1. Assessing environmental suitability for species based on Tukey's depth* Maria João Martins 1 with: J.O.Cerdeira 1,2, T. Monteiro-Henriques 1, P.C. Silva 1, D. Alagador 3, A. Franco 4, M. Campagnolo 1, P. Arsénio 1, F. Aguiar 1, M. Cabeza 5 1 CEF, Instituto Superior de Agronomia, Universidade Técnica de Lisboa, Portugal 2 Dept of Math., Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Portugal 3 Rui Nabeiro Biodiversity Chair, Universidade de Évora, Portugal 4 School of Environment Sciences, University of East Anglia, UK 5 Dept of Biosciences, University of Helsinki, Finland

2. (1957) Formally quantifies the niche concept in terms of geometric space two independent environmental variables v1v1 v2v2 ba c d a, b, c, d define the limiting values permitting a species to survive and reproduce possible environmental state permitting the species to exist indefinitely 2 Assessing environmental suitability for species based on Tukey's depth. IAVS 2013, Tartu, Estonia

3. (1957) Formally quantifies the niche concept in terms of geometric space independence fails v1v1 v2v2 ba c d rectangle → polygon possible environmental state permitting the species to exist indefinitely 3 Assessing environmental suitability for species based on Tukey's depth. IAVS 2013, Tartu, Estonia

4. (1957) Introduce v 3, v 4,..., v n until all the ecological factors relative to the species have been considered n-dimensional hypervolume: every point corresponds to a state of the environment which would permit the species to exist indefinitely Fundamental niche of the species 4 Assessing environmental suitability for species based on Tukey's depth. IAVS 2013, Tartu, Estonia

5. (1957) 5 Assessing environmental suitability for species based on Tukey's depth. IAVS 2013, Tartu, Estonia

6. Theoretical Question: How to differentiate the niche according to the probability of persistence of the species (or environmental suitability)? Independence For each variable individually consider percentiles BIOCLIM (Nix 1986; Busby 1991) (with percentiles) Data: occurrences of the species and environmental variables in occurrence points Practical Question: How to estimate the niche, distinguishing between areas of different suitability? 6 Assessing environmental suitability for species based on Tukey's depth. IAVS 2013, Tartu, Estonia

7. Independence BIOCLIM (with percentiles) as implemented in package dismo x is a quantile of order 1/10 (or 9/10) of (x 1,...,x 10 ) (x,y) or y is a quantile of order 1/10 (or 9/10) of (y 1,...,y 10 ) 7 Assessing environmental suitability for species based on Tukey's depth. IAVS 2013, Tartu, Estonia

8. Independence BIOCLIM (with percentiles) as implemented in package dismo 0.2 0.4 0.6 0.8 1 median 0 8 Assessing environmental suitability for species based on Tukey's depth. IAVS 2013, Tartu, Estonia

9. How to deal with dependence (of the ecological variables on their effect over species) ? Propose: to use a generalization of quantile in multi-dimensional space Tukey depth (Tukey 1975) If P is a set of k points of ℝ n, the Tukey depth of q with respect to P, is d q (P)=m/k, where m is the minimum number of points needed to remove from P so that q is not in the convex-hull of the remaining points of P. The Tukey depth is implemented in package depth 9 Assessing environmental suitability for species based on Tukey's depth. IAVS 2013, Tartu, Estonia

10. Tukey depth If P is a set of k points of ℝ n, the Tukey depth of q with respect to P, is d q (P)=m/k, where m is the minimum number of points needed to remove from P so that q is not in the convex-hull of the remaining points of P. P={ ∙ } n=2, k=10 q d q (P)=1/10 Tukey depth discriminates the convex hull into nested regions of increasing depth for the interior 10 Assessing environmental suitability for species based on Tukey's depth. IAVS 2013, Tartu, Estonia

11. BIOCLIM vs Tukey depth If P is a set of k points of ℝ n, the depth of q with respect to P is m/k, where m is the minimum number of points needed to remove from P so that q is not in the hyper rectangleis not in the convex hull of the remaining points of P. 11 Assessing environmental suitability for species based on Tukey's depth. IAVS 2013, Tartu, Estonia

12. BIOCLIM vs Tukey depth Computational complexity k points of ℝ n, the depth of q with respect to P needs a search in k directions if n=2 k C n-1 directions 2 directions if n=2 n directions 3 directions if n=3k(k-1) directions if n=3 12 Assessing environmental suitability for species based on Tukey's depth. IAVS 2013, Tartu, Estonia

13. Application: main Quercus species of the Iberian Peninsula Data: Occurrences from the Proyecto Anthos (Anthos 2011) and from the Herbarium “João de Carvalho e Vasconcellos” (Lisbon, LISI) Four environmental variables from Worldclim website (Hijmans et al. 2006) – isothermality, the minimum temperature of coldest month, annual precipitation and the precipitation of driest quarter. Methods: For each site within the Iberian Peninsula (ca. 1 km x 1 km resolution) - the binary environmental suitability (suitable vs. unsuitable) was obtained through the BIOCLIM and the Convex Hull. - the environmental suitability was obtained by the Tukey depth Results: Geographical maps of the environmental suitability 13 Assessing environmental suitability for species based on Tukey's depth. IAVS 2013, Tartu, Estonia

14. Application: main Quercus species of the Iberian Peninsula binary environmental suitability BIOCLIMConvex Hull 14 Assessing environmental suitability for species based on Tukey's depth. IAVS 2013, Tartu, Estonia

15. Application: main Quercus species of the Iberian Peninsula binary environmental suitability BIOCLIMConvex Hull 15 Assessing environmental suitability for species based on Tukey's depth. IAVS 2013, Tartu, Estonia

16. Application: main Quercus species of the Iberian Peninsula binary environmental suitability BIOCLIMConvex Hull 16 Assessing environmental suitability for species based on Tukey's depth. IAVS 2013, Tartu, Estonia

17. Application: main Quercus species of the Iberian Peninsula environmental suitability by Tukey depth 17 Assessing environmental suitability for species based on Tukey's depth. IAVS 2013, Tartu, Estonia

18. Application: main Quercus species of the Iberian Peninsula environmental suitability by Tukey depth 18 Assessing environmental suitability for species based on Tukey's depth. IAVS 2013, Tartu, Estonia

19. Using Tukey depth to assess the separability of a partition In a recent paper we propose to assess the separability of clusters based on a general notion of interiority. The separability index (oDSI) lies between zero and one. Taking the Tukey depth as the interiority criterion: oDSI = 1 when every point is outside the convex hulls of all other clusters of the partition oDSI = 0 when every point is in the convex hull of some other cluster of the partition for two sets with the same cardinality and the same statistical distribution, oDSI ≈ 0.5 19 Assessing environmental suitability for species based on Tukey's depth. IAVS 2013, Tartu, Estonia

20. Using Tukey depth to assess the separability of a partition oDSI=0.93oDSI=0.07 20 Assessing environmental suitability for species based on Tukey's depth. IAVS 2013, Tartu, Estonia

21. Main advantages of Tukey depth methodology easy geometric interpretation 21 Assessing environmental suitability for species based on Tukey's depth. IAVS 2013, Tartu, Estonia straightforward relationship with Hutchinson's niche theory results invariant to the scale of ecological variable (do not depend on the units of measurement of the variables) robustness to outliers (for deeper points) THANK YOU FOR YOUR ATTENTION

22. References Busby, J.R. (1991). BIOCLIM - A Bioclimate Analysis and Prediction System. In: Nature Conservation: Cost Effective Biological Surveys and Data Analysis, eds. C.R. Margules & M.P. Austin, 64–68. CSIRO. Cerdeira J.O., Martins M.J. and Silva P.C. (2012). A combinatorial approach to assess the separability of clusters, Journal of Classification, 29: 7-22. Hutchinson, G.E. (1957). Concluding remarks. Cold Spring Harbor Symposia on Quantitative Biology 22: 415–427. Nix, H.A. (1986). A biogeographic analysis of Australian elapid snakes. In: Atlas of elapid snakes of Australia, ed. R. Longmore, 4–15. Canberra: Australian Government Publishing Service. Tukey, J. (1975). Mathematics and the picturing of data. Proc. 1975 Inter. Cong. Math., Vancouver, 523-531. 22 Assessing environmental suitability for species based on Tukey's depth. IAVS 2013, Tartu, Estonia