Bayesian Estimation of Limited Dependent Variable Spatial Autoregressive Models

A Gibbs sampling (Markov chain Monte Carlo) method for estimating spatial autoregressive limited dependent variable models is presented. The method can accommodate data sets containing spatial outliers and general forms of non‐constant variance. It is argued that there are several advantages to the method proposed here relative to that proposed and illustrated in McMillen (1992) for spatial probit models.     

Prediction of Bike-sharing Trip Counts: Comparing Parametric Spatial Regression Models to a Geographically Weighted XGBoost Algorithm

Regression models are commonly applied in the analysis of transportation data. This research aims at broadening the range of methods used for this task by modeling the spatial distribution of bike-sharing trips in Cologne, Germany, applying both parametric regression models and a modified machine learning approach while incorporating measures to account for spatial autocorrelation. Independent variables included in the models consist of land use types, elements of the transport system and sociodemographic characteristics. Out of several regression models with different underlying distributions, a Tweedie generalized additive model is chosen by its values for AIC, RMSE, and sMAPE to be compared to an XGBoost model. To consider spatial relationships, spatial splines are included in the Tweedie model, while the estimations of the XGBoost model are modified using a geographically weighted regression. Both methods entail certain advantages: while XGBoost leads to far better values regarding RMSE and sMAPE and therefore to a better model fit, the Tweedie model allows an easier interpretation of the influence of the independent variables including spatial effects.     

Duration Modeling of Spatial Point Patterns

This paper attempts to further the research by Odland and Ellis (1992) in applying event history methodology to the analysis of spatial point patterns (that is, event patterns). Its empirical focus is the event pattern derived from the adoption of an agricultural innovation, the Harvestore, in southern Ontario, Canada, from 1963 to 1986. Event history analysis involves the use of discrete-state, continuous-time stochastic models to investigate a temporal longitudinal record on discrete variables. Event history models are usually concerned with durations of time between events and the effects of intertemporal time dependencies on future event occurrences. As such, they are often referred to as duration models. Many of the methods used in event history analysis allow the use of other nonnegative interval measurements in place of standard temporal intervals to investigate a series of events. In particular, spatial intervals (or durations) of distances between events may also be accommodated by event history models. Our analysis extends the previous research of Odland and Ellis (1992) by using a wider range of parametric models to explore duration dependence, investigating the role of spatial censoring, and using a more extensive set of explanatory variables. In addition, simulation experiments and graphical tests are used to evaluate the empirical event pattern against one generated from Complete Spatial Randomness. Results indicate that the event pattern formed by the Harvestore adopter farms is clustered (that is, is described by positive duration dependency), the sales agent is a significant factor in the distribution of adopters, and that contrasting results are obtained from the analysis using censored data versus uncensored data.     

Employment Densities, Spatial Autocorrelation, and Subcenters in Large Metropolitan Areas

Employment density functions are estimated for 62 large metropolitan areas. Estimated gradients are statistically significant for distance from the nearest subcenter as well as for distance from the traditional central business district. Lagrange Multiplier (LM) tests imply significant spatial autocorrelation under highly restrictive ordinary least squares (OLS) specifications. The LM test statistics fall dramatically when the models are estimated using flexible parametric and nonparametric methods. The results serve as a warning that functional form misspecification causes spatial autocorrelation.     

Spatial Autoregressive Models for Geographically Hierarchical Data Structures

This article discusses how standard spatial autoregressive models and their estimation can be extended to accommodate geographically hierarchical data structures. Whereas standard spatial econometric models normally operate at a single geographical scale, many geographical data sets are hierarchical in nature—for example, information about houses nested into data about the census tracts in which those houses are found. Here we outline four model specifications by combining different formulations of the spatial weight matrix W and of ways of modeling regional effects. These are (1) groupwise W and fixed regional effects; (2) groupwise W and random regional effects; (3) proximity‐based W and fixed regional effects; and (4) proximity‐based W and random regional effects. We discuss each of these model specifications and their associated estimation methods, giving particular attention to the fourth. We describe this as a hierarchical spatial autoregressive model. We view it as having the most potential to extend spatial econometrics to accommodate geographically hierarchical data structures and as offering the greatest coming together of spatial econometric and multilevel modeling approaches. Subsequently, we provide Bayesian Markov Chain Monte Carlo algorithms for implementing the model. We demonstrate its application using a two‐level land price data set where land parcels nest into districts in Beijing, China, finding significant spatial dependence at both the land parcel level and the district level.     

Applying Measures of Spatial Autocorrelation: Computation and Simulation

Knowing about the challenges and opportunities of spatial autocorrelation is one thing, but applying the measures to one's own data is another matter entirely. While manual computation of the measures for toy data sets is possible, applying them to small data sets required the use of computers and thus software. This article will shed some light on how the measures were and are implemented in software and on implementation issues that are still not fully resolved.     

New Statistical Approaches to Intra‐individual Isotopic Analysis and Modelling of Birth Seasonality in Studies of Herd Animals

This paper introduces improved methods for statistically assessing birth seasonality and intra‐annual variation in δ¹⁸O from faunal tooth enamel. The first method estimates input parameters for use with a previously developed parametric approach by C. Tornero et al. The second method uses a non‐parametric clustering procedure to group individuals with similar time‐series data and estimate birth seasonality. This method was successful in analysing data from a modern sample with known season of birth, as well as two heterogeneous archaeological data sets. Modelling indicates that the non‐parametric approach estimates birth seasonality more successfully than the parametric method when less of the tooth row is preserved. The new approach offers a high level of statistical rigour and flexibility in dealing with the time‐series data produced through intra‐individual sampling in isotopic analysis.     

BORDERS AS BOUNDARIES TO FISCAL POLICY INTERACTIONS? AN EMPIRICAL ANALYSIS OF POLITICIANS’ OPINIONS ON RIVALS IN THE COMPETITION FOR FIRMS

Studies of spatial policy interdependence in (local) public policies usually concentrate on the relations between jurisdictions within a single analyzed region, and disregard possible extraregional effects. However, the theoretical spatial statistics literature shows that biased estimates might emerge if spatial interactions extend beyond the boundaries of the available data (i.e., the boundary value problem). This paper empirically assesses the practical relevance of this concern by studying German local politicians’ assessments of their jurisdictions’ main competitors in the struggle to attract firms. We find that location near a border significantly undermines politicians’ perception that the fiercest competitive pressure derives from jurisdictions within their own state. This effect sets in about 20 km (10.2 km) from a national (international) border. These results indicate that nearest municipalities perceive each other as competitors regardless of the state or country where they are located, which has important implications for estimating spatial dependence models.     

On the similarity of tree‐ring patterns: Assessing the influence of semi‐synchronous growth changes on the Gleichläufigkeitskoeffizient for big tree‐ring data sets

The Gleichläufigkeitskoeffizient (GLK), or the percentage of parallel variation (%PV), is an often used non‐parametric similarity measure in dendrochronological research. However, when analysing big data sets using the GLK, this measure has some issues. The main problem is that it includes not only synchronous but also semi‐synchronous growth changes. These are years in which the growth in one of the compared series does not change in two subsequent years. This influences the GLK, often only slightly, but the larger the data set the stronger the effect. The similarity between tree‐ring series can be more objectively expressed by replacing the GLK with the synchronous (SGC) and semi‐synchronous growth changes (SSGC). The calculation is similar, since GLK = SGC + SSGC/2. Large values of the SSGC are indicative of possible anomalies or even errors. The SGC is much better suited than the GLK to describe similarity. The SGC should therefore be used to analyse big data sets, for clustering and/or dendroprovenance studies. It is recommended to combine the SGC with parametric measures.     

Exploring the Use of Computer Vision Metrics for Spatial Pattern Comparison

Detection of changes in spatial processes has long been of interest to quantitative geographers seeking to test models, validate theories, and anticipate change. Given the current “data-rich” environment of today, it may be time to reconsider the methodological approaches used for quantifying change in spatial processes. New tools emerging from computer vision research may hold particular potential to make significant advances in quantifying changes in spatial processes. In this article, two comparative indices from computer vision, the structural similarity (SSIM) index, and the complex wavelet structural similarity (CWSSIM) index were examined for their utility in the comparison of real and simulated spatial data sets. Gaussian Markov random fields were simulated and compared with both metrics. A case study into comparison of snow water equivalent spatial patterns over northern Canada was used to explore the properties of these indices on real-world data. CWSSIM was found to be less sensitive than SSIM to changing window dimension. The CWSSIM appears to have significant potential in characterizing change and/or similarity; distinguishing between map pairs that possess subtle structural differences. Further research is required to explore the utility of these approaches for empirical comparison cases of different forms of landscape change and in comparison to human judgments of spatial pattern differences.     

Potential Use of Radiometric Data for Wildlife Habitat Modelling

The potential use of existing radiometric data sets, previously collected for prospecting purposes, has very rarely been used as a variable predictor in wildlife habitat modelling. The utility of radiometric data for predicting vegetation community patterns and wildlife habitat was investigated in the Australian arid zone using the Burt Plain bioregion as a case study. Using spatial datasets and a Species Distribution Modelling Toolkit, arid zone vertebrate species were modelled with Generalised Linear Modelling (GLM) regression modelling techniques. These models were used to predict the probability of occurrence of a species at any given location, defined in terms of its environmental attributes. A statistical correlation between the radioactive elements uranium, thorium and potassium, and terrain aspect was found. No statistical correlations were established between the radioactive elements and vegetation patterns; although we suspect these exist at finer scales of mapping. Radiometric data were identified as explanatory variables in the habitat models of all of the 32 vertebrate species examined, and used as illustration in the development of probabilistic spatial predictions of three species (Red Kangaroo, Macropus rufus; Lesser Hairy‐footed Dunnart, Sminthopsis youngsoni; and Rabbit, Oryctolagus cuniculus) in the bioregion. Our analyses suggest that radiometric data sets involving the radioactive elements: (uranium, thorium, and potassium), and vegetation could be used as predictors of biodiversity patterns at the bioregional and landscape level. This is an important finding given the challenges posed in undertaking broad‐scale biological surveys in the arid zone of Australia.     

From Big Noise to Big Data: Toward the Verification of Large Data sets for Understanding Regional Retail Flows

There has been much excitement among quantitative geographers about newly available data sets, characterized by high volume, velocity, and variety. This phenomenon is often labeled as “Big Data” and has contributed to methodological and empirical advances, particularly in the areas of visualization and analysis of social networks. However, a fourth v—veracity (or lack thereof)—has been conspicuously lacking from the literature. This article sets out to test the potential for verifying large data sets. It does this by cross‐comparing three unrelated estimates of retail flows—human movements from home locations to shopping centers—derived from the following geo‐coded sources: (1) a major mobile telephone service provider; (2) a commercial consumer survey; and (3) geotagged Twitter messages. Three spatial interaction models also provided estimates of flow: constrained and unconstrained versions of the “gravity model” and the recently developed “radiation model.” We found positive relationships between all data‐based and theoretical sources of estimated retail flows. Based on the analysis, the mobile telephone data fitted the modeled flows and consumer survey data closely, while flows obtained directly from the Twitter data diverged from other sources. The research highlights the importance of verification in flow data derived from new sources and demonstrates methods for achieving this.     

A Methodology for Neural Spatial Interaction Modeling

This paper attempts to develop a mathematically rigid and unified framework for neural spatial interaction modeling. Families of classical neural network models, but also less classical ones such as product unit neural network ones are considered for the cases of unconstrained and singly constrained spatial interaction flows. Current practice appears to suffer from least squares and normality assumptions that ignore the true integer nature of the flows and approximate a discrete‐valued process by an almost certainly misrepresentative continuous distribution. To overcome this deficiency we suggest a more suitable estimation approach, maximum likelihood estimation under more realistic distributional assumptions of Poisson processes, and utilize a global search procedure, called Alopex, to solve the maximum likelihood estimation problem. To identify the transition from underfitting to overfitting we split the data into training, internal validation, and test sets. The bootstrapping pairs approach with replacement is adopted to combine the purity of data splitting with the power of a resampling procedure to overcome the generally neglected issue of fixed data splitting and the problem of scarce data. In addition, the approach has power to provide a better statistical picture of the prediction variability. Finally, a benchmark comparison against the classical gravity models illustrates the superiority of both, the unconstrained and the origin constrained neural network model versions in terms of generalization performance measured by Kullback and Leibler's information criterion.     

ESTIMATION AND HYPOTHESIS TESTING FOR NONPARAMETRIC HEDONIC HOUSE PRICE FUNCTIONS

ABSTRACT In contrast to the rigid structure of standard parametric hedonic analysis, nonparametric estimators control for misspecified spatial effects while using highly flexible functional forms. Despite these advantages, nonparametric procedures are still not used extensively for spatial data analysis due to perceived difficulties associated with estimation and hypothesis testing. We demonstrate that nonparametric estimation is feasible for large datasets with many independent variables, offering statistical tests of individual covariates and tests of model specification. We show that fixed parameterization of distance to the nearest rapid transit line is a misspecification and that pricing of access to this amenity varies across neighborhoods within Chicago.     

BAYSIAN INFERENCE FOR ORDERED RESPONSE DATA WITH A DYNAMIC SPATIAL‐ORDERED PROBIT MODEL

ABSTRACT Many databases involve ordered discrete responses in a temporal and spatial context, including, for example, land development intensity levels, vehicle ownership, and pavement conditions. An appreciation of such behaviors requires rigorous statistical methods, recognizing spatial effects and dynamic processes. This study develops a dynamic spatial‐ordered probit (DSOP) model in order to capture patterns of spatial and temporal autocorrelation in ordered categorical response data. This model is estimated in a Bayesian framework using Gibbs sampling and data augmentation, in order to generate all autocorrelated latent variables. It incorporates spatial effects in an ordered probit model by allowing for interregional spatial interactions and heteroskedasticity, along with random effects across regions or any clusters of observational units. The model assumes an autoregressive, AR(1), process across latent response values, thereby recognizing time‐series dynamics in panel data sets. The model code and estimation approach is tested on simulated data sets, in order to reproduce known parameter values and provide insights into estimation performance, yielding much more accurate estimates than standard, nonspatial techniques. The proposed and tested DSOP model is felt to be a significant contribution to the field of spatial econometrics, where binary applications (for discrete response data) have been seen as the cutting edge. The Bayesian framework and Gibbs sampling techniques used here permit such complexity, in world of two‐dimensional autocorrelation.     

SELECTION BIAS IN SPATIAL ECONOMETRIC MODELS

ABSTRACT. The problem of spatial autocorrelation has been ignored in selection-bias models estimated with spatial data. Spatial autocorrelation is a serious problem in these models because the heteroskedasticity with which it commonly is associated causes inconsistent parameter estimates in models with discrete dependent variables. This paper proposes estimators for commonly-employed spatial models with selection bias. A maximum-likelihood estimator is applied to data on land use and values in 1920s Chicago. Evidence of significant heteroskedasticity and selection bias is found.     

EFFICIENT ALGORITHMS FOR CONSTRUCTING PROPER HIGHER ORDER SPATIAL LAG OPERATORS*

ABSTRACT. This paper extends the work of Blommestein and Koper (1992)–BK–on the construction of higher-order spatial lag operators without redundant and circular paths. For the case most relevant in spatial econometrics and spatial statistics, i.e., when contiguity between two observations (locations) is defined in a simple binary fashion, some deficiencies of the BK algorithms are outlined, corrected and an improvement suggested. In addition, three new algorithms are introduced and compared in terms of performance for a number of empirical contiguity structures. Particular attention is paid to a graph theoretic perspective on spatial lag operators and to the most efficient data structures for the storage and manipulation of spatial lags. The new forward iterative algorithm which uses a list form rather than a matrix to store the spatial lag information is shown to be several orders of magnitude faster than the BK solution. This allows the computation of proper higher-order spatial lags “on the fly” for even moderately large data sets such as 3,111 contiguous U. S. counties, which is not practical with the other algorithms.     

Bayesian Areal Wombling for Geographical Boundary Analysis

In the analysis of spatially referenced data, interest often focuses not on prediction of the spatially indexed variable itself, but on boundary analysis , that is, the determination of boundaries on the map that separate areas of higher and lower values. Existing boundary analysis methods are sometimes generically referred to as wombling , after a foundational article by Womble (1951). When data are available at point level (e.g., exact latitude and longitude of disease cases), such boundaries are most naturally obtained by locating the points of steepest ascent or descent on the fitted spatial surface (Banerjee, Gelfand, and Sirmans 2003). In this article, we propose related methods for areal data (i.e., data which consist only of sums or averages over geopolitical regions). Such methods are valuable in determining boundaries for data sets that, perhaps due to confidentiality concerns, are available only in ecological (aggregated) format, or are only collected this way (e.g., delivery of health-care or cost information). After a brief review of existing algorithmic techniques (including that implemented in the commercial software BoundarySeer), we propose a fully model-based framework for areal wombling, using Bayesian hierarchical models with posterior summaries computed using Markov chain Monte Carlo methods. We explore the suitability of various existing hierarchical and spatial software packages (notably S-plus and WinBUGS) to the task, and show the approach's superiority over existing nonstochastic alternatives, both in terms of utility and average mean square error behavior. We also illustrate our methods (as well as the solution of advanced modeling issues such as simultaneous inference) using colorectal cancer late detection data collected at the county level in the state of Minnesota.     

New Developments in the Use of Spatial Technology in Archaeology

Spatial technology is integral to how archaeologists collect, store, analyze, and represent information in digital data sets. Recent advances have improved our ability to look for and identify archaeological remains and have increased the size and complexity of our data sets. In this review we outline trends in visualization, data management, archaeological prospecting, modeling, and spatial analysis, as well as key advances in hardware and software. Due to developments in education, information technology, and landscape archaeology, the implementation of spatial technology has begun to move beyond superficial applications and is no longer limited to environmental deterministic approaches. In the future, spatial technology will increasingly change archaeology in ways that will enable us to become better practitioners, scholars, and stewards.