Bill.Shipley@USherbrooke.ca

1. Communityassemblythrough trait selection(CATS): Modellingfromincomplete information Bill.Shipley@USherbrooke.ca

2. A seminar in three parts The ecological concept The maximum entropyformalism Edwin Jaynes The CATS model

3. Part 1 The ecological concept

4. bioticfilters (determined by traits) Trait-basedfiltering Regionalspeciespool: determined by traits + history Immigration rate: determined by abundance in region + traits abioticfilters (determined by traits) Local community Relative abundance in local community

5. communitymean trait value relative abundance~ probability of passing filters A B C D E species tA tB tC td tE traits of species A plant strategyis a suite of physiological, morphological or phenological traits of individualsthat affect probabilities of survival, reproduction or immigration and thatissystematically associatedwithparticularenvironmental conditions The mostcommon trait values in a local communitywillbepossessed by thoseindividualshaving the greatestprobabilites of survival, reproduction and immigration. Philip Grime This average trait value willreflect the selective advantage/disadvantage of this trait in passing through the variousabiotic and bioticfilters

6. Twoconsequences of Grime’sideas for communityassembly… If we know the values of particularabiotic variables determining the filters, thenweshouldbe able to predict the « typical » values of the functional traits found in this local community; i.e. community-weightedmeans. Philip Grime If we know the « typical » values of the functional traits that are found in this community, and we know the actualtrait values of the species in a regional species pool, thenweshouldbe able to predictwhich of thesespecieswillbe dominants, whichwillbesubordinates, and whichwillbe rare or absent.

7. The three basic parts of the CATS model

8. metacommunity metacommunity q : distribution of relative abundance of species in the metacommunity Trait (t) λ: selection on trait in the local community (greater probability of passing if smaller…) p : distribution of relative abundance of species in the local community Local community Grimes’s plant strategy Trait (t)

9. Measuringabundances Local abundance Abundance of plant speciescanbemeasured as - numbers of stems - biomass or indirect measures of this (cover, dbh …) Units of observation are ambiguous - numbers of what? (ramets ≠ genets, biomassunits?) Units of observations are neverindependent Most values will be zero (species present in region but not in local site)

10. Measuringabundances Meta-communityabundances Sometimes no information Sometimes vague information (verycommon, common, rare) Sometimes more quantitative information

11. Part 2 The maximum entropyformalism Edwin Jaynes Jaynes, E.T. 2003. Probability theory. The logic of science. Cambridge U Press.

12. How canwequantifylearning? If the sun does set tonight (and our previous information told us it would,p=0.99999) then we will have learned almost nothing new (almost no new information) If the sun doesn’t set tonight (and our previous information told us it would, 1-p=0.00001) then we will have learned something incredible (lots of new information) The amount of learning (new information): Historical log2, wewill use loge=ln Claude Shannon

13. Average information content (i.e. new information) Specify all of the logically possible states in whichsomephenomenoncanexist (i=1,2,…,S) Based on whatwe know beforeobserving the actual state, assign values between 0 and 1 to each possible state: p=(p1, p2, …, pS) The amount of new information thatwewilllearn if state i occursis The averageamount of new information thatwewilllearnis Information entropy

14. Information content and uncertainty New information = information thatwedon’tyetpossess = uncertainty Information entropymeasures the amount of new information wewill gain once welearn the truth Information entropythereforemeasures the amount of information thatwedon’tyetpossess Information entropyis a measure of the degree of uncertaintythatwe have about the state of a phenomenon Maximum uncertainty = maximum entropy

15. A betting game: game A I bringyou, blindfolded, to a fieldoutside Sherbrooke. I tell youthatthereis a plant atyourfeetthatbelongs to eitherspecies A or species B. You have assignprobabilities to the two possible states (species A or B) and willgetthat proportion of $1,000,000 once youlearn the speciesname. Whatisyouranswer?

16. A betting game: game B New gamewithsome new clues: (i) The site is a former cultivatedfieldthatwasabandoned by a farmer last year. (ii) Species A is an annualherb. Species B is a climax tree. You have assignprobabilities to the two possible states and willgetthat proportion of $1,000,000. Whatisyouranswer? If you changed your bet in this second game then these new clues provided you with some new, but incomplete, information before you learned the answer.

17. Maximum entropy (maximum uncertainty) at p=(0.5,0.5) Amount of information contained in the ecological clues Amount of remaining uncertianty, given the ecological clues p(annualherb)=0.9 p(climax tree)=0.1)

18. Relative entropy (Kullback-Leibler divergence) Relative to a “prior” or reference distribution (q) Let qi=1/S and S be the fixed number of unordered, discrete states; q is a uniform distribution Information entropy α relative entropygiven a uniform distribution Maximizing relative entropy = maximizingentropy

19. Jaynes, E.T. 2003. Probability theory. The logic of science. Cambridge U Press. The maximum entropyformalism in an ecologicalcontext: A three-step program Edwin Jaynes

20. Jaynes, E.T. 2003. Probability theory. The logic of science. Cambridge U Press. Step 1: specifying the relative abundances in the regional meta-community (the prior distribution) Specify a reference (q, prior) distribution that encodes what you know about the relative abundances of each species in the species pool before obtaining any information about the local community. The maximum entropyformalism in an ecologicalcontext Edwin Jaynes “All I know is that there are S species in the meta-community” “All I know is that there are S species in the meta-community and I know their relative abundances in this meta-community” qi

21. Jaynes, E.T. 2003. Probability theory. The logic of science. Cambridge U Press. Step 2: quantifying what we know about trait-based community assembly in the local community The maximum entropyformalism in an ecologicalcontext Edwin Jaynes “I have measured the average value of my functional traits (community-weighted traits values)” “ I have measured the environmental conditions, and know the function linking these environmental conditions to the community-weighted trait values”

22. Jaynes, E.T. 2003. Probability theory. The logic of science. Cambridge U Press. Step 3: choose a probability distribution which agrees with what we know, but doesn’t include any more information (i.e. don’t lie). The maximum entropyformalism in an ecologicalcontext Choose values of p that agree with what we know: Edwin Jaynes And that maximizes the remaining uncertainty

23. Jaynes, E.T. 2003. Probability theory. The logic of science. Cambridge U Press. The solution is a general exponential distribution qi: Prior distribution of species i tij: Trait value of trait j of species i The maximum entropyformalism in an ecologicalcontext Edwin Jaynes λj:The amount by which a one-unit increase in trait j will result in a proportional change in relative abundance (pi)

24. Jaynes, E.T. 2003. Probability theory. The logic of science. Cambridge U Press. Practical considerations Except for maxent models with only a few constraints, we need numerical methods in order to fit them to data. I use a proportionality between the maximum likelihood of a multinomial distribution and the λ values of the solution to the maxentproblem (ImprovedIterativeScaling), available in the maxent() function of the FD library in R. The maximum entropyformalism in an ecologicalcontext Edwin Jaynes Della Pietra, S. Della Pietra, V., Lafferty, J. 1997. Inducing features of random fields. IEEE Transactions Patern Analysis and Machine Intelligence 19:1-13 By permuting trait vectors relative to species’ observed relative abundances, one can develop permutation tests of significance concerning model fit Laliberté, E., Shipley, B. 2009. Measuring functional diversity from multiple traits, and other tools for functional ecology R. package, Vienna, Austria Shipley, B. 2010. Ecology 91:2794-2805

25. CATS There are now many empirical applications of this model in many places around the world, and applied at many geographical scales. Two examples…

26. SHIPLEY et al. 2011. A strong test of a maximum entropy model of trait-basedcommunityassembly. Ecology 92: 507–517. Daniel Laughlin’s PhD thesis 96 1m2 quadrats containing the understory herbaceous plants of ponderosapine forests (Arizona USA). Quadrats were distributed across seven permanent sites within a 120 km2 landscape between2000-2500 m altitude. Available information 12 environmental variables measured in each quadrat 20 functional traits measured per species Relative abundance of each species in each quadrat

27. If species is present, and not rare (>10%), CATS predicts its abundance well After ~7 traits, mostly redundant information If species is present, but rare (<10%), CATS can’t distinguish degrees of rarity If species is absent (X) then CATS will predict it to be rare Significant predictive ability by 3 traits

28. If we only know the environmental conditions of a site, and the general relationship between community-weighted traits and the environment, how well can CATS do? Actual measured community-weighted value for this site = 9.5 Predicted value for community-weighted trait, given that we know the environmental value is 7 = 7.61 General relationship: Y=2.5+0.73X

29. The best possible prediction given the environment would be obtained with 79 separate generalized additive (form-free) regressions – one for each species in the species pool – of relative abundance vs. the environmental variables. gam(S~X)

31. Second example: tropical forests in French Guiana Shipley, Paine & Baraloto. 2012. Quantifying the importance of local niche-based and stochastic processes to tropical tree community assembly. . Ecology93: 760-769 The unifiedneutraltheory of biodiversity and biogeography The per capita probabilities of immigration from the meta-community, and the per capita probabilities of survival and reproduction of all species are equal (demographicneutrality). Subsequent population dynamics in the local community are determined purely by random drift. Stephen Hubbell Local relative abundance ~meta-community relative abundance + drift « Neutralprior » = meta-community relative abundances

32. 1. Fit model using traits but a uniformprior (i.e. no effect of meta-community immigration) 2. Fit model permuting traits but with a neutralprior (i.e. no effects of local trait-basedselection, but contribution from immigration) 3. Fit model with traits and withneutralprior(i.e. locat trait-basedselection plus immigration frommeta-community Partition the total variance explainedinto: That due only to immigration That due only to local trait-basedselection That due jointly to immigration and traits (correlationswithmeta-community Unexplained variation due to demographic stochasticity

33. Wedidthisatthree spatial scales: 1, 0.25 and 0.01 ha. Trait-basedselection Demographic stochasticity Dispersal limitation frommeta- community

35. Practical considerations. How do we fit this model to empirical data? A trick involving likelihood A total of A independent allocations of individuals or units of biomass into each of the S species in the species pool the number of independent allocations to species i The probability of a single allocation going to species i (i.e. the probability of sufficient resources being captured to produce one individual or unit of biomass for species i); a function of its traits (Ti), the strength of the trait on selection (λ), and its meta-community abundance (qi)

36. The likelihood: In practice we can never know this, since neither individuals or resources (biomass) are ever allocated independently… Taking logarithms, dividing both sides by A, and re-arranging, one obtains

37. Will maximise the likelihood of the unknown multinomial distribution Maximising this… But we already know that

38. Choose values of λthat maximise thisfunction, given the vector of traits (ti) for eachspecies in the species pool and theirmeta-communityabundances (qi): The dual solution (Della Pietra et al. 1997): the values of λthat maximise the relative entropy in the maximum entropyformalism are the same as the values that maximise the likelihood of a multinomial distribution. Della Pietra, S. Della Pietra, V., Lafferty, J. 1997. Inducing features of random fields. IEEE Transactions Patern Analysis and Machine Intelligence 19:1-13. Improved Iterative Scaling algorithm maxent() in the FD package of R. Laliberté, E., Shipley, B. 2009. Measuring functional diversity from multiple traits, and other tools for functional ecology R. package, Vienna, Austria

39. Permutation tests for the CATS model Choose values of p that agree with what we know: H0: trait values (tij) are independent of the observed relative abundances (oi) And that maximizes the remaining uncertainty

40. Permutation tests for the CATS model 1. Calculate The smaller the value, the better the fit 2. Randomly permute the vector of trait values (T*i) between the species (so traits are independent of observed relative abundances) 3. Calculate 4. Repeat steps 2 & 3 a large number of times. 5. Count the number of times f*> f. This is an estimate of the null probability. Shipley, B. 2010. Ecology 91:2794-2805 maxent.test() function in the FD library.