Measuring Bandwidth Uncertainty in Multiscale Geographically Weighted Regression Using Akaike Weights

Abstract

Bandwidth, a key parameter in geographically weighted regression models, is closely related to the spatial scale at which the underlying spatially heterogeneous processes being examined take place. Generally, a single optimal bandwidth (geographically weighted regression) or a set of covariate-specific optimal bandwidths (multiscale geographically weighted regression) is chosen based on some criterion, such as the Akaike information criterion (AIC), and then parameter estimation and inference are conditional on the choice of this bandwidth. In this article, we find that bandwidth selection is subject to uncertainty in both single-scale and multiscale geographically weighted regression models and demonstrate that this uncertainty can be measured and accounted for. Based on simulation studies and an empirical example of obesity rates in Phoenix, we show that bandwidth uncertainties can be quantitatively measured by Akaike weights and confidence intervals for bandwidths can be obtained. Understanding bandwidth uncertainty offers important insights about the scales over which different processes operate, especially when comparing covariate-specific bandwidths. Additionally, unconditional parameter estimates can be computed based on Akaike weights accounts for bandwidth selection uncertainty.

带宽是地理加权回归模型中的一个关键参数，此参数与所研究潜在空间异构过程中所发生的空间尺度密切相关。在此过程中，通常会根据某些准则（例如赤池信息准则（AIC））选择单一最佳带宽（地理加权回归）或一组根据特定于协变量的最佳带宽（多尺度地理加权回归），然后以该带宽选择为条件，进行参数估计和推断。本文的作者发现，带宽选择在单尺度和多尺度地理加权回归模型中均受到不确定性的影响。作者还证明了这种不确定性可以被测量和解释。基于凤凰城关于肥胖率的模拟研究和实证举例，作者表明可以通过赤池权重对带宽不确定性进行定量测量，可以获得带宽的置信区间。理解带宽不确定性为不同进程的运行尺度提供了重要见解，尤其是在比较特定于协变量的带宽时更是如此。另外，赤池权重所揭示的带宽选择的不确定性，还可以用于计算无条件参数估值。

La amplitud de banda, un parámetro clave en los modelos de regresión geográficamente ponderada, está estrechamente relacionada con la escala espacial en la cual ocurren los procesos subyacentes con heterogeneidad espacial, bajo escrutinio. En general, una amplitud de banda óptima individual (regresión geográficamente ponderada) o un conjunto óptimo de amplitudes de banda con covariaciones específicas (regresión geográficamente ponderada a multiescala) son escogidas a partir de un criterio determinado, tal como el criterio de información Akaike (AIC), y desde ahí el estimativo e inferencia del parámetro quedan condicionados por la escogencia de esta amplitud de banda. En este artículo, encontramos que la selección de amplitud de banda está sujeta a incertidumbre en los modelos de regresión geográficamente ponderada tanto a escala sencilla como a multiescala, y demostramos que esta incertidumbre puede medirse y explicarse. Con base en estudios de simulación y en un ejemplo empírico de tasas de obesidad en Phoenix, mostramos que las incertidumbres de amplitud de banda pueden medirse cuantitativamente con pesos Akaike, y se pueden derivar los intervalos de confianza para las amplitudes de banda. Entender la incertidumbre de amplitud de banda ofrece perspectivas importantes acerca de las escalas a que operan diferentes procesos, especialmente cuando se comparan amplitudes de banda de covariación específica. Por otro lado, los cálculos incondicionales de parámetro pueden computarse tomando en cuenta los pesos Akaike para la incertidumbre en la selección de amplitud de banda.

Key Words:

• Akaike weight
• bandwidth
• model selection uncertainty
• multiscale geographically weighted regression
• spatial processes scale

关键词:

• 赤池权重
• 带宽
• 模型选择不确定性
• 多尺度地理加权回归
• 空间过程规模。

Palabras clave:

• amplitud de banda
• escala de procesos espaciales
• incertidumbre en la selección del modelo
• peso Akaike
• regresión geográficamente ponderada a multiescala

View correction statement:

Correction

Notes

1 Using a step size smaller than ten will produce more detailed Akaike weight curve but with additional computation.

2 Six sparsely populated tracts are removed in this example.

Additional information

Notes on contributors

Ziqi Li

ZIQI LI is a PhD Candidate in the School of Geographical Sciences and Urban Planning, Arizona State University, Tempe, AZ 85281. E-mail: [email protected]. His research interests include GIScience and spatial data science. He is currently working on the MGWR model, covering aspects from computation and inference to applications.

A. Stewart Fotheringham

A. STEWART FOTHERINGHAM is a Regents’ Professor of Computational Spatial Science in the School of Geographical Sciences and Urban Planning at Arizona State University, Tempe, AZ 85281. E-mail: [email protected]. His research interests include spatial data analytics, spatial processes, and spatial interaction modeling.

Taylor M. Oshan

TAYLOR M. OSHAN is an Assistant Professor in the Center for Geospatial Information Science within the Department of Geographical Sciences at the University of Maryland, College Park, MD 20740. E-mail: [email protected]. His research interests are centered on developing methods to analyze multiscale spatial and temporal processes and applying them in the context of urban health and transportation, as well as building open source tools.

Levi John Wolf

LEVI JOHN WOLF is a Senior Lecturer (Assistant Professor) in the School of Geographical Sciences at the University of Bristol, Bristol BS8 1QU UK. E-mail: [email protected]. His research interests include probabilistic programming and machine learning for applications in urban geography, politics, sociology, and economics.