Performance of a Bayesian state-space model of semelparous species for stock-recruitment data subject to measurement error

Measurement errors in spawner abundance create problems for fish stock assessment scientists. To deal with measurement error, we develop a Bayesian state-space model for stock-recruitment data that contain measurement error in spawner abundance, process error in recruitment, and time series bias. Through extensive simulations across numerous scenarios, we compare the statistical performance of the Bayesian state-space model with that of standard regression for a traditional stock-recruitment model that only considers process error. Performance varies depending on the information content in data, as determined by stock productivity, types of harvest situations, and amount of measurement error. Overall, in terms of estimating optimal spawner abundance S_MSY, the Ricker density-dependence parameter β, and optimal harvest rate h_MSY, the Bayesian state-space model works best for informative data from low and variable harvest rate situations for high-productivity salmon stocks. The traditional stock-recruitment model (TSR) may be used for estimating α and h_MSY for low-productivity stocks from variable and high harvest rate situations. However, TSR can severely overestimate S_MSY when spawner abundance is measured with large error in low and variable harvest rate situations. We also found that there is substantial merit in using h_MSY (or benchmarks derived from it) instead of S_MSY as a management target.     

Models to compare management options for a protogynous fish.

Populations of gag (Mycteroperca microlepis), a hermaphroditic grouper, have experienced a dramatic shift in sex ratio over the past 25 years due to a decline in older age classes. The highly female-skewed sex ratio can be predicted as a consequence of increased fishing mortality that truncates the age distribution, and raises some concern about the overall fitness of the population. Management efforts may need to be directed toward maintenance of sex ratio as well as stock size, with evaluations of recruitment based on sex ratio or male stock size in addition to the traditional female-based stock-recruitment relationship. We used two stochastic, age-structured models to heuristically compare the effects of reducing fishing mortality on different life history stages and the relative impact of reductions in fertilization rates that may occur with highly skewed sex ratios. Our response variables included population size, sex ratio, lost egg fertility, and female spawning stock biomass. Population growth rates were highest for scenarios that reduced mortality for female gag (nearshore closure), while improved sex ratios were obtained most quickly with spawning reserves. The effect of reduced fertility through sex ratio bias was generally low but depended on the management scenario employed. Our results demonstrate the utility of evaluation of fishery management scenarios through model analysis and simulation, the synergistic interaction of life history and response to changes in mortality rates, and the importance of defining management goals.     

A new multimodal and asymmetric bivariate circular distribution

Multimodal and asymmetric bivariate circular data arise in several different disciplines and fitting appropriate distribution plays an important role in the analysis of such data. In this paper, we propose a new bivariate circular distribution which can be used to model both asymmetric and multimodal bivariate circular data simultaneously. In fact the proposed density covers unimodality as well as multimodality, symmetry as well as asymmetry of circular bivariate data. A number of properties of the proposed density are presented. A Bayesian approach with MCMC scheme is employed for statistical inference. Three real datasets and a simulation study are provided to illustrate the performance of the proposed model in comparison with alternative models such as finite mixture Cosine model.     

Interactions between cod, herring and sprat in the changing environment of the Baltic Sea: A dynamic model analysis

The interactions between cod (Gadus morhua), herring (Clupea harengus) and sprat (Sprattus sprattus) in the Central Baltic Sea were examined with a simple dynamic model, an alternative to more complicated and data-demanding multispecies and ecosystem models. The main aims of the study were to compare the effect of alternative structures on the model output and examine the control relationships in the fish assemblage under different environmental conditions. The effect of environmental conditions was modelled using a stock-recruitment equation for cod incorporating an environmental index. The model output was especially sensitive to the functional response in predation by cod on herring and sprat. The type II functional response led to a collapse of the clupeid stocks when cod was abundant, while the type III response produced more realistic stock dynamics. According to the simulations, an abundant cod stock was able to keep the sprat stock at a low level, while the herring stock was less affected and benefited from the decreased density of sprat. Simulation of different fishing scenarios indicated that reducing fishing mortality to the level currently advised by ICES would allow the recovery of the cod stock even in unfavourable environmental conditions.     

Poor-data and data-poor species stock assessment using a Bayesian hierarchical approach

Appropriate inference for stocks or species with low-quality data (poor data) or limited data (data poor) is extremely important. Hierarchical Bayesian methods are especially applicable to small-area, small-sample-size estimation problems because they allow poor-data species to borrow strength from species with good-quality data. We used a hammerhead shark complex as an example to investigate the advantages of using hierarchical Bayesian models in assessing the status of poor-data and data-poor exploited species. The hammerhead shark complex (Sphyrna spp.) along the Atlantic and Gulf of Mexico coasts of the United States is composed of three species: the scalloped hammerhead (S. lewini), the great hammerhead (S. mokarran), and the smooth hammerhead (S. zygaena) sharks. The scalloped hammerhead comprises 70-80% of the catch and has catch and relative abundance data of good quality, whereas great and smooth hammerheads have relative abundance indices that are both limited and of low quality presumably because of low stock density and limited sampling. Four hierarchical Bayesian state-space surplus production models were developed to simulate variability in population growth rates, carrying capacity, and catchability of the species. The results from the hierarchical Bayesian models were considerably more robust than those of the nonhierarchical models. The hierarchical Bayesian approach represents an intermediate strategy between traditional models that assume different population parameters for each species and those that assume all species share identical parameters. Use of the hierarchical Bayesian approach is suggested for future hammerhead shark stock assessments and for modeling fish complexes with species-specific data, because the poor-data species can borrow strength from the species with good data, making the estimation more stable and robust.     

A universal approach to estimate biomass and carbon stock in tropical forests using generic allometric models

Allometric equations allow aboveground tree biomass and carbon stock to be estimated from tree size. The allometric scaling theory suggests the existence of a universal power-law relationship between tree biomass and tree diameter with a fixed scaling exponent close to 8/3. In addition, generic empirical models, like Chave's or Brown's models, have been proposed for tropical forests in America and Asia. These generic models have been used to estimate forest biomass and carbon worldwide. However, tree allometry depends on environmental and genetic factors that vary from region to region. Consequently, theoretical models that include too few ecological explicative variables or empirical generic models that have been calibrated at particular sites are unlikely to yield accurate tree biomass estimates at other sites. In this study, we based our analysis on a destructive sample of 481 trees in Madagascar spiny dry and moist forests characterized by a high rate of endemism (> 95%). We show that, among the available generic allometric models, Chave's model including diameter, height, and wood specific gravity as explicative variables for a particular forest type (dry, moist, or wet tropical forest) was the only one that gave accurate tree biomass estimates for Madagascar (R2 > 83%, bias < 6%), with estimates comparable to those obtained with regional allometric models. When biomass allometric models are not available for a given forest site, this result shows that a simple height-diameter allometry is needed to accurately estimate biomass and carbon stock from plot inventories.     

Regression models for exceedance data via the full likelihood

Many situations in practice require appropriate specification of operating characteristics under extreme conditions. Typical examples include environmental sciences where studies include extreme temperature, rainfall and river flow to name a few. In these cases, the effect of geographic and climatological inputs are likely to play a relevant role. This paper is concerned with the study of extreme data in the presence of relevant auxiliary information. The underlying model involves a mixture distribution: a generalized Pareto distribution is assumed for the exceedances beyond a high threshold and a non-parametric approach is assumed for the data below the threshold. Thus, the full likelihood including data below and above the threshold is considered in the estimation. The main novelty is the introduction of a regression structure to explain the variation of the exceedances through all tail parameters. Estimation is performed under the Bayesian paradigm and includes model choice. This allows for determination of higher quantiles under each covariate configuration and upper bounds for the data, where appropriate. Simulation results show that the models are appropriate and identifiable. The models are applied to the study of two temperature datasets: maxima in the U.S.A. and minima in Brazil, and compared to other related models.     

Lifting a veil on diversity: a Bayesian approach to fitting relative-abundance models.

Bayesian methods incorporate prior knowledge into a statistical analysis. This prior knowledge is usually restricted to assumptions regarding the form of probability distributions of the parameters of interest, leaving their values to be determined mainly through the data. Here we show how a Bayesian approach can be applied to the problem of drawing inference regarding species abundance distributions and comparing diversity indices between sites. The classic log series and the lognormal models of relative- abundance distribution are apparently quite different in form. The first is a sampling distribution while the other is a model of abundance of the underlying population. Bayesian methods help unite these two models in a common framework. Markov chain Monte Carlo simulation can be used to fit both distributions as small hierarchical models with shared common assumptions. Sampling error can be assumed to follow a Poisson distribution. Species not found in a sample, but suspected to be present in the region or community of interest, can be given zero abundance. This not only simplifies the process of model fitting, but also provides a convenient way of calculating confidence intervals for diversity indices. The method is especially useful when a comparison of species diversity between sites with different sample sizes is the key motivation behind the research. We illustrate the potential of the approach using data on fruit-feeding butterflies in southern Mexico. We conclude that, once all assumptions have been made transparent, a single data set may provide support for the belief that diversity is negatively affected by anthropogenic forest disturbance. Bayesian methods help to apply theory regarding the distribution of abundance in ecological communities to applied conservation.     

Impact of a once-through cooling system on the yellow perch stock in the western basin of lake erie

The surplus production model, a conventional fishery stock assessment model, is applied to assess the entrainment and impingement impact of the Monroe Power Plant on the yellow perch standing stock and fishery in the western basni of Lake Erie. Biological parameters of the model are estimated from commercial catch and effort data and entrainment and impingement coefficients are estimated from power plant data. The model is applied to estimate stock biomass, egg production, and larva production; the proportions entrained and impinged are then estimated. The impact of water withdrawal on the equilibrium standing stock and maximum sustainable yield from the fishery is estimated and the impact of increased water withdrawal on the equilibrium standing maximum sustainable yield are larger than the proportion of the standing stock entrained and impinged, but the impact of the Monroe Power Plant is relatively small; it decreases biomass and the maximum sustainable yield of the yellow perch stock by only a few percent. However, there are several power plants impacting the yellow perch stock of the western basin of Lake Erie and the combined impact should be examined.     

Impact of time series data on calibration and prediction uncertainty for a deterministic hydrogeochemical model

Model calibration is fundamental in applications of deterministic process-based models. Uncertainty in model predictions depends much on the input data and observations available for model calibration. Here we explored how model predictions (forecasts) and their uncertainties vary with the length of time series data used in calibration. As an example we used the hydrogeochemical model MAGIC and data from Birkenes, a small catchment in southern Norway, to simulate future water chemistry under a scenario of reduced acid deposition. A Bayesian approach with a Markov Chain Monte Carlo (MCMC) technique was used to calibrate the model to different lengths of observed data (4–29 years) and to estimate the prediction uncertainty each calibration. The results show that the difference between modelled and observed water chemistry (calibration goodness of fit) in general decreases with increasing length of the time series used in calibration. However, there are considerable differences for different time series of the same length. The results also show that the uncertainties in predicted future acid neutralizing capacity were lowest (i.e. the distribution peak narrowest) when using the longest time series for calibration. As for calibration success, there were considerable differences between the future distributions (prediction uncertainty) for the different calibrations.     

A Bayesian approach to evaluating site impairment

In the United States, each state is required to list water resources that are declared to be impaired under guidelines set by the Clean Water Act. Measurements are typically collected on a number of chemical constituents and compared with a standard. If there are too many measurements exceeding the standard, then the site is declared impaired. The approach is non-statistical but similar to a Binomial test. The Binomial approach would convert the measurements to binary data then test if the proportion exceeding the standard is excessive. Both methods convert measurements to binary values hence exclude potentially important information in the data. We present a statistical approach using a Bayesian model that uses the raw data instead of the binary transformed data. The population distribution of a family of location-scale parameter models is studied under the model. Posterior distributions from the Bayesian analysis are used in the decision-making process and error probabilities for the Bayesian and the Binomial approaches are compared for a normal population.     

Population dynamics of splitnose rockfish (Sebastes diploproa) in the Northeast Pacific Ocean

We developed an age-structured population model of splitnose rockfish, Sebastes diploproa, in the Northeast Pacific Ocean. Splitnose rockfish is a bycatch species that co-occurs with several commercially important species that are currently declared overfished. Bycatch species are typically not the focus of stock assessment efforts because of their limited economic importance, but they may suffer the same population declines as species with which they co-occur. To examine the dynamics of splitnose rockfish for the first time, we analyzed data from three groundfish fisheries and four research surveys conducted in the Northeast Pacific Ocean. To develop a model, we used Stock Synthesis, a statistical framework for the construction of a population dynamics models utilizing both fishery-dependent and fishery-independent data. In the model, we reconstructed the total catch of the species back to 1900, estimated the dynamics of the stock spawning output and recruitment and evaluated biomass depletion relative to the stock's unfished state, as well as sources of uncertainty in model outputs. The results indicate that the splitnose rockfish is currently not overfished even though it has experienced several periods of abrupt decline in its biomass. Revisiting age data from earlier years, monitoring fishery discard, and investigating the spatial dynamics of splitnose rockfish is important to further improve the understanding of this species’ population dynamics, and decrease uncertainty in model results.     

Semiparametric space–time survival modeling of chronic wasting disease in deer

In this paper, we propose a semiparametric survival model to investigate the pattern of spatial and temporal variation in disease prevalence of chronic wasting disease (CWD) in wild deer in Wisconsin over the years 2002 and 2006. The semiparametric survival model we suggested allows to build a more flexible model than the parametric model with fewer parametric assumptions by modeling the baseline hazard using a Gamma process prior. Based on the proposed model, we investigate the geographical distribution of CWD, and assess the effect of sex on disease prevalence. We use a Bayesian hierarchical framework where latent parameters capture temporal and spatial trends in disease incidence, incorporating sex and spatially correlated random effects. We also propose bivariate baseline hazard which change over age and time simultaneously to adopt different effects of age and time on the baseline hazard. Inference is carried out by using MCMC simulation techniques in a fully Bayesian framework. Our results suggest that disease has been spreaded mainly in the disease eradication zone and male deer show a significantly higher infection probability than female deer.     

Modelling spatial zero-inflated continuous data with an exponentially compound Poisson process

A parsimonious model is presented as an alternative to delta approaches to modelling zero-inflated continuous data. The data model relies on an exponentially compound Poisson process, also called the law of leaks (LOL). It represents the process of sampling resources that are spatially distributed as Poisson distributed patches, each containing a certain quantity of biomass drawn from an exponential distribution. In an application of the LOL, two latent structures are proposed to account for spatial dependencies between zero values at different scales within a hierarchical Bayesian framework. The LOL is compared to the delta-gamma (ΔΓ) distribution using bottom-trawl survey data. Results of this case study emphasize that the LOL provides slightly better fits to learning samples with a very high proportion of zero values and small strictly positive abundance data. Additionally, it offers better predictions of validation samples.     

A Bayesian approach to retransformation bias in transformed regression

Ecological data analysis often involves fitting linear or nonlinear equations to data after transforming either the response variable, the right side of the equation, or both, so that the standard suite of regression assumptions are more closely met. However, inference is usually done in the natural metric and it is well known that retransforming back to the original metric provides a biased estimator for the mean of the response variable. For the normal linear model, fit under a log-transformation, correction factors are available to reduce this bias, but these factors may not be generally applicable to all model forms or other transformations. We demonstrate that this problem is handled in a straightforward manner using a Bayesian approach, which is general for linear and nonlinear models and other transformations and model error structures. The Bayesian framework provides a predictive distribution for the response variable so that inference can be made at the mean, or over the entire distribution to incorporate the predictive uncertainty.     

Effect of variable fishing strategy on fisheries under changing effort and pressure: An agent-based model application

An agent-based model was used to evaluate the response of a two-species fish community to fishing boat exploration strategies, namely: boats following high-yield boats (Cartesian); boats fishing at random sites (stochast-random); and boats fishing at least exploited sites (stochast-pressure). At low fishing pressure, the stochast-random mode yielded a high average catch per boat while sustaining fish biomass. At high fishing pressure, the Cartesian mode was more effective. For the Cartesian strategy, fish biomass exhibited four distinct behaviors with increasing number of boats. In the first phase, the fish biomass dropped with increasing number of boats due to a corresponding rise in biomass extraction. Rapid exploitation occurred in the second phase, when two or more boats occupied the same initial area, that led to the faster abandonment of those sites which then underwent biomass recovery. In the third phase, adding more boats resulted in a fluctuating stock biomass, where the combined effects of initial spatial distribution of boats and rapid localization led to either full stock recovery when boats were eventually confined to a single location due to spillovers, or stock extirpation when the entire area became fully occupied. Beyond the third phase, stock extirpation was assured. In order to break the pattern of localization (bandwagon effect), we introduced stochast-random intruders in a Cartesian-dominated fishery. Adding a single intruder changed the patchy-structured stock biomass pattern of a purely Cartesian fishery to a uniformly explored stock biomass pattern because of the additional spatial information provided by the intruder. Consequently, the average catch per boat increased but at the expense of a disproportionate decline in equilibrium biomass.     

Analyzing longitudinal circular data by projected normal models: a semi-parametric approach based on finite mixture models

The analysis of circular data has been recently the focus of a wide range of literature, with the general objective of providing reliable parameter estimates in the presence of heterogeneity and/or dependence among observations under a longitudinal setting. In this paper, we extend the variance component model approach to the analysis of longitudinal circular data, defining a mixed effects model for radial projections onto the circle and introducing dependence between projections through a set of correlated random coefficients. Estimation is carried out by numerical integration through an expectation-maximization algorithm without parametric assumptions upon the random coefficients distribution. The resulting model is a finite mixture of projected normal distributions. A simulation study has been carried out to investigate the behavior of the proposed model in a series of empirical situations. The proposed model is computationally parsimonious and, when applied to a real dataset on animal orientation, produces novel results.     

Hierarchical Bayesian Spatial Models for Multispecies Conservation Planning and Monitoring

Abstract: Biologists who develop and apply habitat models are often familiar with the statistical challenges posed by their data's spatial structure but are unsure of whether the use of complex spatial models will increase the utility of model results in planning. We compared the relative performance of nonspatial and hierarchical Bayesian spatial models for three vertebrate and invertebrate taxa of conservation concern (Church's sideband snails [Monadenia churchi], red tree voles [Arborimus longicaudus], and Pacific fishers [Martes pennanti pacifica]) that provide examples of a range of distributional extents and dispersal abilities. We used presence–absence data derived from regional monitoring programs to develop models with both landscape and site‐level environmental covariates. We used Markov chain Monte Carlo algorithms and a conditional autoregressive or intrinsic conditional autoregressive model framework to fit spatial models. The fit of Bayesian spatial models was between 35 and 55% better than the fit of nonspatial analogue models. Bayesian spatial models outperformed analogous models developed with maximum entropy (Maxent) methods. Although the best spatial and nonspatial models included similar environmental variables, spatial models provided estimates of residual spatial effects that suggested how ecological processes might structure distribution patterns. Spatial models built from presence–absence data improved fit most for localized endemic species with ranges constrained by poorly known biogeographic factors and for widely distributed species suspected to be strongly affected by unmeasured environmental variables or population processes. By treating spatial effects as a variable of interest rather than a nuisance, hierarchical Bayesian spatial models, especially when they are based on a common broad‐scale spatial lattice (here the national Forest Inventory and Analysis grid of 24 km² hexagons), can increase the relevance of habitat models to multispecies conservation planning.     

Population dynamics and control of invasive Spartina alterniflora: inference and forecasting under uncertainty

Managing invaded ecosystems entails making decisions about control strategies in the face of scientific uncertainty and ecological stochasticity. Statistical tools such as model selection and Bayesian decision analysis can guide decision-making by estimating probabilities of outcomes under alternative management scenarios, but these tools have seldom been applied in invasion ecology. We illustrate the use of model selection and Bayesian methods in a case study of smooth cordgrass (Spartina alterniflora) invading Willapa Bay, Washington. To address uncertainty in model structure, we quantified the weight of evidence for two previously proposed hypotheses, that S. alterniflora recruitment varies with climatic conditions (represented by sea surface temperature) and that recruitment is subject to an Allee effect due to pollen limitation. By fitting models to time series data, we found strong support for climate effects, with higher per capita seedling production in warmer years, but no evidence for an Allee effect based on either the total area invaded or the mean distance between neighboring clones. We used the best-supported model to compare alternative control strategies, incorporating uncertainty in parameter estimates and population dynamics. For a fixed annual removal effort, the probability of eradication in 10 years was highest, and final invaded area lowest, if removals targeted the smallest clones rather than the largest or randomly selected clones. The relationship between removal effort and probability of eradication was highly nonlinear, with a sharp threshold separating -0% and -100% probability of success, and this threshold was 95% lower in simulations beginning early rather than late in the invasion. This advantage of a rapid response strategy is due to density-dependent population growth, which produces alternative stable equilibria depending on the initial invasion size when control begins. Our approach could be applied to a wide range of invasive species management problems where appropriate data are available.     

A fast Bayesian method for updating and forecasting hourly ozone levels

A Bayesian hierarchical space-time model is proposed by combining information from real-time ambient AIRNow air monitoring data, and output from a computer simulation model known as the Community Multi-scale Air Quality (Eta-CMAQ) forecast model. A model validation analysis shows that the model predicted maps are more accurate than the maps based solely on the Eta-CMAQ forecast data for a 2 week test period. These out-of sample spatial predictions and temporal forecasts also outperform those from regression models with independent Gaussian errors. The method is fully Bayesian and is able to instantly update the map for the current hour (upon receiving monitor data for the current hour) and forecast the map for several hours ahead. In particular, the 8 h average map which is the average of the past 4 h, current hour and 3 h ahead is instantly obtained at the current hour. Based on our validation, the exact Bayesian method is preferable to more complex models in a real-time updating and forecasting environment.