A Note on the Mixed Geographically Weighted Regression Model*

Abstract. A mixed, geographically weighted regression (GWR) model is useful in the situation where certain explanatory variables influencing the response are global while others are local. Undoubtedly, how to identify these two types of the explanatory variables is essential for building such a model. Nevertheless, It seems that there has not been a formal way to achieve this task. Based on some work on the GWR technique and the distribution theory of quadratic forms in normal variables, a statistical test approach is suggested here to identify a mixed GWR model. Then, this note mainly focuses on simulation studies to examine the performance of the test and to provide some guidelines for performing the test in practice. The simulation studies demonstrate that the test works quite well and provides a feasible way to choose an appropriate mixed GWR model for a given data set.     

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.     

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.     

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.     

A SPATIAL HEDONIC APPROACH ON LAND USE CHANGE ANTICIPATIONS*

ABSTRACT This paper estimates land use conversion anticipation when zoning is the primary tool of land use regulation. Extending the Capozza and Helsey model (1989) to the case of uncertainty in future land use zoning, a spatialized hedonic model is proposed to estimate such anticipation phenomena at a fine level (cadastral unit). Estimations use Mixed Geographically Weighted Regression (MGWR) techniques with a two‐stage model that links agricultural and developable land markets. This allows for mapping varying spatial parameters that measure anticipation effects within the theoretical framework. Results confirm the influence of anticipation on agricultural land prices in the Provence region. Moreover, the level of data spatialization allows us to take into account intra‐municipalities' heterogeneity of land use conversion anticipation.     

Geographical and Temporal Weighted Regression (GTWR)

Both space and time are fundamental in human activities as well as in various physical processes. Spatiotemporal analysis and modeling has long been a major concern of geographical information science (GIScience), environmental science, hydrology, epidemiology, and other research areas. Although the importance of incorporating the temporal dimension into spatial analysis and modeling has been well recognized, challenges still exist given the complexity of spatiotemporal models. Of particular interest in this article is the spatiotemporal modeling of local nonstationary processes. Specifically, an extension of geographically weighted regression (GWR), geographical and temporal weighted regression (GTWR), is developed in order to account for local effects in both space and time. An efficient model calibration approach is proposed for this statistical technique. Using a 19‐year set of house price data in London from 1980 to 1998, empirical results from the application of GTWR to hedonic house price modeling demonstrate the effectiveness of the proposed method and its superiority to the traditional GWR approach, highlighting the importance of temporally explicit spatial modeling.     

COINTEGRATION BETWEEN U.S. WHEAT MARKETS

ABSTRACT. Average monthly price data from twelve hinterland markets and the Houston port price for wheat are studied in a cointegration framework using the Engle-Granger "two-step" procedure and Johansen's maximum likelihood procedure. Out-of-sample forecasts from an error correction model are compared to those from a vector autoregression fit to levels and a univariate autoregression fit to first differences. This comparison suggests that modeling these (cointegrated) data as a levels vector autoregression, rather than as an error-correction process, results in significantly higher error bias, but lower error variance, at long horizons.     

Dynamics of Land Surface Temperature in Response to Land‐Use/Cover Change

In this study, we employed Geographical Information Systems and remote sensing techniques to investigate the impact of land‐use/cover change on land surface temperature (LST) in a rapidly urbanisation city, Kunming in south‐west China. Spatial patterns of LST and land use for 1992 and 2006 were derived from Landsat images to examine how LST responded to urban growth. Remote sensing indices were used to quantify land‐use types and employed as explanatory variables in LST modelling. The geographically weighted regression (GWR), a location dependent model, was performed to explore the influences of the spatially varied land‐use conditions on the LST patterns. Results revealed that rapid urbanisation in Kunming altered the local thermal environment, particularly in increasing the LST in the zone surrounding the urban core. Remote sensing indices demonstrated that water and vegetation played an important role in mitigating the urban heat island effect, while built‐up and barren land accounted for the increase in LST. The GWR improved the goodness‐of‐fit for LST modelling and provided insights into the spatially varied relationship between LST and land‐use conditions.     

Exploring the Relations Between Riverbank Erosion and Geomorphological Controls Using Geographically Weighted Logistic Regression

The relations between riverbank erosion and geomorphological variables that are thought to control or influence erosion are commonly modelled using regression. For a given river, a single regression model might befitted to data on erosion and its geomorphological controls obtained along the river's length. However, it is likely that the influence of some variables may vary with geographical location (i.e., distance upstream). For this reason, the spatially stationary regression model should be replaced with a non‐stationary equivalent. Geographically weighted regression (GWR) is a suitable choice. In this paper, GWR is extended to predict the binary presence or absence of erosion via the logistic model. This extended model was applied to data obtained from historical archives and a spatially intensive field survey of a length of 42 km of the Afon Dyfi in West Wales. The model parameters and the residual deviance of the model varied greatly with distance upstream. The practical implication of the result is that different management practices should be implemented at different locations along the river. Thus, the approach presented allowed inference of spatially varying management practice as a consequence of spatially varying geomorphological process.     

Spatial Patterns in the Association between the Prevalence of Asthma and Determinants of Health

The World Health Organization endorses the study of diseases from the perspective of the Determinants of Health (DH), that is, the circumstances in which people are born and raised, the environment in which they grow up and age and their lifestyle. The aim of this study is to analyze the spatial behavior of the prevalence of asthma in Aragon, a Mediterranean region in Spain, under the DH approach. The methodological process entailed building a spatial database collating asthma prevalence as dependent variable, and lifestyle, socioeconomic, and climate indicators as explanatory factors, and then evaluating the spatial variability of the relationships by combining the Principal Component Analysis (PCA), Multiscale Geographically Weighted Regression (MGWR) models and cartographic design techniques. MGWR evidenced spatially varying relationships operating at different scales. Lifestyles seem closely tied to the prevalence of asthma in most of the study area while urban functionality and local climate patterns seem to boost prevalence rates in some specific enclaves. Consequently, the social and environmental conditions that characterize the study area translate into several DH scenarios modulating the spatial distribution of asthma. This differential DH behavior detected by local regression models is relevant to guiding and refining public health decision-making.     

Developing Geostatistical Space–Time Models to Predict Outpatient Treatment Burdens from Incomplete National Data

Basic health system data such as the number of patients utilizing different health facilities and the types of illness for which they are being treated are critical for managing service provision. These data requirements are generally addressed with some form of national Health Management Information System (HMIS), which coordinates the routine collection and compilation of data from national health facilities. HMIS in most developing countries are characterized by widespread underreporting. Here we present a method to adjust incomplete data to allow prediction of national outpatient treatment burdens. We demonstrate this method with the example of outpatient treatments for malaria within the Kenyan HMIS. Three alternative modeling frameworks were developed and tested in which space–time geostatistical prediction algorithms were used to predict the monthly tally of treatments for presumed malaria cases (MC) at facilities where such records were missing. Models were compared by a cross-validation exercise and the model found to most accurately predict MC incorporated available data on the total number of patients visiting each facility each month. A space–time stochastic simulation framework to accompany this model was developed and tested in order to provide estimates of both local and regional prediction uncertainty. The level of accuracy provided by the predictive model, and the accompanying estimates of uncertainty around the predictions, demonstrate how this tool can mitigate the uncertainties caused by missing data, substantially enhancing the utility of existing HMIS data to health-service decision makers.     

Geographically Weighted Discriminant Analysis

In this article, we propose a novel analysis technique for geographical data, Geographically Weighted Discriminant Analysis. This approach adapts the method of Geographically Weighted Regression (GWR), allowing the modeling and prediction of categorical response variables. As with GWR, the relationship between predictor and response variables may alter over space, and calibration is achieved using a moving kernel window approach. The methodology is outlined and is illustrated with an example analysis of voting patterns in the 2005 UK general election. The example shows that similar social conditions can lead to different voting outcomes in different parts of England and Wales. Also discussed are techniques for visualizing the results of the analysis and methods for choosing the extent of the moving kernel window.     

Single and Multiscale Models of Process Spatial Heterogeneity

Recent work in local spatial modeling has affirmed and broadened interest in multivariate local spatial analysis. Two broad approaches have emerged: Geographically Weighted Regression (GWR) which follows a frequentist perspective and Bayesian Spatially Varying Coefficients models. Although several comparisons between the two approaches exist, recent developments, particularly in GWR, mean that these are incomplete and missing some important axes of comparison. Consequently, there is a need for a more thorough comparison of the two families of local estimators, including recent developments in multiscale variants and their relative performance under controlled conditions. We find that while both types of local models generally perform similarly on a series of criteria, some interesting and important differences exist.     

Local indicators for categorical data: impacts of scaling decisions

When the geographic distribution of landscape pattern varies, global indices fail to capture the spatial nonstationarity within the dataset. Methods that measure landscape pattern at a spatially local scale are advantageous, as an index is computed at each point in the dataset. The geographic distribution of local indices is used to discover spatial trends. Local indicators for categorical data (LICD) can be used to statistically quantify local spatial patterns in binary geographic datasets. LICD, like other spatially local methods, are impacted by decisions relating to the spatial scale of the data, such as the data grain (p), and analysis parameters such as the size of the local neighbourhood (m). The goal of this article is to demonstrate how the choice of the m and p parameters impacts LICD analysis. We also briefly discuss the impacts spatial extent can have on analysis; specifically the local composition measure. An example using 2006 forest cover data for a region in British Columbia, Canada where mountain pine beetle mitigation and salvage harvesting has occurred is used to show the impacts of changing m and p. Selection of local window size (m = 3,5,7) impacts the prevalence and interpretation of significant results. Increasing data grain (p) had varying effects on significant LICD results. When implementing LICD the choice of m and p impacts results. Exploring multiple combinations of m and p will provide insight into selection of ideal parameters for analysis.     

A Generalized Randomization Approach to Local Measures of Spatial Association

This article establishes a unified randomization significance testing framework upon which various local measures of spatial association are commonly predicated. The generalized randomization approach presented is composed of two testing procedures, the extended Mantel test and the generalized vector randomization test. These two procedures employ different randomization assumptions, namely total and conditional randomization, according to the way in which they incorporate local measures. By properly specifying necessary matrices and vectors for a particular local measure of spatial association under a particular randomization assumption, the generalized randomization approach as a whole yields a reliable set of equations for expected values and variances, which then is confirmed by a Monte Carlo simulation utilizing random permutations.     

Empirical Zoning Distributions for Small Area Data

It is well established that using data summaries for a set of geographic areas or zones to estimate the parameters of a statistical model, commonly called ecological inference, frequently leads to the modifiable area unit problem (MAUP). In this article, the zoning effect of the MAUP is investigated for a range of scales. A zoning distribution is defined, and then used to characterize the zoning effect for parameter estimates from ecological analyses. Zone‐independent parameter estimates are obtained using the mean of the zoning distribution, and assessed using the variance of the zoning distribution. Zoning distributions are illustrated for parameter estimates from two ecological regression models at multiple scales using Australian National Health Survey data. For both a continuous response variable and a binary response variable, the empirical zoning distributions are unimodal, relatively symmetrical with appreciable variation, even when based on a large number of zones. The “ecological mean,” or expected value of the empirical zoning distribution at each scale, displays systematic variation with scale and the zoning distribution variance also depends on scale. The results demonstrate that the zoning effect should not be ignored, and the sensitivity of ecological analysis results to the analysis zones should be assessed.     

A Hybrid Framework for Space–Time Modeling of Environmental Data. 环境数据时空建模的混合框架

The space–time autoregressive integrated moving average (STARIMA) model family provides useful tools for modeling space–time processes that exhibit stationarity (or near stationarity) in space and time. However, a more general method for routine use and efficient computation is needed to model the nonlinearities and nonstationarities of environmental space–time series. This article presents a hybrid framework combining machine learning and statistical methods to address this issue. It uses an artificial neural network (ANN) to extract global deterministic (nonlinear) space–time trends and a STARIMA model to extract local stochastic space–time variations in data. A four‐stage procedure is proposed for analyzing and modeling space–time series. The proposed framework and procedures are applied to forecast annual average temperature at 137 national meteorological stations in China. The results demonstrate that the hybrid framework achieves better forecasting accuracy than the STARIMA model alone. This finding suggests that the combination of machine learning and statistical methods provides a very powerful tool for analyzing and modeling space–time series of environmental data that have strong spatial nonlinear and nonstationary components.     

Exploring Heterogeneities with Geographically Weighted Quantile Regression: An Enhancement Based on the Bootstrap Approach

Geographically weighted quantile regression (GWQR) has been proposed as a spatial analytical technique to simultaneously explore two heterogeneities, one of spatial heterogeneity with respect to data relationships over space and one of response heterogeneity across different locations of the outcome distribution. However, one limitation of GWQR framework is that the existing inference procedures are established based on asymptotic approximation, which may suffer computation difficulties or yield incorrect estimates with finite samples. In this article, we suggest a bootstrap approach to address this limitation. Our bootstrap enhancement is first validated by a simulation experiment and then illustrated with an empirical U.S. mortality data. The results show that the bootstrap approach provides a practical alternative for inference in GWQR and enhances the utilization of GWQR.     

Force-Based Beam Finite Element (FE) for the Pushover Analysis of Masonry Buildings

A simplified approach for analyzing the nonlinear response of masonry buildings, based on the equivalent frame modeling procedure and on the nonlinear equivalent static analyses, is presented. A nonlinear beam finite element (FE) is formulated in the framework of a force-based approach, where the stress fields are expanded along the beam local axis, and introduced in a global displacement-based FE code. In order to model the nonlinear constitutive response of the masonry material, the lumped hinge approach is adopted and both flexural and shear plastic hinges are located at the two end nodes of the beam. A classical elastic-plastic constitutive relationship describes the nonlinear response of the hinges, the evolution of the plastic variables being governed by the Kuhn-Tucker and consistency conditions. An efficient element state determination procedure is implemented, which condenses the local deformation residual into the global residual vector, thus avoiding the need to perform the inner loops for computing the element nonlinear response. The comparison with some relevant experimental and real full-scale masonry walls is presented, obtaining a very good agreement with the available results, both in terms of global pushover curves and damage distributions.     

Renewable Energy Politics: Policy Typologies,Policy Tools,and State Deployment of Renewables

This paper investigates the competing forces driving the development of renewable energy in the American states. We formulate a framework of state renewable energy politics and develop a set of hypotheses regarding the role of politics, policies, and prices in renewable energy development. We test these hypotheses with a fixed effect vector decomposition model using a panel data set for the U.S. states from 1990 to 2008. The results indicate that renewable energy development is influenced by regulatory institutions, the party affiliations of the governor and legislators, and the professionalism of the legislature, accompanied by the effects of various policy instruments.