Accounting for spatial autocorrelation from model selection to statistical inference: Application to a national survey of a diurnal raptor

Planning actions for species conservation involves working at both an ecologically meaningful spatial scale and a scale suitable for implementing management or conservation plans. Animal populations and conservation policies often operate across wide areas. Large-extent spatial datasets are thus often used, but their analyses rarely deal with problems inherent to spatial datasets such as residual spatial autocorrelation, which can bias or even reverse results. Here we propose a procedure for analysing a large-scale count dataset integrating residual spatial autocorrelation in a Generalized Linear Model framework by combining and extending previously published methods. The first step concerns the selection of the environmental variables by a modified cross-validation procedure allowing for residual spatial autocorrelation. Then the second step consists in evaluating the spatial effect of the model using a spatial filtering approach based on the variogram parameters. We apply this method to the Black kite (Milvus migrans) to estimate the distribution and population size of this species in France. We found some divergence in estimated population size between spatial and non spatial models, as well as in the distribution map. We also found that the uncertainty of the model was underestimated by the residual spatial autocorrelation. Our analysis confirms previous results, that residual spatial autocorrelation should be always accounted for, especially in conservation where false results may lead to poor management decisions.     

Microhabitat analyses support relationships between niche breadth and range size when spatial autocorrelation is strong

Multiple evidence of positive relationships between nice breadth and range size (NB–RS) suggested that this can be a general ecological pattern. However, correlations between niche breadth and range size can emerge as a by-product of strong spatial structure of environmental variables. This can be problematic because niche breadth is often assessed using broad-scale macroclimatic variables, which suffer heavy spatial autocorrelation. Microhabitat measurements provide accurate information on species tolerance, and show limited autocorrelation. The aim of this study was to combine macroclimate and microhabitat data to assess NB–RS relationships in European plethodontid salamanders (Hydromantes), and to test whether microhabitat variables with weak autocorrelation can provide less biased NB–RS estimates across species. To measure macroclimatic niche, we gathered comprehensive information on the distribution of all Hydromantes species, and combined them with broad-scale climatic layers. To measure microhabitat, we recorded salamander occurrence across > 350 caves and measured microhabitat features influencing their distribution: humidity, temperature and light. We assessed NB–RS relationships through phylogenetic regression; spatial null-models were used to test whether the observed relationships are a by-product of autocorrelation. We observed positive relationships between niche breadth and range size at both the macro- and microhabitat scale. At the macroclimatic scale, strong autocorrelation heavily inflated the possibility to observe positive NB–RS. Spatial autocorrelation was weaker for microhabitat variables. At the microhabitat level, the observed NB–RS was not a by-product of spatial structure of variables. Our study shows that heavy autocorrelation of variables artificially increases the possibility to detect positive relationships between bioclimatic niche and range size, while fine-scale data of microhabitat provide more direct measure of conditions selected by ectotherms, and enable less biased measures of niche breadth. Combining analyses performed at multiple scales and datasets with different spatial structure provides more complete niche information and effectively tests the generality of niche breadth–range size relationships.     

Geographical and Temporal Body Size Variation in a Reptile: Roles of Sex,Ecology, Phylogeny and Ecology Structured in Phylogeny

Geographical body size variation has long interested evolutionary biologists, and a range of mechanisms have been proposed to explain the observed patterns. It is considered to be more puzzling in ectotherms than in endotherms, and integrative approaches are necessary for testing non-exclusive alternative mechanisms. Using lacertid lizards as a model, we adopted an integrative approach, testing different hypotheses for both sexes while incorporating temporal, spatial, and phylogenetic autocorrelation at the individual level. We used data on the Spanish Sand Racer species group from a field survey to disentangle different sources of body size variation through environmental and individual genetic data, while accounting for temporal and spatial autocorrelation. A variation partitioning method was applied to separate independent and shared components of ecology and phylogeny, and estimated their significance. Then, we fed-back our models by controlling for relevant independent components. The pattern was consistent with the geographical Bergmann''s cline and the experimental temperature-size rule: adults were larger at lower temperatures (and/or higher elevations). This result was confirmed with additional multi-year independent data-set derived from the literature. Variation partitioning showed no sex differences in phylogenetic inertia but showed sex differences in the independent component of ecology; primarily due to growth differences. Interestingly, only after controlling for independent components did primary productivity also emerge as an important predictor explaining size variation in both sexes. This study highlights the importance of integrating individual-based genetic information, relevant ecological parameters, and temporal and spatial autocorrelation in sex-specific models to detect potentially important hidden effects. Our individual-based approach devoted to extract and control for independent components was useful to reveal hidden effects linked with alternative non-exclusive hypothesis, such as those of primary productivity. Also, including measurement date allowed disentangling and controlling for short-term temporal autocorrelation reflecting sex-specific growth plasticity.     

Less than eight (and a half) misconceptions of spatial analysis

Spatial analyses are indispensable analytical tools in biogeography and macroecology. In a recent Guest Editorial, Hawkins (Journal of Biogeography, 2012, 39 , 1–9) raised several issues related to spatial analyses. While we concur with some points, we here clarify those confounding (1) spatial trends and spatial autocorrelation, and (2) spatial autocorrelation in the response variable and in the residuals. We argue that recognizing spatial autocorrelation in statistical modelling is not only a crucial step in model diagnostics, but that disregarding it is essentially wrong.     

Spatial distribution of communal nests in a colonial breeding bird: benefits without costs?

Abstract The spatial organization of individuals, or groups of individuals, within a population can provide valuable information about social organization and population dynamics. We analysed the spatial distribution of nests of the sociable weaver (Philetairus socius) on two farms in the Kalahari. Sociable weavers build large communal nests on big savannah trees, forming a pattern of trees with and without nests. We used two spatial statistics, Ripley's K and the pair correlation function, to describe characteristics of the point patterns over a range of distances. (i) At distances of 200 and 300 m, communal nests were clustered. (ii) At distances greater than 1000 m, communal nests were regularly distributed. These findings are independent of the spatial distribution of trees. Furthermore, we used Moran's I to analyse spatial autocorrelation of nest sizes. We expected negative autocorrelation because of competition between nests. But on both farms there was no significant autocorrelation of nest sizes for any distance class. The regular distribution observed at larger distances may indicate competition and/or territoriality among different nests, but the lack of spatial autocorrelation between nest sizes suggests that these interactions may happen between nest clusters rather than between single nests. This was confirmed by significant clustering of nests on small scales. We thus suggest, that colonies of P. socius consist of several nests on adjacent trees forming a cluster of subcolonies. The question why sociable weavers establish subcolonies instead of adding more chambers to the natal nest, could not simply be answered by limitation of nesting space. We suggest a strategy to avoid costs due to increasing colony size.     

Size of environmental grain and resource matching

For most animals their foraging environment consists of a patch network. In random environments there are no spatial autocorrelation at all, while in fine-grained systems positive autocorrelations flip to negative ones and back again against distance. With increasing grain size the turnover rate of spatial autocorrelation slows down. Using a cellular automaton with foragers having limited information about their feeding environment we examined how well consumer numbers matched resource availability, also known as the ideal free distribution. The match is the better the smaller the size of the environmental grain. This is somewhat contrary to the observation that in large-grained environments the spatial autocorrelation is high and positive over long distances. In such an environment foragers, by knowing a limited surrounding, should in fact know a much larger area because of the spatially autocorrelated resource pattern. Yet, when foragers have limited knowledge, we observed that the degree of undermatching (i.e., more individuals in less productive patches than expected) increases with increasing grain size.     

Structural heterogeneity and spatial autocorrelation in a natural mature Pinus sylvestris dominated forest

The tree species composition, vertical stratification and patterns of spatial autocorrelation at the tree and quadrate (25 × 25 m) scales were studied in a natural mature PinuS sylvestris dominated forest in eastern Finland. For the analyses we mapped the locations and dimensions of trees taller than 10 m in a 9 ha (300 × 300 m) area, and within this area we mapped all trees taller than 0.3 m on a core plot of 4 ha (200 × 200 m). The overall tree size distribution was bimodal. the dominant layer and the understory forming the peak frequencies. Pinus sylvestris dominated the main canopy, together with scattered Betula pendula and Picea abies. Alnus incana, Populus tremula, Salix caprea, Sorbus aucuparia and Juniperus communis occurred only in the under- and middlestories. Autocorrelation analysis (semivarianee) of tree size variation revealed spatial patterns, which were strongly dependent on the size of trees included in the analysis. When all living trees, including the understory regeneration, were taken into account, the autocorrelation pattern ranged up to 35 m inter-tree distances, reflecting the spatial scale of understory regeneration patches. Competitive interaction among middle- and upperstory trees (height>10 m) had contrasting effects on autocorrelation pattern depending on spatial scale. At the fine scale, dominant trees suppressed their smaller close neighbors (asymmetric competition), which was shown as increased tree size variation at small inter-tree distances (<2 m). At slightly larger inter-tree distances, specifically among large trees of similar size, competition was more symmetrical, which resulted in decreased tree size variation at these inter-tree distances (3–4 m). This effect was seen most clearly in the dominant trees, there being a clear autocorrelation pattern in tree size up to inter-tree distances of ～4 m. At the quadrate scale (25 × 25 m) the analysis revealed high local variation in structural characteristics such as tree height diversity (THD), tree species diversity (H) and autocorrelation of tree height. The analysis suggests that naturally developed P. sylvestris forests exhibit complex small-scale patterns of structural heterogeneity and spatial autocorrelation in tree size. These patterns may be important for stand-scale habitat diversity and can have aggregated effects on ecosystem dynamics at larger spatial scales though their influence on the spread of disturbance and regeneration after disturbance.     

Glycemic Index,Glycemic Load and Mammographic Breast Density: The EPIC Florence Longitudinal Study

A few studies have evaluated the association between diet and mammographic breast density (MBD) and results are inconsistent. MBD, a well-recognized risk factor for breast cancer, has been proposed as a marker of cumulative exposure to hormones and growth factors. Diets with a high glycemic index (GI) or glycemic load (GL) may increase breast cancer risk, via an effect on the insulin-like growth factor axis. We have investigated the association between carbohydrate intake, GI, GL and MBD in a prospective study. We identified a large series of women, in the frame of the EPIC-Florence cohort, with a mammogram taken five years after enrolment, when detailed information on dietary and lifestyle habits and anthropometric measurements had been collected. Mammograms have been retrieved (1,668, 83%) and MBD assessed according to Wolfe’s classification. We compared women with high MBD (P2+DY Wolfe’s categories) with those with low MBD (N1+P1) through logistic models adjusted for age, education, body mass index, menopause, number of children, breast feeding, physical activity, non-alcohol energy, fibers, saturated fat and alcohol. A direct association between GL and high MBD emerged in the highest quintile of intake in comparison with the lowest quintile (OR = 1.73, 95%CI 1.13–2.67, p for trend = 0.048) while no association with glycemic index was evident. These results were confirmed after exclusion of women reporting to be on a diet or affected with diabetes, and when Hormone Replacement Therapy at the date of mammographic examination used to assess MBD was considered. The effect was particularly evident among leaner women, although no interaction was found. A positive association was suggested for increasing simple sugar and total carbohydrates intakes limited to the highest quintiles. In this Italian population we observed an association between glycemic load, total and rapidly absorbed carbohydrates and high MBD. These novel results warrant further investigations.     

Incorporating spatial autocorrelation into species distribution models alters forecasts of climate‐mediated range shifts

Species distribution models (SDMs) are widely used to forecast changes in the spatial distributions of species and communities in response to climate change. However, spatial autocorrelation (SA) is rarely accounted for in these models, despite its ubiquity in broad‐scale ecological data. While spatial autocorrelation in model residuals is known to result in biased parameter estimates and the inflation of type I errors, the influence of unmodeled SA on species' range forecasts is poorly understood. Here we quantify how accounting for SA in SDMs influences the magnitude of range shift forecasts produced by SDMs for multiple climate change scenarios. SDMs were fitted to simulated data with a known autocorrelation structure, and to field observations of three mangrove communities from northern Australia displaying strong spatial autocorrelation. Three modeling approaches were implemented: environment‐only models (most frequently applied in species' range forecasts), and two approaches that incorporate SA; autologistic models and residuals autocovariate (RAC) models. Differences in forecasts among modeling approaches and climate scenarios were quantified. While all model predictions at the current time closely matched that of the actual current distribution of the mangrove communities, under the climate change scenarios environment‐only models forecast substantially greater range shifts than models incorporating SA. Furthermore, the magnitude of these differences intensified with increasing increments of climate change across the scenarios. When models do not account for SA, forecasts of species' range shifts indicate more extreme impacts of climate change, compared to models that explicitly account for SA. Therefore, where biological or population processes induce substantial autocorrelation in the distribution of organisms, and this is not modeled, model predictions will be inaccurate. These results have global importance for conservation efforts as inaccurate forecasts lead to ineffective prioritization of conservation activities and potentially to avoidable species extinctions.     

Distribution of abundance across the range in eastern North American trees

Aim  We analysed spatial datasets of abundance across the entirety, or near entirety, of the geographical ranges of 134 tree species to test macroecological hypotheses concerning the distribution of abundance across geographical ranges.
Location  Our abundance estimates came via the USDA Forest Service Forest Inventory and Analysis Eastwide Database, which contains data for 134 eastern North American tree species.
Methods  We extracted measures of range size and the spatial location of abundance relative to position in the range for each species to test four hypotheses: (a) species occur in low abundance throughout most of their geographical range; (b) there is a positive interspecific relationship between abundance and range size; (c) species are more abundant in the centre of their range; and (d) there is a bimodal distribution of spatial autocorrelation in abundance across a species range.
Results  Our results demonstrate that (a) most species (85%) are abundant somewhere in their geographical range; (b) species achieving relatively high abundance tend to have larger range sizes; (c) the widely held assumption that species exhibit an 'abundant-centre distribution' is not well supported for the majority of species; we suggest 'abundant-core' as a more suitable term; and (d) there is no evidence of a bimodal distribution of spatial autocorrelation in abundance.  

Main Conclusions 

For many tree species, high abundance can be achieved at any position in the range, though suitable sites are found with less frequency towards range edges. Competitive relationships may be involved in the distribution of abundance across tree ranges and species with larger ranges (and possibly broader niches) may be affected more by biotic interactions than smaller ranging species.     

Do vegetation patch spatial patterns disrupt the spatial organization of plant species?

The mechanisms that structure plant diversity and generate long-range correlated spatial patterns have important implications for the conservation of fragmented landscapes. The ability to disperse and persist influences a plant species’ capacity for spatial organization, which can play a critical role in structuring plant diversity in metacommunities. This study examined the spatial patterns of species diversity within a network of patches in Cabo de Gata Natural Park, southeastern Spain. The objectives were to understand how the spatial heterogeneity of species composition (beta diversity) varies in a structured landscape, and how the long-range spatial autocorrelation of plant species is affected by the spatial configuration of patches.The mechanisms underlying the spatial distribution of plants acted at two scales. Between patches, spatial variation in species distributions was greater than that expected based on spatial randomization, which indicated that movement among patches was restricted. Within patches, diffusion processes reduced spatial variability in species distributions, and the effect was more prominent in large patches. Small patch size negatively influenced the long-range spatial autocorrelation of characteristic species, whereas inter-patch distance had a stronger effect on species frequency than it had on the disruption of spatial organized patterns.The long-range spatial autocorrelation was evaluated based on the dispersal abilities of the species. Among the 106 species evaluated, 39% of the woody species, 17% of the forbs, and 12% of the grasses exhibited disrupted long-range spatial autocorrelation where patches were small. The species that are more vulnerable to the effects of fragmentation tended to be those that have restricted dispersal, such as those that have short-range dispersal (atelechoric), e.g., Phlomis purpurea, Cistus albidus, Teucrium pseudochamaepytis, Brachypodium retusum, and the ballistic species, Genista spartioides. Helianthemum almeriense is another vulnerable species that has actively restricted dispersal (antitelechory), which is common in arid regions. Wind dispersers such as Launaea lanifera were less vulnerable to the effects of fragmentation. Long-distance dispersers whose persistence depends on facilitative interactions with other individuals, e.g., allogamous species such as Thymus hyemalis, Ballota hirsuta, and Anthyllis cytisoides, exhibit disrupted long-range spatial autocorrelation when patch size is reduced.     

Combining spatial and phylogenetic eigenvector filtering in trait analysis

Aim To analyse the effects of simultaneously using spatial and phylogenetic information in removing spatial autocorrelation of residuals within a multiple regression framework of trait analysis. Location Switzerland, Europe. Methods We used an eigenvector filtering approach to analyse the relationship between spatial distribution of a trait (flowering phenology) and environmental covariates in a multiple regression framework. Eigenvector filters were calculated from ordinations of distance matrices. Distance matrices were either based on pure spatial information, pure phylogenetic information or spatially structured phylogenetic information. In the multiple regression, those filters were selected which best reduced Moran's I coefficient of residual autocorrelation. These were added as covariates to a regression model of environmental variables explaining trait distribution. Results The simultaneous provision of spatial and phylogenetic information was effectively able to remove residual autocorrelation in the analysis. Adding phylogenetic information was superior to adding purely spatial information. Applying filters showed altered results, i.e. different environmental predictors were seen to be significant. Nevertheless, mean annual temperature and calcareous substrate remained the most important predictors to explain the onset of flowering in Switzerland; namely, the warmer the temperature and the more calcareous the substrate, the earlier the onset of flowering. A sequential approach, i.e. first removing the phylogenetic signal from traits and then applying a spatial analysis, did not provide more information or yield less autocorrelation than simple or purely spatial models. Main conclusions The combination of spatial and spatio‐phylogenetic information is recommended in the analysis of trait distribution data in a multiple regression framework. This approach is an efficient means for reducing residual autocorrelation and for testing the robustness of results, including the indication of incomplete parameterizations, and can facilitate ecological interpretation.     

Methods to account for spatial autocorrelation in the analysis of species distributional data: a review

Species distributional or trait data based on range map (extent‐of‐occurrence) or atlas survey data often display spatial autocorrelation, i.e. locations close to each other exhibit more similar values than those further apart. If this pattern remains present in the residuals of a statistical model based on such data, one of the key assumptions of standard statistical analyses, that residuals are independent and identically distributed (i.i.d), is violated. The violation of the assumption of i.i.d. residuals may bias parameter estimates and can increase type I error rates (falsely rejecting the null hypothesis of no effect). While this is increasingly recognised by researchers analysing species distribution data, there is, to our knowledge, no comprehensive overview of the many available spatial statistical methods to take spatial autocorrelation into account in tests of statistical significance. Here, we describe six different statistical approaches to infer correlates of species’ distributions, for both presence/absence (binary response) and species abundance data (poisson or normally distributed response), while accounting for spatial autocorrelation in model residuals: autocovariate regression; spatial eigenvector mapping; generalised least squares; (conditional and simultaneous) autoregressive models and generalised estimating equations. A comprehensive comparison of the relative merits of these methods is beyond the scope of this paper. To demonstrate each method's implementation, however, we undertook preliminary tests based on simulated data. These preliminary tests verified that most of the spatial modeling techniques we examined showed good type I error control and precise parameter estimates, at least when confronted with simplistic simulated data containing spatial autocorrelation in the errors. However, we found that for presence/absence data the results and conclusions were very variable between the different methods. This is likely due to the low information content of binary maps. Also, in contrast with previous studies, we found that autocovariate methods consistently underestimated the effects of environmental controls of species distributions. Given their widespread use, in particular for the modelling of species presence/absence data (e.g. climate envelope models), we argue that this warrants further study and caution in their use. To aid other ecologists in making use of the methods described, code to implement them in freely available software is provided in an electronic appendix.     

Development of a local size hierarchy causes regular spacing of trees in an even-aged Abies forest: analyses using spatial autocorrelation and the mark correlation function

Background and Aims

During the development of an even-aged plant population, the spatial distribution of individuals often changes from a clumped pattern to a random or regular one. The development of local size hierarchies in an Abies forest was analysed for a period of 47 years following a large disturbance in 1959.

Methods

In 1980 all trees in an 8 × 8 m plot were mapped and their height growth after the disturbance was estimated. Their mortality and growth were then recorded at 1- to 4-year intervals between 1980 and 2006. Spatial distribution patterns of trees were analysed by the pair correlation function. Spatial correlations between tree heights were analysed with a spatial autocorrelation function and the mark correlation function. The mark correlation function was able to detect a local size hierarchy that could not be detected by the spatial autocorrelation function alone.

Key Results

The small-scale spatial distribution pattern of trees changed from clumped to slightly regular during the 47 years. Mortality occurred in a density-dependent manner, which resulted in regular spacing between trees after 1980. The spatial autocorrelation and mark correlation functions revealed the existence of tree patches consisting of large trees at the initial stage. Development of a local size hierarchy was detected within the first decade after the disturbance, although the spatial autocorrelation was not negative. Local size hierarchies that developed persisted until 2006, and the spatial autocorrelation became negative at later stages (after about 40 years).

Conclusions

This is the first study to detect local size hierarchies as a prelude to regular spacing using the mark correlation function. The results confirm that use of the mark correlation function together with the spatial autocorrelation function is an effective tool to analyse the development of a local size hierarchy of trees in a forest.Key words: Abies, local size hierarchy, mark correlation function, pair correlation function, regenerating forest, regular spacing, spatial autocorrelation     

佛坪国家级自然保护区秦岭箭竹克隆结构的SSR分析

利用SSR分子标记技术分析了佛坪国家级自然保护区秦岭箭竹(Fargesia qinlingensis)的克隆多样性和克隆结构,以探讨小尺度范围内秦岭箭竹自然居群遗传变异的分布特征,对该种开花特性、高山地区生态环境维护和大熊猫的保护提供重要依据。结果表明7对SSR引物共扩增出79个位点,其中多态性位点77个,多态位点百分率(PPB)为97.47%。秦岭箭竹的142个分株共形成107个克隆,最大克隆可达5 m。克隆多样性略高于其他克隆植物的平均值(D=0.62,G/N=0.17,E=0.68),基因型比率(G/N)、Simpson指数(D)、平均克隆大小(N/G)和Fager均匀性指数(E)分别为0.7535、0.9680、1.3271和0.5109。克隆空间结构分析表明秦岭箭竹的克隆构型为密集型,各克隆呈镶嵌性分布,同一克隆的分株排列紧密。克隆聚类分析表明各克隆之间聚类不明显,总体上来自同一样地的克隆被聚为一类。空间自相关分析显示在空间距离为36 m范围内,分株比基株有更显著的空间遗传结构,空间自相关系数r的取值范围分别为0.084—0.626和0.024—0.288,说明克隆繁殖在一定程度上限制了空间遗传结构的范围。样地内秦岭箭竹个体在空间距离小于44 m时存在显著的正相关空间结构,特别是在4 m处表现出最大的空间自相关系数(r=0.626),表明空间距离相距4 m内的个体最有可能属于同一克隆,4 m比5 m更能表现出清晰的克隆结构,X-轴截距为52.280,代表了秦岭箭竹不规则克隆的平均最小长度。秦岭箭竹的克隆多样性和克隆结构与初始苗补充、花粉散播方式和微环境差异有关。     

Spatial genetic structure of two sympatric neotropical palms with contrasting life histories

The spatial genetic structure within sympatric populations of two neotropical dioecious palm species with contrasting life histories was characterized to evaluate the influence of life history traits on the extent of genetic isolation by distance. Chamaedorea tepejilote is a common wind-pollinated arboreal understory palm. Chamaedorea elatior is an uncommon climbing subcanopy palm with entomophilous pollination syndrome. A total of 59 allozyme alleles for C. tepejilote and 53 alleles for C. elatior was analyzed using both unweighted (Iu) and weighted (Iw) Moran's I spatial autocorrelation statistics. The spatial genetic structure detected within these populations is consistent with those reported for highly dispersed plants. A significance test for differences between mean Moran's I-coefficients revealed less spatial genetic structure within the C. tepejilote population than that in the C. elatior population. Adjacent individuals of C. elatior exhibited significant spatial genetic autocorrelation (Iu=0.039, Iw=0.034), indicating a Wright's neighborhood size of about 100 individuals. For C. tepejilote, nonrandom genetic distribution among nearest neighbors was detected, even from small spatial autocorrelation values (Iu=0.008, Iw=0.009), consistent with a neighborhood size of about 300 individuals. For both species, seed dispersal, mortality among life cycle stages, overlapping generations, and contrasting traits of mating and reproduction influence the standing spatial genetic structure within populations.     

Analysis of Large-Scale Spatial Heterogeneity in Vegetation Indices among North American Landscapes

We analyzed the spatial heterogeneity in vegetation indices among 13 North American landscapes by using full Landsat Thematic Mapper images. Landscapes varied broadly in the statistical distribution of vegetation indices, but were successfully ordinated by using a measure of central tendency (the mean) and a measure of dispersion (the standard deviation or the coefficient of variation). Differences in heterogeneity among landscapes were explained by their topographic relief and their land cover. Landscape heterogeneity (standard deviation of the Normalized Difference Vegetation Index, NDVI) tended to increase linearly with topographic relief (standard deviation of elevation), but landscapes with low relief were much more heterogeneous than expected from this relationship. The latter were characterized by a large proportion of agricultural land. Percent agriculture, in turn, was inversely related to topographic relief. The strength of these relationships was evaluated against changes in image spatial resolution (grain size). Aggregation of NDVI images to coarser grain size resulted in steady decline of their standard deviation. Although the relationship between landscape heterogeneity and explanatory variables was generally preserved, rates of decrease in heterogeneity with grain size differed among landscapes. A spatial autocorrelation analysis showed that rates of decrease were related to the scale at which pattern is manifested. On one end of the spectrum are agricultural, low-relief landscapes with low spatial autocorrelation and small-scale heterogeneity associated with fields; their heterogeneity decreased sharply as grain size increased. At the other end, desert landscapes were characterized by low small-scale heterogeneity, high spatial autocorrelation, and almost no change in heterogeneity as grain sized was increased—their heterogeneity, associated with land forms, was present at a large scale. Received 1 October 1997; accepted 11 February 1998.     

Patterns of endemism of the eastern North American cave fauna

Aim Over 250 species of obligate terrestrial cave‐dwelling animals (troglobionts) are known from single caves in the eastern United States. We investigate their geographical distribution, especially in relation to other troglobionts. We relate these patterns to taxonomic group, opportunities for dispersal and geographical location. Location Caves of the United States east of the Mississippi River. Methods We associated over 3000 records of more than 450 troglobiotic species and subspecies with hexagons of 1000, 5000 and 10,000 km² in size. We calculated Moran's I, black–white joins and cubic regression of endemics on non‐endemics at all three spatial scales. For 5000 km² hexagons, we modelled the spatial autocorrelation of the residuals of the cubic regression of endemics on non‐endemics. Results Differences among orders in percentage single‐cave endemism were not significant, except for Pseudoscorpionida, which was higher (69%) than any other order. At all three scales, Moran's I and black–white joins were significant, indicating a clumped distribution of both single‐cave endemics and other troglobionts. Spatial patterns were similar at all three scales and Moran's I was highest at 5000 km². The cubic fit of endemics to non‐endemics was consistently better, with less systematic error or residuals, than were linear or quadratic models. Residuals showed a significant geographical pattern with excess endemics in more southerly locations. Main conclusions There was both a non‐spatial and spatial component to the pattern of single‐cave endemism. The non‐spatial component was the association of high levels of single‐cave endemism with areas of high diversity of non‐endemics. It may be that both are high because of high secondary productivity. Spatially, single‐cave endemism is high in central rather than peripheral areas and in the southern part of the range. It is not higher in areas of more dissected limestone, which would reduce migration rates; if anything endemism is lower. Regional spatial effects are important, indicating that cave communities cannot be understood (or protected) in isolation.     

中西太平洋鲣鱼围网渔业资源的热点分析和空间异质性

中西太平洋是世界鲣鱼围网主要作业水域。基于我国渔船2005—2009年的中西太平洋鲣鱼围网生产数据,运用空间统计方法对该水域鲣鱼资源的空间自相关性和空间异质性特征进行分析,并结合海洋环境特征分析资源分布的热点区域。(1)通过常规统计学计算获得鲣鱼资源的偏态Sk、峰态数Ku、变异值Cv、s2/m和全局空间自相关Geary c系数,发现中西太平洋鲣鱼资源总体上是以低密度区域为主,高密度区域较少;鱼类资源密度值差异较大,资源表现出强烈集聚分布,总体的空间自相关性中等偏弱。(2)通过局部空间自相关的热点分析方法计算,发现局部空间自相关性较强,存在多个在统计学上通过显著性检验的资源热点和冷点。(3)通过地统计方法研究鲣鱼资源的空间变异性特征和方向变异时,空间自相关类型上最优模型是球形模型,鲣鱼资源密度各向同性,最大相关距离1000km左右。发现空间自相关引起的差异占整个差异的50%左右,为中等强度变异;在方向性变异上,主要体现在南北向上,其该向上结构性误差占67%,而东西向结构性误差占49%。这一结果和海洋环境的南北向上结构性远好于东西向结构性有关;从各方向的分维数看,数值介于1.876—1.9之间,数值较大,空间自相关较弱。(4)以资源热点区域作为区域性渔场,结合海洋温度和叶绿素场海洋环境特征,将中西太平洋鲣鱼资源分为3个不同的局部渔场,即2个暖池渔场,1个冷舌渔场。冷舌渔场由中东太平洋赤道上升流引起,在锋面地带提供了较为丰富的初级生产力,便于鱼类获得丰富的食物;暖池渔场靠近岛屿和陆地区域,近岸上升流系统提供了丰富的初级生产力。(5)将热点分析和渔场重心方法及栖息地指数的优缺点做了对比,建议以后采用空间残差模型深入研究空间自相关问题。     

Spatial ordination of vegetation data using a generalization of Wartenberg's multivariate spatial correlation

Question: Are there spatial structures in the composition of plant communities? Methods: Identification and measurement of spatial structures is a topic of great interest in plant ecology. Univariate measurements of spatial autocorrelation such as Moran's I and Geary's c are widely used, but extensions to the multivariate case (i.e. multi‐species) are rare. Here, we propose a multivariate spatial analysis based on Moran's I (MULTISPATI) by introducing a row‐sum standardized spatial weight matrix in the statistical triplet notation. This analysis, which is a generalization of Wartenberg's approach to multivariate spatial correlation, would imply a compromise between the relations among many variables (multivariate analysis) and their spatial structure (autocorrelation). MULTISPATI approach is very flexible and can handle various kinds of data (quantitative and/or qualitative data, contingency tables). A study is presented to illustrate the method using a spatial version of Correspondence Analysis. Location: Territoire d'Etude et d'Expérimentation de Trois‐Fontaines (eastern France). Results: Ordination of vegetation plots by this spatial analysis is quite robust with reference to rare species and highlights spatial patterns related to soil properties.