The Bayesian conditional independence model for measurement error: applications in ecology

The measurement error model is a well established statistical method for regression problems in medical sciences, although rarely used in ecological studies. While the situations in which it is appropriate may be less common in ecology, there are instances in which there may be benefits in its use for prediction and estimation of parameters of interest. We have chosen to explore this topic using a conditional independence model in a Bayesian framework using a Gibbs sampler, as this gives a great deal of flexibility, allowing us to analyse a number of different models without losing generality. Using simulations and two examples, we show how the conditional independence model can be used in ecology, and when it is appropriate.     

Assessing conservation status with extensive but low-resolution data: Application of frequentist and Bayesian models to endangered Athabasca River rainbow trout

Use of extensive but low-resolution abundance data is common in the assessment of species at-risk status based on quantitative decline criteria under International Union for Conservation of Nature (IUCN) and national endangered species legislation. Such data can be problematic for 3 reasons. First, statistical power to reject the null hypothesis of no change is often low because of small sample size and high sampling uncertainty leading to a high frequency of type II errors. Second, range-wide assessments composed of multiple site-specific observations do not effectively weight site-specific trends into global trends. Third, uncertainty in site-specific temporal trends and relative abundance are not propagated at the appropriate spatial scale. A common result is the propensity to underestimate the magnitude of declines and therefore fail to identify the appropriate at-risk status for a species. We used 3 statistical approaches, from simple to more complex, to estimate temporal decline rates for a designatable unit (DU) of rainbow trout in the Athabasca River watershed in western Canada. This DU is considered a native species for purposes of listing because of its genetic composition characterized as >0.95 indigenous origin in the face of continuing introgressive hybridization with introduced populations in the watershed. Analysis of abundance trends from 57 time series with a fixed-effects model identified 33 sites with negative trends, but only 2 were statistically significant. By contrast, a hierarchical linear mixed model weighted by site-specific abundance provided a DU-wide decline estimate of 16.4% per year and a 3-generation decline of 93.2%. A hierarchical Bayesian mixed model yielded a similar 3-generation decline trend of 91.3% and the posterior distribution showed that the estimate had a >99% probability of exceeding thresholds for an endangered listing. We conclude that the Bayesian approach was the most useful because it provided a probabilistic statement of threshold exceedance in support of an at-risk status recommendation.     

Estimating a distance dependent contagion function using point sample data

Natural events and human activities cause changes in landscape structure. Landscape metrics are used as a useful tool to study landscape trends and ecological processes related to the landscape structure. These metrics are commonly calculated on wall-to-wall raster data from remote sensing. A recent trend is to use sample data to estimate landscape metrics. In this study, point sampling was used to estimate a vector-based and distance dependent contagion metric. The metric is an extension of the established contagion. The statistical properties, for both unconditional and conditional contagions, were assessed by a point (point pairs) sampling experiment in maps from the National Inventory of landscapes in Sweden. Random and systematic sampling designs were tested for nine point distances and five sample sizes and for two classification systems. The systematic design showed slightly smaller root mean square error (RMSE) and bias than the random design. Both true and estimated values were calculated using computer programs in FORTRAN, which was specifically written for the purpose of the study. For a given sample size, RMSE and bias increased with increasing point distance. The estimator of unconditional contagion had acceptable RMSE and bias for moderate sample sizes, but in the conditional case the bias (and thus the RMSE) was unacceptably large. The main reason for this is that small classes (by area) affect both the true value of the contagion and are often missing in the sample. The method proposed can be adopted in gradient-based model of landscape structure where no distinct border is assumed between polygons. The method can also be applied in field-based inventories.     

Spatiotemporal hierarchical modelling of extreme precipitation in Western Australia using anisotropic Gaussian random fields

We discuss an approach for the statistical modelling of extreme precipitation events in South-West Australia over space and time, using a latent spatiotemporal process where precipitation maxima follow a generalised extreme value distribution. Temporal features are captured by modelling trends on the location and scale parameters. Spatial features are captured using anisotropic Gaussian random fields. Site specific explanatory variables are also incorporated. We fit several models using Bayesian inferential methods to precipitation extremes recorded at 36 weather stations around the Western Australian state capital city of Perth over the period 1907–2009. Model choice is performed using the DIC criterion. The best fitting model shows significant non-stationarity over time, with extreme precipitation events becoming less frequent. Extreme precipitation events are stronger at coastal locations, with the intensity decreasing as we head to the higher and drier areas to the North-East.     

Trends of SO2emission in the European union, 1985–1997

The trends of yearly emission of sulphur dioxide are analysed for the European Union during a period of time from 1985 to 1997. To achieve the above matter the method of the least squares model has been used. Major SO₂emissions were found in Germany, the United Kingdom, Spain, Italy and France. However, high SO₂emissions by km²were found in Germany, the United Kingdom and Belgium. The most remarkable results of the trend analysis appears as follows: 12 countries with significant downward trends, 2 countries with significant upward trends and 1 country with no significant trend. A decreasing trend is evident for the most part of the E.U., although Portugal and Greece generated significant increasing trends of SO₂emission for the mentioned year period.     

A single-chain-based multidimensional Markov chain model for subsurface characterization

Multidimensional Markov chain models in geosciences were often built on multiple chains, one in each direction, and assumed these 1-D chains to be independent of each other. Thus, unwanted transitions (i.e., transitions of multiple chains to the same location with unequal states) inevitably occur and have to be excluded in estimating the states at unobserved locations. This consequently may result in unreliable estimates, such as underestimation of small classes (i.e., classes with smaller than average areas) in simulated realizations. This paper presents a single-chain-based multidimensional Markov chain model for estimation (i.e., prediction and conditional stochastic simulation) of spatial distribution of subsurface formations with borehole data. The model assumes that a single Markov chain moves in a lattice space, interacting with its nearest known neighbors through different transition probability rules in different cardinal directions. The conditional probability distribution of the Markov chain at the location to be estimated is formulated in an explicit form by following the Bayes’ Theorem and the conditional independence of sparse data in cardinal directions. Since no unwanted transitions are involved, the model can estimate all classes fairly. Transiogram models (i.e., 1-D continuous Markov transition probability diagrams) are used to provide transition probability input with needed lags to generalize the model. Therefore, conditional simulation can be conducted directly and efficiently. The model provides an alternative for heterogeneity characterization of subsurface formations.

Interaction structures analysed from water-quality data

Fortnightly observations of water quality parameters, discharge and water temperature along the River Elbe have been subjected to a multivariate data analysis. In a previous study [Petersen, W., Bertino, L., Callies, U., Zorita, E., 2001. Process identification by principal component analysis of river-quality data. Ecol. Model. 138, 193–213] applied principal component analysis (PCA) to show that 60% of variability in the data set can be explained through just two linear combinations of eight original variables. In the present paper more advanced multivariate methods are applied to the same data set, which are supposed to suit better interpretations in terms of the underlying system dynamics.The first method, graphical modelling, represents interaction structures in terms of a set of conditional independence constraints between pairs of variables given the values of all other variables. Assuming data from a multinormal distribution conditional independence constraints are expressed by zero partial correlations. Different graphical structures with nodes for each variable and connecting edges between them can be assessed with regard to their likelihood. The second method, canonical correlation analysis (CCA), is applied for studying the correlation structures of external forcing and water quality parameters.Results of CCA turn out to be consistent with the dominant patterns of variability obtained from PCA. The percentages of variability explained by external forcing, however, are estimated to be smaller. Fitting graphical models allows a more detailed representation of interaction structures. For instance, for given discharge and temperature correlated variations of the concentrations of oxygen and nitrate, respectively, can be modelled as being mediated by variations of pH, which is a representer for algal activity. Considerably simplified graphical models do not much affect the outcomes of both PCA and CCA, and hence it is concluded that these graphical models successfully represent the main interaction structures represented by the covariance matrix of the data. The analysed conditional independence patterns provide constraints to be satisfied by directed probabilistic networks, for instance.     

Declines in migrant shorebird populations from a winter‐quarter perspective

Many long‐distance migrating shorebird (i.e., sandpipers, plovers, flamingos, oystercatchers) populations are declining. Although regular shorebird monitoring programs exist worldwide, most estimates of shorebird population trends and sizes are poor or nonexistent. We built a state‐space model to estimate shorebird population trends. Compared with more commonly used methods of trend estimation, state‐space models are more mechanistic, allow for the separation of observation and state process, and can easily accommodate multivariate time series and nonlinear trends. We fitted the model to count data collected from 1990 to 2013 on 18 common shorebirds at the 2 largest coastal wetlands in southern Africa, Sandwich Harbour (a relatively pristine bay) and Walvis Bay (an international harbor), Namibia. Four of the 12 long‐distance migrant species declined since 1990: Ruddy Turnstone (Arenaria interpres), Little Stint (Calidris minuta), Common Ringed Plover (Charadrius hiaticula), and Red Knot (Calidris canutus). Populations of resident species and short‐distance migrants increased or were stable. Similar patterns at a key South African wetland suggest that shorebird populations migrating to southern Africa are declining in line with the global decline, but local conditions in southern Africa's largest wetlands are not contributing to these declines. State‐space models provide estimates of population levels and trends and could be used widely to improve the current state of water bird estimates.     

A geospatial model of forest dynamics with controlled trend surface

This paper proposes a method of controlled trend surface to simultaneously account for large-scale spatial trends and non-spatial local effects. With this method, a geospatial model of forest dynamics was developed for the Alaska boreal forest from 446 constantly monitored permanent sample plots. The geospatial component of this model represented large-scale spatial trends in recruitment, diameter growth, and mortality. The model was tested on two sets of validation plots which represented temporal and spatial extensions of the current sample coverage. The results suggest that the controlled trend surface model was generally more accurate than both the non-spatial and conventional trend surface models. With this model, we mapped the forest dynamics of the entire Alaska boreal region by aggregating predicted stand states across the region. It was predicted that under current conditions of climate and natural disturbances, most of the Alaska boreal forest region may undergo a major shift from deciduous-dominant to conifer-dominant, with an average increase of 0.33 m² ha year⁻¹ in basal area over the Twenty-First Century.     

Bayesian modelling of extreme wind speed at Cape Town,South Africa

In the framework of generalized extreme value (GEV) distribution, the frequentist and Bayesian methods have been used to analyse the extremes of annual maxima wind speed recorded by automatic weather stations in Cape Town, Western Cape, South Africa. In the frequentist approach, the GEV distribution parameters were estimated using maximum likelihood, whereas in the Bayesian method the Markov Chain Monte Carlo technique with the Metropolis–Hastings algorithm was used. The results show that the GEV model with trend in the location parameter appears to be a better model for annual maxima data. The paper also discusses a method to construct informative priors empirically using historical data of the underlying process from other weather stations. The results from the Bayesian analysis show that posterior inference might be affected by the choice of priors and hence by the distance between a weather station used to formulate the priors and the point of interest.     

Simulation of annual productivity and long-term population trends of white-winged doves in the Tamaulipan Biotic Province

We present a model that synthesizes decades of field data on white-winged doves (Zenaida asiatica asiatica; WWDO) in the Tamaulipan Biotic Province. The model is represented as a discrete-time, deterministic compartment model based on difference equations with a one-week time step designed to simulate annual productivity and long-term trends in abundance. We evaluated the model by comparing simulated annual productivity and long-term population trends to field data. Based on simulation results, we identified apparent inconsistencies in the database; we could not generate the observed annual production index with the model parameterized based on field nest success and survivorship data, nor could we generate a stable long-term population trend with the model parameterized based on suggested sustainable harvest rates and empirically-based estimates of migratory return rates. Simulation results suggest that nest success might be closer to 22% (rather than 35%). A similar trend resulted when simulated hunting pressure was increased by 25% (to 31%), or return rates of migrating juveniles and adults were decreased by 5.5 and 5.0%, to 69 and 77%, respectively, with all other values at the baseline level. For these reasons, until better estimates of nest success and migratory return rates are available, model predictions must be viewed with caution.     

Effects of buoyancy, transparency and zooplankton feeding on surface maxima and deep maxima: Comprehensive mathematical model for vertical distribution in cyanobacterial biomass

Inhomogeneous vertical distributions of the cyanobacterial biomass are widely observed during the summer season in stratified lake ecosystems. Among these are surface maxima characterized by surface scum formation and deep or subsurface maxima also known as deep chlorophyll maxima (DCM). The former occurs at the epilimnion in eutrophic lakes, and are usually caused by colonial cyanobacteria such as Microcystis. On the other hand, the latter occurs at the metalimnion and the upper part of the hypolimnion near the thermocline in oligotrophic lakes, and are referred to filamentous cyanobacteria such as Oscillatoria. The aim of this paper is to present a simple mathematical model that can simultaneously describe these phenomena including the annual and diurnal variations, emphasizing the roles of buoyancy regulation, transparency of the lake and zooplankton feeding on cyanobacteria. According to our computer analyses, the increased buoyancy, the low clarity of the lake and the low rate of zooplankton feeding take significant roles in formation of surface maxima, while the reversal of these factors makes deep maxima predominant. Our two-component model with nutrients and cyanobacteria can distinguish between two phenomena by changing the parameters for these factors, without altering the model itself.     

Inference about the ratio of means from Negative Binomial paired count data

We derive some statistical properties of the distribution of two Negative Binomial random variables conditional on their total. This type of model can be appropriate for paired count data with Poisson over-dispersion such that the variance is a quadratic function of the mean. This statistical model is appropriate in many ecological applications including comparative fishing studies of two vessels and or gears. The parameter of interest is the ratio of pair means. We show that the conditional means and variances are different from the more commonly used Binomial model with variance adjusted for over-dispersion, or the Beta-Binomial model. The conditional Negative Binomial model is complicated because it does not eliminate nuisance parameters like in the Poisson case. Maximum likelihood estimation with the unconditional Negative Binomial model can result in biased estimates of the over-dispersion parameter and poor confidence intervals for the ratio of means when there are many nuisance parameters. We propose three approaches to deal with nuisance parameters in the conditional Negative Binomial model. We also study a random effects Binomial model for this type of data, and we develop an adjustment to the full-sample Negative Binomial profile likelihood to reduce the bias caused by nuisance parameters. We use simulations with these methods to examine bias, precision, and accuracy of estimators and confidence intervals. We conclude that the maximum likelihood method based on the full-sample Negative Binomial adjusted profile likelihood produces the best statistical inferences for the ratio of means when paired counts have Negative Binomial distributions. However, when there is uncertainty about the type of Poisson over-dispersion then a Binomial random effects model is a good choice.     

Impact of climate change on soil moisture dynamics in Brandenburg with a focus on nature conservation areas

Global warming impacts the water cycle not only by changing regional precipitation levels and temporal variability, but also by affecting water flows and soil moisture dynamics. In Brandenburg, increasing average annual temperature and decreasing precipitation in summer have already been observed. For this study, past trends and future effects of climate change on soil moisture dynamics in Brandenburg were investigated, considering regional and specific spatial impacts. Special Areas of Conservation (SACs) were focused on in particular. A decreasing trend in soil water content was shown for the past by analyzing simulation results from 1951 to 2003 using the integrated ecohydrological model SWIM [Krysanova, V., Müller-Wohlfeil, D.-I., Becker, A., 1998. Development and test of a spatially distributed hydrological/water quality model for mesoscale watersheds. Ecol. Model. 106, 261–289]. The trend was statistically significant for some areas, but not for the entire region. Simulated soil water content was particularly low in the extremely dry year 2003. Comparisons of simulated trends in soil moisture dynamics with trends in the average annual Palmer Drought Severity Index for the region showed largely congruent patterns, though the modeled soil moisture trends are characterized by a much higher spatial resolution. Regionally downscaled climate change projections representing the range between wetter and drier realizations were used to evaluate future trends of available soil water. A further decrease of average available soil water ranging from −4% to −15% was projected for all climate realizations up to the middle of the 21st century. An average decrease of more than 25 mm was simulated for 34% of the total area in the dry realization. Available soil water contents in SACs were generally higher and trends in soil moisture dynamics were lower mainly due to their favorable edaphic conditions. Stronger absolute and relative changes in the simulated trends for the past and future were shown for SACs within Brandenburg than for the state as a whole, indicating a high level of risk for many wetland areas. Nonetheless, soil water content in SACs is expected to remain higher than average under climate change conditions as well, and SACs therefore have an important buffer function under the projected climate change. They are thus essential for local climate and water regulation and their status as protected areas in Brandenburg should be preserved.     

Bodennahes Ozon: Trends in Bayern

Long-term ground-based measurements of ozone in Bavaria (Germany) are evaluated in respect of their trend during the last 20 years. First a method is described to characterize the measuring-sites in relation to the levels of precursors: The derived division in three classes proved its worth interpreting the results. Following, the trend of ozone-concentration are calculated by linear regression analysis and are tested in respect of their significance. Generally the Bavarian results fit in the trends of a long-term increase of ozone-concentrations observed at several central European stations. In detail, there are differences between stations situated at elevated sites or at other sites without traffic (class I) and stations situated at sites with significant influence of nearby traffic (classes II and III). At stations of class I an increase of ozone until the mid 80’s is recorded. Afterwards this trend seems to be stopped. But at the stations of classes II and III an increase of ozone is occuring only since the mid 80’s. The observed trends are correlated with the trends of NO_x-emissions.     

Inferential permutation tests for maximum entropy models in ecology

Maximum entropy (maxent) models assign probabilities to states that (1) agree with measured macroscopic constraints on attributes of the states and (2) are otherwise maximally uninformative and are thus as close as possible to a specified prior distribution. Such models have recently become popular in ecology, but classical inferential statistical tests require assumptions of independence during the allocation of entities to states that are rarely fulfilled in ecology. This paper describes a new permutation test for such maxent models that is appropriate for very general prior distributions and for cases in which many states have zero abundance and that can be used to test for conditional relevance of subsets of constraints. Simulations show that the test gives correct probability estimates under the null hypothesis. Power under the alternative hypothesis depends primarily on the number and strength of the constraints and on the number of states in the model; the number of empty states has only a small effect on power. The test is illustrated using two empirical data sets to test the community assembly model of B. Shipley, D. Vile, and E. Garnier and the species abundance distribution models of S. Pueyo, F. He, and T. Zillio.     

Estimating trend precision and power to detect trends across grouped count data

Ecologists commonly use grouped or clustered count data to estimate temporal trends in counts, abundance indices, or abundance. For example, the U.S. Breeding Bird Survey data represent multiple counts of birds from within each of multiple, spatially defined routes. Despite a reliance on grouped counts, analytical methods for prospectively estimating precision of trend estimates or statistical power to detect trends that explicitly acknowledge the characteristics of grouped count data are undescribed. These characteristics include the fact that the sampling variance is an increasing function of the mean, and that sampling and group-level variance estimates are generally estimated on different scales (the sampling and log scales, respectively). We address these issues for repeated sampling of a single population using an analytical approach that has the flavor of a generalized linear mixed model, specifically that of a negative binomial-distributed count variable with random group effects. The count mean, including grand intercept, trend, and random group effects, is modeled linearly on the log scale, while sampling variance of the mean is estimated on the log scale via the delta method. Results compared favorably with those derived using Monte Carlo simulations. For example, at trend = 5% per temporal unit, differences in standard errors and in power were modest relative to those estimated by simulation (< or = /11/% and < or = /16/%, respectively), with relative differences among power estimates decreasing to < or = /7/% when power estimated by simulations was > or = 0.50. Similar findings were obtained using data from nine surveys of fingernail clams in the Mississippi River. The proposed method is suggested (1) where simulations are not practical and relative precision or power is desired, or (2) when multiple precision or power calculations are required and where the accuracy of a fraction of those calculations will be confirmed using simulations.     

A generalized estimating equations approach for capture–recapture closed population models

The estimation of population density animal population parameters, such as capture probability, population size, or population density, is an important issue in many ecological applications. Capture–recapture data may be considered as repeated observations that are often correlated over time. If these correlations are not taken into account then parameter estimates may be biased, possibly producing misleading results. We propose a generalized estimating equations (GEE) approach to account for correlation over time instead of assuming independence as in the traditional closed population capture–recapture studies. We also account for heterogeneity among observed individuals and over-dispersion, modelling capture probabilities as a function of covariates. The GEE versions of all closed population capture–recapture models and their corresponding estimating equations are proposed. We evaluate the effect of accounting for correlation structures on capture–recapture model selection based on the quasi-likelihood information criterion (QIC). An example is used for an illustrative application and for comparison to currently used methodology. A Horvitz–Thompson-like estimator is used to obtain estimates of population size based on conditional arguments. A simulation study is conducted to evaluate the performance of the GEE approach in capture-recapture studies. The GEE approach performs well for estimating population parameters, particularly when capture probabilities are high. The simulation results also reveal that estimated population size varies on the nature of the existing correlation among capture occasions.     

Uncertainty in future soil carbon trends at a central U.S. site under an ensemble of GCM scenario climates

This study uses DAYCENT model to investigate the sensitivity of soil organic carbon (SOC) at an intensely cultivated site in the U.S. Midwest under an ensemble of scenario climates predicted by IPCC models. The model ensemble includes three IPCC models (Canadian, French, German), three emission scenarios (B1, A1B, A2) and three time periods (late 20th, mid-21st, late 21st century). DAYCENT shows that SOC at the site would decline by 0.3-2.6 kg m⁻² (5-35%) depending on the models and scenarios from late 20th to mid-21st century despite a larger increase of future net primary production (NPP) than respiration. The future SOC decrease is mostly attributable to harvest loss. The wide spread in future SOC decline rates are in part because SOC decrease (by respiration) is directly proportional to SOC itself. Any uncertainty in absolute SOC in DAYCENT would translate directly into its trend, unlike other variables such as temperature whose trends are independent of their values themselves, contrasting the reliability of SOC trend with temperature change.     

Predictors of occupancy trend across spatial scale

Many explorations of extinction probability have had a global focus, yet it is unclear whether variables that explain the probability of extinction at large spatial extents are the same as those at small spatial extents. Thus, we used nearly annual presence-absence records for the most recent 40 years of a 110-year data set from Palenque, Mexico, an area with ongoing deforestation, to explore which of >200 species of birds have probabilities of extirpation that are likely to increase. We assessed associations between long-term trends in species presence (i.e., detection in a given year) and body size, geographic range size, diet, dependence on forest cover, taxonomy, and ecological specialization. Our response variable was the estimated slope of a weighted logistic regression for each species. We assessed the relative strength of each predictor by means of a model ranking scheme. Several variables associated with high extinction probability at global extents, such as large body size or small geographic range size, were not associated with occurrence of birds over time at our site. Body size was associated with species loss at Palenque, but occurrence trends of both very large and very small species, particularly the latter, have declined, or the species have been extirpated. We found no association between declining occurrence trend and geographic range size, yet decline correlated with whether a species depends on forest (mean occupancy trend =-0.0380, 0.0263, and 0.0186 for, respectively, species with high, intermediate, or low dependence on forest) and with complex combinations of diet and foraging strata (e.g., occurrence of canopy insectivores and terrestrial omnivores has increased, whereas occurrence of mid-level frugivores and terrestrial granivores has decreased). Our findings emphasize that analyses of local areas are necessary to explicate extirpation risk at various spatial extents.