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Journal Article

Spatial outliers as a pattern determinant for explaining heterogeneity

AuthorsKai Ren*, Yongze Song*, Xinyue Yang, Xi Wang, Min Chen, Qiang Yu

JournalInternational Journal of Geographical Information Science,

Keywords Spatial heterogeneity Spatial outlier patterns Second-dimension outliers Geographical detector Stratification-based analysis
https://doi.org/10.1080/13658816.2026.2682957

Overview

This paper introduces a second-dimension outlier-driven heterogeneity (SOH) model for explaining spatial heterogeneity with local outlier configurations. The model derives multi-scale spatial outlier patterns (SOPs), embeds them in a stratification-detection workflow, and evaluates individual, interaction, and scale-dependent effects in Australian barley production.

Abstract

Explaining spatial heterogeneity typically relies on first-dimension covariates that describe spatial gradients. However, many geographic phenomena also exhibit irregular and locally extreme structures that are difficult to capture using smooth relationships. This study proposed a second-dimension outlier-driven heterogeneity (SOH) model, in which the first dimension refers to covariate variation across space, while the second dimension represents spatial pattern information captured from local outlier configurations. SOH derives multi-scale spatial outlier patterns (SOPs) using a second-dimension outlier model and embeds them in a stratification-detection workflow, where decision tree-based stratification defines strata and the geographical detector evaluates explanatory power via the power of determinant (PD). The model supports evaluation of individual effects, SOP interactions, and SOP-variable interactions, with assessment of scale dependence across neighbourhood buffers. Application of this model to spatial heterogeneity in Australian barley production showed that SOPs strengthened heterogeneity explanation relative to original variables, and that SOP interactions and SOP-variable interactions yielded synergistic gains in PD. A scale threshold around 200 km was identified, beyond which SOP-only models approached the explanatory performance of combined models, indicating that multi-scale SOPs captured broad spatial context. Overall, SOH provides a unified approach for incorporating outlier-driven spatial patterns into spatial heterogeneity analysis.

Method Implementation

SOH explains spatial heterogeneity by adding a second dimension of spatial pattern information to conventional covariates. The first dimension is the original covariate variation across space, while the second dimension is a set of multi-scale spatial outlier patterns (SOPs) derived from local outlier configurations.

\[ X = \{x_i\}_{i=1}^{n}, \qquad Y = \{y_i\}_{i=1}^{n} \]

  1. Prepare spatial variables on a common support

    The response variable \(Y\) and explanatory variables \(X\) are first aligned on the same spatial units so that covariate values, SOPs, and heterogeneity statistics can be compared consistently. In the barley case study, the response was SA2-level barley production normalised by polygon area, and environmental predictors were harmonised to a common coordinate system and analysis grid before being aggregated to the modelling support.

  2. Generate multi-scale spatial outlier patterns

    For each explanatory variable \(X\), SOH uses the second-dimension outlier model to identify positive and negative local deviations relative to neighbourhood distributions. Across a set of buffer radii \(R=\{r_1,r_2,\ldots,r_k\}\), the spatial pattern variable \(\Psi\) is constructed as:

    \[ \Psi = \bigcup_{r \in R} \left\{ \sum_{v \in \mathcal{N}_r(u)} O^{+}(X,v)\,\mathbb{I}\!\left(O^{+}(X,v)>\tau\right), \sum_{v \in \mathcal{N}_r(u)} O^{-}(X,v)\,\mathbb{I}\!\left(O^{-}(X,v)<-\tau\right) \right\} \]

    Here, \(\mathcal{N}_r(u)\) is the neighbourhood around target spatial unit \(u\), \(O^{+}\) and \(O^{-}\) are positive and negative outlier components, and \(\tau\) controls the threshold for retaining pronounced local deviations. In the case study, positive and negative outliers were defined using the \(\bar{x}\pm2\sigma\) criterion, and SOPs were generated from 20 km to 200 km at 20-km intervals.

  3. Build SOP variables for each spatial unit

    For each spatial unit \(i\), positive and negative outlier summaries are retained at each neighbourhood scale to form the second-dimension pattern set:

    \[ \Psi_i = \left\{ O^{(r_1)}_{+,i}, O^{(r_1)}_{-,i}, O^{(r_2)}_{+,i}, O^{(r_2)}_{-,i}, \ldots, O^{(r_k)}_{+,i}, O^{(r_k)}_{-,i} \right\} \]

    These SOP variables describe where an explanatory factor is locally enriched, depleted, or spatially unusual relative to its surrounding context. They are used as explanatory pattern descriptors rather than as residuals or noise filters.

  4. Transform predictors into data-adaptive strata

    SOH uses a stratification-detection strategy. A CART regression tree converts an original variable, an SOP variable, or their combination into categorical strata:

    \[ Z^{(\Psi)} = \{z_i^{(\Psi)}\}_{i=1}^{n}, \qquad z_i^{(\Psi)} \in \{1,2,\ldots,L\} \]

    In the implementation, CART was fitted with the R package rpart using \(cp=0.01\), and the number of strata \(L\) was determined adaptively by the terminal nodes of the fitted tree. The tree is used here as a stratification tool, not as an out-of-sample prediction model.

  5. Measure explanatory power with the geographical detector

    The geographical detector evaluates whether the strata induced by \(\Psi\) explain spatial heterogeneity in \(Y\). The power of determinant (PD) is:

    \[ PD = \Omega_{\Psi} = 1 - \frac{ \sum_{h=1}^{L} N_h^{(\Psi)}\sigma_h^{2(\Psi)} }{ N^{(\Psi)}\sigma^{2(\Psi)} } \]

    A larger PD indicates that the constructed strata produce stronger between-stratum differentiation and lower within-stratum variance, meaning stronger explanatory power for spatial heterogeneity.

  6. Evaluate original variables, SOPs, interactions, and scale effects

    PD is calculated under several predictor configurations: original covariates \(X\), SOP variables \(\Psi\), and combined sets \(X \cup \Psi\). The same framework is then used to test pairwise SOP interactions, interactions between SOPs and original variables, category-level interactions, and the sensitivity of PD to neighbourhood buffer size. In the paper, this workflow showed that SOPs added explanatory power beyond original variables and that the SOH response stabilised around a neighbourhood threshold of approximately 200 km.

Figures and Tables

Figure 1. Conceptual framework of the second-dimension outlier-driven heterogeneity (SOH) model.
Figure 1. Conceptual framework of the second-dimension outlier-driven heterogeneity (SOH) model. 图 1. 第二维度空间异常驱动异质性（SOH）模型的概念框架。
Figure 2. Spatial distribution of barley production normalised by SA2 polygon area across Australia and the study area.
Figure 2. Spatial distribution of barley production normalised by SA2 polygon area (ton/km2) across Australia and the study area. The blue boundary delineates the study area analysed in this study. 图 2. 澳大利亚及研究区内按 SA2 多边形面积归一化的大麦产量空间分布。蓝色边界表示本研究分析的研究区。
Table 1. A summary of explanatory variables that potentially affect spatial disparities of barley production.
Table 1. A summary of explanatory variables that potentially affect spatial disparities of barley production. 表 1. 可能影响大麦产量空间差异的解释变量汇总。
Figure 3. Spatial distribution of explanatory variables for barley production in southeastern Australia.
Figure 3. Spatial distribution of explanatory variables for barley production in southeastern Australia. (a)-(c) Climate: air temperature (AT), total precipitation (TP), and vapour pressure deficit (VPD); (d) Topography: elevation (ELE); (e)-(i) Soil properties: soil organic carbon (SOC), total nitrogen (NTO), total phosphorus (PTO), soil pH (CaCl2) (pHc), and available water capacity (AWC); (j)-(l) Vegetation: normalized difference vegetation index (NDVI), net primary production (NPP), and evapotranspiration (ETa). 图 3. 澳大利亚东南部大麦产量解释变量的空间分布。
Figure 4. Schematic workflow of SOH modelling for spatial heterogeneity analysis.
Figure 4. Schematic workflow of SOH modelling for spatial heterogeneity analysis. 图 4. 用于空间异质性分析的 SOH 建模流程示意图。
Figure 5. Global Moran's I diagnostic for SA2-level barley production.
Figure 5. Global Moran's I diagnostic for SA2-level barley production. (a) Moran scatter plot showing the relationship between standardised barley production and its spatial lag. (b) Reference distribution based on 999 random permutations, indicating significant positive spatial autocorrelation in the response variable. 图 5. SA2 尺度大麦产量的全局 Moran's I 诊断。
Figure 6. Spatial distribution of second-dimension outlier patterns for the three explanatory variables with the highest PD values.
Figure 6. Spatial distribution of second-dimension outlier patterns (SOPs) for the three explanatory variables with the highest PD values in barley production spatial heterogeneity. 图 6. 大麦产量空间异质性中 PD 值最高的三个解释变量对应的第二维度空间异常模式（SOPs）空间分布。
Figure 7. PD values of individual original variables and their corresponding spatial outlier patterns.
Figure 7. PD values of individual original variables and their corresponding spatial outlier patterns (SOPs) for explaining spatial heterogeneity in barley production. (a) shows PD values of individual original variables, while (b) presents PD values of the corresponding individual SOPs. 图 7. 单个原始变量及其对应空间异常模式（SOPs）解释大麦产量空间异质性的 PD 值。
Table 2. Top 10 pairwise interactions among spatial outlier patterns ranked by PD.
Table 2. Top 10 pairwise interactions among spatial outlier patterns (SOPs) ranked by power of determinant (PD). 表 2. 按解释力（PD）排序的空间异常模式（SOPs）前 10 个两两交互。
Figure 8. PD of interactions among SOPs and between SOPs and original spatial variables.
Figure 8. Power of determinant (PD) of interactions among spatial outlier patterns (SOPs) and between SOPs and original spatial variables for explaining spatial heterogeneity in barley production. (a) shows PD values of pairwise interactions among individual SOPs, (b) presents PD values of interactions among SOP categories, (c) illustrates PD values of pairwise interactions between individual original variables and SOPs, and (d) displays PD values of interactions between original variable categories and SOP categories. 图 8. SOPs 之间以及 SOPs 与原始空间变量之间交互作用解释大麦产量空间异质性的 PD 值。
Table 3. Top 30 interactions between original spatial variables and spatial outlier patterns ranked by PD.
Table 3. Top 30 interactions between original spatial variables and spatial outlier patterns (SOPs) ranked by power of determinant (PD). 表 3. 按解释力（PD）排序的原始空间变量与空间异常模式（SOPs）前 30 个交互。
Figure 9. Comparison of PD values between original variables and SOP-enhanced counterparts.
Figure 9. Comparison of power of determinant (PD) values between original variables (or categories) and their SOP-enhanced counterparts. (a) PD values of individual original variables and corresponding Variable+SOP models. (b) PD differences between Variable+SOP and original Variable models, highlighting the contribution of SOPs. (c) PD values of variable categories with and without SOP integration. (d) PD differences between Category+SOP and original Category models. 图 9. 原始变量（或类别）与其 SOP 增强对应模型之间的 PD 值比较。
Figure 10. Overall performance and scale sensitivity of the SOH model.
Figure 10. Overall performance and scale sensitivity of the second-dimension outlier-driven heterogeneity (SOH) model. (a) Comparison of PD values for three scenarios: all SOP variables (A1), all SOP variables combined with original variables (A2), and all original variables alone (A3). (b) Variation of PD values across neighbourhood buffer radius for original variables, SOP-only models, and combined Original+SOP models. 图 10. 第二维度空间异常驱动异质性（SOH）模型的整体表现与尺度敏感性。

Data and Code Availability

The data and code supporting this study are publicly available through both Figshare and GitHub.

Funding

This research was supported by the National Natural Science Foundation of China (Outstanding Young Scholars Program, Grant No. 42325107).