Modeling the Uncertainty of Slope and Aspect Estimates Derived from Spatial Databases

Estimates of slope and aspect are commonly made from digital elevation models (DEMs), and are subject to the uncertainty present in such models. We show that errors in slope and aspect depend on the spatial structure of DEM errors. We propose a general-purpose model of DEM errors in which a spatially auto-regressive random field is added as a disturbance term to elevations. In addition, we propose a general procedure for propagating such errors through GIS operations. In the absence of explicit information on the spatial structure of DEM errors, we demonstrate the potential utility of a worst-case analysis. A series of simulations are used to make general observations about the nature and severity of slope and aspect errors.     

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.     

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.     

Spatial‐temporal patterns of snow cover in western Canada

Snow cover is often measured as snow‐water equivalent (SWE), which refers to the amount of water stored in a snow pack that would be available upon melting. Snow cover and SWE represent a source of local snow‐melt release, and are sensitive to regional and global atmospheric circulation, and changes in climate. Monitoring SWE using satellite‐based passive microwave radiometry has provided nearly three decades of continuous data for North America. The availability of spatially and temporally extensive SWE data enables a better understanding of the nature of space‐time trends in snow cover, changes in these trends and linking these trends to underlying landscape and terrain characteristics. To address these interests, we quantify the spatial pattern of SWE by applying a local measure of spatial autocorrelation to 25 years of mean February SWE derived from passive microwave retrievals. Using a method for characterizing the temporal trends in the spatial pattern of SWE, temporal trends and variability in spatial autocorrelation are quantified. Results indicate that within the Canadian Prairies, extreme values of SWE are becoming more spatially coherent, with potential impacts on water availability, and hazards such as flooding. These results also highlight the need for Canadian ecological management units that consider winter conditions.     

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.     

Measuring Spatial Autocorrelation of Vectors

This article introduces measures to quantify spatial autocorrelation for vectors. In contrast to scalar variables, spatial autocorrelation for vectors involves an assessment of both direction and magnitude in space. Extending conventional approaches, measures of global and local spatial associations for vectors are proposed, and the associated statistical properties and significance testing are discussed. The new measures are applied to study the spatial association of taxi movements in the city of Shanghai. Complications due to the edge effect are also examined.     

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

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

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.     

A Map Overlay Error Model Based on Boundary Geometry

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

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.     

Lagrange Multiplier Test Diagnostics for Spatial Dependence and Spatial Heterogeneity

Several diagnostics for the assessment of model misspecification due to spatial dependence and spatial heterogeneity are developed as an application of the Lagrange Multiplier principle. The starting point is a general model which incorporates spatially lagged dependent variables, spatial residual autocorrelation and heteroskedasticity. Particular attention is given to tests for spatial residual autocorrelation in the presence of spatially lagged dependent variables and in the presence of heteroskedasticity. The tests are formally derived and illustrated in a number of simple empirical examples.     

PROBIT WITH SPATIAL AUTOCORRELATION

ABSTRACT. Commonly-employed spatial autocorrelation models imply heteroskedastic errors, but heteroskedasticity causes probit to be inconsistent. This paper proposes and illustrates the use of two categories of estimators for probit models with spatial autocorrelation. One category is based on the EM algorithm, and requires repeated application of a maximum-likelihood estimator. The other category, which can be applied to models derived using the spatial expansion method, only requires weighted least squares.     

Unconditional Maximum Likelihood Estimation of Linear and Log-Linear Dynamic Models for Spatial Panels

This article hammers out the estimation of a fixed effects dynamic panel data model extended to include either spatial error autocorrelation or a spatially lagged dependent variable. To overcome the inconsistencies associated with the traditional least-squares dummy estimator, the models are first-differenced to eliminate the fixed effects and then the unconditional likelihood function is derived taking into account the density function of the first-differenced observations on each spatial unit. When exogenous variables are omitted, the exact likelihood function is found to exist. When exogenous variables are included, the pre-sample values of these variables and thus the likelihood function must be approximated. Two leading cases are considered: the Bhargava and Sargan approximation and the Nerlove and Balestra approximation. As an application, a dynamic demand model for cigarettes is estimated based on panel data from 46 U.S. states over the period from 1963 to 1992.     

我国装备制造业的创新、知识溢出和产学研合作——基于一个扩展的知识生产函数方法

知识溢出的本地化现象揭示了地理边界的限制机理,然而越来越多的研究表明知识溢出也可发生在远距离的空间范围内。本文以我国装备制造业为例,采用扩展的知识生产函数方法探讨大学和科研机构的知识溢出对区域创新的影响。结果表明,大学和科研机构能够产生强烈的本地和跨区域的知识溢出效应,地理空间和创新网络在跨区域知识溢出中扮演重要角色。其中,产学研合作产生的知识溢出不仅局限于区域尺度,也发生在更高的国家尺度;在地理空间的制约影响下,高素质劳动力的流动对区域创新的影响十分显著且稳定,而大学和科研机构的衍生溢出机制则不那么明显,其原因可能与空间尺度选择有关。     

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.     

Properties of Tests for Spatial Dependence in Linear Regression Models

Based on a large number of Monte Carlo simulation experiments on a regular lattice, we compare the properties of Moran's I and Lagrange multiplier tests for spatial dependence, that is, for both spatial error autocorrelation and for a spatially lagged dependent variable. We consider both bias and power of the tests for six sample sizes, ranging from twenty-five to 225 observations, for different structures of the spatial weights matrix, for several underlying error distributions, for misspecified weights matrices, and for the situation where boundary effects are present. The results provide an indication of the sample sizes for which the asymptotic properties of the tests can be considered to hold. They also illustrate the power of the Lagrange multiplier tests to distinguish between substantive spatial dependence (spatial lag) and spatial dependence as a nuisance (error autocorrelation).     

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.     

The Size-adjusted Critical Region of Moran's I Test Statistics for Spatial Autocorrelation and Its Application to Geographical Areas

In this article, we explore the expression of the asymptotic approximation of the distribution function of Moran's I test statistic for the check of spatial autocorrelation, and we derive a more accurate critical value, which gives the rejection region similar to significant level α to the order of N^?1 (N = sample size). We show that in some cases our procedure effectively finds the significance of positive spatial autocorrelation in the problem testing for the lack of the spatial autocorrelation. Compared with our method, the testing procedure of Cliff and Ord (1971) is clearly ad hoc and should not be applied blindly, as they pointed out. Our procedure is derived from the theory of asymptotic expansion. We numerically analyze four types of county systems with rectangular lattices and three regional areas with irregular lattices.     

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.