Closed‐Form Maximum Likelihood Estimates of Nearest Neighbor Spatial Dependence

Models of n² potential spatial dependencies among n observations spread irregularly over space seem unlikely to yield simple structure. However, the use of the nearest neighbor leads to a very parsimonious eigenstructure of the associated adjacency matrix which results in an extremely simple closed form for the log determinant. In turn, this leads to a closed‐form solution for the maximum likelihood estimates of the spatially autoregressive and mixed regressive spatially autoregressive models. With the closed‐form solution, one can find the neighbors and compute maximum likelihood estimates for 100,000 observations in under one minute. The model has theoretical, pedagogical, diagnostic, modeling, and methodological applications. For example, the model could serve as a more enlightened null hypothesis for geographic data than spatial independence.     

A Geostatistical Linear Regression Model for Small Area Data. 一种适用于小区域数据的地统计线性回归模型

We present a new linear regression model for use with aggregated, small area data that are spatially autocorrelated. Because these data are aggregates of individual‐level data, we choose to model the spatial autocorrelation using a geostatistical model specified at the scale of the individual. The autocovariance of observed small area data is determined via the natural aggregation over the population. Unlike lattice‐based autoregressive approaches, the geostatistical approach is invariant to the scale of data aggregation. We establish that this geostatistical approach also is a valid autoregressive model; thus, we call this approach the geostatistical autoregressive (GAR) model. An asymptotically consistent and efficient maximum likelihood estimator is derived for the GAR model. Finite sample evidence from simulation experiments demonstrates the relative efficiency properties of the GAR model. Furthermore, while aggregation results in less efficient estimates than disaggregated data, the GAR model provides the most efficient estimates from the data that are available. These results suggest that the GAR model should be considered as part of a spatial analyst's toolbox when aggregated, small area data are analyzed. More important, we believe that the GAR model's attention to the individual‐level scale allows for a more flexible and theory‐informed specification than the existing autoregressive approaches based on an area‐level spatial weights matrix. Because many spatial process models, both in geography and in other disciplines, are specified at the individual level, we hope that the GAR covariance specification will provide a vehicle for a better informed and more interdisciplinary use of spatial regression models with area‐aggregated data.     

SPATIAL ECONOMETRIC MODELING OF ORIGIN‐DESTINATION FLOWS*

ABSTRACT Standard spatial autoregressive models rely on spatial weight structures constructed to model dependence among n regions. Ways of parsimoniously modeling the connectivity among the sample of N=n² origin‐destination (OD) pairs that arise in a closed system of interregional flows has remained a stumbling block. We overcome this problem by proposing spatial weight structures that model dependence among the N OD pairs in a fashion consistent with standard spatial autoregressive models. This results in a family of spatial OD models introduced here that represent an extension of the spatial regression models described in Anselin (1988) .     

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.     

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.     

Stochastic Efficiency of Bayesian Markov Chain Monte Carlo in Spatial Econometric Models: An Empirical Comparison of Exact Sampling Methods

Spatial econometric specifications pose unique computational challenges to Bayesian analysis, making it difficult to estimate models efficiently. In the literature, the main focus has been on extending Bayesian analysis to increasingly complex spatial models. The stochastic efficiency of commonly used Markov Chain Monte Carlo (MCMC) samplers has received less attention by comparison. Specifically, Bayesian methods to analyze effective sample size and samplers that provide large effective size have not been thoroughly considered in the literature. Thus, we compare three MCMC techniques: the familiar Metropolis‐within‐Gibbs sampling, Slice‐within‐Gibbs sampling, and Hamiltonian Monte Carlo. The latter two methods, while common in other domains, are not as widely encountered in Bayesian spatial econometrics. We assess these methods across four different scenarios in which we estimate the spatial autoregressive parameter in a mixed regressive, spatial autoregressive specification (or, spatial lag model). We find that off‐the‐shelf implementations of the newer high‐yield simulation techniques require significant adaptation to be viable. We further find that the effective sizes are often significantly smaller than nominal sizes. In addition, we find that stopping simulation early may understate posterior credible interval widths when effective sample size is small. More broadly, we suggest that sample information and stopping rules deserve more attention in both applied and basic Bayesian spatial econometric research.     

A Comparison of Three Exploratory Methods for Cluster Detection in Spatial Point Patterns

This paper compares the performances of three exploratory methods for cluster detection in spatial point patterns where the at-risk population is known. After reviewing two existing methods, Openshaw et al. (1987) and Besag and Newell (1991), an alternative method is introduced. These three methods are then compared empirically using two point patterns drawn from a disaggregate housing database consisting of 28,832 observations. Each observation in the data set contains attributes of single-family detached dwellings in the City of Amherst, New York. This paper provides some new insights into the performance of the three methods, as previous applications have used spatially aggregated (and hence rather inaccurate) data. The paper also demonstrates the utility of GIS for this type of spatial analysis.     

Does Econometric Methodology Matter? An Analysis of Public Policy Using Spatial Econometric Techniques

A popular approach to examining the effects of public policy has been to rely on a spatial data sample of border counties as in Holmes (1998)—border counties from a sample of states that are used in conjunction with least‐squares estimation techniques in an attempt to isolate the policy impact while controlling for spatial dependence that often arises from latent or unobserved variables. This technique is in the spirit of control‐group methodologies from the laboratory sciences. This paper contrasts border‐county estimation results from Holmes' (1998) approach and those from a related methodology set forth in Holcombe and Lacombe (2003), with estimates from a spatial autoregressive model explicitly accounting for within‐state and between‐state public policy effects. As an illustration, the paper examines the effects of Aid to Families with Dependent Children (AFDC) and Food Stamp payments on female‐headed households and female labor force participation using the three different methods.     

EMPIRICAL IMPLICATIONS OF ADVANCES IN CANONICAL THEORY*

Canonical correlation has seen growing acceptance in geographical research as a tool for analysing the interrelationships between two sets of variables.¹ It provides a natural extension to the multivariate case of simple correlation analysis introduced into the discipline in the fifties for measuring the degree of areal association between two individual variables.² It has also proved valuable for forging a link between traditional geographic variables measuring the attributes of places and those indicating interactions among them.³ Recently, major developments in canonical theory have occurred which provide two major benefits for geographical research.⁴ First, asymmetrical regression relationships in addition to symmetrical correlation relationships between two variable sets can be determined. Researchers can use canonical regression to examine the degree to which one variable set is capable of predicting the other, in addition to canonical correlation which examines the symmetrical interrelationships between the two.⁵ Secondly, much improved methods are available for measuring the number, strength, and nature of the interrelationships between the two variable sets, and for assessing the adequacy of the canonical model in general.⁸ The purpose of this paper is to provide an overview of these developments and, more particularly, to explore their implications for the validity of empirical results obtained in earlier applications of canonical analysis. This is not intended as a criticism of these studies but rather as an attempt to further our understanding of spatial structure and process through re-examination of existing data in the light of refined techniques.     

EISCAT observations of ion composition and temperature anisotropy in the high-latitude F-region

The papers by Winseret al. [(1990) J. atmos. terr. Phys.52, 501] and Häggström and Collis [(1990) J. atmos. terr. Phys.52, 519] used plasma flows and ion temperatures, as measured by the EISCAT tristatic incoherent scatter radar, to investigate changes in the ion composition of the ionospheric F-layer at high latitudes, in response to increases in the speed of plasma convection. These studies reported that the ion composition rapidly changed from mainly O⁺ to almost completely (>90%) molecular ions, following rapid increases in ion drift speed by >1 km s⁻¹. These changes appeared inconsisent with theoretical considerations of the ion chemistry, which could not account for the large fractions of molecular ions inferred from the obsevations. In this paper, we discuss two causes of this discrepancy. First, we reevaluate the theoretical calculations for chemical equilibrium and show that, if we correct the derived temperatures for the effect of the molecular ions, and if we employ more realistic dependences of the reaction rates on the ion temperature, the composition changes derived for the faster convection speeds can be explained. For the Winser et al. observations with the radar beam at an aspect angle of ϕ = 54.7° to the geomagnetic field, we now compute a change to 89% molecular ions in < 2 min, in response to the 3 km s⁻¹ drift. This is broadly consistent with the observations. But for the two cases considered by Häggström and Collis, looking along the field line (ϕ = 0°), we compute the proportion of molecular ions to be only 4 and 16% for the observed plasma drifts of 1.2 and 1.6 km s⁻¹, respectively. These computed proportions are much smaller than those derived experimentally (70 and 90%). We attribute the differences to the effects of non-Maxwellian, anisotropic ion velocity distribution functions. We also discuss the effect of ion composition changes on the various radar observations that report anisotropies of ion temperature.     

Local Forms of Spatial Analysis

Local forms of spatial analysis focus on exceptions to the general trends represented by more traditional global forms of spatial analysis. There is currently a rapid expansion in the development of such techniques but their history almost exactly parallels that of Geographical Analysis, with the first examples of local analysis appearing in the late 1960s. Indeed, Geographical Analysis has published many of the significant contributions in this field. This paper reviews the development of local forms of spatial analysis and assesses the current situation. Following a discussion on the nature and importance of local analysis, examples are given of local forms of point pattern analysis; local graphical approaches; local measures of spatial dependency; the spatial expansion method; adaptive filtering; multilevel modeling; geographically weighted regression; random coefficients models; autoregressive models; and local forms of spatial interaction models.     

A SPATIAL AND TEMPORAL AUTOREGRESSIVE LOCAL ESTIMATION FOR THE PARIS HOUSING MARKET

ABSTRACT This original study examines the potential of a spatiotemporal autoregressive Local (LSTAR) approach in modeling transaction prices for the housing market in inner Paris. We use a data set from the Paris Region notary office (Chambre des notaires d’Île‐de‐France) which consists of approximately 250,000 transactions units between the first quarter of 1990 and the end of 2005. We use the exact X–Y coordinates and transaction date to spatially and temporally sort each transaction. We first choose to use the STAR approach proposed by Pace et al., 1998 . This method incorporates a spatiotemporal filtering process into the conventional hedonic function and attempts to correct for spatial and temporal correlative effects. We find significant estimates of spatial dependence effects. Moreover, using an original methodology, we find evidence of a strong presence of both spatial and temporal heterogeneity in the model. It suggests that spatial and temporal drifts in households socio‐economic profiles and local housing market structure effects are certainly major determinants of the price level for the Paris Housing Market.     

The K‐Function Method on a Network and Its Computational Implementation

This paper proposes two statistical methods, called the network K‐function method and the network cross K‐function method, for analyzing the distribution of points on a network. First, by extending the ordinary K‐function method defined on a homogeneous infinite plane with the Euclidean distance, the paper formulates the K‐function method and the cross K‐function method on a finite irregular network with the shortest‐path distance. Second, the paper shows advantages of the network K‐function methods, such as that the network K‐function methods can deal with spatial point processes on a street network in a small district, and that they can exactly take the boundary effect into account. Third, the paper develops the computational implementation of the network K‐functions, and shows that the computational order of the K‐function method is O(n²_Q log n_Q) and that of the network cross K‐function is O(n_Q log U3Q), where n_Q is the number of nodes of a network.     

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.     

Beyond Moran's I: Testing for Spatial Dependence Based on the Spatial Autoregressive Model

The statistic known as Moran's I is widely used to test for the presence of spatial dependence in observations taken on a lattice. Under the null hypothesis that the data are independent and identically distributed normal random variates, the distribution of Moran's I is known, and hypothesis tests based on this statistic have been shown in the literature to have various optimality properties. Given its simplicity, Moran's I is also frequently used outside of the formal hypothesis-testing setting in exploratory analyses of spatially referenced data; however, its limitations are not very well understood. To illustrate these limitations, we show that, for data generated according to the spatial autoregressive (SAR) model, Moran's I is only a good estimator of the SAR model's spatial-dependence parameter when the parameter is close to 0. In this research, we develop an alternative closed-form measure of spatial autocorrelation, which we call APLE , because it is an approximate profile-likelihood estimator (APLE) of the SAR model's spatial-dependence parameter. We show that APLE can be used as a test statistic for, and an estimator of, the strength of spatial autocorrelation. We include both theoretical and simulation-based motivations (including comparison with the maximum-likelihood estimator), for using APLE as an estimator. In conjunction, we propose the APLE scatterplot, an exploratory graphical tool that is analogous to the Moran scatterplot, and we demonstrate that the APLE scatterplot is a better visual tool for assessing the strength of spatial autocorrelation in the data than the Moran scatterplot. In addition, Monte Carlo tests based on both APLE and Moran's I are introduced and compared. Finally, we include an analysis of the well-known Mercer and Hall wheat-yield data to illustrate the difference between APLE and Moran's I when they are used in exploratory spatial data analysis.     

Investigations of ionospheric radio wave absorption processes using imaging riometer techniques

The recent development of imaging riometer techniques has enabled a range of new, interesting observations of the complex dynamics of auroral and polar radio wave absorption events. These events mostly relate to the precipitation of energetic particles, creating enhanced ionization in the D-region. However, E-region heating by large electric fields and F-region electron density enhancements may also—at times—be responsible for observable absorption effects. Observations of ionospheric radio wave absorption processes using imaging riometer techniques may provide detailed characteristics of the spatial and temporal structures of small-scale disturbance events, velocity vectors for drifting features and frequency spectra for modulated events. This presentation will give a brief summary of imaging riometer techniques and a survey of existing and planned imaging riometer installations. Furthermore, the characteristics of frequently occurring absorption event types are summarized. In a companion paper imaging riometer observations are presented for some selected absorption events.     

Boards and Cords: Discriminating Types of Artificial Cranial Deformation in Prehispanic South Central Andean Populations

For over a century, a number of ambiguous typologies have been employed to distinctly categorise types of artificial cranial deformation. This paper provides a quantitative method, based on multiple dimensions and discriminant function analysis, by which to assign skulls not only into discrete categories: deformed or not, but also by type: annular or tabular. A series of prehispanic, adult, human crania (n = 469) from archaeological sites in Argentina, Bolivia, Chile and Peru represented by both normal and artificially deformed specimens, provide craniometric data for four measurements across the vault: maximum cranial length, breadth and height and the frontal chord. These data are used to develop three indices which in turn are used to compute two discriminant functions. Results are plotted on a territorial map whereby the type of deformity can be determined. When these methods were applied to a comparative cranial sample of nondeformed skulls from South America, 100% of the samples was found to be nondeformed. When these methods were applied to the samples which were subjectively classified a priori by the first author as nondeformed, 81.3% of the samples were found to be nondeformed. This study demonstrates the value of a more objective and quantitative method by which to classify artificial cranial deformation, and thus provides a new approach. Copyright © 2011 John Wiley & Sons, Ltd.     

Multi-Scale and Nested-Intensity Sampling Techniques for Archaeological Survey

Abstract

This paper discusses sampling techniques for archaeological survey that are directed toward evaluating the properties of surface artifact distributions. The sampling techniques we experimented with consist of a multi-scale sampling plot developed in plant ecology and the use of a nested-intensity survey design. We present results from the initial application of these methods. The sampling technique we borrowed from plant ecology is the Modified-Whittaker multiscale sampling plot, which gathers observations at the spatial scales of 1 sq m, 10 sq m, 100 sq m, and 1000 sq m. Nested-intensity surveys gather observations on the same sample units at multiple resolutions. We compare the results of a closely-spaced walking survey, a crawling survey, and a test excavation to a depth of 10 cm. These techniques were applied to ten 20 × 50 m survey plots distributed over an area of 418 ha near the Hudson-Meng Bison Bonebed in NW Nebraska. These approaches can significantly improve the accuracy of survey data. Our results show that high-resolution coverage techniques overlook more material than archaeologists have suspected. The combined approaches of multi-scale and nested-intensity sampling provide new tools to improve our ability to investigate the properties of surface records.     

Augmenting the landscape scene: students as participatory evaluators of mobile geospatial technologies

This paper provides a two-phase study to compare alternative techniques for augmenting landscape scenes on geography fieldtrips. The techniques were: a pre-prepared acetate overlay; a custom-designed mobile field guide; a locative media app on a smartphone; a virtual globe on a tablet PC; a head-mounted virtual reality display, and a geo-wand style mobile app. In one field exercise the first five techniques were compared through analysis of interviews and student video diaries, combined with direct observation. This identified a particular challenge of how to direct user attention correctly to relevant information in the field of view. To explore this issue in more detail, a second field exercise deployed “Zapp”, a bespoke geo-wand-style app capable of retrieving information about distant landscape features. This was evaluated using first-person video and spatial logging of in-field interactions. This paper reflects upon the relative merits of these approaches and highlights particular challenges of using technology to mimic a human field guide in pointing out specific aspects of the landscape scene. We also explore the role of students acting as design informants and research co-participants, which can be mutually beneficial in promoting a critical appreciation of the role of technology to support learning about the landscape.     

MORE THAN ONE WAY TO STUDY A BUILDING: APPROACHES TO PREHISTORIC HOUSEHOLD AND SETTLEMENT SPACE

Summary.   This article reviews a number of research methodologies used to record household and settlement architecture and assesses their value in the investigation of the human use of prehistoric built space. It exemplifies, through case studies, five broad approaches to, and research techniques associated with, the investigation of such architecture. These approaches are: architectural form; the spatial distribution of activities; continuity and standardization; the relationship between built and non-built space; and human patterns of movement. Then, drawing mainly on Near Eastern, and particularly Anatolian, material, it shows how a sixth approach, the use of ethnographic observation and analogy, provides insights into functional and seasonal variations in spatial use, patterns of movement and social organization. It identifies seven categories of data collection and nine observations drawn from the ethnographic material which together provide an investigative and interpretative framework for the study of early farming communities in the Near East and elsewhere.