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Journal Article

Second-dimension outliers for spatial prediction

AuthorsKai Ren, Yongze Song*, Qiang Yu*

JournalInternational Journal of Geographical Information Science,

Keywords Spatial outliers Spatial prediction Second-dimension spatial association Agricultural production forecasting
https://doi.org/10.1080/13658816.2025.2580414

Overview

This paper introduces second-dimension outliers (SDO) as a new way to encode the outlier context around unsampled locations and then uses SDO-enhanced machine learning to improve spatial prediction of wheat production across Australia.

Abstract

Spatial prediction aims to accurately estimate attributes at unsampled locations based on spatial dependencies, patterns, variability, and covariates, providing knowledge of complex spatial systems and supporting diverse applications. However, existing methods for spatial prediction ignore geographic and environmental characteristics outside sample locations, particularly spatial outliers, significantly impacting prediction accuracy. This study introduces the concept of second-dimension outliers (SDO) and SDO models that incorporate local outlier information at unsampled locations to enhance prediction accuracy. SDO models generate SDO variables that capture samples' external geographic and environmental characteristics in the spatial prediction. This study develops SDO-based machine learning to predict wheat production in Australia, using cross-validation to evaluate prediction accuracy. Results demonstrate that SDO-based support vector machines (SVM) improve spatial prediction accuracy, with the R2 increasing from 0.555 to 0.671 compared to aspatial SVM, particularly for extreme values. The developed local outlier strength index that examines the strength of SDO ensures more accurate and smooth spatial predictions. The SDO concept provides more in-depth explanatory information from an innovative spatial perspective and a detailed understanding of local outliers for spatial prediction, making it a robust and effective tool for spatial statistical inference and geographic computation across various fields.

Method Implementation

The SDO model improves spatial prediction by extracting local outlier information from explanatory variables around sampled locations and converting that information into second-dimension variables. These variables are then combined with the original covariates in machine-learning models, so outliers are not removed; their local strength is quantified and used as additional spatial context.

  1. Define sampled and unsampled spatial supports

    Let \(u\) denote sampled locations where the dependent variable is observed, and let \(v\) denote unsampled grid locations or surrounding locations outside the dependent-variable samples. The original covariates \(X\) are available over the prediction domain, allowing local outlier information around each sampled location to be extracted from nearby unsampled locations.

  2. Select local buffer ranges

    A set of local ranges is defined around each sample point so that outlier information can be measured at multiple spatial scales:

    \[ b_\alpha,\qquad \alpha = 1,2,\ldots,m \]

    In the simulation experiment, six buffers from 2 to 7 were used. In the wheat production case study, seven buffer distances from 100 km to 700 km at 100-km intervals were used, balancing spatial context, computational efficiency, and statistical reliability.

  3. Identify positive and negative local outliers

    Within each buffer, values at surrounding locations are compared with the local distribution. Positive outliers exceed \(\bar{x}+2\sigma\), while negative outliers fall below \(\bar{x}-2\sigma\). For an explanatory variable, the local outlier vector around location \(u\) is:

    \[ O_s(u),\qquad s=1,2,\ldots,n \]

    Here, \(O_s(u)\) represents outlier values at unsampled locations \(v\) within the buffer around \(u\), and \(n\) is the number of detected outliers in that buffer.

  4. Compute outlier strength indices

    The outlier strength index (OSI) summarises the cumulative magnitude of positive and negative outliers for each explanatory variable and buffer distance:

    \[ \begin{aligned} I^P(X,b_\alpha,v) &= \sum_{s=1}^{m} O^P_s(b_\alpha,v),\\ I^N(X,b_\alpha,v) &= \sum_{t=1}^{n} O^N_t(b_\alpha,v),\\ I(X,b_\alpha,v) &= \left[ I^P(X,b_\alpha,v), I^N(X,b_\alpha,v) \right] \end{aligned} \]

    Each buffer produces one positive and one negative SDO variable for each explanatory variable. In the wheat case, this generated 14 SDO variables per original predictor across the seven buffer distances.

  5. Train SDO-enhanced machine-learning models

    The original covariates and their SDO variables are combined as the model input:

    \[ Y(u) = f\!\left( X,\, I(X,b_\alpha,v) \right) \]

    The function \(f(\cdot)\) can be implemented with machine-learning algorithms such as RF or SVM. In the real-data application, the best-performing model was SDO-SVM with a Gaussian radial basis function kernel:

    \[ K(\vec{x_i},\vec{y_i}) = e^{-\gamma\left\|\vec{x_i}-\vec{y_i}\right\|^2} \]

  6. Validate prediction accuracy and robustness

    The SDO-enhanced models are compared with their aspatial counterparts using five-fold cross-validation and metrics including \(R^2\), RMSE, and MAE. The case study compared SDO-SVM, SDO-RF, SDO-CRM, SDO-XGB, SDO-GBM, SDO-KNN, and SDO-ENR against corresponding aspatial models. Sensitivity analysis then tested buffer threshold, buffer interval, and outlier threshold settings; the preferred case configuration used a 700 km buffer threshold, 100 km interval, and \(\bar{x}\pm2\sigma\) outlier criterion.

Figures and Tables

Figure 1. Schematic overview of second-dimension outliers (SDO) model for examining the relationship between dependent variables and spatial local outliers.
Figure 1. Schematic overview of second-dimension outliers (SDO) model for examining the relationship between dependent variables and spatial local outliers. 图 1. 用于检验因变量与空间异常关系的第二维度空间异常（SDO）模型示意图。
Figure 2. Summary of simulation data including dependent variable (a), explanatory variables (b), and the correlation analysis with them.
Figure 2. Summary of simulation data including dependent variable (a), explanatory variables (b), and the correlation analysis with them. 图 2. 模拟数据概览，包括因变量（a）、解释变量（b）及其相关性分析。
Figure 3. The second-dimension variables of the explanatory variables of the simulated data are generated by the SDO model. The letters p and n mean positive and negative outliers, respectively. The letters b2, b3, ..., b7 mean the buffers with sizes varying from 2 to 7.
Figure 3. The second-dimension variables of the explanatory variables of the simulated data are generated by the SDO model. The letters p and n mean positive and negative outliers, respectively. The letters b2, b3, ..., b7 mean the buffers with sizes varying from 2 to 7. 图 3. 模拟数据解释变量的第二维度变量由 SDO 模型生成。字母 p 和 n 分别表示正异常和负异常；b2、b3、...、b7 表示大小从 2 到 7 的缓冲区。
Figure 4. Results of SDO modeling on simulation data using random forest (RF) and support vector machine model (SVM). (a) SDO prediction result using RF, (b) aspatial prediction results based RF and SVM, and SDO-SVM prediction, (c) Importance of the individual variables for the aspatial (upper) and SDO (lower) RF models.
Figure 4. Results of SDO modeling on simulation data using random forest (RF) and support vector machine model (SVM). (a) SDO prediction result using RF, (b) aspatial prediction results based RF and SVM, and SDO-SVM prediction, (c) Importance of the individual variables for the aspatial (upper) and SDO (lower) RF models. 图 4. 基于随机森林（RF）和支持向量机（SVM）的模拟数据 SDO 建模结果。(a) 基于 RF 的 SDO 预测结果；(b) 基于 RF 和 SVM 的非空间预测结果以及 SDO-SVM 预测结果；(c) 非空间 RF 模型（上）与 SDO-RF 模型（下）的变量重要性。
Figure 5. Annual wheat production in Australia's LGAs within the wheat belt in 2021.
Figure 5. Annual wheat production in Australia's LGAs within the wheat belt in 2021. 图 5. 2021 年澳大利亚小麦带内各地方政府区域（LGA）的小麦年产量。
Figure 6. Spatial distributions of explanatory variables in 2021. (a) and (b) Climate variables, (c)-(e) Environmental variables, (f)-(i) Soil attributes.
Figure 6. Spatial distributions of explanatory variables in 2021. (a) and (b) Climate variables, (c)-(e) Environmental variables, (f)-(i) Soil attributes. 图 6. 2021 年解释变量的空间分布。(a) 和 (b) 为气候变量，(c)-(e) 为环境变量，(f)-(i) 为土壤属性。
Table 1. A summary of explanatory variables that potentially affect spatial disparities of wheat production.
Table 1. A summary of explanatory variables that potentially affect spatial disparities of wheat production. 表 1. 可能影响小麦产量空间差异的解释变量汇总。
Figure 7. Main steps of the second-dimension outliers (SDO) model for predicting wheat production in Australia.
Figure 7. Main steps of the second-dimension outliers (SDO) model for predicting wheat production in Australia. 图 7. 基于第二维度空间异常（SDO）模型预测澳大利亚小麦产量的主要步骤。
Figure 8. Correlation analysis for wheat production and explanatory variables, including air temperature, evapotranspiration, EVI, NDVI, total precipitation, total nitrogen, total phosphorus, silt, sand, and wind speed.
Figure 8. Correlation analysis for wheat production and explanatory variables, including air temperature, evapotranspiration, EVI, NDVI, total precipitation, total nitrogen, total phosphorus, silt, sand, and wind speed. 图 8. 小麦产量与解释变量的相关性分析，包括气温、蒸散发、EVI、NDVI、总降水量、总氮、总磷、粉砂、砂和风速。
Figure 9. The distribution of the second-dimension temperature variable generated by the SDO model under different buffers, which includes positive outliers (red) and negative outliers (blue).
Figure 9. The distribution of the second-dimension temperature variable generated by the SDO model under different buffers, which includes positive outliers (red) and negative outliers (blue). 图 9. SDO 模型在不同缓冲区下生成的第二维度温度变量分布，其中红色表示正异常，蓝色表示负异常。
Figure 10. Spatial distributions of second-dimension outlier variables for total precipitation (TP), normalized difference vegetation index (NDVI), and sand (SND) at buffer distances 100, 400, and 700 km, generated using the SDO model.
Figure 10. Spatial distributions of second-dimension outlier variables for total precipitation (TP), normalized difference vegetation index (NDVI), and sand (SND) at buffer distances 100, 400, and 700 km, generated using the SDO model. 图 10. 由 SDO 模型生成的总降水量（TP）、归一化植被指数（NDVI）和砂（SND）在 100、400 和 700 km 缓冲距离下的第二维度空间异常变量分布。
Table 2. Improvements of model accuracy in machine learning by SDO models compared with aspatial models (bold values indicating the best-performing SDO results).
Table 2. Improvements of model accuracy in machine learning by SDO models compared with aspatial models (bold values indicating the best-performing SDO results). 表 2. 与非空间模型相比，SDO 模型在机器学习中的精度提升（加粗值表示表现最优的 SDO 结果）。
Figure 11. Sensitivity analysis of the SDO model: impact of buffer threshold, buffer interval, and outlier (standard deviation) threshold settings on model accuracy (cross-validation root mean square error, CV-RMSE).
Figure 11. Sensitivity analysis of the SDO model: impact of buffer threshold, buffer interval, and outlier (standard deviation) threshold settings on model accuracy (cross-validation root mean square error, CV-RMSE). 图 11. SDO 模型敏感性分析：缓冲阈值、缓冲间隔和空间异常（标准差）阈值设置对模型精度（交叉验证均方根误差，CV-RMSE）的影响。
Figure 12. Prediction of average wheat production in Australia based on the aspatial model (a) and SDO machine learning model (c). The importance of top 30 SDO variables (d) and aspatial variables (b) in the machine learning model.
Figure 12. Prediction of average wheat production in Australia based on the aspatial model (a) and SDO machine learning model (c). The importance of top 30 SDO variables (d) and aspatial variables (b) in the machine learning model. 图 12. 基于非空间模型（a）和 SDO 机器学习模型（c）的澳大利亚平均小麦产量预测结果，以及机器学习模型中前 30 个 SDO 变量（d）和非空间变量（b）的重要性。
Figure 13. Comparison of SDO and aspatial model predictions along cross-sections in the wheat belt region of New South Wales. (a) Locations of section A (southwest to northeast) and section B (northwest to southeast) in the wheat belt region of New South Wales. (b, c) Predicted wheat production along sections A and B, comparing SDO-SVM and aspatial SVM models. (d) Original wheat production data with cross-sections overlaid. (e, f) Predicted wheat production from the SDO-SVM model (e) and the aspatial SVM model (f), with overlaid cross-sections.
Figure 13. Comparison of SDO and aspatial model predictions along cross-sections in the wheat belt region of New South Wales. (a) Locations of section A (southwest to northeast) and section B (northwest to southeast) in the wheat belt region of New South Wales. (b, c) Predicted wheat production along sections A and B, comparing SDO-SVM and aspatial SVM models. (d) Original wheat production data with cross-sections overlaid. (e, f) Predicted wheat production from the SDO-SVM model (e) and the aspatial SVM model (f), with overlaid cross-sections. 图 13. 新南威尔士州小麦带区域沿剖面的 SDO 模型与非空间模型预测结果比较。(a) 新南威尔士州小麦带内剖面 A（西南至东北）和剖面 B（西北至东南）的位置；(b, c) 沿剖面 A 和 B 的小麦产量预测，对比 SDO-SVM 与非空间 SVM 模型；(d) 叠加剖线的原始小麦产量数据；(e, f) 分别为 SDO-SVM 模型（e）和非空间 SVM 模型（f）的预测结果，并叠加剖线。
Figure 14. Density distribution map of the average wheat production prediction results of the SDO and aspatial models in Australia and each state.
Figure 14. Density distribution map of the average wheat production prediction results of the SDO and aspatial models in Australia and each state. 图 14. 澳大利亚整体及各州范围内 SDO 模型与非空间模型平均小麦产量预测结果的密度分布图。

Data and Codes Availability Statement

The data and codes that support the findings of this study are available through both Figshare and GitHub.

Funding

This research was supported by the China Scholarship Council (Grant No. 202206300058).