Smooth Pycnophylactic Interpolation Produced by Density-Equalising Map Projections

TL;DR

This paper introduces a computationally efficient method for smooth pycnophylactic interpolation using density-equalising map projections, providing an alternative to cellular automaton algorithms for geospatial data smoothing.

Contribution

The paper presents a software implementation of density-equalising map projections for pycnophylactic interpolation, offering a potentially faster alternative to existing methods.

Findings

Method is computationally efficient

Can be combined with other methods for improved results

Provides a non-automaton approach to smoothing

Abstract

A large amount of quantitative geospatial data are collected and aggregated in discrete enumeration units (e.g. countries or states). Smooth pycnophylactic interpolation aims to find a smooth, nonnegative function such that the area integral over each enumeration unit is equal to the aggregated data. Conventionally, smooth pycnophylactic interpolation is achieved by a cellular automaton algorithm that converts a piecewise constant function into an approximately smooth function defined on a grid of coordinates on an equal-area map. An alternative approach, proposed by Tobler in 1976, is to construct a density-equalising map projection in which areas of enumeration units are proportional to the aggregated data. A pycnophylactic interpolation can be obtained from the Jacobian of this projection. Here, we describe a software implementation of this method. Although solutions are not necessarily optimal in terms of predefined quantitative measures of smoothness, our method is computationally efficient and can potentially be used in tandem with other methods to accelerate convergence towards an optimal solution.