Autocorrelation In Model Residuals Research Articles

A statistical evaluation of replicated block designs and spatial variability in sorghum performance trials

ABSTRACTSuccess of hybrid performance trials has always been susceptible to the efficiency at which the experimental design can remove any effect of spatial autocorrelation associated with environmental factors. Blocking in a randomized design is one way of accounting for this. Another method is to have a model of the environmental variability. Measures of soil variability could be useful to represent spatial structure in a trial. Soil apparent electrical conductivity (ECa) measurements can be collected rapidly and non-invasively and have been well documented to be able to map soil variability at the meter scale. We present a statistical evaluation that compares the effectiveness of the traditional replicated block designs with spatially explicit soil ECa measurements. Soil ECa, sorghum (Sorghum bicolor (L.) Moench) grain yield, and plant height were measured from six sorghum hybrid evaluation trials across Texas in 2017. Three linear models were tested to determine the presence or absence of spatial autocorrelation of model residuals within each performance trial. Moran’s I tests on model residuals showed that neither method was consistently effective in accounting for spatial variability. Blocking was more effective at one site for both plant height and grain yield, whereas ECa data were more effective at two sites for grain yield only. Based on these results, and the relatively low cost of using both methods together, we propose that plant breeders interested in addressing spatial autocorrelation in models from trial results may consider using both methods and select the best model, post hoc.

Evaluation of methods to assess the uncertainty in estimated energy savings

In this work we present a methodology to evaluate the accuracy of methods used to quantify the uncertainty in estimated total energy savings. We focus on savings measurement and verification (M&V) approaches that use a baseline model to characterize energy use, and that forward-project the model for a counterfactual to determine avoided energy use. These approaches are common to the International Performance Measurement and Verification Protocol's (IPMVP's) Option C and Option B. This methodology can be used to evaluate the uncertainty in savings estimates that are due to model error. It has been applied to evaluate two uncertainty estimation methods, including the industry standard ASHRAE Guideline 14 approach. The evaluation used data from 69 commercial buildings and four different baseline models that span daily and hourly granularity, as well as linear and non-linear/non-parametric forms. The findings of this work indicate that the standard methods that are widely used by the M&V community for estimating the total savings uncertainty over the post-installation period tend to underestimate the uncertainty. The tendency to underestimate the uncertainty is stronger for hourly models than for daily models, due to stronger autocorrelation in model residuals at the hourly time scale.

Improved Inverse Modeling by Separating Model Structural and Observational Errors

A practical formal likelihood function (L) is developed to separate model structure errors and observation errors by the separation of correlated and uncorrelated model residuals. L overcomes the time-consuming problem of likelihood functions proposed by previous studies, and combines the Mean Square Error (MSE) and first-order Auto-Regression (AR(1)) models. For comparison of the effect of different error models, MSE, AR(1), and L are used as efficiency criteria to calibrate the three-dimensional variably saturated ground-water flow model (MF2K-VSF) based on the soil tank seepages of rainfall–runoff experiments. Results of L are nearly the same as those of AR(1) due to negligible observational errors. Although all calibrated models well mimic the seepage discharges, MF2K-VSF with MSE cannot capture the groundwater level and soil suction processes because of the considerable autocorrelation of model residuals owing to model inadequacies (e.g., neglect of the soil moisture hysteresis), which obviously violates the statistical assumption of MSE. By contrast, L accounts for the model structural errors and thus enhances the reliability of hydrological simulations.

Developing spatial models to guide conservation of grassland birds in the U.S. Northern Great Plains

ABSTRACT Conservation of bird populations is increasingly focused on landscapes. We combined data collected in 2005–2011 from 16,250 North American Breeding Bird Survey (BBS) survey points with local and remotely sensed environmental data to model the distribution of 7 grassland bird species in the Northern Great Plains of the United States. We analyzed data at the survey point level, which is consistent with the scale of conservation treatments that we apply, and avoided information loss caused by pooling data at the BBS route level. By accounting for observer effects, nesting of survey points within routes, and sequence of survey points, we accommodated BBS survey design, refined estimates of important habitat predictors, improved model fit, and reduced or eliminated positive spatial autocorrelation in model residuals. The predictive power of models was greatly increased by including variables that characterized annual and long-term precipitation, as well as local land cover attributes not available fro...

Environment and biogeography drive aquatic plant and cladoceran species richness across Europe

Summary The relative role of environmental and regional processes in determining patterns of biodiversity may vary across spatial scales. In inland aquatic communities, macrophytes and zooplankton have the potential to disperse over large spatial scales (hundreds of kilometres), but the role of environmental and regional processes in determining species richness (SR) at different spatial scales is poorly understood. To assess the relative importance of environmental and regional processes acting on both regional (within‐region) and continental (among‐region) scales, we surveyed 139 waterbodies in five geographic regions (Spain, Italy, the Netherlands, Scotland and Norway) and in each region measured environmental variables, descriptors of connectivity and SR of two aquatic taxa, namely aquatic plants and cladocerans. Analyses using boosted regression trees and generalised linear models, integrated with multiple imputation of missing data, indicated that both environmental and regional processes were related to local SR. Total phosphorus was the main environmental correlate of SR, showing a unimodal relationship with both taxonomic groups. Conductivity and lake depth were also related to cladoceran SR, and Secchi depth was related to aquatic plant richness. The SR of aquatic plants and cladocerans were significantly correlated, but this relationship was mainly indirect (i.e. determined by the effect of common environmental variables). The limited role of connectivity and space (lack of spatial autocorrelation in model residuals) relative to environment suggests that dispersal limitation is not as important as the environment in explaining richness patterns for both groups within regions. However, region identity strongly influenced SR, which suggests an important effect of biogeographic factors (e.g. the level of endemicity; the number of unique species in each region) and/or dispersal limitation at the continental scale. We conclude that environmental factors and biogeographic processes largely determine the patterns of local and regional SR in aquatic plants and zooplankton.

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.

Estimation of sulfur dioxide air pollution concentrations with a spatial autoregressive model

In this paper, we develop a land-use regression model for sulfur dioxide air pollution concentrations. We make use of mobile monitoring data collected in Hamilton, Ontario, Canada, between 2005 and 2010 inclusive. The observed SO2 concentrations are regressed against a comprehensive set of land use and transportation variables. Land use and transportation variables are assessed as the amount of each characteristic within buffers of 50, 100, 200, 400, 800, and 1600 m around pollution observation locations. In the first instance of regression modeling, we apply ordinary least-squares regression. The OLS model R2 for training data was 0.38, and an R2 of 0.3 for a 50% held out cross-validation data set. The residuals are spatially correlated with the OLS model as determined with Moran's I. We thus applied a simultaneous autoregressive model, specifically the spatial error model. The resulting model slightly improved fit as determined by a pseudo R2 = 0.4, improved log-likelihood, and reduced MSE, RMSE, and MAE. The spatial error model residuals were not spatially auto-correlated, resulting in a valid model. SAR modeling is a natural extension to OLS regression models and solves the issue of spatial autocorrelation in model residuals with a one-stage model.

Local spatial modeling of paddy soil landscape patterns in response to urbanization across the urban agglomeration around Hangzhou Bay, China

This study employed geographically weighted regression (GWR), a local spatial modeling approach, to characterize the local varying response of paddy soils to urbanization at the landscape scale in the urban agglomeration around Hangzhou Bay, China. Spatial non-stationary and scale-dependent relationships between three geographical urbanization indices (the common urbanization intensity index, distance to road corridors and distance to urban centers) and four spatial metrics depicting paddy soil landscape patterns (total area, edge density, aggregation index, and perimeter area ratio distribution) were captured at two block scales (4 km × 4 km and 7 km × 7 km) because the predictive ability of GWR and the nature and strength of the identified relationships all varied significantly in space and scale. Moreover, GWR models have achieved better performance than their corresponding global ordinary least squares (OLS) models given their greater ability to (1) describe data (higher adjusted R2 and lower Akaike Information Criterion values) and (2) address spatial autocorrelation in model residuals. This study highlighted scale effects when applying GWR. Both bandwidth and block size can impact the predictive ability of GWR as well as estimated parameters for GWR analysis. This study demonstrates a novel joint application of GWR and associated metrics in applied geography.

Evaluation of structure specification in linear mixed models for modeling the spatial effects in tree height-diamater relationships.

In recent years, linear mixed models (LMM) have become more pop- ular to deal with spatial effects in forestry and ecological data. In this study, different structure specifications of linear mixed model were applied to model tree height-diameter relationships, including LMM with random blocks only (LMM-block), LMM with spatial covariance only (LMM-covariance), and the combination of the last two (LMM-block-covariance). Further, the between- group heterogeneous variances were incorporated into LMM-covariance and LMM-block-covariance. The results indicated that, in general, LMM-covar- iance significantly reduced spatial autocorrelation in model residuals, while LMM-block was effective in dealing with spatial heterogeneity. LMM-block treated the blocks as random effects and avoided the estimation of param- eters of the variogram model. Thus, it produced better model predictions than LMM-covariance. LMM-block-covariance took both block effects and spatial covariance into account, and significantly improve model fitting. However, it did not produce better model predictions due to the increase of model complexity and estimation of the local variogram within each block. Keywords linear mixed model, spatial heterogeneity, spatial autocorrelation, height-diameter equation.

Specialization of Mutualistic Interaction Networks Decreases toward Tropical Latitudes

Species-rich tropical communities are expected to be more specialized than their temperate counterparts. Several studies have reported increasing biotic specialization toward the tropics, whereas others have not found latitudinal trends once accounting for sampling bias or differences in plant diversity. Thus, the direction of the latitudinal specialization gradient remains contentious. With an unprecedented global data set, we investigated how biotic specialization between plants and animal pollinators or seed dispersers is associated with latitude, past and contemporary climate, and plant diversity. We show that in contrast to expectation, biotic specialization of mutualistic networks is significantly lower at tropical than at temperate latitudes. Specialization was more closely related to contemporary climate than to past climate stability, suggesting that current conditions have a stronger effect on biotic specialization than historical community stability. Biotic specialization decreased with increasing local and regional plant diversity. This suggests that high specialization of mutualistic interactions is a response of pollinators and seed dispersers to low plant diversity. This could explain why the latitudinal specialization gradient is reversed relative to the latitudinal diversity gradient. Low mutualistic network specialization in the tropics suggests higher tolerance against extinctions in tropical than in temperate communities.

Geographically Local Linear Mixed Models for Tree Height-Diameter Relationship

A geographically local linear mixed model (GLLMM) was proposed to handle spatial autocorrelation and heterogeneity simultaneously. Under the framework of geographically weight regression (GWR), GLLMM incorporated the spatial dependence among neighboring observations at each location in the study area by modeling local variograms and using spatial weighting matrix. Our results indicated that GLLMM fitted the example data better than GWR as measured by the Akaike information criterion for appropriate bandwidths. We also tested the ability of GWR and GLLMM in spatial interpolation using a subset of data. GLLMM had higher prediction accuracy and smaller spatial autocorrelation in model residuals than did GWR. Further, GLLMM enabled mapping of the geostatistical parameters of local variograms, which were used to identify spots or local areas of high spatial heterogeneity or autocorrelation in the study region. Therefore, GLLMM is a useful local regression technique for modeling the variable relationships in forest stands with heterogeneous micro-site conditions and diverse correlations between neighboring trees.

Overlapping landscapes: A persistent, but misdirected concern when collecting and analyzing ecological data

AbstractA primary focus of wildlife ecology is studying how the arrangement, quality, and distribution of habitat influence wildlife populations at multiple spatial scales. A practical limitation of conducting wildlife–habitat investigations in the field, however, is that sampling points tend to be close to one another, resulting in spatial clustering. Consequently, when ecologists seek to quantify the effects of environmental predictors surrounding their sampling points, they encounter the issue of using landscapes that are partially or completely overlapping. A presumed problem of overlapping landscapes is that data generated from these landscapes, when used as predictors in statistical modeling, might violate the assumption of independence. However, the independence of error is the critical assumption, not the independence of predictor variables. Nonetheless, many researchers strive to avoid such overlaps through sampling design or alternative analytical procedures and specialized software programs have been created to assist with this. We present theoretical arguments and empirical evidence showing that changing the amount of overlap does not alter the degree of spatial autocorrelation. Using data derived from 2 broad‐scaled avian monitoring programs, we quantified the relationship between forest cover and bird abundance and occurrence at multiple landscapes ranging from 100 m to 24 km across. We found no clear evidence that increasing overlap of landscapes increased spatial autocorrelation in model residuals. Our results demonstrate that the concern of overlapping landscapes as a potential cause of violation of spatial independency among sampling units is misdirected and represents an oversimplification of the statistical and ecological issues surrounding spatial autocorrelation. Overlapping landscapes and spatial autocorrelation are separate issues in the modeling of wildlife populations and their habitats; non‐overlapping landscapes do not ensure spatial independency and overlapping landscapes do not necessarily lead to greater spatial autocorrelation in model errors. © 2011 The Wildlife Society.

Multi-scale analysis of spatially varying relationships between agricultural landscape patterns and urbanization using geographically weighted regression

Scientific interpretation of the relationships between agricultural landscape patterns and urbanization is important for ecological planning and management. Ordinary least squares (OLS) regression is the primary statistical method in previous studies. However, this global regression lacks the ability to uncover some local-specific relationships and spatial autocorrelation in model residuals. This study employed geographically weighted regression (GWR) to examine the spatially varying relationships between several urbanization indicators (urbanization intensity index, distance to urban centers and distance to road) and changes in metrics describing agricultural landscape patterns (total area, patch density, perimeter area ratio distribution and aggregation index) at two block scales (5 km and 10 km). Results denoted that GWR was more powerful than OLS in interpreting relationships between agricultural landscape patterns and urbanization, since GWR was characterized by higher adjust R 2, lower Akaike Information Criterion values and reduced spatial autocorrelations in model residuals. Character and strength of the relationships identified by GWR varied spatially. In addition, GWR results were scale-dependent and scale effects were particularly significant in three aspects: kernel bandwidth of weight determination, block scale of pattern analysis, and window size of local variance analysis. Homogeneity and heterogeneity in the relationships between agricultural landscape patterns and urbanization were subject to the coupled influences of the three scale effects. We argue that the spatially varying relationships between agricultural landscape patterns and urbanization are not accidental but nearly universal. This study demonstrated that GWR has the potential to provide references for ecological planners and managers to address agricultural landscapes issues at all scales.

Breeding bird diversity in relation to environmental gradients in China

Geographic variation in species richness has been explained by different theories such as energy, productivity, energy–water balance, habitat heterogeneity, and freezing tolerance. This study determines which of these theories best account for gradients of breeding bird richness in China. In addition, we develop a best-fit model to account for the relationship between breeding bird richness and environment in China. Breeding bird species richness in 207 localities (3271 km 2 per locality on average) from across China was related to thirteen environmental variables after accounting for sampling area. The Akaike's information criterion (AIC) was used to evaluate model performance. We used Moran's I to determine the magnitude of spatial autocorrelation in model residuals, and used simultaneous autoregressive model to determine coefficients of determination and AIC of explanatory variables after accounting for residual spatial autocorrelation. Of all environmental variables examined, normalized difference vegetation index, a measure of plant productivity, is the best variable to explain the variance in breeding bird richness. We found that species richness of breeding birds at the scale examined is best predicted by a combination of plant productivity, elevation range, seasonal variation in potential evapotranspiration, and mean annual temperature. These variables explained 47.3% of the variance in breeding bird richness after accounting for sampling area; most of the explained variance in richness is attributable to the first two of the four variables.

The effect of species response form on species distribution model prediction and inference

Ecological theory and current evidence support the validity of various species response curves according to a variety of environmental gradients. Various methods have been developed for building species distribution models but it is not well known how these methods perform under various assumptions about the form of the underlying species response. It is also not well known how spatial correlation in species occurrence affects model performance. These effects were investigated by applying an environmental envelope method (BIOCLIM) and three regression-based methods: logistic regression (LR), generalized additive modelling (GAM), and classification and regression tree (CART) to simulated species occurrence data. Each simulated species was constructed as a sum of responses with varying weights. Three basic species response curves were assumed: Gaussian (bell-shaped), Beta (skew) and linear. The two non-linear responses conform to standard ecological niche theory. All three responses were applied in turn to three simulated environmental variables, each with varying degrees of spatial autocorrelation. GAM produced the most consistent model performance over all forms of simulated species response. BIOCLIM and CART were inclined to underrate the performance of variables with a linear response. BIOCLIM was less sensitive to data density. LR was susceptible to model misspecification. The use of a linear function in LR underestimated the performance of variables with non-linear species response and contributed to increased spatial autocorrelation in model residuals. Omission of important environmental variables with non-linear species response also contributed to increased spatial autocorrelation in model residuals. Adding a spatial autocovariate term to the LR model (autologistic model) reduced the spatial autocorrelation and improved model performance, but did not correct the misidentification of the dominant environmental determinant. This is to be expected since the autologistic approach was designed primarily for prediction and not for inference. Given that various forms of species response to environmental determinants arise commonly in nature: (1) higher order functions should always be tested when applying LR in modelling species distribution; (2) spatial autocorrelation in species distribution model residuals can indicate that environmental determinants with non-linear response are missing from the model; and (3) deficiencies in LR model performance due to model misspecification can be addressed by adding a spatial autocovariate to the model, but care should be taken when interpreting the coefficients of the model parameters.

Modelling the occurrence and abundance of a colonial species, the arctic tern Sterna paradisaea in the archipelago of SW Finland

Knowledge of the habitat requirements and suitable breeding areas of sea birds is crucial for their management and conservation. However, there are still few studies that have modelled the breeding distribution and abundance of colonial sea birds. In this study, we created predictive distribution models for a colonial species, the arctic tern Sterna paradisaea, using 14 environmental variables calculated for 525 islands in the Archipelago Sea in SW Finland. We modelled the occurrence (presence‐absence) using generalised additive models (GAMs) and abundance (pair numbers/colony size) using hurdle models fitted with GAM. We tested for spatial autocorrelation in model residuals and evaluated the models on independent data. Critical factors influencing the occurrence of the arctic tern were the proportions of boulder or gravel and forest of island area, as well as island maximum elevation and area, such that the species seemed to prefer large and low islands with sparse vegetation. Abundance was influenced by the proportions of boulder or gravel and bare rock of island area, as well as exposure and island area. To some extent, different factors influenced the occurrence and the abundance. The evaluation results of the models were good, with an AUC value of 0.91 for the most accurate presence‐absence model and a Pearson's correlation coefficient of 0.60 for the most accurate hurdle model. The predictive ability of the models increased when we removed islands with single or few breeding pairs from the data set. Although the hurdle models did not produce accurate pair number estimates, they indicated which islands are suitable for larger colonies. Abundance is a crucial factor for colonial species. This modelling technique can therefore be of great value for the conservation and management of the arctic tern and similar colonial species.

Effects of historical and contemporary factors on global patterns in avian species richness

AbstractAim The aim of this study was to determine how regional and historical factors influence global patterns in avian species richness.Location Global.Methods Using a comprehensive data set including 710 World Wildlife Fund terrestrial ecoregions covering nearly all the land surface of the Earth, avian species richness was compared among six biogeographical regions after accounting for sample area, elevational range and climate. Analysis of variance and multiple regressions were used. Spatial autocorrelation in model residuals was accounted for.Results Significant effects of region on avian species richness were found in nearly all comparisons between biogeographical regions.Main conclusions Regional and historical processes have played a role in regulating large‐scale avian species richness patterns across the globe. Avian species richness in different regions of the world cannot be accurately predicted by a single global model. Avian species richness in areas of similar environments may differ substantially between regions, and thus avian species richness in one biogeographical region cannot be predicted using the richness–environment relationship derived from the data of another biogeographical region, even one with similar environments.

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.

Identifying the Spatial Structure in Error Terms with Spatial Covariance Models: A Case Study on Urbanization Influence in Chaparral Bird Species

This study uses various spatial statistical methods to examine and model large- and small-scale spatial structure in bird abundance and urbanization. A set of chaparral-vegetated points across an urbanizing landscape in the Santa Monica Mountains of southern California was surveyed for birds in 1997 and mapped in a GIS. For each sample location, GIS landuse data were used to calculate surrounding urbanization proportion. We first used semivariograms and correlograms to detect large-scale trends, and Moran's I statistic to test for small-scale spatial autocorrelation in bird abundance. Relationships between bird abundance and surrounding urbanization levels were then analyzed using ordinary least-squares (OLS) regression. Upon detection of spatial autocorrelation in model residuals, spatial covariance models were constructed to incorporate this small-scale spatial dependence statistically and obtain non-biased estimates of urbanization influence. As a conservative comparison, we also removed the large-scale spatial trends from bird abundance and regressed the detrended model residuals on surrounding urbanization proportion to examine the effects of local variations in urbanization on bird abundance. Results of spatial covariance models, as well as detrended non-spatial models, indicate that chaparral bird populations are affected by changes in surrounding landscape composition, regardless of spatial location. Semivariogram and correlogram analysis provided further insight into the spatial structures of the bird populations examined.