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For decades, conventional wisdom maintained that binary 0–1 Bernoulli random variables cannot contain extra-binomial variation. Taking an unorthodox stance, Hilbe actively disagreed, especially for correlated observation instances, arguing that the universally adopted diagnostic Pearson or deviance dispersion statistics are insensitive to a variance anomaly in a binary context, and hence simply fail to detect it. However, having the intuition and insight to sense the existence of this departure from standard mathematical statistical theory, but being unable to effectively isolate it, he classified this particular over-/under-dispersion phenomenon as implicit. This paper explicitly exposes his hidden quantity by demonstrating that the variance in/deflation it represents occurs in an underlying predicted beta random variable whose real number values are rounded to their nearest integers to convert to a Bernoulli random variable, with this discretization masking any materialized extra-Bernoulli variation. In doing so, asymptotics linking the beta-binomial and Bernoulli distributions show another conventional wisdom misconception, namely a mislabeling substitution involving the quasi-Bernoulli random variable; this undeniably is not a quasi-likelihood situation. A public bell pepper disease dataset exhibiting conspicuous spatial autocorrelation furnishes empirical examples illustrating various features of this advocated proposition.

Eigenvector spatial filtering (ESF) is becoming a popular way to address spatial dependence. Recently, a random effects specification of ESF (RE-ESF) is receiving considerable attention because of its usefulness for spatial dependence analysis considering spatial confounding. The objective of this study was to analyze theoretical properties of RE-ESF and extend it to overcome some of its disadvantages. We first compare the properties of RE-ESF and ESF with geostatistical and spatial econometric models. There, we suggest two major disadvantages of RE-ESF: it is specific to its selected spatial connectivity structure, and while the current form of RE-ESF eliminates the spatial dependence component confounding with explanatory variables to stabilize the parameter estimation, the elimination can yield biased estimates. RE-ESF is extended to cope with these two problems. A computationally efficient residual maximum likelihood estimation is developed for the extended model. Effectiveness of the extended RE-ESF is examined by a comparative Monte Carlo simulation. The main findings of this simulation are as follows: Our extension successfully reduces errors in parameter estimates; in many cases, parameter estimates of our RE-ESF are more accurate than other ESF models; the elimination of the spatial component confounding with explanatory variables results in biased parameter estimates; efficiency of an accuracy maximization-based conventional ESF is comparable to RE-ESF in many cases.

Virtually all remotely sensed data contain spatial autocorrelation, which impacts upon their statistical features of uncertainty through variance inflation, and the compounding of duplicate information. Estimating the nature and degree of this spatial autocorrelation, which is usually positive and very strong, has been hindered by computational intensity associated with the massive number of pixels in realistically-sized remotely-sensed images, a situation that more recently has changed. Recent advances in spatial statistical estimation theory support the extraction of information and the distilling of knowledge from remotely-sensed images in a way that accounts for latent spatial autocorrelation. This paper summarizes an effective methodological approach to achieve this end, illustrating results with a 2002 remotely sensed-image of the Florida Everglades, and simulation experiments. Specifically, uncertainty of spatial autocorrelation parameter in a spatial autoregressive model is modeled with a beta-beta mixture approach and is further investigated with three different sampling strategies: coterminous sampling, random sub-region sampling, and increasing domain sub-regions. The results suggest that uncertainty associated with remotely-sensed data should be cast in consideration of spatial autocorrelation. It emphasizes that one remaining challenge is to better quantify the spatial variability of spatial autocorrelation estimates across geographic landscapes.

Gravity-type spatial interaction models have been popularly utilized in modeling cross-sectional migration data, but their misspecification also has been raised in the literature. This misspecification issue principally concerns an insufficient accounting of underlying effects of spatial structure, including the presence of network autocorrelation among migration flows. Recent studies reveal that spatial interaction models are significantly improved by incorporating network autocorrelation in log-linear or Poisson regression estimation techniques, which are common estimation methods for spatial interaction models. However, when migration flows are structured as a panel data set from multiple time periods, the data set is likely to display temporal correlation within each measurement unit (here, each flow between a dyad of an origin and a destination) as well as network autocorrelation within each time period. Hence, spatial interaction models should be explicitly specified to account for these two different types of correlation structure. Using the eigenvector spatial filtering technique, this article outlines how to model network autocorrelation among migration flows structured through multiple time spans in either a linear or a generalized linear mixed model. An analysis of annual U.S. interstate migration data reported by the U.S. Internal Revenue Service shows that incorporation of two different types of autocorrelation leads to an improvement of model fitting and more intuitive parameter estimates.

Both historically and in terms of practiced academic organization, the anticipation should be that a flourishing synergistic interface exists between statistics and operations research in general, and between spatial statistics/econometrics and spatial optimization in particular. Unfortunately, for the most part, this expectation is false. The purpose of this paper is to address this existential missing link by focusing on the beneficial contributions of spatial statistics to spatial optimization, via spatial autocorrelation (i.e., dis/similar attribute values tend to cluster together on a map), in order to encourage considerably more future collaboration and interaction between contributors to their two parent bodies of knowledge. The key basic statistical concept in this pursuit is the median in its bivariate form, with special reference to the global and to sets of regional spatial medians. One-dimensional examples illustrate situations that the narrative then extends to two-dimensional illustrations, which, in turn, connects these treatments to the spatial statistics centrography theme. Because of computational time constraints (reported results include some for timing experiments), the summarized analysis restricts attention to problems involving one global and two or three regional spatial medians. The fundamental and foundational spatial, statistical, conceptual tool employed here is spatial autocorrelation: geographically informed sampling designs—which acknowledge a non-random mixture of geographic demand weight values that manifests itself as local, homogeneous, spatial clusters of these values—can help spatial optimization techniques determine the spatial optima, at least for location-allocation problems. A valuable discovery by this study is that existing but ignored spatial autocorrelation latent in georeferenced demand point weights undermines spatial optimization algorithms. All in all, this paper should help initiate a dissipation of the existing isolation between statistics and operations research, hopefully inspiring substantially more collaborative work by their professionals in the future.

Variable transformations have a long and celebrated history in statistics, one that was rather academically glamorous at least until generalized linear models theory eclipsed their nurturing normal curve theory role. Still, today it continues to be a covered topic in introductory mathematical statistics courses, offering worthwhile pedagogic insights to students about certain aspects of traditional and contemporary statistical theory and methodology. Since its inception in the 1930s, it has been plagued by a paucity of adequate back-transformation formulae for inverse/reciprocal functions. A literature search exposes that, to date, the inequality E(1/X) ≤ 1/(E(X), which often has a sizeable gap captured by the inequality part of its relationship, is the solitary contender for solving this problem. After documenting that inverse data transformations are anything but a rare occurrence, this paper proposes an innovative, elegant back-transformation solution based upon the Kummer confluent hypergeometric function of the first kind. This paper also derives formal back-transformation formulae for the Manly transformation, something apparently never done before. Much related future research remains to be undertaken; this paper furnishes numerous clues about what some of these endeavors need to be.

Matrix/linear algebra continues bestowing benefits on theoretical and applied statistics, a practice it began decades ago (re Fisher used the word matrix in a 1941 publication), through a myriad of contributions, from recognition of a suite of matrix properties relevant to statistical concepts, to matrix specifications of linear and nonlinear techniques. Consequently, focused parts of matrix algebra are topics of several statistics books and journal articles. Contributions mostly have been unidirectional, from matrix/linear algebra to statistics. Nevertheless, statistics offers great potential for making this interface a bidirectional exchange point, the theme of this review paper. Not surprisingly, regression, the workhorse of statistics, provides one tool for such historically based recompence. Another prominent one is the mathematical matrix theory eigenfunction abstraction. A third is special matrix operations, such as Kronecker sums and products. A fourth is multivariable calculus linkages, especially arcane matrix/vector operators as well as the Jacobian term associated with variable transformations. A fifth, and the final idea this paper treats, is random matrices/vectors within the context of simulation, particularly for correlated data. These are the five prospectively reviewed discipline of statistics subjects capable of informing, inspiring, or otherwise furnishing insight to the far more general world of linear algebra.

The existing quantitative geography literature contains a dearth of articles that span spatial autocorrelation (SA), a fundamental property of georeferenced data, and spatial optimization, a popular form of geographic analysis. The well-known location–allocation problem illustrates this state of affairs, although its empirical geographic distribution of demand virtually always exhibits positive SA. This latent redundant attribute information alludes to other tools that may well help to solve such spatial optimization problems in an improved, if not better than, heuristic way. Within a proof-of-concept perspective, this paper articulates connections between extensions of the renowned Majority Theorem of the minisum problem and especially the local indices of SA (LISA). The relationship articulation outlined here extends to the p = 2 setting linkages already established for the p = 1 spatial median problem. In addition, this paper presents the foundation for a novel extremely efficient p = 2 algorithm whose formulation demonstratively exploits spatial autocorrelation.

Police patrolling intends to enhance traffic safety by mitigating the risks associated with vehicle crashes and accidents. From a view of operations, patrolling requires an effective distribution of resources and often involves area delineations for this distribution purpose. Given constraints such as budget and human resources for traffic safety, delineating geographic areas optimally for police patrol areas is an important agenda item. This paper considers two p-median location models using segments on a street network as observational units on which traffic issues such as vehicle crashes occur. It also uses two weight sets to construct an enhanced delineation of police patrol areas in the City of Plano, Texas. The first model for the standard p-median formulation gives attention to the cumulative number of motor vehicle crashes from 2011 to 2021 on the major transportation networks in Plano. The second model, an extension of this first p-median one, uses balancing constraints to achieve balanced spatial coverage across patrol areas. These two models are also solved with network kernel density count estimates (NKDCE) instead of crash counts. These smoothed densities on a network enable consideration of uncertainty affiliated with this aggregation. The analysis results of this paper suggest that the p-median models provide effective specifications, including their capability to define patrol areas that encompass the entire study region while minimizing distance costs. The inclusion of balancing constraints ensures a more equitable distribution of workloads among patrol areas, improving overall efficiency. Additionally, the model with NKDCE results in an improved workload balance among delineated areas for police patrolling activities, thus supporting more informed spatial decision-making processes for public safety.

Fundamental to most classical data collection sampling theory development is the random drawings assumption requiring that each targeted population member has a known sample selection (i.e., inclusion) probability. Frequently, however, unrestricted random sampling of spatially autocorrelated data is impractical and/or inefficient. Instead, randomly choosing a population subset accounts for its exhibited spatial pattern by utilizing a grid, which often provides improved parameter estimates, such as the geographic landscape mean, at least via its precision. Unfortunately, spatial autocorrelation latent in these data can produce a questionable mean and/or standard error estimate because each sampled population member contains information about its nearby members, a data feature explicitly acknowledged in model-based inference, but ignored in design-based inference. This autocorrelation effect prompted the development of formulae for calculating an effective sample size (i.e., the equivalent number of sample selections from a geographically randomly distributed population that would yield the same sampling error) estimate. Some researchers recently challenged this and other aspects of spatial statistics as being incorrect/invalid/misleading. This paper seeks to address this category of misconceptions, demonstrating that the effective geographic sample size is a valid and useful concept regardless of the inferential basis invoked. Its spatial statistical methodology builds upon the preceding ingredients.