Statistical Ecology

Upcoming short course entitled “Introductory R” at The Wildlife Society annual meeting in Albuquerque, New Mexico

This workshop, aimed at new R users, provides an introduction to object manipulation, data visualization and analysis, and basic programming in R. The R computing environment is a free, flexible, and open-source tool for data management, visualization, and analysis. We will cover a broad spectrum of topics including: how to import and export data, data types and object classes, simple R functions for querying, summarizing, plotting, and analyzing data, exploring spatial data, basic programming (e.g., for loops, writing R functions), and how to find and use R help resources. No prior experience with R is necessary, but participants should be comfortable with introductory statistics. The workshop will alternate between interactive instruction and guided exercises where participants will implement their new skills. Participants should bring their own laptop and have recent versions of R and RStudio installed prior to the workshop.

Co-taught with Drs. Frances Buderman and Brittany Mosher

Upcoming short course entitled “Spatio-temporal dynamic statistical modeling in practice”

Short course on spatio-temporal dynamic statistical modeling at the Western North American Region of the International Biometric Society. The course is scheduled for June 25th, in Santa Fe, New Mexico from 1-5 pm. The course will provide an overview  spatio-temporal statistical modeling,
efficient computation and implementation of ecological diffusion models, and embedding S-T dynamic models into integrated population models for ecological data.

Co-taught  with Drs. Mevin Hooten and Trevor Hefley.

New manuscript accepted in Methods in Ecology and Evolution

Williams, P. J., M. B. Hooten, J. N. Womble, and M. R. Bower. In Press. Estimating occupancy and abundance using aerial images with imperfect detection. Methods in Ecology and Evolution 00:00–00.


Species distribution and abundance are critical population characteristics for efficient management, conservation, and ecological insight. Point process models are a powerful tool for modeling distribution and abundance, and can incorporate many data types, including count data, presence-absence data, and presence-only data. Aerial photographic images are a natural tool for collecting data to fit point process models, but aerial images do not always capture all animals that are present at a site. Methods for estimating detection probability for aerial surveys usually include collecting auxiliary data to estimate the proportion of time animals are available to be detected.

We developed an approach for fitting point process models using an N-mixture model framework to estimate detection probability for aerial occupancy and abundance surveys. Our method uses multiple aerial images taken of animals at the same spatial location to provide temporal replication of sample sites. The intersection of the images provide multiple counts of individuals at different times. We examined this approach using both simulated and real data of sea otters (Enhydra lutris kenyoni) in Glacier Bay National Park, southeastern Alaska.

Using our proposed methods, we estimated detection probability of sea otters to be 0.80, which was an improvement compared to visual aerial surveys that have been used in the past (p=0.76). Further, simulations demonstrated that our approach is a promising tool for estimating occupancy, abundance, and detection probability from aerial photographic surveys.

Our methods can be readily extended to data collected using unmanned aerial vehicles, as technology and regulations permit. The generality of our methods for other aerial surveys depends on how well surveys can be designed to meet the assumptions of N-mixture models.

New manuscript accepted in Ecology on using basis functions for modeling spatial and temporal autocorrelation

Hefley, T.J., K.M. Broms, B.M. Brost, F.E. Buderman, S. Kay, H.R. Scharf, J.R. Tipton, P.J. Williams, and M.B. Hooten. (In Press). The basis function approach to modeling autocorrelation in ecological data.  Ecology.

Abstract: Analyzing ecological data often requires modeling the autocorrelation created by spatial  and temporal processes. Many seemingly disparate statistical methods used to account for  autocorrelation can be expressed as regression models that include basis functions. Basis functions also enable ecologists to modify a wide range of existing ecological models to account for autocorrelation, which can improve inference and predictive accuracy. Furthermore, understanding the properties of basis functions is essential for evaluating the fit of spatial or time-series models, detecting a hidden form of collinearity, and analyzing large data sets. We present important concepts and properties related to basis functions and illustrate several tools and techniques ecologists can use when modeling autocorrelation in ecological data


New manuscript accepted in Ecological Modelling on methods for combining multiple objectives

Williams, P. J, and W. L. Kendall. 2017. A guide to multiobjective optimization for ecological problems. Ecological Modelling 343:54-67. 

Abstract: Choices in ecological research and management are the result of balancing multiple, often competing, objectives. Multi-objective optimization (MOO) is a formal decision-theoretic framework for solving multiple objective problems. MOO is used extensively in other fields including engineering, economics, and operations research. However, its application for solving ecological problems has been sparse, perhaps due to a lack of widespread understanding. Thus, our objective was to provide an accessible primer on MOO, including a review of methods common in other fields, a review of their application in ecology, and a demonstration to an applied resource management problem.

A large class of methods for solving MOO problems can be separated into two strategies: modelling preferences pre-optimization (the a priori strategy), or modelling preferences post-optimization (the a posteriori strategy). The a priori strategy requires describing preferences among objectives without knowledge of how preferences affect the resulting decision. In the a posteriori strategy, the decision maker simultaneously considers a set of solutions (the Pareto optimal set) and makes a choice based on the trade-offs observed in the set. We describe several methods for modelling preferences pre-optimization, including: the bounded objective function method, the lexicographic method, and the weighted-sum method. We discuss modelling preferences post-optimization through examination of the Pareto optimal set. We applied each MOO strategy to the natural resource management problem of selecting a population target for cackling goose (Branta hutchinsii minima) abundance. Cackling geese provide food security to Native Alaskan subsistence hunters in the goose’s nesting area, but depredate crops on private agricultural fields in wintering areas. We developed objective functions to represent the competing objectives related to the cackling goose population target and identified an optimal solution first using the a priori strategy, and then by examining trade-offs in the Pareto set using the a posteriori strategy. We used four approaches for selecting a final solution within the a posteriori strategy; the most common optimal solution, the most robust optimal solution, and two solutions based on maximizing a restricted portion of the Pareto set. We discuss MOO with respect to natural resource management, but MOO is sufficiently general to cover any ecological problem that contains multiple competing objectives that can be quantified using objective functions.

Below is a figure from the manuscript showing the optimality frontier related to three objectives: 1) maximizing subsistence harvest of cackling geese, 2) minimizing agricultural depredation from cackling geese, and 3) balancing competing stakeholder objectives.




New manuscript accepted in Ecology on spatio-temporal methods for estimating colonization dynamics

Williams, P. J., M. B. Hooten, J. N. Womble, G. G. Esslinger, M. R. Bower, and T. J. Hefley. In Press. An integrated data model to estimate spatio-temporal occupancy, abundance, and colonization dynamics. Ecology.

Abstract: Ecological invasions and colonizations occur dynamically through space and time. Estimating the distribution and abundance of colonizing species is critical for efficient management or conservation. We describe a statistical framework for simultaneously estimating spatio-temporal occupancy and abundance dynamics of a colonizing species. Our method accounts for several issues that are common when modeling spatio-temporal ecological data including: multiple levels of detection probability, multiple data sources, and computational limitations that occur when making fine-scale inference over a large spatio-temporal domain. We apply the model to estimate the colonization dynamics of sea otters (Enhydra lutris) in Glacier Bay, in southeastern Alaska.

Statistical decision theory manuscript in print at Ecological Applications

Abstract: Statistical decision theory (SDT) is a sub-field of decision theory that formallincorporates statistical investigation into a decision-theoretic framework to account for uncertainties in a decision problem. SDT provides a unifying analysis of three types of information: statistical results from a data set, knowledge of the consequences of potential choices (i.e., loss), and prior beliefs about a system. SDT links the theoretical development of a large body of statistical methods, including point estimation, hypothesis testing, and confidence interval estimation. The theory and application of SDT have mainly been developed and published in the fields of mathematics, statistics, operations research, and other decision sciences, but have had limited exposure in ecology. Thus, we provide an introduction to SDT for ecologists and describe its utility for linking the conventionallseparate tasks of statistical investigation and decision making in a single framework. We describe the basic framework of both Bayesian and frequentist SDT, its traditional use in statistics, and discuss its application to decision problems that occur in ecologyWe demonstrate SDT with two types of decisions: Bayesian point estimation and an applied management problem of selecting a prescribed fire rotation for managing a grassland birspecies. Central to SDT, and decision theory in general, are loss functions. Thus, we also provide basic guidance and references for constructing loss functions for an SDT problem.