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.     

The consequences of spatial structure for the design and analysis of ecological field surveys

In ecological field surveys, observations are gathered at different spatial locations. The purpose may be to relate biological response variables (e.g., species abundances) to explanatory environmental variables (e.g., soil characteristics). In the absence of prior knowledge, ecologists have been taught to rely on systematic or random sampling designs. If there is prior knowledge about the spatial patterning of the explanatory variables, obtained from either previous surveys or a pilot study, can we use this information to optimize the sampling design in order to maximize our ability to detect the relationships between the response and explanatory variables?
The specific questions addressed in this paper are: a) What is the effect (type I error) of spatial autocorrelation on the statistical tests commonly used by ecologists to analyse field survey data? b) Can we eliminate, or at least minimize, the effect of spatial autocorrelation by the design of the survey? Are there designs that provide greater power for surveys, at least under certain circumstances? c) Can we eliminate or control for the effect of spatial autocorrelation during the analysis? To answer the last question, we compared regular regression analysis to a modified t‐test developed by Dutilleul for correlation coefficients in the presence of spatial autocorrelation.
Replicated surfaces (typically, 1000 of them) were simulated using different spatial parameters, and these surfaces were subjected to different sampling designs and methods of statistical analysis. The simulated surfaces may represent, for example, vegetation response to underlying environmental variation. This allowed us 1) to measure the frequency of type I error (the failure to reject the null hypothesis when in fact there is no effect of the environment on the response variable) and 2) to estimate the power of the different combinations of sampling designs and methods of statistical analysis (power is measured by the rate of rejection of the null hypothesis when an effect of the environment on the response variable has been created).
Our results indicate that: 1) Spatial autocorrelation in both the response and environmental variables affects the classical tests of significance of correlation or regression coefficients. Spatial autocorrelation in only one of the two variables does not affect the test of significance. 2) A broad‐scale spatial structure present in data has the same effect on the tests as spatial autocorrelation. When such a structure is present in one of the variables and autocorrelation is found in the other, or in both, the tests of significance have inflated rates of type I error. 3) Dutilleul's modified t‐test for the correlation coefficient, corrected for spatial autocorrelation, effectively corrects for spatial autocorrelation in the data. It also effectively corrects for the presence of deterministic structures, with or without spatial autocorrelation.
The presence of a broad‐scale deterministic structure may, in some cases, reduce the power of the modified t‐test.     

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.     

Carbon Emission Performance of China's Power Industry: Regional Disparity and Spatial Analysis

This study employs an undesirable‐output–oriented data envelopment analysis model to measure the carbon emission performance of the power industry throughout China's 30 administrative regions during the period of 2003–2012. Also, it further studies the regional disparity and spatial correlation of the carbon emission performance of China's power industry. The main findings are as follows: (1) The carbon emission performance of China's power industry is at a relatively low level, but shows a rising trend. (2) The regional carbon emission performance of China's power industry is extremely unbalanced: The eastern area ranks first, with the highest performance of 0.851, followed by the central area, whereas the western area falls behind, with the lowest performance of 0.760. Provinces in the eastern area generally perform better than those in other areas. (3) According to spatial analysis, the global Moran's I values of carbon emission performance are significantly positive during the sample period, which indicates that the carbon emission performance is a positive spatial correlation and has an obvious clustering effect. The estimate of the local spatial autocorrelation index confirms the imbalance of spatial distribution of the power sector's carbon emission performance. Based on the above findings, several policy suggestions are presented in this article.     

Genetic structure of the indigenous populations of Siberia

This study explores the genetic structure of Siberian indigenous populations on the basis of standard blood group and protein markers and DNA variable number of tandem repeats (VNTR) variation. Four analytical methods were utilized in this study: Harpending and Jenkin's R-matrix; Harpending and Ward's method of correlating genetic heterozygosity (H) to the distance from the centroid of the gene frequency array (r_ii); spatial autocorrelation, and Mantel tests. Because of the underlying assumptions of the various methods, the numbers of populations used in the analyses varied from 15 to 62. Since spatial autocorrelation is based upon separate correlations between alleles, a larger number of standard blood markers and populations were used. Fewest Siberian populations have been sampled for VNTRs, thus, only a limited comparison was possible. The four analytical procedures employed in this study yielded complementary results suggestive of the effects of unique historical events, evolutionary forces, and geography on the distribution of alleles in Siberian indigenous populations. The principal components analysis of the R-matrix demonstrated the presence of populational clusters that reflect their phylogenetic relationship. Mantel comparisons of matrices indicate that an intimate relationship exists between geography, languages, and genetics of Siberian populations. Spatial autocorrelation patterns reflect the isolation-by-distance model of Malecot and the possible effects of long-distance migration. Am J Phys Anthropol 104:177–192, 1997. © 1997 Wiley-Liss, Inc.     

Spatially autocorrelated sampling falsely inflates measures of accuracy for presence‐only niche models

Aim Environmental niche models that utilize presence‐only data have been increasingly employed to model species distributions and test ecological and evolutionary predictions. The ideal method for evaluating the accuracy of a niche model is to train a model with one dataset and then test model predictions against an independent dataset. However, a truly independent dataset is often not available, and instead random subsets of the total data are used for ‘training’ and ‘testing’ purposes. The goal of this study was to determine how spatially autocorrelated sampling affects measures of niche model accuracy when using subsets of a larger dataset for accuracy evaluation. Location The distribution of Centaurea maculosa (spotted knapweed; Asteraceae) was modelled in six states in the western United States: California, Oregon, Washington, Idaho, Wyoming and Montana. Methods Two types of niche modelling algorithms – the genetic algorithm for rule‐set prediction (GARP) and maximum entropy modelling (as implemented with Maxent) – were used to model the potential distribution of C. maculosa across the region. The effect of spatially autocorrelated sampling was examined by applying a spatial filter to the presence‐only data (to reduce autocorrelation) and then comparing predictions made using the spatial filter with those using a random subset of the data, equal in sample size to the filtered data. Results The accuracy of predictions from both algorithms was sensitive to the spatial autocorrelation of sampling effort in the occurrence data. Spatial filtering led to lower values of the area under the receiver operating characteristic curve plot but higher similarity statistic (I) values when compared with predictions from models built with random subsets of the total data, meaning that spatial autocorrelation of sampling effort between training and test data led to inflated measures of accuracy. Main conclusions The findings indicate that care should be taken when interpreting the results from presence‐only niche models when training and test data have been randomly partitioned but occurrence data were non‐randomly sampled (in a spatially autocorrelated manner). The higher accuracies obtained without the spatial filter are a result of spatial autocorrelation of sampling effort between training and test data inflating measures of prediction accuracy. If independently surveyed data for testing predictions are unavailable, then it may be necessary to explicitly account for the spatial autocorrelation of sampling effort between randomly partitioned training and test subsets when evaluating niche model predictions.     

Spatial autocorrelation and red herrings in geographical ecology

Aim Spatial autocorrelation in ecological data can inflate Type I errors in statistical analyses. There has also been a recent claim that spatial autocorrelation generates ‘red herrings’, such that virtually all past analyses are flawed. We consider the origins of this phenomenon, the implications of spatial autocorrelation for macro‐scale patterns of species diversity and set out a clarification of the statistical problems generated by its presence. Location To illustrate the issues involved, we analyse the species richness of the birds of western/central Europe, north Africa and the Middle East. Methods Spatial correlograms for richness and five environmental variables were generated using Moran's I coefficients. Multiple regression, using both ordinary least‐squares (OLS) and generalized least squares (GLS) assuming a spatial structure in the residuals, were used to identify the strongest predictors of richness. Autocorrelation analyses of the residuals obtained after stepwise OLS regression were undertaken, and the ranks of variables in the full OLS and GLS models were compared. Results Bird richness is characterized by a quadratic north–south gradient. Spatial correlograms usually had positive autocorrelation up to c. 1600 km. Including the environmental variables successively in the OLS model reduced spatial autocorrelation in the residuals to non‐detectable levels, indicating that the variables explained all spatial structure in the data. In principle, if residuals are not autocorrelated then OLS is a special case of GLS. However, our comparison between OLS and GLS models including all environmental variables revealed that GLS de‐emphasized predictors with strong autocorrelation and long‐distance clinal structures, giving more importance to variables acting at smaller geographical scales. Conclusion Although spatial autocorrelation should always be investigated, it does not necessarily generate bias. Rather, it can be a useful tool to investigate mechanisms operating on richness at different spatial scales. Claims that analyses that do not take into account spatial autocorrelation are flawed are without foundation.     

Unifying conspecific-encounter index v and Moran's I index

Single-species distributional patterns have been well studied and quantified using a negative binomial model. However, it is still unclear how to quantify multi-species distribution patterns. A previous study (Chen et al. 2019) developed a simple conspecific-encounter index for quantifying multi-species distributional aggregation patterns when biodiversity data are collected in a consecutive scheme along line transects. Here, through simple derivation, we reported that the previously proposed conspecific-encounter index had a close association with Moran's I index, which has been widely applied in spatial ecology for describing spatial autocorrelation of ecological data. Our proof unified the conspecific-encounter index that is applied under the context of line-transect sampling and Moran's I index that is applied under the context of spatial ecology. The unification sheds light on the development of new statistical metrics for quantifying biological diversity and distribution patterns when line transects are used in field surveys.     

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.     

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.     

HOW SEDENTARY ARE GREYWING FRANCOLINS (FRANCOLINUS AFRICANUS)?

Phasianids are considered to be sedentary birds with limited dispersal so that populations may be expected to show genetic isolation by distance. To test this, we examined genetic variability in 618 greywing francolins (Francolinus africanus) at 24 localities over a 1,500 km² area. We subdivided the samples to measure genetic population structure among localities separated by 6–60 km, and among coveys separated by 0.1–6 km. Thirteen of 30 (43%) allozyme loci were polymorphic, and heterozygosity ranged from 5.3 to 8.5% over 24 localities and averaged 7.0%, a value much larger than that found for other phasianids. Significant allele-frequency heterogeneity was detected among localities and among coveys at several localities for several loci. Mantel's test, however, showed that there was no correlation between geographical distance and the allele-frequency difference between localities for all but one allele. Although spatial autocorrelation was detected with Moran's I and Geary's c for two alleles, the geographical patterns of I in correlograms of 18 independent alleles showed a “crazy-quilt” pattern of allele-frequency patches. This shows that the isolation-by-distance model of subpopulation structure is inappropriate for these birds. Individuals, therefore, appear to disperse far beyond neighboring populations. “Private-allele” and F_ST estimates of migration under the island model were 8–9 individuals between localities of each generation. Allele-frequency heterogeneity, large amounts of gene flow, and the general lack of spatial autocorrelation imply that the small, socially-structured populations of greywing are subject to high rates of turnover, founder effects, and random drift.     

Host plant associations and geography interact to shape diversification in a specialist insect herbivore

Disentangling the processes underlying geographic and environmental patterns of biodiversity challenges biologists as such patterns emerge from eco‐evolutionary processes confounded by spatial autocorrelation among sample units. The herbivorous insect, Belonocnema treatae (Hymenoptera: Cynipidae), exhibits regional specialization on three plant species whose geographic distributions range from sympatry through allopatry across the southern United States. Using range‐wide sampling spanning the geographic ranges of the three host plants and genotyping‐by‐sequencing of 1,217 individuals, we tested whether this insect herbivore exhibited host plant‐associated genomic differentiation while controlling for spatial autocorrelation among the 58 sample sites. Population genomic structure based on 40,699 SNPs was evaluated using the hierarchical Bayesian model entropy to assign individuals to genetic clusters and estimate admixture proportions. To control for spatial autocorrelation, distance‐based Moran's eigenvector mapping was used to construct regression variables summarizing spatial structure inherent among sample sites. Distance‐based redundancy analysis (dbRDA) incorporating the spatial variables was then applied to partition host plant‐associated differentiation (HAD) from spatial autocorrelation. By combining entropy and dbRDA to analyse SNP data, we unveiled a complex mosaic of highly structured differentiation within and among gall‐former populations finding evidence that geography, HAD and spatial autocorrelation all play significant roles in explaining patterns of genomic differentiation in B. treatae. While dbRDA confirmed host association as a significant predictor of patterns of genomic variation, spatial autocorrelation among sites explained the largest proportion of variation. Our results demonstrate the value of combining dbRDA with hierarchical structural analyses to partition spatial/environmental patterns of genomic variation.     

Spatial Autocorrelation Approaches to Testing Residuals from Least Squares Regression

In geo-statistics, the Durbin-Watson test is frequently employed to detect the presence of residual serial correlation from least squares regression analyses. However, the Durbin-Watson statistic is only suitable for ordered time or spatial series. If the variables comprise cross-sectional data coming from spatial random sampling, the test will be ineffectual because the value of Durbin-Watson’s statistic depends on the sequence of data points. This paper develops two new statistics for testing serial correlation of residuals from least squares regression based on spatial samples. By analogy with the new form of Moran’s index, an autocorrelation coefficient is defined with a standardized residual vector and a normalized spatial weight matrix. Then by analogy with the Durbin-Watson statistic, two types of new serial correlation indices are constructed. As a case study, the two newly presented statistics are applied to a spatial sample of 29 China’s regions. These results show that the new spatial autocorrelation models can be used to test the serial correlation of residuals from regression analysis. In practice, the new statistics can make up for the deficiencies of the Durbin-Watson test.     

Genetic spatial autocorrelation can readily detect sex-biased dispersal

Sex-biased dispersal is expected to generate differences in the fine-scale genetic structure of males and females. Therefore, spatial analyses of multilocus genotypes may offer a powerful approach for detecting sex-biased dispersal in natural populations. However, the effects of sex-biased dispersal on fine-scale genetic structure have not been explored. We used simulations and multilocus spatial autocorrelation analysis to investigate how sex-biased dispersal influences fine-scale genetic structure. We evaluated three statistical tests for detecting sex-biased dispersal: bootstrap confidence intervals about autocorrelation r values and recently developed heterogeneity tests at the distance class and whole correlogram levels. Even modest sex bias in dispersal resulted in significantly different fine-scale spatial autocorrelation patterns between the sexes. This was particularly evident when dispersal was strongly restricted in the less-dispersing sex (mean distance <200 m), when differences between the sexes were readily detected over short distances. All tests had high power to detect sex-biased dispersal with large sample sizes (n ≥ 250). However, there was variation in type I error rates among the tests, for which we offer specific recommendations. We found congruence between simulation predictions and empirical data from the agile antechinus, a species that exhibits male-biased dispersal, confirming the power of individual-based genetic analysis to provide insights into asymmetries in male and female dispersal. Our key recommendations for using multilocus spatial autocorrelation analyses to test for sex-biased dispersal are: (i) maximize sample size, not locus number; (ii) concentrate sampling within the scale of positive structure; (iii) evaluate several distance class sizes; (iv) use appropriate methods when combining data from multiple populations; (v) compare the appropriate groups of individuals.     

Assessing non-linearity in European temperature-sensitive tree-ring data

We test the application of parametric, non-parametric, and semi-parametric calibration models for reconstructing summer (June–August) temperature from a set of tree-ring width and density data on the same dendro samples from 40 sites across Europe. By comparing the performance of the three calibration models on pairs” of tree-ring width (TRW) and maximum density (MXD) or maximum blue intensity (MXBI), we test whether a non-linear temperature response is more prevalent in TRW or MXD (MXBI) data, and whether it is associated with the temperature sensitivity and/or autocorrelation structure of the dendro parameters. We note that MXD (MXBI) data have a significantly stronger temperature response than TRW data as well as a lower autocorrelation that is more similar to that of the instrumental temperature data, whereas TRW exhibits a redder” variability continuum. This study shows that the use of non-parametric calibration models is more suitable for TRW data, while parametric calibration is sufficient for both MXD and MXBI data – that is, we show that TRW is by far the more non-linear proxy.     

Evaluating,partitioning, and mapping the spatial autocorrelation component in ecological niche modeling: a new approach based on environmentally equidistant records

Most species data display spatial autocorrelation that can affect ecological niche models (ENMs) accuracy‐statistics, affecting its ability to infer geographic distributions. Here we evaluate whether the spatial autocorrelation underlying species data affects accuracy‐statistics and map the uncertainties due to spatial autocorrelation effects on species range predictions under past and future climate models. As an example, ENMs were fitted to Qualea grandiflora (Vochysiaceae), a widely distributed plant from Brazilian Cerrado. We corrected for spatial autocorrelation in ENMs by selecting sampling sites equidistant in geographical (GEO) and environmental (ENV) spaces. Distributions were modelled using 13 ENMs evaluated by two accuracy‐statistics (TSS and AUC), which were compared with uncorrected ENMs. Null models and the similarity statistics I were used to evaluate the effects of spatial autocorrelation. Moreover, we applied a hierarchical ANOVA to partition and map the uncertainties from the time (across last glacial maximum, pre‐insustrial, and 2080 time periods) and methodological components (ENMs and autocorrelation corrections). The GEO and ENV models had the highest accuracy‐statistics values, although only the ENV model had values higher than expected by chance alone for most of the 13 ENMs. Uncertainties from time component were higher in the core region of the Brazilian Cerrado where Q. grandiflora occurs, whereas methodological components presented higher uncertainties in the extreme northern and southern regions of South America (i.e. outside of Brazilian Cerrado). Our findings show that accounting for autocorrelation in environmental space is more efficient than doing so in geographical space. Methodological uncertainties were concentrated in outside the core region of Q. grandiflora's habitat. Conversely, uncertainty due to time component in the Brazilian Cerrado reveals that ENMs were able to capture climate change effects on Q. grandiflora distributions.     

Spatial autocorrelation as a tool for identifying the geographical patterns of aphid annual abundance

Abstract 1 A spatial autocorrelation analysis was undertaken to investigate the spatial structure of annual abundance for the pest aphid Myzus persicae collected in suction traps distributed across north‐west Europe. 2 The analysis was applied at two different scales. The Moran index was used to estimate the degree of spatial autocorrelation at all sites within the study area (global level). The contributions of each site to the global index were identified by the use of a local indicator of spatial autocorrelation (LISA). A hierarchical cluster analysis was undertaken to highlight differences between groups of resulting correlograms. 3 Similarity between traps was shown to occur over large geographical distances, suggesting an impact of phenomena such as climatic gradients or land use types. 4 The presence of outliers and zones of similarity (hot‐spots) and of dissimilarity (cold‐spots) were identified indicating a strong impact of local effects. 5 Several groups of traps characterized by similarities in their local spatial structure (correlograms, value of Moran's I_i) also had similar values for land use variables (the area occupied by agricultural zones, forest and sea). 6 It is concluded that trap data can provide information about Myzus persicae that is representative of large geographical areas. Thus, trap data can be used to estimate the aerial abundance of this species, even if the suction traps are not regularly and densely distributed.     

Environmental steepness,tolerance gradient,and ecogeographical rules in glassfrogs (Anura: Centrolenidae)

Spatial variation in biological traits reflects evolutionary and biogeographical processes of the history of clades, and patterns of body size and range size can be suitable to recover such processes. In the present study, we test for latitudinal and altitudinal gradients in both body and range sizes in an entire family of tropical anurans, Centrolenidae. We partition the species latitudinal, and altitudinal distributions into an indirect measure of tolerance, and then test its effect on the body size gradient. We use an assemblage‐based approach to correlate the traits with altitudinal and latitudinal axes, taking into account both phylogenetic and spatial autocorrelation in data. Centrolenids lack any gradient in range size but show a positive cline of both body size and adaptive body enlargement with altitude. This pattern is also positively correlated with an altitudinal gradient of cold tolerance, thus lending support to the heat balance hypothesis as an explanation of the body size cline. By using an entire Neotropical clade of anurans, we add further support for Bergmann's rule in ectotherms, warn for a likely effect of environmental steepness in fashioning the gradient, and offer evidence for an historical scenario (the Oligocene–Eocene Andean uplift) as its likely trigger. © 2013 The Linnean Society of London