Niches, distributions… and dataNiches, distributions… and data

Miguel NakamuraMiguel Nakamura

Centro de Investigación en Matemáticas (CIMAT), Centro de Investigación en Matemáticas (CIMAT), Guanajuato, MexicoGuanajuato, Mexico

nakamura@cimat.mxnakamura@cimat.mx

Warsaw, November 2007Warsaw, November 2007

““Data”Data”

PresencesPresences

Environmental Environmental layerslayers

Niche and distribution concepts

Ecological theory

conceived in

Nature

realized in

Niche models

inferred using

Data

useproduces

Theoretical niche

defines

Distribution

Premise #1: an observation is the result of at Premise #1: an observation is the result of at least two, multi-factor processesleast two, multi-factor processes

Biology: the fundamental niche, biotic conditions, sink Biology: the fundamental niche, biotic conditions, sink populations, populations, etcetc..

Humans: the collector introduces bias, methods used Humans: the collector introduces bias, methods used determine detection, determine detection, etcetc..

Premise #2: randomness involvedPremise #2: randomness involved

If sites 1 and 2 both have equal conditions If sites 1 and 2 both have equal conditions XX as far as as far as we can see, it does NOT necessarily follow thatwe can see, it does NOT necessarily follow that

““species present at site 1 implies species present at site 2”species present at site 1 implies species present at site 2” Reason: apart from conditions Reason: apart from conditions XX, there may be other, , there may be other,

non-visualized conditions, non-visualized conditions, ZZ, that also influence , that also influence presence. These may differ between site 1 and site 2.presence. These may differ between site 1 and site 2.

Refer to Refer to probabilitiesprobabilities of presence at a site having of presence at a site having conditions conditions XX, instead of a deterministic statement, , instead of a deterministic statement, “species is present at a site”.“species is present at a site”.

““Probability trees”Probability trees”

Graphical devices for tracking random experiments, Graphical devices for tracking random experiments, especially when sequential processes or stages are especially when sequential processes or stages are involved.involved.

Probabilities can be assigned to branches, for Probabilities can be assigned to branches, for calculations.calculations.

Example: die is cast to observe number of spots (Example: die is cast to observe number of spots (NN), ), then then NN coins are tossed and number of heads counted. coins are tossed and number of heads counted.

Die

1

2

Begin

3

4

5

6

1/6

1/6

1/6

1/6

1/6

1/6

# Heads

1

0

1

0

2

1

0

2

3

.50

.50

.25

.25

.50

.125

.375

.375

.125

etc.

Probability of this branch=(1/6)×(.50)

Probability of this cluster=sum of probabilities of individual branches

Species present?

Site visited?

Species detect?

True presenceTrue presence

True absenceTrue absence

A site

Elementary probability tree for describing occurrence dataElementary probability tree for describing occurrence data

False absenceFalse absence

False presenceFalse presence

False presenceFalse presence

False absenceFalse absence

False presenceFalse presence

False absenceFalse absence

Abiotic OK?

Site visited?

Species detect?

Biotic OK?

Species moved?

More-elaborate probability tree: “biological presence” has been expandedMore-elaborate probability tree: “biological presence” has been expanded

Presence-only data: the probability of this branch is product of all probabilities in its path. This is data niche models will use.

Abiotic OK?

Site visited?

Species detect?

Biotic OK?

Species moved?

Filling-in probabilities in the treeFilling-in probabilities in the tree

AA

BB

CC

DD

EE AA×B×B×C×D×E×C×D×E

Interpreting the probabilitiesInterpreting the probabilities

Abiotic OK?

Site visited?

Species detect?

Biotic OK?

Species moved?

AA

BB

CC

DD

EE AA×B×B×C×D×E×C×D×E

Motility of species Motility of species (history, barriers, (history, barriers,

dispersal dispersal capacities, capacities, etcetc.).)

Suitability of biotic Suitability of biotic conditions conditions

(competitors, (competitors, predators, predators,

mutualists, mutualists, etcetc.).)

Suitability of abiotic Suitability of abiotic conditions conditions

(resistance to (resistance to temperature temperature

extremes, water extremes, water stress, stress, etcetc.).)

Sampling bias Sampling bias (accessibility, (accessibility, roads, roads, etcetc.).)

Probability of Probability of detection (methods detection (methods

and effort of and effort of collection)collection)

Spatial sampling biasSpatial sampling bias

Occurrence Data (reptiles)Occurrence Data (reptiles)

Spatial sampling bias Environmental sampling bias?

e1

e2

Geographical space Environmental space

Abiotic OK?

Site visited?

Species detect?

Biotic OK?

Species moved?

A pet exampleA pet example

11

11

.20.20

11 Prob=.20Prob=.20×.80=.16×.80=.16

.80.80

.32.32

.50.50

Prob=.32Prob=.32×.50=.16×.50=.16

Important conclusion:

Factors can combine in different ways and still produce the same observed presence rate!

Two different Two different speciesspecies

Two different Two different sampling schemessampling schemes

Issue raised by pet exampleIssue raised by pet example

Distribution of presence-only data is a function of all Distribution of presence-only data is a function of all factors in the tree. Factors can combine in different ways factors in the tree. Factors can combine in different ways and still produce the same observed presence rate!and still produce the same observed presence rate!

Since observed data is probabilistically identical, any Since observed data is probabilistically identical, any method that uses observed data only, is unable to method that uses observed data only, is unable to discern between Species #1 and Species #2.discern between Species #1 and Species #2.

Sampling bias and other conditions become crucial.Sampling bias and other conditions become crucial.

Abiotic OK?

Site visited?

Species detect?

Biotic OK?

Species moved?

In general, areas of distributionIn general, areas of distribution≠≠datadata

AA

BB

CC

DD

EE Data=AData=A×B×B×C×D×E×C×D×E

Occupied Occupied area=Aarea=A×B×B×C×C

Colonizable Colonizable

area=Barea=B××CC

Abiotically suitable Abiotically suitable area=Carea=C

MotilityMotility BioticBiotic AbioticAbiotic SamplingSampling DetectionDetection DataData Occupied areaOccupied area Colonizable Colonizable areaarea

Abiotically Abiotically suitablesuitable

00 General caseGeneral case AA BB CC DD EE ABCDEABCDE ABCABC BCBC CC

11 Full motility, Full motility, biotic biotic irrelevant, irrelevant, well-sampled, well-sampled, sure detectionsure detection

11 11 CC 11 11 CC CC CC CC

22 Full motility, Full motility, well-sampled, well-sampled, sure detectionsure detection

11 BB CC 11 11 BCBC BCBC BCBC CC

33 Full motility, Full motility, abiotic abiotic irrelevant, irrelevant, well-sampled, well-sampled, sure detectionsure detection

11 BB 11 11 11 BB BB BB 11

44 Partial Partial motility, biotic motility, biotic irrelevantirrelevant

AA 11 CC 11 11 ACAC ACAC CC CC

55 Well-sampled, Well-sampled, sure detectionsure detection

AA BB CC 11 11 ABCABC ABCABC BCBC CC

66 Full motility, Full motility, biotic biotic irrelevant, irrelevant, sampling biassampling bias

11 11 CC DD 11 CDCD CC CC CC

77 Cosmopolitan Cosmopolitan speciesspecies

11 11 11 DD EE DEDE 11 11 11

Some special casesSome special cases

ConclusionsConclusions

One thing is One thing is distribution of speciesdistribution of species, and another issue is , and another issue is distribution of observed datadistribution of observed data. Relationship between data . Relationship between data and the niche must be understood.and the niche must be understood.

Previous tree diagram is far more complicated:Previous tree diagram is far more complicated: Interactions.Interactions. Sink populations.Sink populations. Grid resolution (more on this shortly).Grid resolution (more on this shortly). Recording errors, classification errors.Recording errors, classification errors.

Some special cases allow for simplifications:Some special cases allow for simplifications: Uniform sampling.Uniform sampling. Sure detection.Sure detection. Unrestricted species motility.Unrestricted species motility.

ConclusionsConclusions

Algorithms use observed data. They will all try to fit Algorithms use observed data. They will all try to fit observed dataobserved data to environmental variables. to environmental variables.

This may or may not produce what you are interested in. This may or may not produce what you are interested in. It may if you are willing to make some assumptions It may if you are willing to make some assumptions regarding data.regarding data.

It is your responsibility to determine if these assumptions It is your responsibility to determine if these assumptions are met and to interpret results accordingly. A modeling are met and to interpret results accordingly. A modeling algorithm will not know better.algorithm will not know better.

It is useful to think of “data” as including operational It is useful to think of “data” as including operational assumptions, not merely “numbers”.assumptions, not merely “numbers”.

Probability trees used to understand Probability trees used to understand changes in grid resolutionchanges in grid resolution

1km2km

Merging two sitesMerging two sites

Site 1

Site 2

Site 1-2

AA11

BB11

CC11

DD11

EE11

AA22

BB22

CC22

DD22

EE22

AA1212

BB1212

CC1212

DD1212

EE1212

Is there a relationship between AIs there a relationship between A11, B, B11, C, C11, D, D11, E, E11, ,

AA22, B, B22, C, C22, D, D22, E, E22 and A and A1212, B, B1212, C, C1212, D, D1212 , D , D1212??

If new probabilities are derived from the pair of old sets, If new probabilities are derived from the pair of old sets, then merged tree is function of components.then merged tree is function of components.

Will show that this cannot be done coherently.Will show that this cannot be done coherently.

12 12 12

If "species present at site 1-2" means

"either present at site 1 or present at site 2 or present at both",

then

(present site1-2) (present site 1) (present site2)

(present sites 1 and 2)

A B C P P P

P

1 1 1 2 2 2 (present sites 1 and 2)ABC A B C P

12

1 2

If "accessing site 1-2" means

"either site 1 or site 2 or both are accesed",

then

(access 1-2) (access 1) (access 2) (access sites 1 and 2)

(access sites 1 and 2)

A P P P P

A A P

12

1 2

If "abiotic at site 1-2 OK" means

"abiotic conditions OK at site 1 or site 2",

then

(abiotic OK 1-2) (abiotic OK 1) (abiotic OK 1)

(abiotic OK 1 and 2) (abiotic OK 1 and 2)

C P P P

P C C P

1 1 1 2 2 212

1 2 1 2

One then would conclude

(present sites 1 and 2)

{ (access sites 1 and 2)}{ (abiotic OK 1 and 2)}

ABC A B C PB

A A P C C P

• To produce coherent interpretations for the new tree, the “biotic” probability in the new tree must necessarily depend on accessibility, biotic, and abiotic terms of the original trees.

• Since this interpretation is senseless, the conclusion is that a change in resolution implies a new description of niches/distributions.