Local Spatial Autocorrelation Statistics: Distributional Issues and an Application

The statistics G_i(d) and G_i*(d), introduced in Getis and Ord (1992) for the study of local pattern in spatial data, are extended and their properties further explored. In particular, nonbinary weights are allowed and the statistics are related to Moran's autocorrelation statistic, I. The correlations between nearby values of the statistics are derived and verified by simulation. A Bonferroni criterion is used to approximate significance levels when testing extreme values from the set of statistics. An example of the use of the statistics is given using spatial-temporal data on the AIDS epidemic centering on San Francisco. Results indicate that in recent years the disease is intensifying in the counties surrounding the city.     

Local Spatial Autocorrelation in a Biological Model

We review the recently developed local spatial autocorrelation statistics I_i, c_i, G_i, and G_i*. We discuss two alternative randomization assumptions, total and conditional, and then newly derive expectations and variances under conditional randomization for I_i and c_i, as well as under total randomization for c_i. The four statistics are tested by a biological simulation model from population genetics in which a population lives on a 21 × 21 lattice of stepping stones (sixty-four individuals per stone) and reproduces and disperses over a number of generations. Some designs model global spatial autocorrelation, others spatially random surfaces. We find that spatially random designs give reliable test results by permutational methods of testing significance. Globally autocorrelated designs do not fit expectations by any of the three tests we employed. Asymptotic methods of testing significance failed consistently, regardless of design. Because most biological data sets are autocorrelated, significance testing for local spatial autocorrelation is problematic. However, the statistics are informative when employed in an exploratory manner. We found that hotspots (positive local autocorrelation) and coldspots (negative local autocorrelation) are successfully distinguished in spatially autocorrelated, biologically plausible data sets.     

Comparison of Moran's I and Geary's c in Multivariate Spatial Pattern Analysis

This article compares multivariate spatial analysis methods that include not only multivariate covariance, but also spatial dependence of the data explicitly and simultaneously in model design by extending two univariate autocorrelation measures, namely Moran's I and Geary's c. The results derived from the simulation datasets indicate that the standard Moran component analysis is preferable to Geary component analysis as a tool for summarizing multivariate spatial structures. However, the generalized Geary principal component analysis developed in this study by adding variance into the optimization criterion and solved as a trace ratio optimization problem performs as well as, if not better than its counterpart the Moran principal component analysis does. With respect to the sensitivity in detecting subtle spatial structures, the choice of the appropriate tool is dependent on the correlation and variance of the spatial multivariate data. Finally, the four techniques are applied to the Social Determinants of Health dataset to analyze its multivariate spatial pattern. The two generalized methods detect more urban areas and higher autocorrelation structures than the other two standard methods, and provide more obvious contrast between urban and rural areas due to the large variance of the spatial component.     

The Modifiable Areal Unit Problem,in a Regression Model Whose Independent Variable Is a Distance from a Predetermined Point

This paper deals with the modifiable areal unit problem in the context of a regression model where a dependent variable is an attribute value (say, income) of an atomic data unit (say, a household) and an independent variable is a distance from a predetermined point (say, a central business district) to the atomic data unit (a disaggregated model). We apply this disaggregated model to spatially aggregated data in which the dependent variable is the average income over a spatial unit and the independent variable is the average distance from each household in a spatial unit to the predetermined point (an aggregated model). First, estimating the slope coefficient by the least squares method, we prove that the variance of the estimator for the slope coefficient in the aggregated model is larger than that in the disaggregated model. Second, focusing on variations in the variance of the estimator for the slope coefficient in the aggregated model with respect to the number of zones, we obtain the number of zones in which the variance is close to that in the disaggregated model. Third, we obtain the zoning system that has the minimum variance for a fixed number of zones. We also calculate the maximum variance in order to examine the range of the variance.     

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.     

Analysis of a Distribution of Point Events Using the Network‐Based Quadrat Method

This study proposes a new quadrat method that can be applied to the study of point distributions in a network space. While the conventional planar quadrat method remains one of the most fundamental spatial analytical methods on a two‐dimensional plane, its quadrats are usually identified by regular, square grids. However, assuming that they are observed along a network, points in a single quadrat are not necessarily close to each other in terms of their network distance. Using planar quadrats in such cases may distort the representation of the distribution pattern of points on a network. The network‐based units used in this article, on the other hand, consist of subsets of the actual network, providing more accurate aggregation of the data points along the network. The performance of the network‐based quadrat method is compared with that of the conventional quadrat method through a case study on a point distribution on a network. The χ² statistic and Moran's I statistic of the two quadrat types indicate that (1) the conventional planar quadrat method tends to overestimate the overall degree of dispersion and (2) the network‐based quadrat method derives a more accurate estimate on the local similarity. The article also performs sensitivity analysis on network and planar quadrats across different scales and with different spatial arrangements, in which the abovementioned statistical tendencies are also confirmed.     

Within-development density and housing prices in Singapore

This paper measures how much more households pay for less density in their immediate surroundings. Using transaction and administrative data and exploiting the introduction of a regulation that restricted the number of housing units for certain land lots, we find that households discount density: a 10% increase in within-development density decreases the price per square meter by 5%. Further, the mean price per square meter of the average development increased by 1%–3% after the regulation was introduced, while the amount of built-up space remained constant. The increase in total revenue suggests developers may underestimate the externality caused by density.     

Preservation of old‐growth forests: a case study of Big Timber Park,Whistler, BC

This paper explores the concepts of old‐growth forests, preservation and natural disturbance and demonstrates how contemporary biogeographic theory contributes to successful preservation of old‐growth forests. The case study analyzes the composition, structure, age and growth histories of trees in Big Timber Park, Whistler, British Columbia. Composition. The structure of the forest was diverse and tree ages ranged from 122 to 305 years. The densities of large old Douglas‐fir, shade‐tolerant trees and all snags are indicative of old‐growth forests; however, maximum tree diameters and densities of large snags and logs do not meet quantitative criteria for old‐growth forests. The abundance of tree regeneration and understory vegetation are relatively low. We conclude that some aspects of Big Timber Park are transitional between the mature and old‐growth stages of forest development. In the future, fine‐scale canopy gaps caused by tree senescence and interactions among insects, pathogens and wind are expected to dominate stand dynamics. Forest structure will become increasingly complex, exemplifying the dynamic nature of old‐growth forests. To successfully preserve Big Timber Park and La préservation des forêts other old‐growth forests in Canada requires understanding of natural disturbances. Preservation and the criteria and indicators used to measure successful preservation must explicitly include elements of natural change.     

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.     

Power of the Rank Adjacency Statistic to Detect Spatial Clustering in a Small Number of Regions

The rank adjacency statistic D is a statistical method for assessing spatial autocorrelation or clustering of geographical data. It was originally proposed for summarizing the geographical patterns of cancer data in Scotland (IARC 1985). In this paper, we investigate the power of the rank adjacency statistic to detect spatial clustering when a small number of regions is involved. The investigations were carried out using Monte Carlo simulations, which involved generating patterned/clustered values and computing the power with which the D statistic would detect it. To investigate the effects of region shapes, structure of the regions, and definition of weights, simulations were carried out using two different region shapes, binary and nonhinary weights, and three different lattice structures. The results indicate that in the typical example of considering Canadian total mortality at the electoral district level, the D statistic had adequate power to detect general spatial autocorrelation in twenty‐five or more regions. There was an inverse relationship between power and the level of connectedness of the regions, which depends on the weighting function, shape, and arrangement of the regions. The power of the D statistic was also found to compare favorably with that of Moran's I statistic.     

Bias in Fish Bone Recovery from Differential Collection Techniques and Its Impact on Cultural Interpretation in California

ABSTRACT

While studies on recovery bias are numerous in the literature, the current research focuses on defining best practices in field collection techniques to provide a more accurate representative of site constituents, particularly fish remains. Methods to determine best practices included use of 1/4-inch, 1/8-inch, and 1/16-inch mesh to process excavated material from 10 × 10?cm and 40 × 40 cm-sized bulk column samples and from a 1 × 1 m test unit. Fish remains were compared to determine density and diversity differences between the samples and how these different data sets may impact cultural interpretations. This research shows that while collection of larger column samples processed using a smaller mesh size provides a more robust data set, the taxonomic richness present in a 1 × 1 m test unit provides complimentary data that are critical to understanding subsistence trends as a whole.     

Dendrochronological Dating of Wooden Artefacts using Photography

Core sampling is normally used for dendrochronological age determination of both timber and living trees. As core sampling can seriously damage artefacts, a photographic method was tried on 29 wooden objects made of Scots pine (Pinus sylvestris L.). These were photographed and tree‐ring sequences for dendrochronological analysis were measured on the photographs in the laboratory. Nine objects could be dated. The undatable objects had few tree‐rings or a non‐matching tree‐ring pattern. The region in Norway where the material had grown could be determined by matching against various regional chronologies.     

Migrant household homeownership outcomes in large Chinese cities - the sustained impact of hukou

Homeownership by migrant households in large Chinese cities is increasingly evident and sometimes blamed for driving up local housing prices. Hukou reform grants local hukou to migrants in small cities while it allows large cities to set up hukou entry barriers to control migration. This paper explores whether migrant households’ micro-level characteristics and macro-level urban policies relating to hukou reform have any impact on their housing tenure choices. Using data from a 2009 survey of migrant households in six large cities, this study adopts logistic regression models to examine factors influencing labor migrant households’ tenure decisions. We find that household income and head of household hukou type, contribution to pension fund, and enrollment in health insurance all positively predict homeownership. Migrant households are more likely to be found in cities with larger population size. We argue that city entry barriers create new forms of institutional haves and have-nots and new forms of inequality. The sustained impact of previous hukou types suggests inherited inequalities through intergenerational wealth transfer, which calls for targeted welfare policies to mediate.     

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.     

INVESTIGATING SEASONALITY AND SEASON OF BIRTH IN PAST HERDS: A REFERENCE SET OF SHEEP ENAMEL STABLE OXYGEN ISOTOPE RATIOS

Intra‐tooth sequential analysis of enamel δ¹⁸O is currently used to investigate birth seasonality in past animal populations, offering new insights into seasonal availability of animal resources, herd management and seasonality of site occupation. Reference data sets are still required to address two major difficulties: (1) that inter‐individual variability in the record of the seasonal cycle is affected by tooth size; and (2) that the season of birth cannot be directly estimated from the timing of tooth growth, because of a delay in enamel mineralization. We present a data set acquired on the lower second molar of 10 modern sheep from Rousay (Orkney) born within a few weeks of each other in April/May and submitted to the same environmental conditions until death. All sheep have recorded a sinusoidal pattern of δ¹⁸O variation spanning approximately a year. From the difference between the expected and the measured time sequence, the delay of enamel mineralization is estimated to be 5–6 months. The data set is then described using a model mainly based on a cosine function. The period, corresponding to the length of the M2 crown formed over a year, averaged 35.8 mm. A very slight variation of tooth growth rate with time and no attenuation of the isotopic signal towards the cervical margin of the crown could be detected in this data set. The lowest δ¹⁸O values, corresponding to the sheep's first winter, were tracked at a distance from the enamel/root junction that varied between 23.0 and 30.3 mm (x_min mean = 27.6 mm); the highest δ¹⁸O values, corresponding to the sheep's second summer, were between 6.3 and 11.6 mm (x_max mean = 9.9 mm). Most of the variability can be attributed to tooth size. When normalized on the period, x_min and x_max are 0.28 (± 0.05) and 0.78 (± 0.05) on average, meaning that the Rousay sheep have recorded the minimum and maximum δ¹⁸O values on average at 78% and 28%, respectively, of the end of the periodic cycle recorded in the second molar.     

Gravity waves from O2 nightglow during the AIDA '89 campaign III: effects of gravity wave saturation

The O₂ atmospheric (0–1) night airglow emitted within the gravity wave saturation region at ∼90–100 km can serve as a means of studying the wave activity. In this analysis, the atmospheric motions were described by a mean spectral model and an algorithm was developed to infer the wave kinetic energy density and momentum flux from variations in O₂ (0–1) airglow emission rate and rotational temperature. The method was applied to eight nights of data collected by MORTI, a mesopause oxygen rotational temperature imager, during the AIDA campaign of 1989 in Puerto Rico (18°N, 67°W). The observed r.m.s. fractional fluctuations of airglow emission rate and rotational temperature were of the order of ∼0.07–0.15 and ∼0.02–0.04, respectively, and the characteristic vertical wavelengths were estimated at ∼10 2 -20 km. The inferred r.m.s. horizontal velocities and velocity variances were found to be ∼12–25 m/s and ∼150–600 m²/s², with the majority of the horizontal velocity and its variance associated with low-frequency, large-scale wave motions. The estimated momentum fluxes, mainly contributed by high-frequency, small-scale waves, were ∼2–10 m²/s². These results are in good agreement with those obtained from other measurements using different observational methods at low and mid-latitudes.     

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.     

The Majority Theorem for the Single (p = 1) Median Problem and Local Spatial Autocorrelation

Except for about a half dozen papers, virtually all (co)authored by Griffith, the existing literature lacks much content about the interface between spatial optimization, a popular form of geographic analysis, and spatial autocorrelation, a fundamental property of georeferenced data. The popular p-median location-allocation problem highlights this situation: the empirical geographic distribution of demand virtually always exhibits positive spatial autocorrelation. This property of geospatial data offers additional overlooked information for solving such spatial optimization problems when it actually relates to their solutions. With a proof-of-concept outlook, this paper articulates connections between the well-known Majority Theorem of the 1-median minisum problem and local indices of spatial autocorrelation; the LISA statistics appear to be the more useful of these later statistics because they better embrace negative spatial autocorrelation. The relationship articulation outlined here results in the positing of a new proposition labeled the egalitarian theorem.     

A new Pliocene tree kangaroo species (Marsupialia,Macropodinae) from the Chinchilla Local Fauna,southeastern Queensland

A new species of very large tree kangaroo, Bohra wilkinsonorum, is described from a maxillary fragment from the Pliocene Chinchilla Sands of southeastern Queensland. Allocation to Bohra, which has previously been known from postcranial material only, is suggested on the basisof its similar size and stage of evolution to Bohra paulae. Both species of Bohra are plesiomorphic with respect to species of Dendrolagus, and are much larger than any known species of Dendrolagus. This new taxon from Chinchilla has expanded the tree kangaroo record from the Pliocene of southeastern Australia, supporting the hypothesis that the group originated in the late Miocene of ‘mainland’ Australia, finding refuge in north-eastern Queensland and New Guinea as climate became drier in the Quaternary. Fossil tree kangaroos are unknown from the Pliocene of Papua New Guinea where most living species now occur.     

The Assessment of Spatial Autocorrelation through Constrained Multiple Regression

Most published measures of spatial autocorrelation (SA) can be recast as a (normalized) cross-product statistic that indexes the degree of relationship between corresponding entries from two matrices—one specifying the spatial connections among a set of n locations, and the other reflecting a very explicit definition of similarity between the set of values on some variable x realized over the n locations. We first give a very brief sketch of the basic cross-product approach to the evaluation of SA, and then generalize this strategy to include less restrictive specifications for the notion of similarity between the values on x. Using constrained multiple regression, the characterization of variate similarity basic to any assessment of SA can itself be framed according to the information present in the measure of spatial separation. These extensions obviate the inherent arbitrariness in how SA is usually evaluated, which now results from the requirement of a very restrictive definition of variate similarity before a cross-product index can be obtained.