# Fitting a von Mises Distribution over the Global Population

For a statistics course I'm taking this semester, we decided to conduct a worldwide survey. One of our questions, is an approximate coordinate of where the respondent lives. Our instructor recently introduced me to the circular von Mises distribution and it has been fun seeing how we can wrap a normal distribution around a circle or sphere.

But something I have been wondering is, if it is possible to fit a von Mises distribution, or a mixture thereof, on the global population. The tricky thing is that population statistics are usually recorded on a per-country basis, and projected onto 2D Mercator-Projection maps. So the first problem to solve is finding adequate "samples".

Something that I thought of, would be to derive a sample from each country, and to center these samples on the "center of gravity" for said country's population (ex. the most populous city or capital).

Ex. To represent Singapore: $$X_i = [1°19′24.96″N, 103°55′38.42″E],\ Y_i=545$$ Million, for some index $$i$$, where $$X_i$$ are the coordinates for Bedok, the most populous town. Coordinates $$X_i$$ can be transformed into $$[1.81 \text{rad}, 0.0231\text{rad}]$$ to represent them as radians instead.

Has no one done anything like this before? Is it just too impractical, or tedious? And at the very least, is there adequate justification to use this "center of gravity" approach to sample each country as a population center? (Would it make more sense to work on a city/town's population rather than overloading the entire country's population on one point?)

• the population distribution is highly multimodal (most people live in big cities) so it wouldn't make sense to fit a single von mises. it might be better to instead fit a separate distribution to each city of sufficient size; but then you might as well use a normal rather than a von mises, because you are dealing with spatial scales where the topology of the earth is a negligible consideration. Jan 12, 2022 at 21:41
• Yeah, after looking at population distribution datasets like World Pop and GSFC, I have also concluded that one-size fitting all is unfeasible, a mixture is certainly needed. And that so-called "negligibility" has been eating at me from the back of my mind for about a month now. Human settlement won't loop around the Poles, that's for sure. But West-to-East migration patterns are a whole other story. The Pacific does stand out as an anomaly which can't be represented accurately with a Normal Distribution, at least without playing with the projection of the earth. Jan 13, 2022 at 4:17
• Using a Von Mises distribution here or there, might not be the only solution; but somehow a concept of angularness is relevant at some point. Maybe a layered model; with Normal Distributions around clusters of big city population centers could be the basis, and a series of mixed-in Von Mises distr.'s could function as a regulariser. But more than anything right now, I want to see if anyone else has tried something on this scale before. Because this feels like a problem that should not be uncharted territory. Jan 13, 2022 at 4:20
• could you say a bit more about what exactly your goals are in doing this fit, and how you envision it differing from existing population density maps such as luminocity3d.org/WorldPopDen/#3/19.23/35.42 ? Jan 17, 2022 at 4:59
• Well simple, grid-maps are only intuitive to look at, but not very descriptive. f.x. To describe where in the world population is concentrated, you have to list arbitrary geographic regions (which btw, different countries label differently). Similar for urbanization rates. It's un-systemic, and that limits our ability to generalize local observations to global trends. The next best thing, I guess, is to work with a time series of grid-maps, because that at least gives some sense of momentum for flows and growths of population. That seems to be the preferred approach if Covid has shown Jan 17, 2022 at 8:01

Fitting the distribution is not so difficult. For the sphere you can use the von Mises-Fisher distribution) which expresses the probability in terms of the 3d Cartesian coordinates and the estimation of the parameters (when using maximum likelihood) is not so difficult.

Has no one done anything like this before? Is it just too impractical, or tedious?

I have come across descriptions of population as a function of latitude and longitude. I don't know a good image with the proper usage rights for sharing here, but you can see many in a google search.

The mean of those two (when you apply the summation in Cartesian coordinates) would coincide with the estimate of the mean for your von Mises distribution.

The variance might also make some sense for the distribution in the latitude, which has several noisy peaks but is roughly distributed in a single bulge. For the longitude there are however several peaks and the distribution is multimodal. The variance might not describe the entire distribution very well.

This also means that you might wish to use the Kent distribution, which is a generalization that allows a different degree of spread in different directions.

And at the very least, is there adequate justification to use this "center of gravity" approach to sample each country as a population center?

Using the center of gravity would be like having more coarse observations. You could do it but it would be less precise. It is the same for non-spherical distributions.

For the statistics course you might do the exercise by taking the data from counties population and center of gravities. And compare it with the result from more fine grained distributions/datasets that are available.

Using the data for the marginal distribution of latitude and longitude you should be able to fit the von Mises-Fisher distribution, but to fit the Kent distribution you would need to have the joint distribution.