Quantifying conservation concern - Bayesian statistics, birds and the red lists

For species that are still widespread, obtaining accurate and precise measures of population change inevitably means gathering representative sample data rather than undertaking a complete census. In the UK, a system of raising ‘alerts’ utilises stochastic models for such data to identify species in rapid (>50%) or moderate (25-50%) decline across various temporal and spatial scales. Considerable improvements in interpretation can be made by explaining annual fluctuations in terms of explicit population models (rather than trends of an arbitrary mathematical form); through the simultaneous modelling of data from a complete or partial census with those providing information on the demographic rates employed in these models; and through adopting a Bayesian rather than a frequentist statistical approach. A Bayesian approach is natural for quantifying, in the form of a probability, the support provided by the data for assigning a species to each of the categories. Based on territory mapping and ringing data for the lapwing Vanellus vanellus, we describe such an approach. Trends are estimated more precisely than those under models previously employed in the alerts context. Some smoothing is induced, but realistic responses to years of severe weather are retained, and these are expressed also via model-averaged trends in key demographic parameters. We discuss the conservation implications for this declining species, and the wider potential arising from the ability to quantify confidence that population change has exceeded a threshold either generating conservation concern or justifying a subsequent programme of action for recovery.     

Can spatio-temporal geostatistical methods improve high resolution regionalisation of meteorological variables?

Ecological and hydrological models applied over regional domains generally require the input of spatial meteorological time series. We investigate the potential improvements to space–time regionalisations of sparse meteorological data sets when including information on temporal correlations between successive measurements of minimum temperature (T_min), maximum temperature (T_max) and precipitation (P) from 112 stations across Central Oregon. We compared a number of increasingly complex geostatistical models based on Kriging with a baseline inverse distance weighting algorithm. We varied the number of interpolation data used in both space and time and assessed the impact on interpolation skill. Furthermore, we assessed the error and bias reduction resulting from aggregating estimates over increasingly large temporal supports. We hypothesised that incorporating temporal information would decrease errors, and that error and bias would be reduced when considering estimates aggregated over longer time periods. We found that, contrary to our expectations, incorporation of information on temporal autocorrelation decreased interpolation skill by ～5% for T_min and T_max. However, inclusion of temporal autocorrelation improved results for P by ～10%. Increasing the temporal aggregation of estimates was shown to decrease error by up to 50% and bias by up to 30% (daily vs. annual support). These results indicate that instantaneous error may be diluted for phase lagged or integrating elements of the state vector, such as soil moisture, when implementing such surfaces in modelling applications. Results were more successful for temperature than precipitation (daily % error for jack-knife estimates of T_min = 52, T_max = 13, P = 97), reflecting the stochastic nature of precipitation, and problems with non-linearity for the Kriging algorithm.     

Identification of model components for a class of continuous spatiotemporal models

Environmental data routinely are collected at irregularly spaced monitoring stations and at intermittent times, times which may differ by location. This article introduces a class of continuous-time, continuous-space statistical models that can accommodate many of these more complex environmental processes. This class of models in corporates temporal and spatial variability in a cohesive manner and is broad enough to include temporal processes that are assumed to be generated by stochastic differential equations with possibly temporally and spatially correlated errors. A wide range of ARIMA temporal models and geostatistical spatial models are included in the class of models investigated. Techniques for identifying the structure of the temporal and spatial components of this class of models are detailed. Point estimates of model parameters, asymptotic distributions, and Kalman-filter prediction methods are discussed.     

Incorporating Telemetry Error into Hidden Markov Models of Animal Movement Using Multiple Imputation

When data streams are observed without error and at regular time intervals, discrete-time hidden Markov models (HMMs) have become immensely popular for the analysis of animal location and auxiliary biotelemetry data. However, measurement error and temporally irregular data are often pervasive in telemetry studies, particularly in marine systems. While relatively small amounts of missing data that are missing-completely-at-random are not typically problematic in HMMs, temporal irregularity can result in few (if any) observations aligning with the regular time steps required by HMMs. Fitting HMMs that explicitly account for uncertainty attributable to location measurement error, temporally irregular observations, or other forms of missing data typically requires computationally demanding techniques, such as Markov chain Monte Carlo (MCMC). Using simulation and a real-world bearded seal (Erignathus barbatus) example, I investigate a practical alternative to incorporating measurement error and temporally irregular observations into HMMs based on multiple imputation of the position process drawn from a single-state continuous-time movement model. This two-stage approach is relatively simple, performed with existing software using efficient maximum likelihood methods, and completely parallelizable. I generally found the approach to perform well across a broad range of simulated measurement error and irregular sampling rates, with latent states and locations reliably recovered in nearly all simulated scenarios. However, high measurement error coupled with low sampling rates often induced bias in both the estimated probability distributions of data streams derived from the imputed position process and the estimated effects of spatial covariates on state transition probabilities. Results from the two-stage analysis of the bearded seal data were similar to a more computationally intensive single-stage MCMC analysis, but the two-stage analysis required much less computation time and no custom model-fitting algorithms. I thus found the two-stage multiple-imputation approach to be promising in terms of its ease of implementation, computation time, and performance. Code for implementing the approach using the R package “momentuHMM” is provided.Supplementary materials accompanying this paper appear online.     

Spatial autocorrelation and statistical tests: Some solutions

Spatial dependence or spatial autocorrelation often occurs in ecological data and can be a serious problem in analysis, affecting the significance rates of statistical tests, making them too liberal when the dependence is positive. Ecological phenomena often are patchy and give data with a wave structure, producing autocorrelation that cycles between positive and negative with increasing distance, further complicating the situation. This article describes the essentials of dealing with this problem as commonly encountered in analyzing ecological data for two variables. We investigated two related approaches to correcting statistical tests for data with spatial autocorrelation from one-dimensional sampling schemes like the transects used in plant ecology, the example of interest here. Both approaches estimate the “effective sample size” based on the observed autocorrelation structures of the variables. We examined tests of correlation and bivariate goodness-of-fit tests, as well as extensions beyond both of these test classes. The correction methods prove to be robust for a wide range of spatial autocorrelation structures in one-dimensional data and provide reliable corrections in most cases. They fail only when the data have strong and consistent waves that cause persistent cycles in the autocorrelation as a function of distance. By examining the spatial autocorrelation structure of the ecological data, we can predict the likelihood of successful correction for these bivariate tests.     

Using forest structure and composition to predict the occurrence of vertebrate species in Douglas-Fir forests of British Columbia

Forest managers require an understanding of how vertebrate species respond and persist within the dynamics of changing forest environments so that management strategies can retain and recruit structural aspects necessary for the persistence of populations. Species-habitat models are often used to understand these relationships and are subsequently used to manage landscapes. We tested several species-habitat models to predict the presence or absence of a range of vertebrate species (n = 55) and to determine the potential of using vertical and horizontal measures of forest structure as a surrogate of species occurrence. We validated models with temporally and spatially independent data. Some of the models had good predictive accuracy that was retained when validated and thus have application in terms of implementation as management tools. Modelling success varied, however, depending on whether plot or stand data were used. Many models included variables related to spatial relationships of structures. Few models were reliable when applied to independent data; therefore, our results indicate that models cannot be assumed to be applicable in different years or applied outside the area where the model was developed, even with similar spatial and temporal contexts. Overall, we did not find robust relationships necessary to guide management targets for retention and recruitment of specific forest structures. Therefore, using these habitat models as surrogates for monitoring species occurrence is limited. Monitoring aspects of habitat should still be included as part of biodiversity monitoring programs because preservation of structures known to be negatively affected by harvesting (e.g., dead wood, large trees, closed canopies, continuous forests) contributes to local and landscape heterogeneity and has been shown to affect species presence in this study and others.     

Design of ecoregional monitoring in conservation areas of high-latitude ecosystems under contemporary climate change

Land ownership in Alaska includes a mosaic of federally managed units. Within its agency’s context, each unit has its own management strategy, authority, and resources of conservation concern, many of which are migratory animals. Though some units are geographically isolated, many are nevertheless linked by paths of abiotic and biotic flows, such as rivers, air masses, flyways, and terrestrial and aquatic migration routes. Furthermore, individual land units exist within the context of a larger landscape pattern of shifting conditions, requiring managers to understand at larger spatial scales the status and trends in the synchrony and spatial concurrence of species and associated suitable habitats. Results of these changes will determine the ability of Alaska lands to continue to: provide habitat for local and migratory species; absorb species whose ranges are shifting northward; and experience mitigation or exacerbation of climate change through positive and negative atmospheric feedbacks. We discuss the geographic and statutory contexts that influence development of ecological monitoring; argue for the inclusion of significant amounts of broad-scale monitoring; discuss the importance of defining clear programmatic and monitoring objectives; and draw from lessons learned from existing long-term, broad-scale monitoring programs to apply to the specific contexts relevant to high-latitude protected areas such as those in Alaska. Such areas are distinguished by their: marked seasonality; relatively large magnitudes of contemporary change in climatic parameters; and relative inaccessibility due to broad spatial extent, very low (or zero) road density, and steep and glaciated areas. For ecological monitoring to effectively support management decisions in high-latitude areas such as Alaska, a monitoring program ideally would be structured to address the actual spatial and temporal scales of relevant processes, rather than the artificial boundaries of individual land-management units. Heuristic models provide a means by which to integrate understanding of ecosystem structure, composition, and function, in the midst of numerous ecosystem drivers.     

Temporal Synchronization Analysis for Improving Regression Modeling of Fecal Indicator Bacteria Levels

Multiple linear regression models are often used to predict levels of fecal indicator bacteria (FIB) in recreational swimming waters based on independent variables (IVs) such as meteorologic, hydrodynamic, and water-quality measures. The IVs used for these analyses are traditionally measured at the same time as the water-quality sample. We investigated the improvement in empirical modeling performance by using IVs that had been temporally synchronized with the FIB response variable. We first examined the univariate relationship between multiple ??aspects?? of each IV and the response variable to find the single aspect of each IV most strongly related to the response. Aspects are defined by the temporal window and lag (relative to when the response is measured) over which the IV is averaged. Models were then formed using the ??best?? aspects of each IV. Employing iterative cross-validation, we examined the average improvement in the mean squared error of prediction, MSEP, for a testing dataset after using our temporal synchronization technique on the training data. We compared the MSEP values of three methodologies: predictions made using unsynchronized IVs (UNS), predictions made using synchronized IVs where aspects were chosen using a Pearson correlation coefficient (PCC), and predictions using IV aspects chosen using the PRESS statistic (PRS). Averaging over 500 randomly generated testing datasets, the MSEP values using the PRS technique were 50?% lower (p?<?0.001) than the MSEP values of the UNS technique. The average MSEP values of the PCC technique were 26?% lower (p?<?0.001) than the MSEP values of the UNS technique. We conclude that temporal synchronization is capable of significantly improving predictive models of FIB levels in recreational swimming waters.     

Overcoming the rare species modelling paradox: A novel hierarchical framework applied to an Iberian endemic plant

Rare species have restricted geographic ranges, habitat specialization, and/or small population sizes. Datasets on rare species distribution usually have few observations, limited spatial accuracy and lack of valid absences; conversely they provide comprehensive views of species distributions allowing to realistically capture most of their realized environmental niche. Rare species are the most in need of predictive distribution modelling but also the most difficult to model. We refer to this contrast as the “rare species modelling paradox” and propose as a solution developing modelling approaches that deal with a sufficiently large set of predictors, ensuring that statistical models are not over-fitted. Our novel approach fulfils this condition by fitting a large number of bivariate models and averaging them with a weighted ensemble approach. We further propose that this ensemble forecasting is conducted within a hierarchic multi-scale framework. We present two ensemble models for a test species, one at regional and one at local scale, each based on the combination of 630 models. In both cases, we obtained excellent spatial projections, unusual when modelling rare species. Model results highlight, from a statistically sound approach, the effects of multiple drivers in a same modelling framework and at two distinct scales. From this added information, regional models can support accurate forecasts of range dynamics under climate change scenarios, whereas local models allow the assessment of isolated or synergistic impacts of changes in multiple predictors. This novel framework provides a baseline for adaptive conservation, management and monitoring of rare species at distinct spatial and temporal scales.     

An Efficient Approach to Spatiotemporal Analysis and Modeling of Air Pollution Data

A statistically efficient approach is adopted for modeling spatial time-series of large data sets. The estimation of the main diagnostic tool such as the likelihood function in Gaussian spatiotemporal models is a cumbersome task when using extended spatial time-series such as air pollution. Here, using the Innovation Algorithm, we manage to compute it for many spatiotemporal specifications. These specifications refer to the spatial periodic-trend, the spatial autoregressive moving average, the spatial autoregressive integrated and fractionally integrated moving average Gaussian models. Our method is applied to daily pollutants over a large metropolitan area like Athens. In the applied part of our paper, we first diagnose temporal and spatial structures of data using non-likelihood based criteria, such as the empirical autocorrelation and covariance functions. Second, we use likelihood and non-likelihood based criteria to select a spatiotemporal model among various specifications. Finally, using kriging we regionalize the resulting parameter estimates of the best-fitted model in space at any unmonitored location in the Athens region. The results show that a specific autoregressive integrated moving average spatiotemporal model can optimally perform in within and out of spatial sample estimation. Supplemental materials for this article are available from the journal website.     

改进型空间洛伦茨曲线在土地利用结构研究中的应用

洛伦茨曲线和基尼系数是土地利用结构研究的常用方法,现有方法存在统计单元数量少、面积差异较大的问题,使得拟合的洛伦茨曲线不够平滑,计算的基尼系数不够准确。利用网格法对现有方法进行改进,以规则网格对研究区进行分割,增加了统计单元的数量,且各统计单元面积相等,可使拟合点在洛伦茨曲线上均匀分布,提高了洛伦茨曲线的平滑程度以及基尼系数的计算精度。另外,以妫水河流域为应用案例,以1998年和2009年的两期Landsat TM影像为基础数据,对改进型空间洛伦茨曲线在土地利用结构研究方面进行案例应用,结果表明该方法可以有效表达不同土地利用类型之间的结构差异及其时空变化规律。耕地始终是分布最均匀的地类,1998年分布最不均匀的地类是园地,2009年分布最不均匀的地类是水域;1998—2009年,林地和水域等生态用地的分布均匀性下降,耕地和建设用地等人工用地分布的均匀性提高。     

Modeling Complex Spatial Dependencies: Low-Rank Spatially Varying Cross-Covariances With Application to Soil Nutrient Data

Advances in geo-spatial technologies have created data-rich environments which provide extraordinary opportunities to understand the complexity of large and spatially indexed data in ecology and the natural sciences. Our current application concerns analysis of soil nutrients data collected at La Selva Biological Station, Costa Rica, where inferential interest lies in capturing the spatially varying relationships among the nutrients. The objective here is to interpolate not just the nutrients across space, but also associations among the nutrients that are posited to vary spatially. This requires spatially varying cross-covariance models. Fully process-based specifications using matrix-variate processes are theoretically attractive but computationally prohibitive. Here we develop fully process-based low-rank but non-degenerate spatially varying cross-covariance processes that can effectively yield interpolate cross-covariances at arbitrary locations. We show how a particular low-rank process, the predictive process, which has been widely used to model large geostatistical datasets, can be effectively deployed to model non-degenerate cross-covariance processes. We produce substantive inferential tools such as maps of nonstationary cross-covariances that constitute the premise of further mechanistic modeling and have hitherto not been easily available for environmental scientists and ecologists.     

Impacts of climate change on Swiss biodiversity: An indicator taxa approach

We present a new indicator taxa approach to the prediction of climate change effects on biodiversity at the national level in Switzerland. As indicators, we select a set of the most widely distributed species that account for 95% of geographical variation in sampled species richness of birds, butterflies, and vascular plants. Species data come from a national program designed to monitor spatial and temporal trends in species richness. We examine some opportunities and limitations in using these data. We develop ecological niche models for the species as functions of both climate and land cover variables. We project these models to the future using climate predictions that correspond to two IPCC 3rd assessment scenarios for the development of ‘greenhouse’ gas emissions. We find that models that are calibrated with Swiss national monitoring data perform well in 10-fold cross-validation, but can fail to capture the hot-dry end of environmental gradients that constrain some species distributions. Models for indicator species in all three higher taxa predict that climate change will result in turnover in species composition even where there is little net change in predicted species richness. Indicator species from high elevations lose most areas of suitable climate even under the relatively mild B2 scenario. We project some areas to increase in the number of species for which climate conditions are suitable early in the current century, but these areas become less suitable for a majority of species by the end of the century. Selection of indicator species based on rank prevalence results in a set of models that predict observed species richness better than a similar set of species selected based on high rank of model AUC values. An indicator species approach based on selected species that are relatively common may facilitate the use of national monitoring data for predicting climate change effects on the distribution of biodiversity.     

Analysis of spatial point patterns using bundles of product density LISA functions

The analysis of a spatial point pattern is often in volved with looking for spatial structure, such as clustering or regularity in the points (or events). For example, it is of biological interest to characterize the pattern of tree locations in a forest. This has traditionally been done using global summaries, such as the K-function or its differential, the product density function. In this article, we define a local version of the product density function for each event, derived under a definition of a local indicator of spatial association (LISA). These product density LISA functions can then be grouped into bundles of similar functions using multivariate hierarchical clustering techniques. The bundles can then be visualized by a replotting of the data, obtained via classical multidimensional scaling of the statistical distances between functions. Thus, we propose a different way of looking for structure based on how an event relates to nearby events. We apply this method to a point pattern of pine saplings in a Finnish forest and show remarkable, heretofore undiscovered, spatial structure in the data.     

Dynamic Modelling of Spatially Variable Catchment Hydrochemistry for Critical Loads Assessment

Concern about acidification in upland areas has brought about the need to model the stream hydrochemical response to deposition and land-use changes and calculate critical loads. Application of dynamic models such as MAGIC are preferable to steady-state methods, since they are able to produce an estimate of the time scale required to meet some water chemistry target given a reduction in acid deposition. These models typically consider annual changes in stream chemistry at one point. However, in order to protect biota from 'acid episodes', quantification of temporal variability needs to encompass event responses; in addition spatial variability across the catchment also needs to be considered. In this paper, modelling of both spatial and temporal variability is combined in a new framework which enables quantification of catchment hydrochemical variability in time and space. Both low and high flow hydro-chemical variability are quantified in terms of statistical distributions of ANC (Acid Neutralisation Capacity). These are then input as stochastic variables to an EMMA (End-Member Mixing Analysis) model which accounts for temporal variability and ANC is hence predicted as a function of time and space across the whole catchment using Monte-Carlo simulation. The method is linked to MAGIC to predict future scenarios and may be used by iteration to calculate critical loads. The model is applied to the headwaters of the River Severn at Plynlimon, Wales, to demonstrate its capabilities.     

Imputation Approaches for Animal Movement Modeling

The analysis of telemetry data is common in animal ecological studies. While the collection of telemetry data for individual animals has improved dramatically, the methods to properly account for inherent uncertainties (e.g., measurement error, dependence, barriers to movement) have lagged behind. Still, many new statistical approaches have been developed to infer unknown quantities affecting animal movement or predict movement based on telemetry data. Hierarchical statistical models are useful to account for some of the aforementioned uncertainties, as well as provide population-level inference, but they often come with an increased computational burden. For certain types of statistical models, it is straightforward to provide inference if the latent true animal trajectory is known, but challenging otherwise. In these cases, approaches related to multiple imputation have been employed to account for the uncertainty associated with our knowledge of the latent trajectory. Despite the increasing use of imputation approaches for modeling animal movement, the general sensitivity and accuracy of these methods have not been explored in detail. We provide an introduction to animal movement modeling and describe how imputation approaches may be helpful for certain types of models. We also assess the performance of imputation approaches in two simulation studies. Our simulation studies suggests that inference for model parameters directly related to the location of an individual may be more accurate than inference for parameters associated with higher-order processes such as velocity or acceleration. Finally, we apply these methods to analyze a telemetry data set involving northern fur seals (Callorhinus ursinus) in the Bering Sea. Supplementary materials accompanying this paper appear online.     

A Spatio-Temporal Downscaler for Output From Numerical Models

Often, in environmental data collection, data arise from two sources: numerical models and monitoring networks. The first source provides predictions at the level of grid cells, while the second source gives measurements at points. The first is characterized by full spatial coverage of the region of interest, high temporal resolution, no missing data but consequential calibration concerns. The second tends to be sparsely collected in space with coarser temporal resolution, often with missing data but, where recorded, provides, essentially, the true value. Accommodating the spatial misalignment between the two types of data is of fundamental importance for both improved predictions of exposure as well as for evaluation and calibration of the numerical model. In this article we propose a simple, fully model-based strategy to downscale the output from numerical models to point level. The static spatial model, specified within a Bayesian framework, regresses the observed data on the numerical model output using spatially-varying coefficients which are specified through a correlated spatial Gaussian process.     

Continuous-Time Proportional Hazards Regression for Ecological Monitoring Data

We consider a continuous-time proportional hazards model for the analysis of ecological monitoring data where subjects are monitored at discrete times and fixed sites across space. Since the exact time of event occurrence is not directly observed, we rely on dichotomous event indicators observed at monitoring times to make inference about the model parameters. We use autoregression on the response at neighboring sites from a previous time point to take into account spatial dependence. The interesting fact is utilized that the probability of observing an event at a monitoring time when the underlying hazards is proportional falls under the class of generalized linear models with binary responses and complementary log-log link functions. Thus, a maximum likelihood approach can be taken for inference and the computation can be carried out using standard statistical software packages. This approach has significant computational advantages over some of the existing methods that rely on Monte Carlo simulations. Simulation experiments are conducted and demonstrate that our method has sound finite-sample properties. A real dataset from an ecological study that monitored bark beetle colonization of red pines in Wisconsin is analyzed using the proposed models and inference. Supplementary materials that contain technical details are available online.     

A resource selection model for analyzing pseudoreplicated data due to grouping behavior of animals

Quantifying resource selection is of primary interest in animal ecology. Many analyses of resource selection assume spatial and temporal independence of the sampling unit. Autocorrelation between observations, which is a general property of ecological variables, causes difficulties for most standard statistical procedures of resource selection because autocorrelated data violate the assumption of independence. To overcome this problem, we develop a mixed-effects model to estimate resource selection functions from data that are autocorrelated because of unobserved grouping behavior by animals. In the application of the expectation-maximization (EM) algorithm, the computation of the conditional expectation of the complete-data log-likelihood function does not have a closed-form solution requiring numerical integration. A Monte Carlo EM algorithm with Gibbs sampling can be used effectively in such situations to find exact maximum likelihood estimates. We propose a simple automated Monte Carlo EM algorithm with multiple sequences in which the Monte Carlo sample size is increased when the EM step is swamped by Monte Carlo errors.We demonstrate that the model can detect inherent autocorrelation and provide reasonable variance estimates when applied to nocturnal bird migration data. This approach could also be applied to ecological processes with various types of spatially and temporally autocorrelated data, circumventing serious problems caused by dangerous pseudoreplication.