Area-to-point Kriging with inequality-type data

Abstract

In practical applications of area-to-point spatial interpolation, inequality constraints, such as non-negativity, or more general constraints on the maximum and/or minimum allowable value of the resulting predictions, should be taken into account. The geostatistical framework proposed in this paper deals with area-to-point interpolation problems under such constraints, while: (i) explicitly accounting for support differences between sample data and unknown values, (ii) guaranteeing coherent predictions, and (iii) providing a measure of reliability for the resulting predictions. The analogy between the dual form of area-to-point interpolation and a spline allows to solve constrained area-to-point interpolation problems via a constrained quadratic minimization algorithm, after accounting for the following three issues: (i) equality and inequality constraints could be applied to different supports, and such support differences should be considered explicitly in the problem formulation, (ii) if inequality constraints are enforced on the entire set of points discretizing the areal data, it is impossible to obtain a solution of the quadratic programming problem, and (iii) the uniqueness and existence of the solution has to be diagnosed. In this work, stable and efficient computation of point predictions is achieved through the following two steps: (i) initial prediction at all locations via unconstrained area-to-point interpolation, and (ii) constrained area-to-point interpolation with inequality information only at those points whose initial predicted values violate the inequality constraints. Last, the application of the proposed method to area-to-point spatial interpolation with inequality constraints in one and two dimensions is demonstrated using realistically simulated data.

Dasymetric Modeling and Uncertainty

“…The log-linear assumption is convenient because it is guaranteed to produce non-negative population estimates. This is not true, for example, with common geostatistical techniques, where ad hoc fixes are sometimes needed to enforce non-negativity (Yoo and Kyriakidis 2006). Additionally, geostatistical models are tailored to specific sub-populations; there is no guarantee, for instance, that the spatial structure of low-income Black households is the same as the spatial structure of other household types.…”

Geostatistical Smoothing of Areal Data: Mapping Employment Density with Factorial Kriging. 面状数据的地统计滤波：采用因子克里格方法制作就业密度图

“…The possibility of negative predictions is one practical limitation of the factorial kriging method for the application here. One solution might be the inequality constrained kriging approach developed by Yoo and Kyriakidis (2006). That method, however, does not easily handle global non‐negativity constraints, that is, non‐negativity constraints at all locations simultaneously.…”

Reconstructing Population Density Surfaces from Areal Data: A Comparison of Tobler's Pycnophylactic Interpolation Method and Area‐to‐Point Kriging. 面状数据的人口密度面重构：Tobler’s Pycnophylactic 插值法和面到点克里金插值法的对比

“…In the geostatistical framework, inequality and equality constraints can be accounted for by ATP kriging. Yoo and Kyriakidis (2006), for example, applied quadratic programming algorithms to ATP kriging with various types of inequality constraints, while accounting for support differences between source data and target predictions and satisfying the pycnophylactic constraint.…”