Transcript Assessing environmental suitability for species based on Tukey`s

Assessing environmental
suitability for species based on
Tukey's depth*
Maria João Martins1
with: J.O.Cerdeira1,2, T. Monteiro-Henriques1, P.C. Silva1, D. Alagador3, A. Franco4,
M. Campagnolo1, P. Arsénio1, F. Aguiar1, M. Cabeza5
1CEF,
Instituto Superior de Agronomia, Universidade Técnica de Lisboa, Portugal
2Dept of Math., Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Portugal
3Rui Nabeiro Biodiversity Chair, Universidade de Évora, Portugal
4School of Environment Sciences, University of East Anglia, UK
5Dept of Biosciences, University of Helsinki, Finland
(1957)
Formally quantifies the niche concept in terms of geometric space
v2
two independent environmental variables
d
a, b, c, d define the limiting values permitting
a species to survive and reproduce
c
a
b
v1
possible environmental state permitting the species
to exist indefinitely
2
Assessing environmental suitability for species based on Tukey's depth. IAVS 2013, Tartu, Estonia
(1957)
Formally quantifies the niche concept in terms of geometric space
v2
independence fails
d
rectangle → polygon
c
a
b
v1
possible environmental state permitting the species
to exist indefinitely
3
Assessing environmental suitability for species based on Tukey's depth. IAVS 2013, Tartu, Estonia
(1957)
Introduce v3, v4, ..., vn 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
(1957)
5
Assessing environmental suitability for species based on Tukey's depth. IAVS 2013, Tartu, Estonia
Theoretical Question: How to differentiate the niche according
to the probability of persistence of the species (or environmental suitability)?
Data: occurrences of the species
and environmental variables in occurrence points
Practical Question: How to estimate the niche, distinguishing between areas of
different suitability?
Independence
For each variable individually consider percentiles
BIOCLIM (Nix 1986; Busby 1991) (with percentiles)
6
Assessing environmental suitability for species based on Tukey's depth. IAVS 2013, Tartu, Estonia
Independence
package dismo
0.6
0.8
1.0
BIOCLIM (with percentiles) as implemented in
x is a quantile of order
1/10 (or 9/10)
of (x1,...,x10)
_
(x,y)
_
_
0.0
0.2
0.4
y
_
or
y is a quantile of order
1/10 (or 9/10)
of (y1,...,y10)
_
_
_
_
_
_
|
0.0
0.2
| |
|
| || |
0.4
0.6
|
|
0.8
1.0
x
7
Assessing environmental suitability for species based on Tukey's depth. IAVS 2013, Tartu, Estonia
Independence
package dismo
1.0
BIOCLIM (with percentiles) as implemented in
0.6
0.8
0
_
0.2
_
_
0.6
0.0
0.2
0.4
y
_
_
0.4
0.8
_
_
_
1 median
_
_
|
0.0
0.2
| |
|
| || |
0.4
0.6
|
|
0.8
1.0
x
8
Assessing environmental suitability for species based on Tukey's depth. IAVS 2013, Tartu, Estonia
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 ℝn, the Tukey depth of q with respect to P, is
dq(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
9
package depth
Assessing environmental suitability for species based on Tukey's depth. IAVS 2013, Tartu, Estonia
Tukey depth
1.0
If P is a set of k points of ℝn, the Tukey depth of q with respect to P, is
dq(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.
n=2, k=10
0.8
P={∙}
dq(P)=1/10
0.0
0.2
0.4
y
0.6
q
0.0
0.2
0.4
0.6
0.8
1.0
Tukey depth discriminates
the convex hull into nested
regions of increasing depth
for the interior
x
10
Assessing environmental suitability for species based on Tukey's depth. IAVS 2013, Tartu, Estonia
1.0
0.8
0.6
0.0
0.2
0.4
y
0.0
0.2
0.4
y
0.6
0.8
1.0
BIOCLIM vs Tukey depth
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
x
0.4
0.6
0.8
1.0
x
If P is a set of k points of ℝn, 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 rectangle
is 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
1.0
1.0
1.0
1.0
0.2 0.2
0.4 0.4
x
0.8
0.2
0.2
0.0
0.4
y
0.6
0.8
0.0 0.0
0.0
0.4
y
0.6
0.8
0.6
0.2
0.0
0.4
y
0.4
0.2
0.0
y
0.6
0.8
BIOCLIM vs Tukey depth
0.8 0.8 1.0 1.0
0.0 0.0 0.2 0.2
Computational
complexity
0.6 0.6
x
0.4 0.4
x
0.6 0.6
0.8 0.8
1.0 1.0
x
k points of ℝn, the depth of q with respect to P needs a search in
12
2 directions if n=2
k directions if n=2
3 directions if n=3
k(k-1) directions if n=3
n directions
kC
n-1
directions
Assessing environmental suitability for species based on Tukey's depth. IAVS 2013, Tartu, Estonia
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
Application: main Quercus species of the Iberian Peninsula
binary environmental suitability
BIOCLIM
Convex Hull
Quercus faginea subsp. broteroi
Quercus faginea subsp. broteroi
1.0
0.8
0.6
0.4
0.2
0.0
Quercus faginea subsp. faginea
Quercus faginea subsp. faginea
1.0
0.8
0.6
0.4
0.2
0.0
14
1.0
0.8
0.6
0.4
0.2
0.0
1.0
0.8
0.6
0.4
0.2
0.0
Assessing environmental suitability for species based on Tukey's depth. IAVS 2013, Tartu, Estonia
Application: main Quercus species of the Iberian Peninsula
binary environmental suitability
BIOCLIM
Convex Hull
Quercus rotundifolia
Quercus rotundifolia
1.0
0.8
0.6
0.4
0.2
0.0
Quercus pyrenaica
Quercus pyrenaica
1.0
0.8
0.6
0.4
0.2
0.0
15
1.0
0.8
0.6
0.4
0.2
0.0
1.0
0.8
0.6
0.4
0.2
0.0
Assessing environmental suitability for species based on Tukey's depth. IAVS 2013, Tartu, Estonia
Application: main Quercus species of the Iberian Peninsula
binary environmental suitability
BIOCLIM
Convex Hull
Quercus robur
Quercus robur
1.0
0.8
0.6
0.4
0.2
0.0
Quercus suber
Quercus suber
1.0
0.8
0.6
0.4
0.2
0.0
16
1.0
0.8
0.6
0.4
0.2
0.0
1.0
0.8
0.6
0.4
0.2
0.0
Assessing environmental suitability for species based on Tukey's depth. IAVS 2013, Tartu, Estonia
Application: main Quercus species of the Iberian Peninsula
environmental suitability by Tukey depth
Quercus faginea subsp. broteroi
Quercus faginea subsp. faginea
0.20
0.15
0.10
0.05
0.00
0.30
0.25
0.20
0.15
0.10
0.05
0.00
Quercus pyrenaica
Quercus rotundifolia
0.30
0.25
0.20
0.15
0.10
0.05
0.00
17
0.30
0.25
0.20
0.15
0.10
0.05
0.00
Assessing environmental suitability for species based on Tukey's depth. IAVS 2013, Tartu, Estonia
Application: main Quercus species of the Iberian Peninsula
environmental suitability by Tukey depth
Quercus pyrenaica
Quercus suber
0.30
0.25
0.20
0.15
0.10
0.05
0.00
18
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
Assessing environmental suitability for species based on Tukey's depth. IAVS 2013, Tartu, Estonia
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
Using Tukey depth to assess the separability of a partition
-50
0
50
100
Min Temperature of Coldest Month (ºC *10)
20
250
150
0 50
Quercus robur
Quercus faginea subsp. broteroi
-100
oDSI=0.07
Precipitation of Driest Quarter (mm)
250
150
0 50
Precipitation of Driest Quarter (mm)
oDSI=0.93
Quercus rotundifolia
Quercus suber
-100
-50
0
50
100
Min Temperature of Coldest Month (ºC *10)
Assessing environmental suitability for species based on Tukey's depth. IAVS 2013, Tartu, Estonia
Main advantages of Tukey depth methodology
easy geometric interpretation
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
21
Assessing environmental suitability for species based on Tukey's depth. IAVS 2013, Tartu, Estonia
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