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The calculation formula of geographical concentration is as follows:

$$G=100\times\sqrt{\sum_{i=1}^n\left(\frac{n_i}{N}\right)^2}$$

The explanation of the formula is something I don't understand:

In the formula, $G$ represents the concentration of a matter, through the calculation of concentration $G$ can fully reflect the concentration of the surname in any period; you represent the number of individuals occupied by the $i$-th marker; $N$ is the total of individuals. $G$ value is greater than or equal to $0$, $G$ value tends to $0$ indicating that the source of the item is more dispersed; The greater the value of $G$, the greater the concentration of the source of the item.

What is $n_i$, explained simply in relation to the statistics for individuals with a particular surname? Are there multiple $n_i$ in the summation, and if so, how does that affect the calculations for the geographic concentration?

COOLSerdash
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LÜHECCHEgon
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1 Answers1

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From your description you must have $N = \sum_i n_i$ so that the ratios $p_i \equiv n_i/N$ represent the proportions of individuals falling within each marker $i$. Consequently, your function $G$ is a scaled version of the Euclidean norm of the proportion vector $(p_1,...,p_n)$. Now, the Euclidean norm is a strictly convex function of the input vector, so this means that:

$$G(\lambda \mathbf{p} + (1-\lambda) \mathbf{p}') < \lambda G(\mathbf{p}) + (1-\lambda) G(\mathbf{p}'),$$

for any proportion value $0 < \lambda < 1$. Moreover, it can easily be shown that the function is maximised when all individuals occupy the same marker, and minimised when individuals are equally distributed over all the markers. These properties validate the idea that the function $G$ is giving you a measure of the concentration of individuals over the markers.

(From Nick Cox in comments: Such measures have a history at least a century long. Gini, Friedman, Turing. Yule, Hirschman, Simpson, Herfindahl, Good, Greenberg and several others all used some variant of the idea that given proportions $p_i$ then $\sum p_i^2$ measures homogeneity (as this answer implies, if everything is in one category, it equals 1) and thus that its complement or reciprocal measures heterogeneity, diversity, inequality, or whatever you want to call it. Taking the square root or multiplying by 100 don't change the basic idea.)

Ben
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    Such measures have a history at least a century long. Gini, Friedman, Turing. Yule, Hirschman, Simpson, Herfindahl, Good, Greenberg and several others all used some variant of the idea that given proportions $p_i$ then $\sum p_i^2$ measures homogeneity (as @Ben implies, if everything is in one category, it equals 1) and thus that its complement or reciprocal measures heterogeneity, diversity, inequality, or whatever you want to call it. Taking the square root or multiplying by 100 don't change the basic idea. – Nick Cox Aug 22 '21 at 09:56
  • Could there be a non-convex function that measures homogeneity, or is the convexity essential? – eric_kernfeld Aug 22 '21 at 11:51
  • @eric Convexity is not essential. It makes it easy to find global extrema, though. (A simple and potentially practical example of a non-convex function is one that is capped at some value. A cap might be imposed to make the measure more robust, for instance.) – whuber Aug 22 '21 at 14:38