go back, for a moment, to the coefficient of overlapping. It is not difficult to see that for two non-negative real numbers $ a,b\in \mathbb R_+ $,

\[\min\{a,b\} = \frac{1}{2}(a+b - |a-b|).\]

So rewriting $ OVL(f,g) $ using this formula, we obtain

\[OVL(f,g) = \int_X \frac{1}{2}\Big(f(x)+g(x)-|f(x)-g(x)|\Big)dx = 1 - \frac{1}{2}\int_X|f(x)-g(x)|dx,\]

with discrete analog

\[OVL(f,g) = 1-\frac{1}{2}\sum_{x\in X}|f(x)-g(x)|.\]

Thus, it turns out that $ D(f,g) = 1-OVL(f,g) $.

Another interesting property of $ OVL(f,g) $ is the following. Note that

\[OVL(f,g) =\int_X \min\{f,g\}dx = \int_X \mathbb I(f<g)dF + \int_X\mathbb I(g<f)dG\]

Thus, we have

\[OVL(f,g) = E_F[\mathbb I(f(X)<g(X))] + E_G[\mathbb I(g(X)<f(X))],\]

where $ E_H(W) $ is the expectation of the random variable $ W $ under the distribution $ H $. Thus, the coefficient of overlapping shows the error rate when we would infer the true distribution of an observed data point $ x $, amongst the two candidate distributions $ f $ and $ g $, based on the decision rule of choosing the distribution with a higher density at each point.