This talk will discuss data consisting of the times of road culvert passage attempts by fish on a spawning run. Interest lies in quantifying the effects of fish, culvert, and stream characteristics on the probability of passing through culverts. I propose a preliminary Bayesian hierarchical model for the data with the intent to foster discussion in the spirit of the workshop.
We are developing a landscape simulation model of the Everglades landscape that couples vegetation dynamics with fire, hydrology, and climate. A large number of parameters must be estimated, some of which are obtainable from pre-existing empirical data. Other parameters must be fitted dynamically from simulations. This presents two challenges: First, the simulation model generates data on many aspects of the ecosystem and parameter estimates are likely to be sensitive to the choice of metric(s) to measure model fit. Second, the link between model parameters and the state of the ecosystem is stochastic, but this stochasticity is generated internally in the simulation model. We do not know the "correct" underlying stochastic model and we can only generate distributions of results from summaries provided by multiple individual realizations. This complicates the process of parameter estimation. We describe our model, illustrate these challenges, and discuss potential solutions.
If one wore to condense the last decades of environmental research into two observations, the following would represent a candidate: (1) humans are changing both terrestrial and marine biota at an ever increasing pace, and (2) ecological networks are sufficiently complicated that the current generation of detailed mechanistic models are as likely to fail as succeed in predicting the consequences of climate or habitat change. In light of this, ecologists rightly and prudently lean on more correlational trend analyses for early warning signs of ecosystem change and degradation. The precautionary line of reasoning is that ecological interactions such as competition or predation are likely to result in variation at the 'few-generation' time-scale. A trend that represent variation (or changes) at the 'many-generation' time-scale could thus be interpreted as systems transition likely reflecting human alterations of habitat or climate. We will discuss how this may be true for deterministic models. However, stochastic age-structured models predicts apparent trends in abundance that are superimposed on the 'deterministic variability' (high-frequency variability) through the cohort resonance effect. We detail the anatomy of these internally generated trends through transfer functions, and thus revisit the rule-of-thumb that long-term changes in abundance can be used as indicators of anthropogenic (or other external) forcing.
REF: Bjornstad, O. N., R. M. Nisbet, and J.-M. Fromentin. 2004. Trends and cohort resonant effects in age-structured populations. Journal of Animal Ecology 73:: 1157-1167.
Soil organic matter is an important sink for carbon, and the size of this sink is impacted by agricultural management such as organic amendments and tillage practices. Appropriate management can increase carbon sequestration in soils and mitigate greenhouse emissions of carbon dioxide to the atmosphere. Accounting for the amount of carbon sequestration is difficult due to the long temporal scales and fine spatial scales of interest and the complexity of the dynamics of carbon as it cycles between the atmosphere and biosphere, including soil organic matter pools. CENTURY, a biogeophysical process model, is used to model carbon dynamics in a national-level inventory of soil organic carbon stock changes. Inputs to CENTURY include weather, soils data, cropping history, tillage practices, fertilizer usage, and organic amendments, all of which are available to some degree from different national databases at different spatial scales. CENTURY is used to simulate soil organic matter dynamics at points used by the National Resources Inventory (NRI), a nationally-representative two-stage area sample, for which a standard design-based variance estimator provides a consistent estimate of uncertainty. At these NRI points, detailed soils data and cropping history are available, but tillage, mineral fertilizer use and organic amendments are not. Monte Carlo methods are used in conjunction with the design-based methods to provide accounting for both the sampling uncertainty of NRI and the uncertainty of inputs not available from NRI. Finally, external validation data of actual soil organic carbon measurements are used to account for model uncertainty related to imperfections in the structural relationships represented in CENTURY.
This is joint work with Stephen M. Ogle and Keith Paustian, Natural Resources Ecology Laboratory, Colorado State University.
Effective management of wild animal populations requires reliable mathematical models, so that the effects of management action can be predicted, and the uncertainty in these predictions quantified. These models must be able to predict the response of populations to change, while handling the major sources of uncertainty. We describe an approach for formulating and fitting complex discrete-time models. We show how Bayesian methods (sequential Monte Carlo and MCMC) can account for observation error, model uncertainty and process variation (demographic and environmental stochasticity).
This work was done in collaboration with K.B. Newman, C. Fernandez, L. Thomas and J. Harwood.
Ecological processes evolve dynamically and spatially. Uncertainties abound as science tries to describe, and then explain, these processes. Statistical modeling and analysis offers a framework within which scientists can work, but the spatio-temporal setting offers special challenges. This talk will address some of these and pay particular attention to one of the most fascinating: change of support.
Various sampling protocols (e.g., repeated point counts, mark-recapture, or multi-observer sampling) are used to estimate the abundance of a demographically-closed population of individuals (animals or species) that cannot be captured or detected with absolute certainty. In some populations heterogeneity in local abundance or detectability is thought to exist but the sources of heterogeneity are either poorly understood or unobservable. These latent sources of variation are often modeled using simple distributional assumptions, and the quantities of scientific interest are computed based on estimates of the assumed distribution's parameters. This approach, though often satisfactory, is vulnerable to errors in model specification that may be difficult or impossible to assess. An alternative approach is to assume a Dirichlet Process prior on the distribution of latent heterogeneity, thereby allowing for model uncertainty and robust estimation of abundance and detection. The benefits of this approach are illustrated in an analysis of removal counts observed while sampling an endangered population of fishes.
Complex deterministic models are often used to shape public policy regarding outbreaks of highly infectious diseases. These models often forecast disease epidemics and subsequent policy responses based on a single point estimate of the disease reproductive rate (i.e., the number of newly infected individuals arising from a single infected individual). By combining a Markov chain Monte Carlo (MCMC) simulation with a simple set of differential equations, the SEIR model, we were able to estimate the distribution for the reproductive rate of spread of smallpox. Given this distribution, a more informed set of decisions can be reached with regards to the public policy of smallpox inoculation. In general, this method can be widely applied and used to forecast disease outbreaks for other epidemics besides those related to human health.
Community ecologists, ecosystem scientists, and restoration ecologists all recognize that ecological systems are rarely in equilibrium, but they may exist in more than one state for long periods of time. Community ecologists recognize "alternative community states", and ecosystem ecologists recognize "regime shifts"; restoration ecologists have adopted these terms in attempts to manage the long-term dynamics of communities and ecosystems. Statistical discrimination of alternative community states or regime shifts has proven difficult. Community ecologists use ANOVA and its relatives to determine if different community configurations have significantly different responses (state variables) for a single parameter set. Ecosystem ecologists use time-series analysis to determine if parameters are varying through time, and how these parameter shifts can result in new ecosystem dynamics, given a set of state variables. Common statistical methods are needed to bring unity to these different perspectives; restoration ecologists could use these methods to develop statistical benchmarks that can be used to determine if and when a new community state has been reached. Three datasets will be presented for detailed analysis by workshop participants: an experimental study of alternative community states in the Gulf of Maine; a simulation study of phosphorus dynamics in lakes; and a rehabilitation scheme of abandoned mines in western Australia.
Our problem is to generate a simple but informative display of genetic relatedness across a community of islands, to help highlight geographical factors affecting genetic isolation such as ocean distance or cliffs. The quality of genetic data from each island location depends on the number of individuals sampled from that island: for example, a rare genetic allele might not be included in a small sample despite being present at that location. We discuss how a Bayesian resampling process may be used to alleviate the difficulties of small samples. This raises the new problem of how to display the information gathered from resampled genetic distances. Some ideas will be discussed.
Ecological processes (demography, behavioral, disturbance) create a wide range of spatial patterns at multiple scales. To quantify and characterize these spatial patterns several spatial statistics are available. Here I present some of the conceptual and statistical challenges that ecologists face while analyzing spatially heterogeneous dynamic landscapes while trying to disentangle the key scales at which ecological processes occur.
Statistical models to describe species distributions need to be able to accommodate features such as irregular sampling intensity, transformed landscape, and spatial dependence. We show how to build hierarchical models incorporating spatial random effects that capture these features, enable spatial prediction and adequately quantify uncertainty. We illustrate with a large dataset from the Cape Floristic Region in South Africa.
Dynamical systems have long been used to characterize invasions in terms of population growth. This talk provides an example of the statistical estimation and prediction of such spatio-temporal phenomena in the presence of data by utilizing a scientifically based dynamic process while accounting for imperfect knowledge of the process. This specific application involves the ongoing invasion of the Eurasion Collared-Dove.
The use of AIC for model selection, and averaging, has become nearly ubiquitous in ecological studies. The Bayesian alternative has been largely ignored for regular use. Bayesian model selection implemented via MCMC has some attractive properties that that make it a worthy competitor to AIC. For example, Bayesian MCMC methods allow a priori unequal weighting of predictors in a regression model. In some common cases, the method can also be extended to hierarchical models with relative ease. MCMC model selection and parameter inference is demonstrated on a spatial data set concerning fish abundance.
Models for the uptake or release of H2O and CO2 from terrestrial ecosystems are now needed for developing relationships between anthropogenic perturbations, atmospheric CO2 levels, and alterations to the water cycle. What makes modeling these exchange rates in forested ecosystems a complex task is that key processes relevant to water and carbon transfer and storage can vary over many space and time scales. In time, these fluxes are influenced by fast processes such as turbulent transport mechanics (often measured in seconds) and slow process such forest growth (often measured in years to decades). In space, photosynthesis and water uptake occurs at the leaf scale (often measured in millimeter) while stand level variables such as tree density are often measured in kilometers. To date, a nesting framework that couples fast and slow processes has not been developed for this problem though such a framework may be logical precisely because of the wide scale separation between variability at diurnal scales (important for radiative transfer) and monthly scales (important for carbon allocation and leaf area). In this talk, we report on developments of a multiscale approach that reproduces the spectral features of biosphere-atmosphere exchange of carbon, and water from minutes to multiple years. The testing is conducted for a maturing loblolly pine plantation in the Southern Piedmont region of North Carolina. We use pine plantations as a case study because, in addition to their economic importance, they represent a "simple" (in terms of dominant species) yet a non-equilibrium system that has been studied both extensively and intensively. Four-year eddy-covariance data set along with ecological measurements collected at the Duke Forest pine site is used for this purpose. Based on the annual precipitation measurements, the selected four years include a mild drought at the beginning of the growing season, a severe drought (5th largest on record), and two wet years.
Co-authors: Siqueira, M., Stoy, P., Juang, J., Palmroth, S., McCarthy, H., and Oren, R. from Nicholas School of the Environment and Earth Sciences, Duke University
West Nile virus (WNV) is an emergent disease that spread rapidly through-out the contiguous United States following introduction into New York City in 1999. Although the virus is known to have caused several severe epizootics and result in high mortality for some species in lab challenges, there is little known regarding its implications for wild birds. To date, annual censuses collected by "citizen scientists" are the source of avian data with the most extensive temporal and spatial coverage. It is unknown how these data, which include multiple sources of error, perform as a means for evaluating mortality associated with an emerging pathogen. We use hierarchical techniques to evaluate the utility of bird census data in evaluating population effects associated with the introduction and spread of West Nile virus.
Count surveys like the North American Breeding Bird Survey (BBS) and the Christmas Bird Count (CBC) have been much reviled on the grounds that they produce unreliable indices of population size, rather than population estimates. Counts, the argument has gone, can be expressed as C = N p, so that temporal or spatial change in population size N is confounded with changes in detection probability p; what was needed was a means for estimating detection rate p, and obtaining "adjusted counts" to be used as the basis of population inference. In this talk, I argue that what is really needed is model based control for factors influencing detetectability; this, whether one estimates detection rates or not. Estimation of population change is inevitably and inescapably model-based, with inference relying on untestable model assumptions. I illustrate the potential for model based analysis of count survey data using a combined analysis of CBC and BBS data to examine seasonal components of population change without estimating abundance. I illustrate the inevitability of untestable model assumptions using simple examples from closed population mark-recapture.
I will provide an idosyncratic history of uncertainty in ecology, using the classic experiments of Thomas Park on flour beetles (and their analysis by Neyman, Park and Scott, Leslie and Gower, Barnett, and Bartlett) as a motivational framework. This will suggest how we can make the greatest progress going forward and I will connect the challenges of the future with the presentations at the workshop.
Likelihood-based methods have been widely and successfully used for the estimation of wild animal abundance and survival. In this talk we review the many advantages of this classical approach, and also outline a number of recent developments.
We now have tools for choosing between models, for constructing confidence intervals, and for variable selection in appropriate regressions. We can evaluate methods through repeated simulations, and if necessary we can average predictions over models. In addition, we can combine likelihoods if we have relevant different samples of data, and a particular case of this arises when there are data that allow the estimation of demographic rates, as well as data providing appropriate census information.
We shall consider the possible advantages of reparameterisation, and the importance of testing for parameter redundancy, which arises when it is not possible to estimate all of the parameters in a model. An interesting application involves the use of covariates in survival, which may relate to climate and population density, or attributes of individuals, when the covariates may be constant or time-varying. We shall mention a new approach to modelling time-varying individual covariates, and we shall also describe the use of splines for modelling covariates in general.
This talk will be illustrated by a range of examples, involving red-deer (Cervus elaphus), grey herons (Ardea cinerea), dippers (Cinclus cinclus), snow petrels (Pagodroma nivea) and Soay sheep (Ovis aries).
Key Words: AIC; BIC; conditional inference; integrated population modelling; Kalman Filter; likelihood ratio tests; MARK; M-SURGE; multinomial distribution; numerical optimisation; P-splines; random effects; score tests; simulated annealing; trinomial methodology.
Belowground ecosystem processes (e.g., soil, root, microbial respiration; uptake of water by plant roots) are hidden from view and difficult to measure, thus ecologists know very little about belowground compared to aboveground processes. Understanding the belowground system and how it is coupled to aboveground components is essential to developing general theories of ecosystem dynamics, yet we lack sufficient quantitative methods for disentangling the belowground component. To address this problem, I will present a general framework for inferring belowground processes that combines above- and belowground data, semi-mechanistic models of key ecological processes, and hierarchical Bayesian statistical tools.
Population ecology is largely concerned with understanding spatial and temporal variation in abundance and occurrence of species. Consistent with this view, many monitoring programs and smaller-scale population studies adopt an explicit focus on estimating or modeling abundance and understanding factors that influence abundance. One important consideration in the conduct of inference about abundance is an acute inability to observe the state variable of interest in most animal sampling problems. That is, individuals in the sampled population may go undetected by sampling and so observations of putative abundance are intrinsically biased. Historically this issue of non-detection bias (or "detectability") has been viewed as being paramount in the conduct of inference about abundance and other demographic parameters, and it provides a conceptual unification of a large and diverse body of methodology dealing with animal sampling.
There are several prevailing views on modeling abundance in the presence of imperfect detection. The classical view adopts a strong focus on modeling the detection process, and subsequent adjustment of sample counts to obtain abundance, or a second stage of modeling in which parameters of the detection process are fixed. An equally prevalent view is that focused on developing complex models directly from sample counts, absent any explicit consideration of detectability. The conceptual middle ground is occupied by a number of related views that share a common methodological formulation as hierarchical models.
In this paper, I advocate this conceptual middle ground, arguing that many estimation and inference problems (and sampling designs) yield naturally to formulation as hierarchical models. These hierarchical models are comprised of component models describing (1) variation in the observations conditional on the latent state variable (spatially and temporally indexed abundance), and (2) variation in the latent state variable, usually expressing the ecological structure that is the focus of inference. A few brief examples of hierarchical models applied to avian survey data will be given.
Ecological processes often encompass a very extensive range of spatial and temporal scales of variability, and include complicated interactions across domains, variables, and systems. To understand and eventually predict such complicated processes, we must make use of available scientific knowledge, as well as honestly account for uncertainties in that knowledge. A general hierarchical framework is presented for spatio-temporal dynamical processes in which the parameterizations are motivated by classical deterministic models.
Ecosystem analysis is prone to the most serious types of hidden errors, particulary when exploring issues surounding shifts in ecosystem composition or climate change hypotheses, i.e. data and modeling purporting to characterize large spatial scales or long time intervals. Using examples drawn from actual data measured or encountered in my work, I will examine several classes of errors in this area of science, and give examples where the nature and magnitude of the errors could be determined.
Spatially distributed data: representation error, spatial correlation, transport error, ecosystem model error (fitting to a wrong model).
Time series data: serial correlation (tree rings vs. climate), overestimation of the # degrees of freedom, accepting a false hypothesis (trend data for ecosystem function debunked with new types of observations), experimental design error (especially sensor drift, e.g. soil T data, psychrometers).
I conclude that effective steps in reducing hidden errors involve (1) improving the experimental design to investigate sources of bias and artifacts, (2) critically selecting, rejecting, and re-selecting underlying models for data analysis, and (3) liberally experimenting with stochastic simulations (especially simple Markovian models) to assess confidence intervals and test null hypotheses.