1. BIOE 293 Quantitative ecology seminar Marm Kilpatrick Steve Munch Spring Quarter 2015

2. Seminar Goal For 5-10 quantitative methods: To understand how the approach works by reading a “methods” paper or book chapter on it. This includes the assumptions, strengths, weaknesses, and limitations. Read papers more critically. Assess whether an approach is potentially useful for your own work Try analyzing data using the approach and discuss challenges. Sadly, you likely won’t be an expert in any of the topics at the end, but you’ll have a start at becoming one

3. Potential Topics & Voting scheme! 0. Generic intro stuff Statistical approaches Maximum Entropy I. Linear models and apps Generalized linear models Path analysis/SEM Correlated data Phylogenetic methods II. Linear, multivariate PCA, MCA, CCA III. Multi-layer models Hierarchical models Multi-state mark recapture IV. Nonlinear from linear GAMs Wavelets V. Potpourri Meta-analyses Isotope mixing models Ecological Niche models Kriging Machine learning Occupancy modeling

4. Statistical approaches Frequentist, AIC, Bayesian – what questions are they answering, what advantages/disadvantages do each them have? Frequentist: P-values AIC: Best fitting model(s) Bayesian: Descriptions of posterior distributions Maximum Entropy (a different way of thinking about the same stuff, with different pros/cons) p(x) is probability density; m(x) is “background probability distribution”

5. Most statistical methods start with a model for the probability of data (x) given parameters (q) P(x|q) a.k.a. the ‘Likelihood’ It’s what happens next that gets people so worked up:

6. FrequentistBayesian Information theoretic Maximum Entropy -Think of q as fixed, but unknown -Find parameters that maximize P(x|q). -Derive bounds on these estimates that should provide good coverage in repeated sampling. -Hypothesis tests compare fit against some null distribution. -Model selection based on goodness of fit. -Frequently only asymptotically correct -Treat all unknowns as ‘random’ -Use Bayes rule to find P(q|x). -Intervals based directly on P(q|x). -Model selection and Hypothesis tests usually based on P(model|data). PPL also used. -Need to specify P(q) and P(model). -Choose amongst set of candidate models based on some ‘Information criterion.’ -All various attempts to choose model that comes closest to ‘truth’ -Derive probability model that contains smallest amount of ‘extra’ information. -Introduced by ET Jaynes as a way to specify minimally informative priors, later expanded into its own inferential tool. -Current applications in ecology range from purely statistical (e.g. MAxEnt for SDM) to purely theoretical (Harte’s applications to size, area, density distributions) Bayes’ rule P(q|x)=P(x|q)P(q)/P(x) Data: (x) parameters: (q)

7. Linear Models and applications Generalized linear models and data transformations: distributions, links, leverage and more Correlated data – GLS for time series, spatial data

8. Phylogenetic methods (for analyses where species are data points) Felsenstein 1985 Am Nat

9. Linear Models and applications Path analysis/Structural equation modeling Hypotheses The data Wootton 1994 Ecology

10. Multivariate correlational approaches Principal components analysis (PCA), MCA (PCA for categorical data), CCA (for exploring correlations between 2 sets of predictors (matrices)) What people often do after they’ve collected lots of data but don’t know what to do with it

11. III. Multi-layer models (usually linear, but not necessarily) Hierarchical models (Mixed effects models, nested models, random effects models) For analyzing data that is influenced by variables that differ at more than one “level” Multi-state mark recapture models Survival analyses Allow for temporary emigration (temporary movement to unvisited locations) Allow for variable states/traits of individuals to influence survival

12. Hierarchical models Finite mixture models Mixed effects models Hidden Markov models State-space models P(x|q) P(q|r) P(r) Likelihood Prior Hyperprior Introduce ‘hidden’ or ‘latent’ variable to account for heterogeneity among individuals Capture nonstandard distributional shapes Treat some estimated effects (i.e. parameters) as ‘random’ (i.e. variable) Separate observation and process models Allow for imperfect observations of dynamical systems

13. Ecological Niche Models Occurrence data Environmental variables Probability of occurrence (Because everyone loves maps)

14. IV. Nonlinear models (out of linear ones) Generalized additive models (GAMs) Where each f is represented as a ‘basis expansion’ h j (x) are fixed ‘basis functions’ and a j are coefficients to be estimated. Has same structure as a linear model

15. Wavelets

16. Potpourri (Other topics)

17. Meta-analyses Assessing bias, modeling heterogeneity A method for combining results from multiple studies Salkeld et al 2013 Ecol Lett

18. Isotope mixing models Estimate the proportions of different food items in your diet

19. Kriging

20. Machine learning approaches – regression trees, random forests Regression tree: split data into successive groups Random forests: Lots of regression trees to minimize overfitting De’ath&Fabricius 2000 Ecology

21. Occupancy modeling Measuring the occupancy and distribution of an organism when accounting for imperfect detection With additional assumptions, can be used to estimate abundance Uses repeated visitation of locations and presence/absence of species of interest