Environmental and Ecological Statistics

, Volume 22, Issue 3, pp 513–534 | Cite as

Robust Bayesian model averaging for the analysis of presence–absence data

Article

Abstract

When developing a species distribution model, usually one tests several competing models such as logistic regressions characterized by different sets of covariates. Yet, there is an exponential number of subsets of covariates to choose from. This generates the problem of model uncertainty. Bayesian model averaging (BMA) is a state-of-the-art approach to deal with model uncertainty. BMA weights the inferences of multiple models. However, the results yielded by BMA depend on the prior probability assigned to the models. Credal model averaging (CMA) extends BMA towards robustness. It substitutes the single prior over the models by a set of priors. The CMA inferences (e.g., posterior probability of inclusion of a covariate, coefficient of a covariate, probability of presence) are intervals. The interval shows the sensitivity of the BMA estimate on the prior over the models. CMA detects the prior-dependent instances, namely cases in which the most probable outcome becomes presence or absence depending on the adopted prior over the models. On such prior-dependent instances, BMA behaves almost as a random guesser. The weakness of BMA on the prior-dependent instances is to our knowledge pointed out for the first time in the ecological literature. On the prior-dependent instances CMA avoids random guessing acknowledging undecidability. In this way it stimulates the decision maker to convey further information before taking the decision. We provide thorough experiments on different data sets.

Keywords

Bayesian model averaging Credal model averaging Imprecise probability Logistic regression Presence–absence Robust Bayesian analysis Species distribution model  

References

  1. Araùjo MB, Williams PH (2000) Selecting areas for species persistence using occurrence data. Biol Conserv 96:331–345CrossRefGoogle Scholar
  2. Benavoli A, Zaffalon M (2012) A model of prior ignorance for inferences in the one-parameter exponential family. J Stat Plan Inference 142:1960–1979CrossRefGoogle Scholar
  3. Berger JO, Moreno E, Pericchi LR, Bayarri MJ, Bernardo JM, Cano JA, De la Horra J, Martín J, Ríos-Insúa D, Betrò B et al (1994) An overview of robust Bayesian analysis. Test 3:5–124CrossRefGoogle Scholar
  4. Borgo A (2003) Habitat requirements of the Alpine marmot Marmota mar-mota in re-introduction areas of the Eastern Italian Alps. Formulation and validation of habitat suitability models. Acta Theriologica 48:557–569CrossRefGoogle Scholar
  5. Burnham KP, Anderson DR (2002) Model selection and multi-model inference: a practical information-theoretic approach. Springer, BerlinGoogle Scholar
  6. Cantini M, Bianchi C, Bovone N, Preatoni D (1997) Suitability study for the alpine marmot (Marmota marmota marmota) reintroduction on the Grigne massif. Hystrix—Ital J Mammal 9:65–70Google Scholar
  7. Clyde M (2000) Model uncertainty and health effect studies for particulate matter. Environmetrics 11:745–763CrossRefGoogle Scholar
  8. Clyde M, George EI (2004) Model uncertainty. Stat Sci 19:81–94Google Scholar
  9. Corani G, Mignatti A (2013) Credal model averaging of logistic regression for modeling the distribution of marmot burrows. In: Cozman F, Denoeux T, Destercke S, Seidenfeld T (eds) ISIPTA’13: proceedings of the eighth international symposiumon imprecise probability: theories and applications, pp 233–243Google Scholar
  10. Corani G, Mignatti A (2015) Credal model averaging for classification: representing prior ignorance and expert opinions. Int J Approx Reason 56:264–277Google Scholar
  11. Corani G, Zaffalon M (2008a) Credal model averaging: an extension of Bayesian model averaging to imprecise probabilities. In: Proceedings of the ECML-PKDD 2008 (European conference on machine learning and knowledge discovery in databases), pp 257–271Google Scholar
  12. Corani G, Zaffalon M (2008b) Learning reliable classifiers from small or incomplete data sets: the naive credal classifier 2. J Mach Learn Res 9:581–621Google Scholar
  13. Cozman FG (2000) Credal networks. Artif Intell 120:199–233CrossRefGoogle Scholar
  14. Destercke S, Dubois D, Chojnacki E (2008) Unifying practical uncertainty representations I: generalized p-boxes. Int J Approx Reason 49:649–663CrossRefGoogle Scholar
  15. Elith J, Leathwick JR (2009) Species distributionmodels: ecological explanation and prediction across space and time. Annu Rev Ecol Evol Syst 40:677–697CrossRefGoogle Scholar
  16. Friedman JH (1991) Multivariate adaptive regression splines. Ann Stat 19:1–67Google Scholar
  17. Goodwin B, McAllister A, Fahrig L (1999) Predicting invasiveness of plant species based on biological information. Conserv Biol 13:422–426CrossRefGoogle Scholar
  18. Graf RF, Bollmann K, Suter W, Bugmann H (2005) The importance of spatial scale in Habitat models: capercaillie in the Swiss Alps. Landsc Ecol 20:703–717CrossRefGoogle Scholar
  19. Guisan A, Thuiller W (2005) Predicting species distribution: offering more than simple habitat models. Ecol Lett 8:993–1009CrossRefGoogle Scholar
  20. Guisan A, Graham CH, Elith J, Huettmann F (2007) Sensitivity of predictive species distribution models to change in grain size. Divers Distrib 13:332–340CrossRefGoogle Scholar
  21. Herbei R, Wegkamp MH (2006) Classification with reject option. Can J Stat 34:709–721CrossRefGoogle Scholar
  22. Herrero J, Zima J, Coroiu I (2008) Marmota marmota. In: IUCN Red List of Threatened Species. Version 2013.1. www.iuncredlist.org. Downloaded 19 July 2013
  23. Hoeting J, Madigan D, Raftery A, Volinsky C (1999) Bayesian model averaging: a tutorial. Stat Sci 44:382–417Google Scholar
  24. Kavanagh RP, Bamkin KL (1995) Distribution of nocturnal forest birds and mammals in relation to the logging mosaic in south-eastern New South Wales, Australia. Biol Conserv 71:41–53CrossRefGoogle Scholar
  25. Lenti Boero D (2003) Long-term dynamics of space and summer resource use in the alpine marmot (Marmota marmota L.). Ethol Ecol Evol 15:309–327CrossRefGoogle Scholar
  26. Ley E, Steel MF (2009) On the effect of prior assumptions in Bayesian model averaging with applications to growth regression. J Appl Econom 24:651–674CrossRefGoogle Scholar
  27. Li H, Calder CA, Cressie N (2007) Beyond Moran’s i: testing for spa tial dependence based on the spatial autoregressive model. Geogr Anal 39:357–375CrossRefGoogle Scholar
  28. Link W, Barker R (2006) Model weights and the foundations of multimodel inference. Ecology 87:2626–2635CrossRefPubMedGoogle Scholar
  29. Lóopez B, Figueroa I, Pino J, Lóopez A, Potrony D (2009) Potential distribution of the alpine marmot in Southern Pyrenees. Ethol Ecol Evol 21:225–235CrossRefGoogle Scholar
  30. Lóopez B, Pino J, Lóopez A (2010) Explaining the successful introduction of the alpine marmot in the Pyrenees. Biol Invasions 12:3205–3217CrossRefGoogle Scholar
  31. Olsson O, Rogers DJ (2009) Predicting the distribution of a suitable habitat for the white stork in Southern Sweden: identifying priority areas for reintroduction and habitat restoration. Anim Conserv 12:62–70CrossRefGoogle Scholar
  32. Perrin C, Berre D (1993) Socio-spatial organization and activity distribution of the Alpine Marmot Marmota marmota: preliminary results. Ethology 93:21–30CrossRefGoogle Scholar
  33. Peterson AT (2003) Predicting the geography of species’ invasions via ecological niche modeling. Q Rev Biol 78:419–433CrossRefPubMedGoogle Scholar
  34. Raftery AE (1995) Bayesian model selection in social research. Sociol Methodol 25:111–164CrossRefGoogle Scholar
  35. Riley SJ, DeGloria S, Elliot R (1999) A terrain ruggedness index that quantifies topographic heterogeneity. Intermt J Sci 5:23–27Google Scholar
  36. Schweiger AKA, Nopp-Mayr U, Zohmann M (2012) Small-scale habitat use of black grouse (Tetrao tetrix L.) and rock ptarmigan (Lagopus muta helvetica Thienemann) in the Austrian Alps. Eur J Wildl Res 58:35–45CrossRefGoogle Scholar
  37. St-Louis V, Clayton MK, Pidgeon AM, Radeloff VC (2012) An evaluation of prior in uence on the predictive ability of Bayesian model averaging. Oecologia 168:719–726CrossRefPubMedGoogle Scholar
  38. Thomson JR, Mac Nally R, Fleishman E, Horrocks G (2007) Predicting bird species distributions in reconstructed landscapes. Conserv Biol 21:752–766CrossRefPubMedGoogle Scholar
  39. Walley P (1991) Statistical reasoning with imprecise probabilities. Chapman and Hall London, LondonCrossRefGoogle Scholar
  40. Walley P (1996) Inferences from multinomial data: learning about a bag of marbles. J Roy Stat Soc B 58:3–57Google Scholar
  41. Wilson KA, Westphal MI, Possingham HP, Elith J (2005) Sensitivity of conservation planning to different approaches to using predicted species distribution data. Biol Conserv 122:99–112CrossRefGoogle Scholar
  42. Wintle B, McCarthy M, Volinsky C, Kavanagh R (2003) The use of Bayesian model averaging to better represent uncertainty in ecological models. Conserv Biol 17:1579–1590CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Istituto Dalle Molle di studi sull’Intelligenza Artificiale (IDSIA)Scuola universitaria professionale della Svizzera italiana (SUPSI), Università della Svizzera italiana (USI)LuganoSwitzerland
  2. 2.Dipartimento di Elettronica, Informazione e BioingegneriaPolitecnico di MilanoMilanItaly

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