Fuzzy geographical detector: enhancing spatial attribution analysis for continuous geographical variables

The geographical detector (GD) is a widely applied method that uses the q-statistic to assess the explanatory power of variables and identify their interactions. However, GD relies on deterministic spatial stratification, which ignores the fine-grained variation within strata and the gradual transitions between strata, resulting in information loss. While some enhanced methods have incorporated fuzzy spatial stratification, they still assume that each spatial unit belongs to a single stratum, limiting their ability to capture multiple memberships and potentially biasing the detection results. Therefore, we propose a fuzzy geographical detector (FGD) to improve spatial attribution and interaction analysis. For spatial attribution, FGD evaluates the explanatory power using the coefficient of determination, with the within-stratum sum of squares used in GD replaced by the sum of squared errors from a membership-weighted reconstruction, thereby capturing both within-stratum variation and gradual transitions between strata. For interaction analysis, FGD replaces spatial overlays with fuzzy intersections to detect fuzzy interactions among explanatory variables while retaining the interaction criteria of GD. Simulation experiments showed that FGD produced more accurate and stable attribution results than the comparison methods, while the empirical case study illustrated its ability to represent within-stratum variation and fuzzy boundaries.