Area-to-point parameter estimation with geographically weighted regression

Abstract

The modifiable areal unit problem (MAUP) is a problem by which aggregated units of data influence the results of spatial data analysis. Standard GWR, which ignores aggregation mechanisms, cannot be considered to serve as an efficient countermeasure of MAUP. Accordingly, this study proposes a type of GWR with aggregation mechanisms, termed area-to-point (ATP) GWR herein. ATP GWR, which is closely related to geostatistical approaches, estimates the disaggregate-level local trend parameters by using aggregated variables. We examine the effectiveness of ATP GWR for mitigating MAUP through a simulation study and an empirical study. The simulation study indicates that the method proposed herein is robust to the MAUP when the spatial scales of aggregation are not too global compared with the scale of the underlying spatial variations. The empirical studies demonstrate that the method provides intuitively consistent estimates.

Appendix 1: ATP GWR for the extensive variables

A must be determined by considering the volume-preserving property for the extensive variables. In other words, ${\bar{\mathbf{y}}}$ = Ay must hold under the condition that the elements in y are extensive variables. For example, suppose that the population in an aggregated unit a is $\bar{y}_{a}$ and unit a comprises two disaggregated units d and d′; then, ${\bar{\mathbf{y}}}$ = Ay is expressed as

$$ \begin{aligned} & \bar{y}_{a} = \left[ {\begin{array}{*{20}l} {A_{a,d} } \hfill & {A_{{a,d^{\prime}}} } \hfill \\ \end{array} } \right]\left[ {\begin{array}{*{20}l} {Y_{d} } \hfill \\ {Y_{{d^{\prime}}} } \hfill \\ \end{array} } \right], \\ & Y_{d} + Y_{{d^{{\prime }} }} = A_{a,d} Y_{d} + A_{{a,d^{{\prime }} }} Y_{{d^{{\prime }} }} . \\ \end{aligned} $$

(18)

Equation (6) merely states that $\bar{y}_{a} = Y_{d} + Y_{{d^{{\prime }} }}$ must equal A _a,d Y _d  + A _a,d′ Y _d′. This is satisfied by defining A _d and A _d′ as ones. In general, ${\bar{\mathbf{y}}}$ = Ay is fulfilled by defining A _a,d as follows:

$$ A_{a,d} = \left\{ {\begin{array}{*{20}l} 1 \hfill & {{\text{if}}\;d \subseteq a} \hfill \\ 0 \hfill & {\text{otherwise}} \hfill \\ \end{array} } \right.. $$

(19)

ATP GWR for the extensive variables is thus defined by defining A _a,d , which was defined in Eq. (7), as shown in Eq. (19). In this case, the variance of the error term is scaled by the number of disaggregated units in each aggregated unit.