Recursive residuals and model diagnostics for normal and non-normal state space models

Model diagnostics for normal and non-normal state space models are based on recursive residuals which are defined from the one-step ahead predictive distribution. Routine calculation of these residuals is discussed in detail. Various diagnostic tools are suggested to check, for example, for wrong observation distributions and for autocorrelation. The paper also discusses such topics as model diagnostics for discrete time series and model discrimination via Bayes factors. The case studies cover environmental applications such as analysing a time series of the number of daily rainfall occurrences and a time series of daily sulfur dioxide emissions.     

Estimation methods for nonlinear state-space models in ecology

The use of nonlinear state-space models for analyzing ecological systems is increasing. A wide range of estimation methods for such models are available to ecologists, however it is not always clear, which is the appropriate method to choose. To this end, three approaches to estimation in the theta logistic model for population dynamics were benchmarked by Wang (2007). Similarly, we examine and compare the estimation performance of three alternative methods using simulated data. The first approach is to partition the state-space into a finite number of states and formulate the problem as a hidden Markov model (HMM). The second method uses the mixed effects modeling and fast numerical integration framework of the AD Model Builder (ADMB) open-source software. The third alternative is to use the popular Bayesian framework of BUGS. The study showed that state and parameter estimation performance for all three methods was largely identical, however with BUGS providing overall wider credible intervals for parameters than HMM and ADMB confidence intervals.     

Space-time modeling of environmental monitoring data

This study describes and applies statistical methods for space-time modeling of data from environmental monitoring programs, e.g., within areas such as climate change, air pollution and aquatic environment. Such data are often characterized by sparse sampling in both the temporal and spatial dimensions. In order to improve the amount of information on the physical system in question we suggest using statistical modeling methods for monitoring data. Model predictions combined with observations could be analyzed directly to assess the environmental state or as forcing functions for time series models and deterministic, hydrodynamic models. To illustrate the approach we applied the proposed modeling methods to data from the Danish and Swedish marine monitoring programs. Time series with a weekly resolution were predicted from observations of dissolved inorganic nitrogen (DIN) from the Kattegat basin (1993–1997). DIN observations were sparse, irregularly distributed and comprised approximately 10% of the generated time series.     

Predicting abundance of desert riparian birds: validation and calibration of the Effective Area Model.

Reliable prediction of the effects of landscape change on species abundance is critical to land managers who must make frequent, rapid decisions with long-term consequences. However, due to inherent temporal and spatial variability in ecological systems, previous attempts to predict species abundance in novel locations and/or time frames have been largely unsuccessful. The Effective Area Model (EAM) uses change in habitat composition and geometry coupled with response of animals to habitat edges to predict change in species abundance at a landscape scale. Our research goals were to validate EAM abundance predictions in new locations and to develop a calibration framework that enables absolute abundance predictions in novel regions or time frames. For model validation, we compared the EAM to a null model excluding edge effects in terms of accurate prediction of species abundance. The EAM outperformed the null model for 83.3% of species (N=12) for which it was possible to discern a difference when considering 50 validation sites. Likewise, the EAM outperformed the null model when considering subsets of validation sites categorized on the basis of four variables (isolation, presence of water, region, and focal habitat). Additionally, we explored a framework for producing calibrated models to decrease prediction error given inherent temporal and spatial variability in abundance. We calibrated the EAM to new locations using linear regression between observed and predicted abundance with and without additional habitat covariates. We found that model adjustments for unexplained variability in time and space, as well as variability that can be explained by incorporating additional covariates, improved EAM predictions. Calibrated EAM abundance estimates with additional site-level variables explained a significant amount of variability (P < 0.05) in observed abundance for 17 of 20 species, with R2 values >25% for 12 species, >48% for six species, and >60% for four species when considering all predictive models. The calibration framework described in this paper can be used to predict absolute abundance in sites different from those in which data were collected if the target population of sites to which one would like to statistically infer is sampled in a probabilistic way.     

Time-varying models for extreme values

We propose a new approach for modeling extreme values that are measured in time and space. First we assume that the observations follow a Generalized Extreme Value (GEV) distribution for which the location, scale or shape parameters define the space–time structure. The temporal component is defined through a Dynamic Linear Model (DLM) or state space representation that allows to estimate the trend or seasonality of the data in time. The spatial element is imposed through the evolution matrix of the DLM where we adopt a process convolution form. We show how to produce temporal and spatial estimates of our model via customized Markov Chain Monte Carlo (MCMC) simulation. We illustrate our methodology with extreme values of ozone levels produced daily in the metropolitan area of Mexico City and with rainfall extremes measured at the Caribbean coast of Venezuela.     

Sensitivity of a coupled physical–biological model to turbulence: high-frequency simulations in a northern Adriatic station

This paper investigates the impacts of different turbulence models on the biological state at an ocean station in the northern Adriatic sea, named S3, comparing them with other uncertainties inherent to coupled physical–biological simulations. The numerical tool is a 1-D model resulting from the coupling of two advanced numerical models. The hydrodynamic part is modelled using the General Ocean Turbulence Model (www.gotm.net), in a version adopting state-of-the-art second-moment Turbulence Closure Models (TCMs). Marine biogeochemistry is parameterized with the Biogeochemical Flux Model (http://www.bo.ingv.it/bfm), which is a direct descendant of ERSEM (European Regional Sea Ecosystem Model). Results, obtained by forcing the model with hourly wind and solar radiation data and assimilating salinity casts, are compared against monthly observations made at the station during 2000–2001. Provided that modern second-moment TCMs are employed, the comparisons indicate that both the physical and the biological dynamics are relatively insensitive to the choice of the particular scheme adopted, suggesting that TCMs have finally ‘converged’ in recent years. As a further example, the choice of the nutrient boundary conditions has an impact on the system evolution that is more significant than the choice of the specific TCM, therefore representing a possible limitation of the 1-D model applied to stations located in a Region of Freshwater Influence. The 1-D model simulates the onset and intensity of the spring–summer bloom quite well, although the duration of the bloom is not as prolonged as in the data. Since local dynamics appears unable to sustain the bloom conditions well into summer, phytoplankton at the station was most likely influenced by river input or advection processes, an aspect that was not found when the S3 behaviour was adequately modelled using climatological forcings. When the focus is in predicting high-frequency dynamics, it is more likely that lateral advection cannot be neglected. While the physical state can be satisfactorily estimated at these short time scales, the accurate estimation of the biological state in coastal regions still appears as rather elusive.     

Identifying fishing trip behaviour and estimating fishing effort from VMS data using Bayesian Hidden Markov Models

Recent advances in technologies have lead to a vast influx of data on movements, based on discrete recorded position of animals or fishing boats, opening new horizons for future analyses. However, most of the potential interest of tracking data depends on the ability to develop suitable modelling strategies to analyze trajectories from discrete recorded positions. A serious modelling challenge is to infer the evolution of the true position and the associated spatio-temporal distribution of behavioural states using discrete, error-prone and incomplete observations. In this paper, a Bayesian Hierarchical Model (HBM) using Hidden Markov Process (HMP) is proposed as a template for analyzing fishing boats trajectories based on data available from satellite-based vessel monitoring systems (VMS). The analysis seeks to enhance the definition of the fishing pressure exerted on fish stocks, by discriminating between the different behavioural states of a fishing trip, and also by quantifying the relative importance of each of these states during a fishing trip. The HBM approach is tested to analyse the behaviour of pelagic trawlers in the Bay of Biscay. A hidden Markov chain with a regular discrete time step is used to model transitions between successive behavioural states (e.g., fishing, steaming, stopping (at Port or at sea)) of each vessel. The parameters of the movement process (speed and turning angles) are defined conditionally upon the behavioural states. Bayesian methods are used to integrate the available data (typically VMS position recorded at discrete time) and to draw inferences on any unknown parameters of the model. The model is first tested on simulated data with different parameters structures. Results provide insights on the potential of HBM with HMP to analyze VMS data. They show that if VMS positions are recorded synchronously with the instants at which the process switch from one behavioural state to another, the estimation method provides unbiased and precise inferences on behavioural states and on associated movement parameters. However, if the observations are not gathered with a sufficiently high frequency, the performance of the estimation method could be drastically impacted when the discrete observations are not synchronous with the switching instants. The model is then applied to real pathways to estimate variables of interest such as the number of operations per trip, time and distance spent fishing or travelling.     

Description and preliminary validation of the PMSS fast response parallel atmospheric flow and dispersion solver in complex built-up areas

Noxious atmospheric releases may originate from both accidents and malicious activities. They are a major concern for public authorities or first responders who may wish to have the most accurate situational awareness. Nonetheless, it is difficult to reliably and accurately model the flow, transport, and dispersion processes in large complex built-up environments in a limited amount of time and resources compatible with operational needs. The parallel version of Micro-SWIFT-SPRAY (PMSS) is an attempt to propose a physically sound and fast response modelling system applicable to complicated industrial or urban sites in case of a hazardous release. This paper presents and justifies the choice of the diagnostic flow and Lagrangian dispersion models in PMSS. Then, it documents in detail the development of the parallel algorithms used to reduce the computational time of the models. Finally, the paper emphasizes the preliminary model validation and parallel performances of PMSS based on data from both wind tunnel (Evaluation of Model Uncertainty) and in-field reduced-scale (Mock Urban Setting Test) and real-scale (Oklahoma City) experimental campaigns.     

Intercomparison of Two Models,ETA and RAMS,with TRACT Field Campaign Data

In this work a model intercomparison between RAMS and ETA models is carried out, with the aim of evaluating the quality and accuracy of these mesoscale models in reproducing the time evolution of the meteorology in real complex terrain. This is of great importance not only for meteorological forecast but also for air quality assessment. Numerical simulations are performed to reproduce the mean variables' fields and to compare them with measurements collected during the field campaign TRACT. The domain covers the Rhine valley and surrounding mountainous region and we consider a time period of two days. Results from simulations are compared to observations relative to ground stations and radiosoundings. A qualitative analysis is joined to a quantitative estimation of some reference statistical indexes. Both RAMS and ETA models performances are satisfactory when compared to the measured data and also their relative agreement is good. The mean variable fields are reproduced with a satisfactory degree of reliability, even if the simulated profiles are not able to describe the largest fluctuations of the variables. At the surface stations, the best agreement between predictions and observations is obtained for the wind velocity, while the quality of the results is lower for temperature and humidity.     

Testing the Generality of Bird-Habitat Models

Bird-habitat models are frequently used as predictive modeling tools—for example, to predict how a species will respond to habitat modifications. We investigated the generality of the predictions from this type of model. Multivariate models were developed for Golden Eagle (Aquila chrysaetos), Raven (Corvus corax), and Buzzard (Buteo buteo) living in northwest Scotland. Data were obtained for all habitat and nest locations within an area of 2349 km². This assemblage of species is relatively static with respect to both occupancy and spatial positioning. The area was split into five geographic subregions: two on the mainland and three on the adjacent Island of Mull, which has one of United Kingdom's richest raptor fauna assemblages. Because data were collected for all nest locations and habitats, it was possible to build models that did not incorporate sampling error. A range of predictive models was developed using discriminant analysis and logistic regression. The models differed with respect to the geographical origin of the data used for model development. The predictive success of these models was then assessed by applying them to validation data. The models showed a wide range of predictive success, ranging from only 6% of nest sites correctly predicted to 100% correctly predicted. Model validation techniques were used to ensure that the models' predictions were not statistical artefacts. The variability in prediction success seemed to result from methodological and ecological processes, including the data recording scheme and interregional differences in nesting habitat. The results from this study suggest that conservation biologists must be very careful about making predictions from such studies because we may be working with systems that are inherently unpredictable.     

Modeling carcass removal time for avian mortality assessment in wind farms using survival analysis

In monitoring studies at wind farms, the estimation of bird and bat mortality caused by collision must take into account carcass removal by scavengers or decomposition. In this paper we propose the use of survival analysis techniques to model the time of carcass removal. The proposed method is applied to data collected in ten Portuguese wind farms. We present and compare results obtained from semiparametric and parametric models assuming four main competing lifetime distributions (exponential, Weibull, log-logistic and log-normal). Both homogeneous parametric models and accelerated failure time models were used. The fitted models enabled the estimation of the carcass persistence rates and the calculation of a scavenging correction factor for avian mortality estimation. Additionally, we discuss the impact that the distributional assumption can have on parameter estimation. The proposed methodology integrates the survival probability estimation problem with the analysis of covariate effects. Estimation is based on the most suitable model while simultaneously accounting for censored observations, diminishing scavenging rate estimation bias. Additionally, the method establishes a standardized statistical procedure for the analysis of carcass removal time in subsequent studies.     

Testing ecological models: the meaning of validation

The ecological literature reveals considerable confusion about the meaning of validation in the context of simulation models. The confusion arises as much from semantic and philosophical considerations as from the selection of validation procedures. Validation is not a procedure for testing scientific theory or for certifying the ‘truth’ of current scientific understanding, nor is it a required activity of every modelling project. Validation means that a model is acceptable for its intended use because it meets specified performance requirements.Before validation is undertaken, (1) the purpose of the model, (2) the performance criteria, and (3) the model context must be specified. The validation process can be decomposed into several components: (1) operation, (2) theory, and (3) data. Important concepts needed to understand the model evaluation process are verification, calibration, validation, credibility, and qualification. These terms are defined in a limited technical sense applicable to the evaluation of simulation models, and not as general philosophical concepts. Different tests and standards are applied to the operational, theoretical, and data components. The operational and data components can be validated; the theoretical component cannot.The most common problem with ecological and environmental models is failure to state what the validation criteria are. Criteria must be explicitly stated because there are no universal standards for selecting what test procedures or criteria to use for validation. A test based on comparison of simulated versus observed data is generally included whenever possible. Because the objective and subjective components of validation are not mutually exclusive, disagreements over the meaning of validation can only be resolved by establishing a convention.     

Probability distributions and statistical inference for axial data

Observations on axes which lack information on the direction of propagation are referred to as axial data. Such data are often encountered in enviromental sciences, e.g. observations on propagations of cracks or on faults in mining walls. Even though such observations are recorded as angles, circular probability models are inappropriate for such data since the constraint that observations lie only in [0, π) needs to be enforced. Probability models for such axial data are argued here to have a general structure stemming from that of wrapping a circular distribution on a semi-circle. In particular, we consider the most popular circular model, the von Mises or circular normal distribution, and derive the corresponding axial normal distribution. Certain properties of this distribution are established. Maximum likelihood estimation of its parameters are shown to be surprisingly, in contrast to trigonometric moment estimation, numerically quite appealing. Finally we illustrate our results by several real life axial data sets. Received: September 2004/ Revised: December 2004     

Quantifying parameter uncertainty in a coral reef model using Metropolis-Coupled Markov Chain Monte Carlo

Coral reefs are threatened ecosystems, so it is important to have predictive models of their dynamics. Most current models of coral reefs fall into two categories. The first is simple heuristic models which provide an abstract understanding of the possible behaviour of reefs in general, but do not describe real reefs. The second is complex simulations whose parameters are obtained from a range of sources such as literature estimates. We cannot estimate the parameters of these models from a single data set, and we have little idea of the uncertainty in their predictions.We have developed a compromise between these two extremes, which is complex enough to describe real reef data, but simple enough that we can estimate parameters for a specific reef from a time series. In previous work, we fitted this model to a long-term data set from Heron Island, Australia, using maximum likelihood methods. To evaluate predictions from this model, we need estimates of the uncertainty in our parameters. Here, we obtain such estimates using Bayesian Metropolis-Coupled Markov Chain Monte Carlo. We do this for versions of the model in which corals are aggregated into a single state variable (the three-state model), and in which corals are separated into four state variables (the six-state model), in order to determine the appropriate level of aggregation. We also estimate the posterior distribution of predicted trajectories in each case.In both cases, the fitted trajectories were close to the observed data, but we had doubts about the biological plausibility of some parameter estimates. We suggest that informative prior distributions incorporating expert knowledge may resolve this problem. In the six-state model, the posterior distribution of state frequencies after 40 years contained two divergent community types, one dominated by free space and soft corals, and one dominated by acroporid, pocilloporid, and massive corals. The three-state model predicts only a single community type. We conclude that the three-state model hides too much biological heterogeneity, but we need more data if we are to obtain reliable predictions from the six-state model. It is likely that there will be similarly large, but currently unevaluated, uncertainty in the predictions of other coral reef models, many of which are much more complex and harder to fit to real data.     

Generalized site occupancy models allowing for false positive and false negative errors

Site occupancy models have been developed that allow for imperfect species detection or "false negative" observations. Such models have become widely adopted in surveys of many taxa. The most fundamental assumption underlying these models is that "false positive" errors are not possible. That is, one cannot detect a species where it does not occur. However, such errors are possible in many sampling situations for a number of reasons, and even low false positive error rates can induce extreme bias in estimates of site occupancy when they are not accounted for. In this paper, we develop a model for site occupancy that allows for both false negative and false positive error rates. This model can be represented as a two-component finite mixture model and can be easily fitted using freely available software. We provide an analysis of avian survey data using the proposed model and present results of a brief simulation study evaluating the performance of the maximum-likelihood estimator and the naive estimator in the presence of false positive errors.     

Does scale matter in predicting species distributions? case study with the Marbled Murrelet.

Hierarchical selection orders (selection of microsite, patch, home range, population block, and geographic range) are ideal for dictating spatial grain and extent of animal habitat models, but the resultant conditional models are poor for creating predictive maps. I proposed a two-step approach for accurately mapping probability of animal use that incorporates a single-grain analysis of each selection order in the first step and creates a multi-grain model that combines key variables from each selection order in the second step. Such two-step multi-grain models are strongly recommended because they allow interpretation of the scale of selection for a variable. Using a large data set for the Marbled Murrelet (Brachyramphus marmoratus) as a case study and five selection orders, information theory criteria provided strong support that such models are superior to simpler one-step single-grain models for the murrelet. However, a single-grain model can produce high classification accuracy if it represents the most limiting scale. Notably, accuracy of the two-step multi-grain model was no better than a traditional one-step multi-grain model that ignores selection orders, indicating the advantage of two-step modeling is in elucidating scaling effects, not necessarily in improving accuracy of species distribution maps.     

A Bayesian hierarchical framework for calibrating aquatic biogeochemical models

Model practitioners increasingly place emphasis on rigorous quantitative error analysis in aquatic biogeochemical models and the existing initiatives range from the development of alternative metrics for goodness of fit, to data assimilation into operational models, to parameter estimation techniques. However, the treatment of error in many of these efforts is arguably selective and/or ad hoc. A Bayesian hierarchical framework enables the development of robust probabilistic analysis of error and uncertainty in model predictions by explicitly accommodating measurement error, parameter uncertainty, and model structure imperfection. This paper presents a Bayesian hierarchical formulation for simultaneously calibrating aquatic biogeochemical models at multiple systems (or sites of the same system) with differences in their trophic conditions, prior precisions of model parameters, available information, measurement error or inter-annual variability. Our statistical formulation also explicitly considers the uncertainty in model inputs (model parameters, initial conditions), the analytical/sampling error associated with the field data, and the discrepancy between model structure and the natural system dynamics (e.g., missing key ecological processes, erroneous formulations, misspecified forcing functions). The comparison between observations and posterior predictive monthly distributions indicates that the plankton models calibrated under the Bayesian hierarchical scheme provided accurate system representations for all the scenarios examined. Our results also suggest that the Bayesian hierarchical approach allows overcoming problems of insufficient local data by “borrowing strength” from well-studied sites and this feature will be highly relevant to conservation practices of regions with a high number of freshwater resources for which complete data could never be practically collected. Finally, we discuss the prospect of extending this framework to spatially explicit biogeochemical models (e.g., more effectively connect inshore with offshore areas) along with the benefits for environmental management, such as the optimization of the sampling design of monitoring programs and the alignment with the policy practice of adaptive management.     

Composite likelihood approach to the regression analysis of spatial multivariate ordinal data and spatial compositional data with exact zero values

In many environmental and ecological studies, it is of interest to model compositional data. One approach is to consider positive random vectors that are subject to a unit-sum constraint. In landscape ecological studies, it is common that compositional data are also sampled in space with some elements of the composition absent at certain sampling sites. In this paper, we first propose a practical spatial multivariate ordered probit model for multivariate ordinal data, where the response variables can be viewed as the discretized non-negative compositions without the unit-sum constraint. We then propose a novel two-stage spatial mixture Dirichlet regression model. The first stage models the spatial dependence and the presence of exact zero values, and the second stage models all the non-zero compositional data. A maximum composite likelihood approach is developed for parameter estimation and inference in both the spatial multivariate ordered probit model and the two-stage spatial mixture Dirichlet regression model. The standard errors of the parameter estimates are computed by an estimate of the Godambe information matrix. A simulation study is conducted to evaluate the performance of the proposed models and methods. A land cover data example in landscape ecology further illustrates that accounting for spatial dependence can improve the accuracy in the prediction of presence/absence of different land covers as well as the magnitude of land cover compositions.     

Model averaging,missing data and multiple imputation: a case study for behavioural ecology

Model averaging, specifically information theoretic approaches based on Akaike’s information criterion (IT-AIC approaches), has had a major influence on statistical practices in the field of ecology and evolution. However, a neglected issue is that in common with most other model fitting approaches, IT-AIC methods are sensitive to the presence of missing observations. The commonest way of handling missing data is the complete-case analysis (the complete deletion from the dataset of cases containing any missing values). It is well-known that this results in reduced estimation precision (or reduced statistical power), biased parameter estimates; however, the implications for model selection have not been explored. Here we employ an example from behavioural ecology to illustrate how missing data can affect the conclusions drawn from model selection or based on hypothesis testing. We show how missing observations can be recovered to give accurate estimates for IT-related indices (e.g. AIC and Akaike weight) as well as parameters (and their standard errors) by utilizing ‘multiple imputation’. We use this paper to illustrate key concepts from missing data theory and as a basis for discussing available methods for handling missing data. The example is intended to serve as a practically oriented case study for behavioural ecologists deciding on how to handle missing data in their own datasets and also as a first attempt to consider the problems of conducting model selection and averaging in the presence of missing observations.     

Identification of pollution sources in rivers using a hydrodynamic diffusion wave model and improved Bayesian-Markov chain Monte Carlo algorithm

● A hydrodynamic-Bayesian inference model was developed for water pollution tracking. ● Model is not stuck in local optimal solutions for high-dimensional problem. ● Model can estimate source parameters accurately with known river water levels. ● Both sudden spill incident and normal sewage inputs into the river can be tracked. ● Model is superior to the traditional approaches based on the test cases. Water quality restoration in rivers requires identification of the locations and discharges of pollution sources, and a reliable mathematical model to accomplish this identification is essential. In this paper, an innovative framework is presented to inversely estimate pollution sources for both accident preparedness and normal management of the allowable pollutant discharge. The proposed model integrates the concepts of the hydrodynamic diffusion wave equation and an improved Bayesian-Markov chain Monte Carlo method (MCMC). The methodological framework is tested using a designed case of a sudden wastewater spill incident (i.e., source location, flow rate, and starting and ending times of the discharge) and a real case of multiple sewage inputs into a river (i.e., locations and daily flows of sewage sources). The proposed modeling based on the improved Bayesian-MCMC method can effectively solve high-dimensional search and optimization problems according to known river water levels at pre-set monitoring sites. It can adequately provide accurate source estimation parameters using only one simulation through exploration of the full parameter space. In comparison, the inverse models based on the popular random walk Metropolis (RWM) algorithm and microbial genetic algorithm (MGA) do not produce reliable estimates for the two scenarios even after multiple simulation runs, and they fall into locally optimal solutions. Since much more water level data are available than water quality data, the proposed approach also provides a cost-effective solution for identifying pollution sources in rivers with the support of high-frequency water level data, especially for rivers receiving significant sewage discharges.