The Importance of Scale in Spatially Varying Coefficient Modeling

Abstract

Although spatially varying coefficient (SVC) models have attracted considerable attention in applied science, they have been criticized as being unstable. The objective of this study is to show that capturing the “spatial scale” of each data relationship is crucially important to make SVC modeling more stable and, in doing so, adds flexibility. Here, the analytical properties of six SVC models are summarized in terms of their characterization of scale. Models are examined through a series of Monte Carlo simulation experiments to assess the extent to which spatial scale influences model stability and the accuracy of their SVC estimates. The following models are studied: (1) geographically weighted regression (GWR) with a fixed distance or (2) an adaptive distance bandwidth (GWRa); (3) flexible bandwidth GWR (FB-GWR) with fixed distance or (4) adaptive distance bandwidths (FB-GWRa); (5) eigenvector spatial filtering (ESF); and (6) random effects ESF (RE-ESF). Results reveal that the SVC models designed to capture scale dependencies in local relationships (FB-GWR, FB-GWRa, and RE-ESF) most accurately estimate the simulated SVCs, where RE-ESF is the most computationally efficient. Conversely, GWR and ESF, where SVC estimates are naïvely assumed to operate at the same spatial scale for each relationship, perform poorly. Results also confirm that the adaptive bandwidth GWR models (GWRa and FB-GWRa) are superior to their fixed bandwidth counterparts (GWR and FB-GWR). Key Words: flexible bandwidth geographically weighted regression, Monte Carlo simulation, nonstationarity, random effects eigenvector spatial filtering, spatial scale.

尽管空间变异係数（SVC）模型已吸引了应用科学的大量关注, 但却仍被批评不够平稳。本研究的目标在于展现, 捕捉各数据关系的“空间尺度”, 是让SVC模式化更为平稳的重要关键, 这麽做并可增加弹性。本研究于此摘要六大SVC模型在尺度特徵化上的分析属性。本研究通过一系列的蒙特卡罗模拟实验检视模型, 以评估空间尺度影响模型稳定度的程度, 及其SVC评估的精确度。本文研究下列模型：（1）距离固定下的地理加权迴归（GWR）；（2）自适应距离的带宽（GWRa）；（3）固定距离下的弹性带宽GWR（FB-GWR）；或（4）自适应距离带宽（FB-GWRa）；（5）特徵质空间过滤法（ESF）；以及（6）随机效应 ESF （RE-ESF）。研究结果揭露, 设计用来捕捉地方关系中的尺度依赖之SVC模型（FB-GWR、FB-GWRa和RE-ESF）, 最正确地估计模拟的SVCs, 而REESF则在计算上最有效率。反之, SVC估计被天真地假设在各关系中皆在相同的空间尺度上操作的GWR和ESF表现差劲。研究结果同时确认自适应带宽GWR模型（GWRa和FB-GWRa）较其固定带宽的对照组（GWR和FB-GWR）而言更为优越。 关键词： 弹性带宽地理加权迴归, 蒙特卡罗模拟, 非平稳性, 随机效应特徵质空间过滤法, 空间尺度。

Aunque los modelos de coeficiente espacialmente cambiante (SVC) han atraído mucha atención en ciencia aplicada, se les critica de ser inestables. El objetivo del presente estudio es mostrar que la captura de la “escala espacial” de cada relación de datos es crucialmente importante para hacer el modelado del SVC más estable y de ese modo agregarle flexibilidad. Se resumen en este artículo las propiedades analíticas de seis modelos de SVC en términos de su caracterización por escala. Los modelos son examinados a través de una serie de experimentos de simulación de Monte Carlo para evaluar el alcance con que la escala espacial influye sobre la estabilidad del modelo y en la exactitud de las estimaciones SVC. Se estudian los siguientes modelos: (1) regresión geográficamente ponderada (GWR) con una distancia fija, o (2) un ancho de banda de distancia adaptable (GWRa); (3) ancho de banda flexible GWR (FB-GWR) con distancia fija, o (4) anchuras de banda de distancia adaptable (FB-GWRa); (5) filtrado espacial eigenvector (ESF); y (6) efectos aleatorios ESF (RE-ESF). Los resultados revelan que los modelos SVC diseñados para captar las dependencias de escala en las relaciones locales (FB-GWR, FB-GWRa, y RE-ESF) calculan con la mayor exactitud los SVC simulados, donde REESF es el de mayor eficiencia computacional. Por el contrario, GWR y ESF, donde ingenuamente se asume que las estimaciones operan a la misma escala espacial para cada relación, registran un desempeño pobre. Los resultados también confirman que los modelos GWR del ancho de banda adaptable (GWRa y FB-GWRa) son superiores a sus contrapartes de ancho de banda fija (GWR y FB-GWR). Palabras clave: escala espacial, filtrado espacial eigenvector de efectos aleatorios, no-estacionalidad, regresión geográficamente ponderada con ancho de banda flexible, simulación Monte Carlo.

Key Words:

• flexible bandwidth geographically weighted regression
• Monte Carlo simulation
• nonstationarity
• random effects eigenvector spatial filtering
• spatial scale.

Notes

1 Griffith (2017) shows that the MC-based ESF is superior to the Geary’s ratio-based ESF (Geary 1954), which could be used.

2 This problem, which is purely due to the local collinearity between x₁ and x_k, appears irrespective of the scale of the true spatially varying associations, β_ks.

3 The predictor variables x₁ and x₂ are generated mutually independently. It would be an interesting topic for future work to evaluate SVC estimation accuracy by varying scales and the degree of multicollinearity simultaneously (e.g., Páez, Farber, and Wheeler 2011; Fotheringham and Oshan 2016; Oshan and Fotheringham 2017).

4 Estimation instability does not appear in this section because SVCs are implicitly assumed known and not estimated.

Additional information

Funding

 This work was supported by the National Natural Science Foundation of China (41401455, U1533102), the Japan Society for the Promotion of Science (17K12974, 17K14738, 15H04054), and the Biotechnology and Biological Sciences Research Council grants – BBS/E/C/000J0100, BBS/E/C/000I03320 and BBS/E/C/000I0330. The contribution of Science Foundation Ireland (Investigators Programme Grant 15/IA/3090 – Building City Dashboards) is gratefully acknowledged.

Notes on contributors

Daisuke Murakami

DAISUKE MURAKAMI is an Assistant Professor in the Department of Data Science, Institute of Statistical Mathematics, Tachikawa, Tokyo 190–8562, Japan. E-mail: [email protected]. His research interests include spatial and temporal statistics, quantitative geography, and regional science.

Binbin Lu

BINBIN LU is a Lecturer in the School of Remote Sensing and Information Engineering, Wuhan University, Wuhan, Hubei, China. E-mail: [email protected]. His research interests include spatial statistics, geographically weighted modeling, open-source geographic information systems, and R coding.

Paul Harris

PAUL HARRIS is a Senior Research Scientist at Rothamsted Research, North Wyke, Okehampton, Devon EX20 2SB, UK. E-mail: [email protected]. His research focuses on the development and application of spatial statistics to agricultural, ecological, and environmental data.

Chris Brunsdon

CHRIS BRUNSDON is Professor of Geocomputation and Director of the National Centre for Geocomputation, National University of Ireland Maynooth, Maynooth, Kildare, Ireland. E-mail: [email protected]. His research interests include spatial data analysis and statistical modeling, computational data science, spatial data visualization, and the analysis of social and economic data.

Martin Charlton

MARTIN CHARLTON is Senior Lecturer in the National Centre for Geocomputation, Maynooth University, Maynooth, County Kildare, Ireland. E-mail: [email protected]. His research interests include spatial modeling, quantitative geography, geographic information science, and spatial epidemiology.

Tomoki Nakaya

TOMOKI NAKAYA is Professor of Environmental Geography, Graduate School of Environmental Sciences, Tohoku University, 468-1, Aoba, Aramaki, Aoba-ku, Sendai, 980-0845, Japan. E-mail: [email protected]. His research interests include quantitative geography, spatial statistics, and spatial epidemiology.

Daniel A. Griffith

DANIEL A. GRIFFITH is Ashbel Smith Professor of Geospatial Information Sciences in the School of Economic, Political, and Policy Sciences at the University of Texas at Dallas, Richardson, TX 75080. E-mail: [email protected]. His research interests include quantitative geography, spatial statistics, urban economics, and urban public health.