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1.  A review of techniques for spatial modeling in geographical, conservation and landscape genetics 
Genetics and Molecular Biology  2009;32(2):203-211.
Most evolutionary processes occur in a spatial context and several spatial analysis techniques have been employed in an exploratory context. However, the existence of autocorrelation can also perturb significance tests when data is analyzed using standard correlation and regression techniques on modeling genetic data as a function of explanatory variables. In this case, more complex models incorporating the effects of autocorrelation must be used. Here we review those models and compared their relative performances in a simple simulation, in which spatial patterns in allele frequencies were generated by a balance between random variation within populations and spatially-structured gene flow. Notwithstanding the somewhat idiosyncratic behavior of the techniques evaluated, it is clear that spatial autocorrelation affects Type I errors and that standard linear regression does not provide minimum variance estimators. Due to its flexibility, we stress that principal coordinate of neighbor matrices (PCNM) and related eigenvector mapping techniques seem to be the best approaches to spatial regression. In general, we hope that our review of commonly used spatial regression techniques in biology and ecology may aid population geneticists towards providing better explanations for population structures dealing with more complex regression problems throughout geographic space.
PMCID: PMC3036944  PMID: 21637669
autocorrelation; geographical genetics; isolation-by-distance; landscape genetics; spatial regression
2.  A heteroskedastic error covariance matrix estimator using a first-order conditional autoregressive Markov simulation for deriving asympotical efficient estimates from ecological sampled Anopheles arabiensis aquatic habitat covariates 
Malaria Journal  2009;8:216.
Autoregressive regression coefficients for Anopheles arabiensis aquatic habitat models are usually assessed using global error techniques and are reported as error covariance matrices. A global statistic, however, will summarize error estimates from multiple habitat locations. This makes it difficult to identify where there are clusters of An. arabiensis aquatic habitats of acceptable prediction. It is therefore useful to conduct some form of spatial error analysis to detect clusters of An. arabiensis aquatic habitats based on uncertainty residuals from individual sampled habitats. In this research, a method of error estimation for spatial simulation models was demonstrated using autocorrelation indices and eigenfunction spatial filters to distinguish among the effects of parameter uncertainty on a stochastic simulation of ecological sampled Anopheles aquatic habitat covariates. A test for diagnostic checking error residuals in an An. arabiensis aquatic habitat model may enable intervention efforts targeting productive habitats clusters, based on larval/pupal productivity, by using the asymptotic distribution of parameter estimates from a residual autocovariance matrix. The models considered in this research extends a normal regression analysis previously considered in the literature.
Field and remote-sampled data were collected during July 2006 to December 2007 in Karima rice-village complex in Mwea, Kenya. SAS 9.1.4® was used to explore univariate statistics, correlations, distributions, and to generate global autocorrelation statistics from the ecological sampled datasets. A local autocorrelation index was also generated using spatial covariance parameters (i.e., Moran's Indices) in a SAS/GIS® database. The Moran's statistic was decomposed into orthogonal and uncorrelated synthetic map pattern components using a Poisson model with a gamma-distributed mean (i.e. negative binomial regression). The eigenfunction values from the spatial configuration matrices were then used to define expectations for prior distributions using a Markov chain Monte Carlo (MCMC) algorithm. A set of posterior means were defined in WinBUGS 1.4.3®. After the model had converged, samples from the conditional distributions were used to summarize the posterior distribution of the parameters. Thereafter, a spatial residual trend analyses was used to evaluate variance uncertainty propagation in the model using an autocovariance error matrix.
By specifying coefficient estimates in a Bayesian framework, the covariate number of tillers was found to be a significant predictor, positively associated with An. arabiensis aquatic habitats. The spatial filter models accounted for approximately 19% redundant locational information in the ecological sampled An. arabiensis aquatic habitat data. In the residual error estimation model there was significant positive autocorrelation (i.e., clustering of habitats in geographic space) based on log-transformed larval/pupal data and the sampled covariate depth of habitat.
An autocorrelation error covariance matrix and a spatial filter analyses can prioritize mosquito control strategies by providing a computationally attractive and feasible description of variance uncertainty estimates for correctly identifying clusters of prolific An. arabiensis aquatic habitats based on larval/pupal productivity.
PMCID: PMC2760564  PMID: 19772590
3.  Enhanced spatial models for predicting the geographic distributions of tick-borne pathogens 
Disease maps are used increasingly in the health sciences, with applications ranging from the diagnosis of individual cases to regional and global assessments of public health. However, data on the distributions of emerging infectious diseases are often available from only a limited number of samples. We compared several spatial modelling approaches for predicting the geographic distributions of two tick-borne pathogens: Ehrlichia chaffeensis, the causative agent of human monocytotropic ehrlichiosis, and Anaplasma phagocytophilum, the causative agent of human granulocytotropic anaplasmosis. These approaches extended environmental modelling based on logistic regression by incorporating both spatial autocorrelation (the tendency for pathogen distributions to be clustered in space) and spatial heterogeneity (the potential for environmental relationships to vary spatially).
Incorporating either spatial autocorrelation or spatial heterogeneity resulted in substantial improvements over the standard logistic regression model. For E. chaffeensis, which was common within the boundaries of its geographic range and had a highly clustered distribution, the model based only on spatial autocorrelation was most accurate. For A. phagocytophilum, which has a more complex zoonotic cycle and a comparatively weak spatial pattern, the model that incorporated both spatial autocorrelation and spatially heterogeneous relationships with environmental variables was most accurate.
Spatial autocorrelation can improve the accuracy of predictive disease risk models by incorporating spatial patterns as a proxy for unmeasured environmental variables and spatial processes. Spatial heterogeneity can also improve prediction accuracy by accounting for unique ecological conditions in different regions that affect the relative importance of environmental drivers on disease risk.
PMCID: PMC2373776  PMID: 18412972
4.  Geographical and Temporal Body Size Variation in a Reptile: Roles of Sex, Ecology, Phylogeny and Ecology Structured in Phylogeny 
PLoS ONE  2014;9(8):e104026.
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.
PMCID: PMC4121295  PMID: 25090025
5.  Effects on population persistence: the interaction between environmental noise colour, intraspecific competition and space 
It is accepted that accurate estimation of risk of population extinction, or persistence time, requires prediction of the effect of fluctuations in the environment on population dynamics. Generally, the greater the magnitude, or variance, of environmental stochasticity, the greater the risk of population extinction. Another characteristic of environmental stochasticity, its colour, has been found to affect population persistence. This is important because real environmental variables, such as temperature, are reddened or positively temporally autocorrelated. However, recent work has disagreed about the effect of reddening environmental stochasticity. Ripa and Lundberg (1996) found increasing temporal autocorrelation (reddening) decreased the risk of extinction, whereas a simple and powerful intuitive argument (Lawton 1988) predicts increased risk of extinction with reddening. This study resolves the apparent contradiction, in two ways, first, by altering the dynamic behaviour of the population models. Overcompensatory dynamics result in persistence times increasing with increased temporal autocorrelation; undercompensatory dynamics result in persistence times decreasing with increased temporal autocorrelation. Secondly, in a spatially subdivided population, with a reasonable degree of spatial heterogeneity in patch quality, increasing temporal autocorrelation in the environment results in decreasing persistence time for both types of competition. Thus, the inclusion of coloured noise into ecological models can have subtle interactions with population dynamics.
PMCID: PMC1688739
Extinction Coloured Noise Competition Space
6.  Quantitative approaches in climate change ecology 
Global Change Biology  2011;17(12):3697-3713.
Contemporary impacts of anthropogenic climate change on ecosystems are increasingly being recognized. Documenting the extent of these impacts requires quantitative tools for analyses of ecological observations to distinguish climate impacts in noisy data and to understand interactions between climate variability and other drivers of change. To assist the development of reliable statistical approaches, we review the marine climate change literature and provide suggestions for quantitative approaches in climate change ecology. We compiled 267 peer-reviewed articles that examined relationships between climate change and marine ecological variables. Of the articles with time series data (n = 186), 75% used statistics to test for a dependency of ecological variables on climate variables. We identified several common weaknesses in statistical approaches, including marginalizing other important non-climate drivers of change, ignoring temporal and spatial autocorrelation, averaging across spatial patterns and not reporting key metrics. We provide a list of issues that need to be addressed to make inferences more defensible, including the consideration of (i) data limitations and the comparability of data sets; (ii) alternative mechanisms for change; (iii) appropriate response variables; (iv) a suitable model for the process under study; (v) temporal autocorrelation; (vi) spatial autocorrelation and patterns; and (vii) the reporting of rates of change. While the focus of our review was marine studies, these suggestions are equally applicable to terrestrial studies. Consideration of these suggestions will help advance global knowledge of climate impacts and understanding of the processes driving ecological change.
PMCID: PMC3597248
7.  Time Series Analysis of Particle Tracking Data for Molecular Motion on the Cell Membrane 
Bulletin of mathematical biology  2009;71(8):1967-2024.
Biophysicists use single particle tracking (SPT) methods to probe the dynamic behavior of individual proteins and lipids in cell membranes. The mean squared displacement (MSD) has proven to be a powerful tool for analyzing the data and drawing conclusions about membrane organization, including features like lipid rafts, protein islands, and confinement zones defined by cytoskeletal barriers. Here, we implement time series analysis as a new analytic tool to analyze further the motion of membrane proteins. The experimental data track the motion of 40 nm gold particles bound to Class I major histocompatibility complex (MHCI) molecules on the membranes of mouse hepatoma cells.
Our first novel result is that the tracks are significantly autocorrelated. Because of this, we developed linear autoregressive models to elucidate the autocorrelations. Estimates of the signal to noise ratio for the models show that the autocorrelated part of the motion is significant. Next, we fit the probability distributions of jump sizes with four different models. The first model is a general Weibull distribution that shows that the motion is characterized by an excess of short jumps as compared to a normal random walk. We also fit the data with a chi distribution which provides a natural estimate of the dimension d of the space in which a random walk is occurring. For the biological data, the estimates satisfy 1 < d < 2, implying that particle motion is not confined to a line, but also does not occur freely in the plane. The dimension gives a quantitative estimate of the amount of nanometer scale obstruction met by a diffusing molecule. We introduce a new distribution and use the generalized extreme value distribution to show that the biological data also have an excess of long jumps as compared to normal diffusion. These fits provide novel estimates of the microscopic diffusion constant.
Previous MSD analyses of SPT data have provided evidence for nanometer-scale confinement zones that restrict lateral diffusion, supporting the notion that plasma membrane organization is highly structured. Our demonstration that membrane protein motion is autocorrelated and is characterized by an excess of both short and long jumps reinforces the concept that the membrane environment is heterogeneous and dynamic. Autocorrelation analysis and modeling of the jump distributions are powerful new techniques for the analysis of SPT data and the development of more refined models of membrane organization.
The time series analysis also provides several methods of estimating the diffusion constant in addition to the constant provided by the mean squared displacement. The mean squared displacement for most of the biological data shows a power law behavior rather the linear behavior of Brownian motion. In this case, we introduce the notion of an instantaneous diffusion constant. All of the diffusion constants show a strong consistency for most of the biological data.
PMCID: PMC3672074  PMID: 19657701
Time series analysis; Single particle tracking; Cell membrane; Mean squared displacement
8.  Temporal autocorrelation functions for movement rates from global positioning system radiotelemetry data 
Autocorrelation has been viewed as a problem in telemetry studies because sequential observations are not independent in time or space, therefore violating assumptions for statistical inference. Yet nearly all ecological and behavioural data are autocorrelated in both space and time. We argue that there is much to learn about the structure of ecological and behavioural data from patterns of autocorrelation. Such patterns include periodicity in movement and patchiness in spatial data, which can be characterized by an autocorrelogram, semivariogram or spectrum. We illustrate the utility of temporal autocorrelation functions (ACFs) for analysing step-length data from GPS telemetry of wolves (Canis lupus), cougars (Puma concolor), grizzly bears (Ursus arctos) and elk (Cervus elaphus) in western Alberta. ACFs often differ by season, reflecting differences in foraging behaviour. In wilderness landscapes, step-length ACFs for predators decay slowly to apparently random patterns, but sometimes display strong daily rhythms in areas of human disturbance. In contrast, step lengths of elk are consistently periodic, reflecting crepuscular activity.
PMCID: PMC2894959  PMID: 20566498
Alberta; autocorrelation; GPS radiotelemetry; movement; periodicity; step length
9.  Spatio-Temporal Patterns of Barmah Forest Virus Disease in Queensland, Australia 
PLoS ONE  2011;6(10):e25688.
Barmah Forest virus (BFV) disease is a common and wide-spread mosquito-borne disease in Australia. This study investigated the spatio-temporal patterns of BFV disease in Queensland, Australia using geographical information system (GIS) tools and geostatistical analysis.
Methods/Principal Findings
We calculated the incidence rates and standardised incidence rates of BFV disease. Moran's I statistic was used to assess the spatial autocorrelation of BFV incidences. Spatial dynamics of BFV disease was examined using semi-variogram analysis. Interpolation techniques were applied to visualise and display the spatial distribution of BFV disease in statistical local areas (SLAs) throughout Queensland. Mapping of BFV disease by SLAs reveals the presence of substantial spatio-temporal variation over time. Statistically significant differences in BFV incidence rates were identified among age groups (χ2 = 7587, df = 7327,p<0.01). There was a significant positive spatial autocorrelation of BFV incidence for all four periods, with the Moran's I statistic ranging from 0.1506 to 0.2901 (p<0.01). Semi-variogram analysis and smoothed maps created from interpolation techniques indicate that the pattern of spatial autocorrelation was not homogeneous across the state.
This is the first study to examine spatial and temporal variation in the incidence rates of BFV disease across Queensland using GIS and geostatistics. The BFV transmission varied with age and gender, which may be due to exposure rates or behavioural risk factors. There are differences in the spatio-temporal patterns of BFV disease which may be related to local socio-ecological and environmental factors. These research findings may have implications in the BFV disease control and prevention programs in Queensland.
PMCID: PMC3192738  PMID: 22022430
10.  Clonal and Fine-scale Genetic Structure in Populations of a Restricted Korean Endemic, Hosta jonesii (Liliaceae) and the Implications for Conservation 
Annals of Botany  2005;96(2):279-288.
• Background and Aims In plant populations the magnitude of spatial genetic structure of apparent individuals (including clonal ramets) can be different from that of sexual individuals (genets). Thus, distinguishing the effects of clonal versus sexual individuals in population genetic analyses could provide important insights for evolutionary biology and conservation. To investigate the effects of clonal spread on the fine-scale spatial genetic structure within plant populations, Hosta jonesii (Liliaceae), an endemic species to Korea, was chosen as a study species.
• Methods Using allozymes as genetic markers, spatial autocorrelation analysis of ramets and of genets was conducted to quantify the spatial scale of clonal spread and genotype distribution in two populations of H. jonesii.
• Key Results Join-count statistics revealed that most clones are significantly aggregated at <3-m interplant distance. Spatial autocorrelation analysis of all individuals resulted in significantly higher Moran's I values at 0–3-m interplant distance than analyses of population samples in which clones were excluded. However, significant fine-scale genetic structure was still observed when clones were excluded.
• Conclusions These results suggest that clones enhance the magnitude of spatial autocorrelation due to localized clonal spread. The significant fine-scale genetic structure detected in samples excluding clones is consistent with the biological and ecological traits exhibited by H. jonesii including bee pollination and limited seed dispersal. For conservation purposes, genetic diversity would be maximized in local populations of H. jonesii by collecting or preserving individuals that are spaced at least 5 m apart.
PMCID: PMC4246876  PMID: 15928007
Hosta jonesii; allozymes; clonal structure; conservation; fine-scale genetic structure; Korean endemic; Liliaceae; sampling strategies
11.  Controlling for localised spatio-temporal autocorrelation in long-term air pollution and health studies 
Estimating the long-term health impact of air pollution using an ecological spatio-temporal study design is a challenging task, due to the presence of residual spatio-temporal autocorrelation in the health counts after adjusting for the covariate effects. This autocorrelation is commonly modelled by a set of random effects represented by a Gaussian Markov random field (GMRF) prior distribution, as part of a hierarchical Bayesian model. However, GMRF models typically assume the random effects are globally smooth in space and time, and thus are likely to be collinear to any spatially and temporally smooth covariates such as air pollution. Such collinearity leads to poor estimation performance of the estimated fixed effects, and motivated by this epidemiological problem, this paper proposes new GMRF methodology to allow for localised spatio-temporal smoothing. This means random effects that are either geographically or temporally adjacent are allowed to be autocorrelated or conditionally independent, which allows more flexible autocorrelation structures to be represented. This increased flexibility results in improved fixed effects estimation compared with global smoothing models, which is evidenced by our simulation study. The methodology is then applied to the motivating study investigating the long-term effects of air pollution on respiratory ill health in Greater Glasgow, Scotland between 2007 and 2011.
PMCID: PMC4272194  PMID: 24648100
air pollution and health studies; Gaussian Markov random fields; spatio-temporal autocorrelation
12.  Explaining Local-Scale Species Distributions: Relative Contributions of Spatial Autocorrelation and Landscape Heterogeneity for an Avian Assemblage 
PLoS ONE  2013;8(2):e55097.
Understanding interactions between mobile species distributions and landcover characteristics remains an outstanding challenge in ecology. Multiple factors could explain species distributions including endogenous evolutionary traits leading to conspecific clustering and endogenous habitat features that support life history requirements. Birds are a useful taxon for examining hypotheses about the relative importance of these factors among species in a community. We developed a hierarchical Bayes approach to model the relationships between bird species occupancy and local landcover variables accounting for spatial autocorrelation, species similarities, and partial observability. We fit alternative occupancy models to detections of 90 bird species observed during repeat visits to 316 point-counts forming a 400-m grid throughout the Patuxent Wildlife Research Refuge in Maryland, USA. Models with landcover variables performed significantly better than our autologistic and null models, supporting the hypothesis that local landcover heterogeneity is important as an exogenous driver for species distributions. Conspecific clustering alone was a comparatively poor descriptor of local community composition, but there was evidence for spatial autocorrelation in all species. Considerable uncertainty remains whether landcover combined with spatial autocorrelation is most parsimonious for describing bird species distributions at a local scale. Spatial structuring may be weaker at intermediate scales within which dispersal is less frequent, information flows are localized, and landcover types become spatially diversified and therefore exhibit little aggregation. Examining such hypotheses across species assemblages contributes to our understanding of community-level associations with conspecifics and landscape composition.
PMCID: PMC3564802  PMID: 23393564
13.  Effect of Site Level Environmental Variables, Spatial Autocorrelation and Sampling Intensity on Arthropod Communities in an Ancient Temperate Lowland Woodland Area 
PLoS ONE  2013;8(12):e81541.
The interaction of arthropods with the environment and the management of their populations is a focus of the ecological agenda. Spatial autocorrelation and under-sampling may generate bias and, when they are ignored, it is hard to determine if results can in any way be trusted. Arthropod communities were studied during two seasons and using two methods: window and panel traps, in an area of ancient temperate lowland woodland of Zebracka (Czech Republic). The composition of arthropod communities was studied focusing on four site level variables (canopy openness, diameter in the breast height and height of tree, and water distance) and finally analysed using two approaches: with and without effects of spatial autocorrelation. I found that the proportion of variance explained by space cannot be ignored (≈20% in both years). Potential bias in analyses of the response of arthropods to site level variables without including spatial co-variables is well illustrated by redundancy analyses. Inclusion of space led to more accurate results, as water distance and tree diameter were significant, showing approximately the same ratio of explained variance and direction in both seasons. Results without spatial co-variables were much more disordered and were difficult to explain. This study showed that neglecting the effects of spatial autocorrelation could lead to wrong conclusions in site level studies and, furthermore, that inclusion of space may lead to more accurate and unambiguous outcomes. Rarefactions showed that lower sampling intensity, when appropriately designed, can produce sufficient results without exploitation of the environment.
PMCID: PMC3857189  PMID: 24349087
14.  Spatial Analysis of Eco-environmental Risk Factors of Cutaneous Leishmaniasis in Southern Iran 
Despite the advances in the diagnosis and treatment of leishmaniasis, it is still considered as a severe public health problem particularly in developing countries and a great economic burden on the health resources. The present study was designed and conducted to determine the eco-environmental characteristics of the leishmaniasis disease by spatial analysis.
Materials and Methods:
In an ecological study, data were collected on eco-environmental factors of Fars province in Iran and on cutaneous leishmaniasis (CL) cases from 2002 to 2009. geographic weighted regression (GWR) was used to analyse the data and compare them with ordinary least square (OLS) regression model results. Moran's Index was applied for analysis of spatial autocorrelation in residual of OLS. P value less than 0.05 was considered as significant and adjusted R2 was used for model preferences.
There was a significant spatial autocorrelation in the residuals of OLS model (Z=2.45, P=0.014). GWR showed that rainy days, minimum temperature, wind velocity, maximum relative humidity and population density were the most important eco-environmental risk factors and explained 0.388 of the associated factors of CL.
Spatial analysis can be a good tool for detection and prediction of CL disease. In autocorrelated and non-stationary data, GWR model yields a better fitness than OLS regression model. Also, population density can be used as a surrogate variable of acquired immunity and increase the adjusted R2.
PMCID: PMC3339126  PMID: 22557853
Ecological study; environmental factors; geographic information systems; geographic weighted regression; leishmaniasis; spatial analysis
15.  Associations between street connectivity and active transportation 
Past studies of associations between measures of the built environment, particularly street connectivity, and active transportation (AT) or leisure walking/bicycling have largely failed to account for spatial autocorrelation of connectivity variables and have seldom examined both the propensity for AT and its duration in a coherent fashion. Such efforts could improve our understanding of the spatial and behavioral aspects of AT. We analyzed spatially identified data from Los Angeles and San Diego Counties collected as part of the 2001 California Health Interview Survey.
Principal components analysis indicated that ~85% of the variance in nine measures of street connectivity are accounted for by two components representing buffers with short blocks and dense nodes (PRIN1) or buffers with longer blocks that still maintain a grid like structure (PRIN2). PRIN1 and PRIN2 were positively associated with active transportation (AT) after adjustment for diverse demographic and health related variables. Propensity and duration of AT were correlated in both Los Angeles (r = 0.14) and San Diego (r = 0.49) at the zip code level. Multivariate analysis could account for the correlation between the two outcomes.
After controlling for demography, measures of the built environment and other factors, no spatial autocorrelation remained for propensity to report AT (i.e., report of AT appeared to be independent among neighborhood residents). However, very localized correlation was evident in duration of AT, particularly in San Diego, where the variance of duration, after accounting for spatial autocorrelation, was 5% smaller within small neighborhoods (~0.01 square latitude/longitude degrees = 0.6 mile diameter) compared to within larger zip code areas. Thus a finer spatial scale of analysis seems to be more appropriate for explaining variation in connectivity and AT.
Joint analysis of the propensity and duration of AT behavior and an explicitly geographic approach can strengthen studies of the built environment and physical activity (PA), specifically AT. More rigorous analytical work on cross-sectional data, such as in the present study, continues to support the need for experimental and longitudinal study designs including the analysis of natural experiments to evaluate the utility of environmental interventions aimed at increasing PA.
PMCID: PMC2876088  PMID: 20412597
16.  Interactive effects of temporal correlations, spatial heterogeneity and dispersal on population persistence 
It is an ecological truism that population persistence depends on a population's growth rate when rare. To understand the interplay between temporal correlations, spatial heterogeneity and dispersal on persistence, an analytic approximation for this growth rate is derived for partially mixing populations. Partial mixing has two effects on population growth. In the absence of temporal correlations in relative fitness, greater movement to patches with, on average, higher relative fitness increases population growth rates. In the absence of spatial heterogeneity in the average relative fitnesses, lower dispersal rates enhance population growth when temporal autocorrelations of relative fitness within a patch exceed temporal cross-correlations in relative fitness between patches. This approximation implies that metapopulations whose expected fitness in every patch is less than 1 can persist if there are positive temporal autocorrelations in relative fitness, sufficiently weak spatial correlations and the population disperses at intermediate rates. It also implies that movement into lower quality habitats increases the population growth rate whenever the net temporal variation in per capita growth rates is sufficiently larger than the difference in the means of these per capita growth rates. Moreover, temporal autocorrelations, whether they be negative or positive, can enhance population growth for optimal dispersal strategies.
PMCID: PMC2871869  PMID: 20164099
metapopulation persistence; spatial heterogeneity; temporal correlations; source-sink dynamics
17.  From spatial ecology to spatial epidemiology: modeling spatial distributions of different cancer types with principal coordinates of neighbor matrices 
Epidemiology and ecology share many fundamental research questions. Here we describe how principal coordinates of neighbor matrices (PCNM), a method from spatial ecology, can be applied to spatial epidemiology. PCNM is based on geographical distances among sites and can be applied to any set of sites providing a good coverage of a study area. In the present study, PCNM eigenvectors corresponding to positive autocorrelation were used as explanatory variables in linear regressions to model incidences of eight most common cancer types in Finnish municipalities (n = 320). The dataset was provided by the Finnish Cancer Registry and it included altogether 615,839 cases between 1953 and 2010.
PCNM resulted in 165 vectors with a positive eigenvalue. The first PCNM vector corresponded to the wavelength of hundreds of kilometers as it contrasted two main subareas so that municipalities located in southwestern Finland had the highest positive site scores and those located in midwestern Finland had the highest negative scores in that vector. Correspondingly, the 165th PCNM vector indicated variation mainly between the two small municipalities located in South Finland. The vectors explained 13 - 58% of the spatial variation in cancer incidences. The number of outliers having standardized residual > |3| was very low, one to six per model, and even lower, zero to two per model, according to Chauvenet’s criterion. The spatial variation of prostate cancer was best captured (adjusted r2 = 0.579).
PCNM can act as a complementary method to causal modeling to achieve a better understanding of the spatial structure of both the response and explanatory variables, and to assess the spatial importance of unmeasured explanatory factors. PCNM vectors can be used as proxies for demographics and causative agents to deal with autocorrelation, multicollinearity, and confounding variables. PCNM may help to extend spatial epidemiology to areas with limited availability of registers, improve cost-effectiveness, and aid in identifying unknown causative agents, and predict future trends in disease distributions and incidences. A large advantage of using PCNM is that it can create statistically valid reflectors of real predictors for disease incidence models with only little resources and background information.
PMCID: PMC4131804  PMID: 25126105
Cancer incidence; Finland; Principal coordinates of neighbor matrices; Spatial epidemiology
18.  Joint Effects of Habitat Heterogeneity and Species’ Life-History Traits on Population Dynamics in Spatially Structured Landscapes 
PLoS ONE  2014;9(9):e107742.
Both habitat heterogeneity and species’ life-history traits play important roles in driving population dynamics, yet there is little scientific consensus around the combined effect of these two factors on populations in complex landscapes. Using a spatially explicit agent-based model, we explored how interactions between habitat spatial structure (defined here as the scale of spatial autocorrelation in habitat quality) and species life-history strategies (defined here by species environmental tolerance and movement capacity) affect population dynamics in spatially heterogeneous landscapes. We compared the responses of four hypothetical species with different life-history traits to four landscape scenarios differing in the scale of spatial autocorrelation in habitat quality. The results showed that the population size of all hypothetical species exhibited a substantial increase as the scale of spatial autocorrelation in habitat quality increased, yet the pattern of population increase was shaped by species’ movement capacity. The increasing scale of spatial autocorrelation in habitat quality promoted the resource share of individuals, but had little effect on the mean mortality rate of individuals. Species’ movement capacity also determined the proportion of individuals in high-quality cells as well as the proportion of individuals experiencing competition in response to increased spatial autocorrelation in habitat quality. Positive correlations between the resource share of individuals and the proportion of individuals experiencing competition indicate that large-scale spatial autocorrelation in habitat quality may mask the density-dependent effect on populations through increasing the resource share of individuals, especially for species with low mobility. These findings suggest that low-mobility species may be more sensitive to habitat spatial heterogeneity in spatially structured landscapes. In addition, localized movement in combination with spatial autocorrelation may increase the population size, despite increased density effects.
PMCID: PMC4169469  PMID: 25232739
19.  Spatio-temporal analysis to identify determinants of Oncomelania hupensis infection with Schistosoma japonicum in Jiangsu province, China 
Parasites & Vectors  2013;6:138.
With the successful implementation of integrated measures for schistosomiasis japonica control, Jiangsu province has reached low-endemicity status. However, infected Oncomelania hupensis snails could still be found in certain locations along the Yangtze river until 2009, and there is concern that they might spread again, resulting in the possible re-emergence of infections among people and domestic animals alike. In order to establish a robust surveillance system that is able to detect the spread of infected snails at an early stage, sensitive and reliable methods to identify risk factors for the establishment of infected snails need to be developed.
A total of 107 villages reporting the persistent presence of infected snails were selected. Relevant data on the distribution of infected snails, and human and livestock infection status information for the years 2003 to 2008 were collected. Spatio-temporal pattern analysis including spatial autocorrelation, directional distribution and spatial error models were carried out to explore spatial correlations between infected snails and selected explanatory factors.
The area where infected snails were found, as well as their density, decreased significantly between 2003 and 2008. Changes in human and livestock prevalences were less pronounced. Three statistically significant spatial autocorrelations for infected snails were identified. (i) The Moran’s I of infected snails increased from 2004 to 2007, with the snail density increasing and the area with infected snails decreasing. (ii) The standard deviations of ellipses around infected snails were decreasing and the central points of the ellipses moved from West to East. (iii) The spatial error models indicated no significant correlation between the density of infected snails and selected risk factors.
We conclude that the contribution of local infection sources including humans and livestock to the distribution of infected snails might be relatively small and that snail control may limit infected snails to increasingly small areas ecologically most suitable for transmission. We provide a method to identify these areas and risk factors for persistent infected snail presence through spatio-temporal analysis, and a suggested framework, which could assist in designing evidence based control strategies for schistosomiasis japonica elimination.
PMCID: PMC3654978  PMID: 23648203
Schistosomiasis; Infected snails; Determinants; Spatio-temporal analysis; China
20.  Environmental factors associated with the malaria vectors Anopheles gambiae and Anopheles funestus in Kenya 
Malaria Journal  2009;8:268.
The Anopheles gambiae and Anopheles funestus mosquito species complexes are the primary vectors of Plasmodium falciparum malaria in sub-Saharan Africa. To better understand the environmental factors influencing these species, the abundance, distribution and transmission data from a south-eastern Kenyan study were retrospectively analysed, and the climate, vegetation and elevation data in key locations compared.
Thirty villages in Malindi, Kilifi and Kwale Districts with data on An. gambiae sensu strict, Anopheles arabiensis and An. funestus entomological inoculation rates (EIRs), were used as focal points for spatial and environmental analyses. Transmission patterns were examined for spatial autocorrelation using the Moran's I statistic, and for the clustering of high or low EIR values using the Getis-Ord Gi* statistic. Environmental data were derived from remote-sensed satellite sources of precipitation, temperature, specific humidity, Normalized Difference Vegetation Index (NDVI), and elevation. The relationship between transmission and environmental measures was examined using bivariate correlations, and by comparing environmental means between locations of high and low clustering using the Mann-Whitney U test.
Spatial analyses indicated positive autocorrelation of An. arabiensis and An. funestus transmission, but not of An. gambiae s.s., which was found to be widespread across the study region. The spatial clustering of high EIR values for An. arabiensis was confined to the lowland areas of Malindi, and for An. funestus to the southern districts of Kilifi and Kwale. Overall, An. gambiae s.s. and An. arabiensis had similar spatial and environmental trends, with higher transmission associated with higher precipitation, but lower temperature, humidity and NDVI measures than those locations with lower transmission by these species and/or in locations where transmission by An. funestus was high. Statistical comparisons indicated that precipitation and temperatures were significantly different between the An. arabiensis and An. funestus high and low transmission locations.
These finding suggest that the abundance, distribution and malaria transmission of different malaria vectors are driven by different environmental factors. A better understanding of the specific ecological parameters of each malaria mosquito species will help define their current distributions, and how they may currently and prospectively be affected by climate change, interventions and other factors.
PMCID: PMC2793260  PMID: 19941637
21.  Exploring the effects of spatial autocorrelation when identifying key drivers of wildlife crop-raiding 
Ecology and Evolution  2014;4(5):582-593.
Few universal trends in spatial patterns of wildlife crop-raiding have been found. Variations in wildlife ecology and movements, and human spatial use have been identified as causes of this apparent unpredictability. However, varying spatial patterns of spatial autocorrelation (SA) in human–wildlife conflict (HWC) data could also contribute. We explicitly explore the effects of SA on wildlife crop-raiding data in order to facilitate the design of future HWC studies. We conducted a comparative survey of raided and nonraided fields to determine key drivers of crop-raiding. Data were subsampled at different spatial scales to select independent raiding data points. The model derived from all data was fitted to subsample data sets. Model parameters from these models were compared to determine the effect of SA. Most methods used to account for SA in data attempt to correct for the change in P-values; yet, by subsampling data at broader spatial scales, we identified changes in regression estimates. We consequently advocate reporting both model parameters across a range of spatial scales to help biological interpretation. Patterns of SA vary spatially in our crop-raiding data. Spatial distribution of fields should therefore be considered when choosing the spatial scale for analyses of HWC studies. Robust key drivers of elephant crop-raiding included raiding history of a field and distance of field to a main elephant pathway. Understanding spatial patterns and determining reliable socio-ecological drivers of wildlife crop-raiding is paramount for designing mitigation and land-use planning strategies to reduce HWC. Spatial patterns of HWC are complex, determined by multiple factors acting at more than one scale; therefore, studies need to be designed with an understanding of the effects of SA. Our methods are accessible to a variety of practitioners to assess the effects of SA, thereby improving the reliability of conservation management actions.
PMCID: PMC4098139  PMID: 25035800
Elephant; Generalized Linear Model; Human–wildlife conflict; Okavango Delta; spatial scale
22.  Weak Isolation by Distance in Diaperis boleti, a Fungivorous Saproxylic Beetle 
Living in unstable habitats is expected to decrease the intensity of isolation by distance in populations through the need for frequent movements of individuals. Insects associated with fruiting bodies of fungi therefore are supposed to have weak spatial genetic structure of populations compared with those living in more stable habitats. With the use of an amplified fragment length polymorphism technique, this study investigated the isolation by distance, inbreeding, and genetic diversity in Diaperis boleti (L.) (Coleoptera: Tenebrionidae), a fungivorous saproxylic beetle that inhabits sporocarps of Laetiporus sulphureus (Bulliard) Murrill (Polyporales) on trees growing in highly-fragmented agricultural landscapes. Isolation by distance was tested with spatial autocorrelation analysis of kinship (individual-based approach) and correlating matrices of genetic and geographic distances with the Mantel test (population-based approach). These results were compared with the results obtained for saproxylic beetles living in the same landscape but differing in ecological preferences. It was shown that the species dependent on sporocarps of wooddecomposing fungi had higher variability, lower individual inbreeding, and less intensive isolation by distance pattern than saproxylic beetles living in tree hollows. It was also demonstrated that spatial autocorrelation analysis of kinship is a more sensitive approach for detecting finescale spatial genetic structure than the Mantel test.
PMCID: PMC4212856  PMID: 25204855
spatial genetic structure; spatial autocorrelation; AFLP; inbreeding; saproxylic beetles
23.  A study protocol to evaluate the relationship between outdoor air pollution and pregnancy outcomes 
BMC Public Health  2010;10:613.
The present study protocol is designed to assess the relationship between outdoor air pollution and low birth weight and preterm births outcomes performing a semi-ecological analysis. Semi-ecological design studies are widely used to assess effects of air pollution in humans. In this type of analysis, health outcomes and covariates are measured in individuals and exposure assignments are usually based on air quality monitor stations. Therefore, estimating individual exposures are one of the major challenges when investigating these relationships with a semi-ecologic design.
Semi-ecologic study consisting of a retrospective cohort study with ecologic assignment of exposure is applied. Health outcomes and covariates are collected at Primary Health Care Center. Data from pregnant registry, clinical record and specific questionnaire administered orally to the mothers of children born in period 2007-2010 in Portuguese Alentejo Litoral region, are collected by the research team. Outdoor air pollution data are collected with a lichen diversity biomonitoring program, and individual pregnancy exposures are assessed with spatial geostatistical simulation, which provides the basis for uncertainty analysis of individual exposures. Awareness of outdoor air pollution uncertainty will improve validity of individual exposures assignments for further statistical analysis with multivariate regression models.
Exposure misclassification is an issue of concern in semi-ecological design. In this study, personal exposures are assigned to each pregnant using geocoded addresses data. A stochastic simulation method is applied to lichen diversity values index measured at biomonitoring survey locations, in order to assess spatial uncertainty of lichen diversity value index at each geocoded address. These methods assume a model for spatial autocorrelation of exposure and provide a distribution of exposures in each study location. We believe that variability of simulated exposure values at geocoded addresses will improve knowledge on variability of exposures, improving therefore validity of individual exposures to input in posterior statistical analysis.
PMCID: PMC2964632  PMID: 20950449
24.  Dry weather induces outbreaks of human West Nile virus infections 
Since its first occurrence in the New York City area during 1999, West Nile virus (WNV) has spread rapidly across North America and has become a major public health concern in North America. By 2002, WNV was reported in 40 states and the District of Columbia with 4,156 human and 14,539 equine cases of infection. Mississippi had the highest human incidence rate of WNV during the 2002 epidemic in the United States. Epidemics of WNV can impose enormous impacts on local economies. Therefore, it is advantageous to predict human WNV risks for cost-effective controls of the disease and optimal allocations of limited resources. Understanding relationships between precipitation and WNV transmission is crucial for predicting the risk of the human WNV disease outbreaks under predicted global climate change scenarios.
We analyzed data on the human WNV incidences in the 82 counties of Mississippi in 2002, using standard morbidity ratio (SMR) and Bayesian hierarchical models, to determine relationships between precipitation and human WNV risks. We also entertained spatial autocorrelations of human WNV risks with conditional autocorrelative (CAR) models, implemented in WinBUGS 1.4.3.
We observed an inverse relationship between county-level human WNV incidence risk and total annual rainfall during the previous year. Parameters representing spatial heterogeneity in the risk of human exposure to WNV improved model fit. Annual precipitation of the previous year was a predictor of spatial variation of WNV risk.
Our results have broad implications for risk assessment of WNV and forecasting WNV outbreaks. Assessing risk of vector-born infectious diseases will require understanding of complex ecological relationships. Based on the climatologically characteristic drought occurrence in the past and on climate model predictions for climate change and potentially greater drought occurrence in the future, we suggest that the frequency and relative risk of WNV outbreaks could increase.
PMCID: PMC2841181  PMID: 20181272
25.  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 
Annals of Botany  2008;102(3):435-441.
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.
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).
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.
PMCID: PMC2701798  PMID: 18599560
Abies; local size hierarchy; mark correlation function; pair correlation function; regenerating forest; regular spacing; spatial autocorrelation

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