Using Spatial Filters and Exploratory Data Analysis to Enhance Regression Models of Spatial Data

Residual spatial autocorrelation is a situation frequently encountered in regression analysis of spatial data. The statistical problems arising due to this phenomenon are well‐understood. Original developments in the field of statistical analysis of spatial data were meant to detect spatial pattern, in order to assess whether corrective measures were required. An early development was the use of residual autocorrelation as an exploratory tool to improve regression analysis of spatial data. In this note, we propose the use of spatial filtering and exploratory data analysis as a way to identify omitted but potentially relevant independent variables. We use an example of blood donation patterns in Toronto, Canada, to demonstrate the proposed approach. In particular, we show how an initial filter used to rectify autocorrelation problems can be progressively replaced by substantive variables. In the present case, the variables so retrieved reveal the impact of urban form, travel habits, and demographic and socio‐economic attributes on donation rates. The approach is particularly appealing for model formulations that do not easily accommodate positive spatial autocorrelation, but should be of interest as well for the case of continuous variables in linear regression.     

A Route Map for Successful Applications of Geographically Weighted Regression

Geographically Weighted Regression (GWR) is increasingly used in spatial analyses of social and environmental data. It allows spatial heterogeneities in processes and relationships to be investigated through a series of local regression models rather than a single global one. Standard GWR assumes that relationships between the response and predictor variables operate at the same spatial scale, which is frequently not the case. To address this, several GWR variants have been proposed. This paper describes a route map to decide whether to use a GWR model or not, and if so which of three core variants to apply: a standard GWR, a mixed GWR or a multiscale GWR (MS-GWR). The route map comprises 3 primary steps that should always be undertaken: (1) a basic linear regression, (2) a MS-GWR, and (3) investigations of the results of these in order to decide whether to use a GWR approach, and if so for determining the appropriate GWR variant. The paper also highlights the importance of investigating a number of secondary issues at global and local scales including collinearity, the influence of outliers, and dependent error terms. Code and data for the case study used to illustrate the route map are provided.     

DERIVATIVES OF FLOWS IN A DOUBLY CONSTRAINED GRAVITY MODEL

For many practical applications it is important to know how the flows in a doubly constrained gravity model react to slight variations in the predetermined marginal totals. The first-order approximation of these variations is a linear function on the set of feasible variations of marginal totals, i.e., the set of variations not violating the consistency constraint of the model. Several methods to find a matrix describing this linear function are developed and compared with former contributions to this issue. Finally, applicability of the methods to sensitivity and error propagation analysis is demonstrated.     

A Simulation Study on Specifying a Regression Model for Spatial Data: Choosing between Autocorrelation and Heterogeneity Effects

In this simulation study, regressions specified with autocorrelation effects are compared against those with relationship heterogeneity effects, and in doing so, provides guidance on their use. Regressions investigated are: (1) multiple linear regression, (2) a simultaneous autoregressive error model, and (3) geographically weighted regression. The first is nonspatial and acts as a control, the second accounts for stationary spatial autocorrelation via the error term, while the third captures spatial heterogeneity through the modeling of nonstationary relationships between the response and predictor variables. The geostatistical‐based simulation experiment generates data and coefficients with known multivariate spatial properties, all within an area‐unit spatial setting. Spatial autocorrelation and spatial heterogeneity effects are varied and accounted for. On fitting the regressions, that each have different assumptions and objectives, to very different geographical processes, valuable insights to their likely performance are uncovered. Results objectively confirm an inherent interrelationship between autocorrelation and heterogeneity, that results in an identification problem when choosing one regression over another. Given this, recommendations on the use and implementation of these spatial regressions are suggested, where knowledge of the properties of real study data and the analytical questions being posed are paramount.     

Crime Risk Maps: A Multivariate Spatial Analysis of Crime Data

In crime analyses, maps showing the degree of risk help police departments to make decisions on operational matters, such as where to patrol or how to deploy police officers. This study statistically models spatial crime data for multiple crime types in order to produce joint crime risk maps. To effectively model and map the spatial crime data, we consider two important characteristics of crime occurrences: the spatial dependence between sites, and the dependence between multiple crime types. We reflect both characteristics in the model simultaneously using a generalized multivariate conditional autoregressive model. As a real‐data application, we examine the number of incidents of vehicle theft, larceny, and burglary in 83 census tracts of San Francisco in 2010. Then, we employ a Bayesian approach using a Markov chain Monte Carlo method to estimate the model parameters. Based on the results, we detect the crime hotspots, thus demonstrating the advantage of using a multivariate spatial analysis for crime data.     

Developing Spatial Weight Matrices for Incorporation into Multiple Linear Regression Models: An Example Using Grizzly Bear Body Size and Environmental Predictor Variables

In this study, we develop spatial autoregressive (SAR) models relating grizzly bear body length to environmental predictor variables in the Alberta Rocky Mountains. We examine the ability of several different spatial neighborhoods to model spatial dependence and compare the estimated parameters and residuals from a standard linear regression model (LRM) with those from three types of SAR models: error, lag, and Durbin. Further, we examine variable selection in the presence of negative dependence by repeating the modeling process using a SAR model. Two findings are that significant negative spatial dependence was present in the residuals of the LRM and that the choice of spatial neighborhood greatly affects the ability to detect spatial dependence. The incorporation of appropriate spatial weights into SAR models improves the fit and increases the significance of the parameter estimates vis‐à‐vis the linear model. The results of this study indicate that negative dependence may not have as severe negative effects on variable selection and parameter estimation as positive dependence. An examination of spatial dependence in regression modeling appears to be an important means of exploring the appropriateness of a sampling framework, predictor variables, and model form. En este estudio desarrollamos modelos espaciales autorregresivos (SAR) que vinculan la longitud del cuerpo de osos grizzli con variables predictivas ambientales en las montañas rocosas de Alberta, Canadá. Examinamos la capacidad de varias vecindades espaciales para modelar la dependencia espacial y la comparación de los parámetros estimados, así como los residuos de un modelo de regresión lineal estándar (LRM) versus tres tipos de modelos SAR: error, retraso (lag) y Durbin. Además, se examina la selección de variables en la presencia de dependencia negativa mediante la repetición del proceso de modelado con un modelo de SAR. El estudio concluye que: 1) existe dependencia espacial negativa significativa en los residuos de la LRM y; 2) la selección de la vecindad espacial afecta en gran medida la capacidad de detectar la dependencia espacial. La incorporación de ponderaciones espaciales correspondientes a los modelos SAR mejora el ajuste y aumenta la importancia de los parámetros estimados versus el modelo lineal. Los resultados de este estudio indican que la dependencia negativa puede no tener los graves efectos negativos en la selección de variables y la estimación de parámetros si se comparan dichos efectos con = la dependencia positiva. Los autores recomiendan un examen de la dependencia espacial en modelos de regresión como medio importante para explorar la conveniencia de un marco de muestreo, de variables de predicción, y de la forma del modelo. 本文构建了阿尔伯达省落基山脉地区的灰熊体态大小与环境预测变量之间的空间自回归模型（SAR）,检验了几种以不同空间邻域矩阵拟合变量的空间相关性,并比较了标准回归模型(LRM)与几种不同类型的SAR模型（空间残差模型、空间滞后模型和空间杜宾模型）的估计参数和残差大小。进而利用一种SAR模型重复模拟过程,进一步测试变量选择对负相关性存在的影响。研究表明,显著的空间负相关存在于LRM的残差中,且空间邻域权重的选择很大程度上影响模型空间相关性的探测能力。将适当的空间权重引入SAR模型中可提高拟合精度,增加相对于线性模型参数估计的显著性。研究结果表明,负相关性在变量选择和参数估计上严重负影响的程度不如正相关性强。回归模型中空间相关性检验似乎是采样结构、预测变量和模型形式适用性分析的一个重要途径。     

现代服务业集聚形成机理空间计量分析

在纳入空间效应前提下,构建现代服务业集聚形成机理空间面板计量模型,对我国28个省域相关数据实证研究表明:我国现代服务业集聚在省域之间有较强的空间依赖性和正的空间溢出效应。技术差异在时间维度上对现代服务业集聚促进作用显著,在空间维度上并不显著;交易费用与现代服务业集聚有显著的负相关性;知识溢出、规模经济、政府行为对现代服务业集聚促进作用显著。     

DIAGNOSTICS FOR REGRESSION MODELING IN SPATIAL ECONOMETRICS*

ABSTRACT. This paper describes statistics for model criticism in spatial econometrics. The purpose of these statistics is to evaluate how well a chosen model fits the data and to identify influential cases and how they affect the aggregate picture. The paper reviews results in Martin (1992) for the regression model with correlated errors where the coefficients of the variance matrix are assumed either known or fixed. The problems of applying the statistics in spatial econometric modeling are discussed. An application is reported which considers diagnostics for the mean function and highlights cases that might influence estimates of the parameter of the error model. Different ways of assessing the influence of cases are also described.     

A Structural Equation Approach to Models with Spatial Dependence

We introduce the class of structural equation models (SEMs) and corresponding estimation procedures into a spatial dependence framework. SEM allows both latent and observed variables within one and the same (causal) model. Compared with models with observed variables only, this feature makes it possible to obtain a closer correspondence between theory and empirics, to explicitly account for measurement errors, and to reduce multicollinearity. We extend the standard SEM maximum likelihood estimator to allow for spatial dependence and propose easily accessible SEM software like LISREL 8 and Mx. We present an illustration based on Anselin's Columbus, OH, crime data set. Furthermore, we combine the spatial lag model with the latent multiple-indicators–multiple-causes model and discuss estimation of this latent spatial lag model. We present an illustration based on the Anselin crime data set again.     

PANEL DATA MODELS WITH SPATIALLY DEPENDENT NESTED RANDOM EFFECTS

This paper focuses on panel data models combining spatial dependence with a nested (hierarchical) structure. We use a generalized moments estimator to estimate the spatial autoregressive parameter and the variance components of the disturbance process. A spatial counterpart of the Cochrane‐Orcutt transformation leads to a feasible generalized least squares procedure to estimate the regression parameters. Monte Carlo simulations show that our estimators perform well in terms of root mean square error compared to the maximum likelihood estimator. The approach is applied to English house price data for districts nested within counties.     

The effect of a priori knowledge on parameter estimation errors with applications to incoherent scatter

The dependence of the width of the a posteriori distribution (i.e. parameter estimation error) in a multiparameter fit on the a priori width of another parameter is studied in the framework of linear statistical inversion theory. An exact formula is given which makes it possible to estimate how accurate the a priori information on a given parameter must be in order to obtain the other parameter with a prescribed accuracy. These estimates are useful in those cases where some errors are strongly correlated, like the ion temperature and the O⁺ content in incoherent scatter measurements with the EISCAT UHF radar. The dependence of the solution on the accuracy of the power profile estimates is given. Likewise, the behaviour of the error when a linear combination of the parameters is known exactly is given. The formulae given have a very simple geometrical meaning, and together they give a complete solution to the problem of estimating the accuracies in a multiparameter fit when a linear combination of the variables is known with a prescribed accuracy. It is shown that under the conditions considered in this paper, oxygen content can be determined to an accuracy of 10 % when the ion temperature is known to within 7 %, if we can assume that the collision frequency is zero.     

The Effect on Attribute Prediction of Location Uncertainty in Spatial Data

A datum is considered spatial if it contains location information. Typically, there is also attribute information, whose distribution depends on its location. Thus, error in location information can lead to error in attribute information, which is reflected ultimately in the inference drawn from the data. We propose a statistical model for incorporating location error into spatial data analysis. We investigate the effect of location error on the spatial lag, the covariance function, and optimal spatial linear prediction (that is, kriging). We show that the form of kriging after adjusting for location error is the same as that of kriging without adjusting for location error. However, location error changes entries in the matrix of explanatory variables, the matrix of co‐variances between the sample sites, and the vector of covariances between the sample sites and the prediction location. We investigate, through simulation, the effect that varying trend, measurement error, location error, range of spatial dependence, sample size, and prediction location have on kriging after and without adjusting for location error. When the location error is large, kriging after adjusting for location error performs markedly better than kriging without adjusting for location error, in terms of both the prediction bias and the mean squared prediction error.     

Application of LASSO to the Eigenvector Selection Problem in Eigenvector‐based Spatial Filtering

Eigenvector‐based spatial filtering is one of the often used approaches to model spatial autocorrelation among the observations or errors in a regression model. In this approach, a subset of eigenvectors extracted from a modified spatial weight matrix is added to the model as explanatory variables. The subset is typically specified via the selection procedure of the forward stepwise model, but it is disappointingly slow when the observations n take a large number. Hence, as a complement or alternative, the present article proposes the use of the least absolute shrinkage and selection operator (LASSO) to select the eigenvectors. The LASSO model selection procedure was applied to the well‐known Boston housing data set and simulation data set, and its performance was compared with the stepwise procedure. The obtained results suggest that the LASSO procedure is fairly fast compared with the stepwise procedure, and can select eigenvectors effectively even if the data set is relatively large (n = 10⁴), to which the forward stepwise procedure is not easy to apply.     

A Map Overlay Error Model Based on Boundary Geometry

An error model for quantifying the magnitudes and variability of errors generated in the areas of polygons during spatial overlay of vector geographic information system layers is presented. Numerical simulation of polygon boundary displacements was used to propagate coordinate errors to spatial overlays. The model departs from most previous error models in that it incorporates spatial dependence of coordinate errors at the scale of the boundary segment. It can be readily adapted to match the scale of error–boundary interactions responsible for error generation on a given overlay. The area of error generated by overlay depends on the sinuosity of polygon boundaries, as well as the magnitude of the coordinate errors on the input layers. Asymmetry in boundary shape has relatively little effect on error generation. Overlay errors are affected by real differences in boundary positions on the input layers, as well as errors in the boundary positions. Real differences between input layers tend to compensate for much of the error generated by coordinate errors. Thus, the area of change measured on an overlay layer produced by the XOR overlay operation will be more accurate if the area of real change depicted on the overlay is large. The model presented here considers these interactions, making it especially useful for estimating errors studies of landscape change over time.     

Some Notes on Parametric Significance Tests for Geographically Weighted Regression

The technique of geographically weighted regression (GWR) is used to model spatial 'drift' in linear model coefficients. In this paper we extend the ideas of GWR in a number of ways. First, we introduce a set of analytically derived significance tests allowing a null hypothesis of no spatial parameter drift to be investigated. Second, we discuss 'mixed' GWR models where some parameters are fixed globally but others vary geographically. Again, models of this type may be assessed using significance tests. Finally, we consider a means of deciding the degree of parameter smoothing used in GWR based on the Mallows C_p statistic. To complete the paper, we analyze an example data set based on house prices in Kent in the U.K. using the techniques introduced.     

Model Uncertainty in Matrix Exponential Spatial Growth Regression Models

This article considers the most important aspects of model uncertainty for spatial regression models, namely, the appropriate spatial weight matrix to be employed and the appropriate explanatory variables. We focus on the spatial Durbin model (SDM) specification in this study that nests most models used in the regional growth literature, and develop a simple Bayesian model‐averaging approach that provides a unified and formal treatment of these aspects of model uncertainty for SDM growth models. The approach expands on previous work by reducing the computational costs through the use of Bayesian information criterion model weights and a matrix exponential specification of the SDM model. The spatial Durbin matrix exponential model has theoretical and computational advantages over the spatial autoregressive specification due to the ease of inversion, differentiation, and integration of the matrix exponential. In particular, the matrix exponential has a simple matrix determinant that vanishes for the case of a spatial weight matrix with a trace of zero. This allows for a larger domain of spatial growth regression models to be analyzed with this approach, including models based on different classes of spatial weight matrices. The working of the approach is illustrated for the case of 32 potential determinants and three classes of spatial weight matrices (contiguity‐based, k‐nearest neighbor, and distance‐based spatial weight matrices), using a data set of income per capita growth for 273 European regions.     

Railways as a Factor of Change in the Distribution of Population in Spain, 1900–1970

Abstract

The central focus of this work is to test a new methodology to measure the impact of the railway on the distribution of population, in this case in Spain. To achieve this, it was necessary to previously integrate data relating to population and railway lines into a geographical information system. The result was a spatial database that includes population data from homogeneous census series obtained for the municipal scale and the evolution of the railway network in service at corresponding points in time. This allowed the authors to apply spatial-temporal analysis. By so doing, this work constitutes an analysis of a new methodology, as they used exploratory spatial data analysis and geographically weighted regression to detect spatial patterns and estimate the influence of the railway and distance from the coast on population change. The results obtained show that the influence of the railway was very pronounced in some areas, while in others it was just one of the factors that could explain major changes in population distribution.     

Geostatistical Prediction and Simulation of Point Values from Areal Data

The spatial prediction and simulation of point values from areal data are addressed within the general geostatistical framework of change of support (the term support referring to the domain informed by each measurement or unknown value). It is shown that the geostatistical framework (i) can explicitly and consistently account for the support differences between the available areal data and the sought-after point predictions, (ii) yields coherent (mass-preserving or pycnophylactic) predictions, and (iii) provides a measure of reliability (standard error) associated with each prediction. In the case of stochastic simulation, alternative point-support simulated realizations of a spatial attribute reproduce (i) a point-support histogram (Gaussian in this work), (ii) a point-support semivariogram model (possibly including anisotropic nested structures), and (iii) when upscaled, the available areal data. Such point-support-simulated realizations can be used in a Monte Carlo framework to assess the uncertainty in spatially distributed model outputs operating at a fine spatial resolution because of uncertain input parameters inferred from coarser spatial resolution data. Alternatively, such simulated realizations can be used in a model-based hypothesis-testing context to approximate the sampling distribution of, say, the correlation coefficient between two spatial data sets, when one is available at a point support and the other at an areal support. A case study using synthetic data illustrates the application of the proposed methodology in a remote sensing context, whereby areal data are available on a regular pixel support. It is demonstrated that point-support (sub-pixel scale) predictions and simulated realizations can be readily obtained, and that such predictions and realizations are consistent with the available information at the coarser (pixel-level) spatial resolution.     

A Comparison of Spatially Varying Regression Coefficient Estimates Using Geographically Weighted and Spatial‐Filter‐Based Techniques

Geographically weighted regression (GWR) is a technique that explores spatial nonstationarity in data‐generating processes by allowing regression coefficients to vary spatially. It is a widely applied technique across domains because it is intuitive and conforms to the well‐understood framework of regression. An alternative method to GWR that has been suggested is spatial filtering, which it has been argued provides a superior alternative to GWR by producing spatially varying regression coefficients that are not correlated with each other and which display less spatial autocorrelation. It is, therefore, worthwhile to examine these claims by comparing the output from both methods. We do this by using simulated data that represent two sets of spatially varying processes and examining how well both techniques replicate the known local parameter values. The article finds no support that spatial filtering produces local parameter estimates with superior properties. The results indicate that the original spatial filtering specification is prone to overfitting and is generally inferior to GWR, while an alternative specification that minimizes the mean square error (MSE) of coefficient estimates produces results that are similar to GWR. However, since we generally do not know the true coefficients, the MSE minimizing specification is impractical for applied research.