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.     

A New Perspective about Moran's Coefficient: Spatial Autocorrelation as a Linear Regression Problem. Moran系数的新视角：空间自相关视为线性回归问题

The computation of Moran's index of spatial autocorrelation requires the definition of a spatial weighting matrix. The eigendecomposition of this doubly centered matrix (i.e., one that forces the sums of all rows and columns to equal zero) has interesting properties that have been exploited in various contexts: distribution properties of the Moran coefficient (MC), spatial filtering in linear models, generalized linear models, and multivariate analysis. In this article, this eigendecomposition is used to propose a new view of MC based on its interpretation in the simple context of linear regression. I use this interpretation to demonstrate the different properties of MC and also the inefficiency of this index in some situations involving simultaneous positive and negative spatial autocorrelation. I propose some new statistics and procedures for testing spatial autocorrelation, and conduct a simulation study to evaluate these new approaches.     

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.     

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.     

Spatial autocorrelation tests and the Classic Maya collapse: Methods and inferences

Spatial autocorrelation, resulting in pattern or structure in geographically distributed data, is discussed in theoretical and practical terms. Tests for spatial autocorrelation are presented, along with an explication of the relationship between autocorrelation models, the product-moment correlation coefficient and the spatial autocorrelation test statistic. Two archaeological examples illustrate the application of the auto-correlation test statistic. The first uses a hypothetical data set, which shows the type of map patterns that appear with various levels of spatial autocorrelation, and the second examines the terminal distribution of long-count-dated monuments at lowland Classic Maya sites. The results of the second example fail to support arguments for simple patterning in the cessation of the erection of these monuments and, by inference, in the Maya collapse itself. Finally, it is argued that while the identification of spatial autocorrelation is often the goal of spatial analyses, the presence of autocorrelation violates the assumptions of certain statistics used in such analyses.     

A CONDITIONALLY PARAMETRIC PROBIT MODEL OF MICRODATA LAND USE IN CHICAGO

Spatial data sets pose challenges for discrete choice models because the data are unlikely to be independently and identically distributed. A conditionally parametric spatial probit model is amenable to very large data sets while imposing far less structure on the data than conventional parametric models. We illustrate the approach using data on 474,170 individual lots in the City of Chicago. The results suggest that simple functional forms are not appropriate for explaining the spatial variation in residential land use across the entire city.     

Evaluating the Temporal and Spatial Heterogeneity of the European Convergence Process, 1980–1999*

Abstract. In this paper, we suggest a framework that allows testing simultaneously for temporal heterogeneity, spatial heterogeneity, and spatial autocorrelation in β‐convergence models. Based on a sample of 145 European regions over the 1980–1999 period, we estimate a Seemingly Unrelated Regression Model with spatial regimes and spatial autocorrelation for two sub‐periods: 1980–1989 and 1989–1999. The assumption of temporal independence between the two periods is rejected, and the estimation results point to the presence of spatial error autocorrelation in both sub‐periods and spatial instability in the second sub‐period, indicating the formation of a convergence club between the peripheral regions of the European Union.     

Comparative Spatial Filtering in Regression Analysis

One approach to dealing with spatial autocorrelation in regression analysis involves the filtering of variables in order to separate spatial effects from the variables’ total effects. In this paper we compare two filtering approaches, both of which allow spatial statistical analysts to use conventional linear regression models. Getis’ filtering approach is based on the autocorrelation observed with the use of the G_i local statistic. Griffith's approach uses an eigenfunction decomposition based on the geographic connectivity matrix used to compute a Moran's I statistic. Economic data are used to compare the workings of the two approaches. A final comparison with an autoregressive model strengthens the conclusion that both techniques are effective filtering devices, and that they yield similar regression models. We do note, however, that each technique should be used in its appropriate context.     

"The Problem of Spatial Autocorrelation" and Local Spatial Statistics

This article examines the relationship between spatial dependency and spatial heterogeneity, two properties unique to spatial data. The property of spatial dependence has led to a large body of research into spatial autocorrelation and also, largely independently, into geostatistics. The property of spatial heterogeneity has led to a growing awareness of the limitation of global statistics and the value of local statistics and local statistical models. The article concludes with a discussion of how the two properties can be accommodated within the same modelling framework.     

Robustness of Spatial Autocorrelation Specifications: Some Monte Carlo Evidence

Abstract This paper examines the robustness of various models of spatial autocorrelation through a series of Monte Carlo experiments in which each model takes a turn at the data generator. The generated data are then used to estimate all of the models. The estimated models are evaluated primarily on their predictive power.     

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.     

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.     

A Wavelet‐Based Extension of Generalized Linear Models to Remove the Effect of Spatial Autocorrelation. 基于小波扩展广义线性模型消除空间自相关的影响

Biogeographical studies are often based on a statistical analysis of data sampled in a spatial context. However, in many cases standard analyses such as regression models violate the assumption of independently and identically distributed errors. In this article, we show that the theory of wavelets provides a method to remove autocorrelation in generalized linear models (GLMs). Autocorrelation can be described by smooth wavelet coefficients at small scales. Therefore, data can be decomposed into uncorrelated and correlated parts. Using an appropriate linear transformation, we are able to extend GLMs to autocorrelated data. We illustrate our new method, called the wavelet‐revised model (WRM), by applying it to multiple regression with response variables conforming to various distributions. Results are presented for simulated data and real biogeographical data (species counts of the plant genus Utricularia [bladderworts] in grid cells throughout Germany). The results of our WRM are compared with those of GLMs and models based on generalized estimating equations. We recommend WRMs, especially as a method that allows for spatial nonstationarity. The technique developed for lattice data is applicable without any prior knowledge of the real autocorrelation structure.     

Testing for Local Spatial Autocorrelation in the Presence of Global Autocorrelation

A fundamental concern of spatial analysts is to find patterns in spatial data that lead to the identification of spatial autocorrelation or association. Further, they seek to identify peculiarities in the data set that signify that something out of the ordinary has occurred in one or more regions. In this paper we provide a statistic that tests for local spatial autocorrelation in the presence of the global autocorrelation that is characteristic of heterogeneous spatial data. After identifying the structure of global autocorrelation, we introduce a new measure that may be used to test for local structure. This new statistic Oi is asymptotically normally distributed and allows for straightforward tests of hypotheses. We provide several numerical examples that illustrate the performance of this statistic and compare it with another measure that does not account for global structure.     

A FURTHER EXPLORATION INTO THE ROBUSTNESS OF SPATIAL AUTOCORRELATION SPECIFICATIONS*

ABSTRACT In a recent study, the robustness of linear models with various spatial autocorrelation specifications was assessed through Monte Carlo experiments, and the geostatistical models were concluded to dominate the weight matrix models in the prediction. The present study tests the soundness of this conclusion with a different framework for prediction and presents some experimental results that can call into doubt the dominance of the geostatistical models over the weight matrix models.     

Income Diffusion and Spatial Econometric Models

This paper considers the development of linked small-area spatial econometric models in which income flows between the areas constitute an important part of model specification. The attributes of income diffusion processes are considered in detail. Both static models and temporal income diffusion models are discussed. The spatial and spatial-temporal autocorrelation functions are derived which provide a measure of the spatial distribution of income. The Green function is also derived as a measure of income impulses through the system of regions. An important aspect of the paper is to relate properties of these functions to key economic parameters in particular propensities to spend locally and to spend nonlocally.     

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.     

Toward a Theory of Spatial Statistics: Another Step Forward

An update is presented to an earlier piece outlining some steps that needed to be taken toward the establishment of a theory of spatial statistics. Findings that have appeared since this first paper are summarized and interpreted, and extensions and suggestions are offered for the further establishment of a basis for a theory of spatial statistics. Topics include boundary considerations, the role of latent spatial dependencies, and small-sample-size issues. These topics embrace the problems of data transformations, edge effect bias, reference sampling distributions, multivariate autocorrelation models, conditional expectations, and higher-order autoregressive structures. In part, a course is charted for the next step to be taken.     

On the Performance of the Subtour Elimination Constraints Approach for the p‐Regions Problem: A Computational Study

The p‐regions is a mixed integer programming (MIP) model for the exhaustive clustering of a set of n geographic areas into p spatially contiguous regions while minimizing measures of intraregional heterogeneity. This is an NP‐hard problem that requires a constant research of strategies to increase the size of instances that can be solved using exact optimization techniques. In this article, we explore the benefits of an iterative process that begins by solving the relaxed version of the p‐regions that removes the constraints that guarantee the spatial contiguity of the regions. Then, additional constraints are incorporated iteratively to solve spatial discontinuities in the regions. In particular we explore the relationship between the level of spatial autocorrelation of the aggregation variable and the benefits obtained from this iterative process. The results show that high levels of spatial autocorrelation reduce computational times because the spatial patterns tend to create spatially contiguous regions. However, we found that the greatest benefits are obtained in two situations: (1) when ; and (2) when the parameter p is close to the number of clusters in the spatial pattern of the aggregation variable.     

Exploring Spatial Patterns Using an Expanded Spatial Autocorrelation Framework

Spatial autocorrelation (SA) is regarded as an important dimension of spatial pattern. SA measures usually consist of two components: measuring the similarity of attribute values and defining the spatial relationships among observations. The latter component is often represented by a spatial weights matrix that predefines spatial relationship between observations in most measures. Therefore, SA measures, in essence, are measures of attribute similarity, conditioned by spatial relationship. Another dimension of spatial pattern can be explored by controlling observations to be compared based upon the degree of attribute similarity. The resulting measures are spatial proximity measures of observations, meeting predefined attribute similarity criteria. Proposed measures reflect degrees of clustering or dispersion for observations meeting certain levels of attribute similarity. An existing spatial autocorrelation framework is expanded to a general framework to evaluate spatial patterns and can accommodate the proposed approach measuring proximity. Analogous to the concept of variogram, clustergram is proposed to show the levels of spatial clustering over a range of attribute similarity, or attribute lags. Specific measures based on the proposed approach are formulated and applied to a hypothetical landscape and an empirical example, showing that these new measures capture spatial pattern information not reflected by traditional spatial autocorrelation measures.