discrete talk

Let’s not be discrete with our SDMs

Let’s not be discrete with our SDMsSlides on Slideshare:

http://www.slideshare.net/oharar/discrete-talk

Bob O’Hara1

1BiK-F, Biodiversity and Climate Change Research CentreFrankfurt am Main

GermanyTwitter:

@bobohara


A ”Real” Curve

0 20 40 60 80 100

020

4060

80

Curve


Approximated with a Discretised Curve

0 20 40 60 80 100

020

4060

80

CurveDiscrete


Better: linear interpolation

0 20 40 60 80 100

020

4060

80

CurveDiscreteInterpolated


With more points, the approximations improve

0 20 40 60 80 100

020

4060

80

CurveDiscreteInterpolated


What does this have to do with distribution models?


What does this have to do with distribution models?

This is how SDMs see the world:

source: http://bit.ly/1l8sG7M

Map produced by Peter Blancher, Science and Technology Branch, Environment Canada, based on data from the

North American Breeding Bird Survey


Problems: scale, within-grid heterogeneity


Let’s sidestep the whole problem

Work in continuous space insteadThe maths will let us work on different scales

I e.g. Renner & Warton (2013) doi:10.1111/j.1541-0420.2012.01824.x

Lets us deal with points & irregular shapesMakes it straightforward to include different sorts of data


Practical Motivation

Map Of Life

www.mol.org/

Different data sources

I GBIF

I expert range maps

I eBird and similarcitizen science efforts

I organised surveys(BBS, BMSs)

I Regional checklists


A Unified Model

There is a single state - density of the species

Actual State

PresenceAbsence

PresenceOnly

ExpertRangeMaps

�� ?

HHHj


Point Processes: Model

Each point in space, ξ, has anintensity, ρ(ξ)

log(ρ(ξ)) = η(ξ) =∑

βX (ξ)+ν(ξ)

The number of individuals in anarea A follows a Poissondistibution with mean

λ(A) =

∫Aρ(ξ)ds


Point Processes: Reality

Approximate λ(ξ) numerically:select some integration points,and sum over those

λ(A) ≈N∑

s=1

|A(s)|eη(s)


Observation Models

Presence only pointsAbundancePresence/AbsenceExpert range


Observation Models: point counts

Assume a small area A: η(ξ) is constant, and observation for atime t, then

n(A, t) ∼ Po(eµ(A,t))

with

µA(A, t) = η(A) + log(|A|) + log(t) + log(p)

where p is the probability of observing each individual.Don’t know all of |A|, t and p, so estimate an interceptCan also add a sampling model to log(p)


Observation Models: presence/absence

This is just the abundance model simplified so that we observezero or more than zero individuals

Pr(n(A, t) > 0) = 1− eeµ(A,t)

or

cloglog(Pr(n(A, t) > 0)) = µ(A, t)


Presence only: point process

log Gaussian Cox ProcessLikelihood is a Poisson GLM (but with non-integer response)


Areal Presence/absence

If an area is large enough, we can’t assume constant covariates, so

Pr(n(A) > 0) = 1− e∫A eρ(ξ)dξ

in practice this is calculated as

1− e∑

s |A(s)|eρ(s)

which causes problems with the fitting


Expert Range Maps

Not the same as areal presence.Instead, use distance to range asa covariate

I within range, this is 0.

I Have to estimate the slopefor outside the range

Use informative priors to forcethe slope to be negative 0 20 40 60 80 100

0.0

0.2

0.4

0.6

0.8

1.0

Space (1d)

Inte

nsity

Species'Range


Put these together

Data likelihoods: P(Xi |λ) for data Xi . Total likelihood is

P(X) =∏i

P(Xi |λ)P(λ)

Where P(λ) is the actual distribution model, and will depend onenvironmental and other covariates


In practice

Be Bayesian. Could use MCMC, but this is quicker in INLA

SolTim.res <- inla(SolTim.formula,

family=c('poisson','binomial'),

data=inla.stack.data(stk.all),

control.family = list(list(link = "log"),

list(link = "cloglog")),

control.predictor=list(A=inla.stack.A(stk.all)),

Ntrials=1, E=inla.stack.data(stk.all)$e, verbose=FALSE)


Red-naped sapsucker

source: Len Blumin,https://flic.kr/p/iTaNSx


Data

−120 −100 −90 −80 −70

2030

4050

60

x

Breeding Bird Survey

−120 −100 −90 −80 −70

2030

4050

60

x

y

eBird

−120 −100 −90 −80 −70

2030

4050

60

GBIF

−120 −100 −90 −80 −70

2030

4050

60

y

Range Maps

Sphyrapicus nuchalis

I expert range

I 2 pointprocesses(eBird, GBIF)

I NorthAmerican BBS


A Fitted Model

mean sd Lower CI Upper CI

altitude -0.20 0.02 -0.24 -0.17annual pre 0.24 0.02 0.21 0.28

annual tem -0.06 0.02 -0.10 -0.02prec seas -0.03 0.01 -0.06 -0.00

temp seas 0.15 0.02 0.12 0.18forest 0.00 0.02 -0.04 0.05

pine -0.00 0.02 -0.03 0.03fir -0.05 0.01 -0.08 -0.03

spruce 0.00 0.01 -0.01 0.02iucnbirds -0.25 0.02 -0.28 -0.22jetzmaps -0.06 0.02 -0.10 -0.02


A Pretty Map

−120 −110 −100 −90 −80 −70

2030

4050

60

−0.99

−0.49

0.01

0.51

1.01


Individual Data Types

−1.5 −1.0 −0.5 0.0 0.5

Estimate

altitude

annual pre

annual tem

prec seas

temp seas

forest

pine

fir

spruce

eBirdGBIFBBS


Individual Data Types: The Maps

−120 −100 −80 −70

2030

4050

60

−0.99

−0.49

0.01

0.51

1.01

All Data

−120 −100 −80 −70

2030

4050

60−0.180.020.220.420.62

eBird

−120 −100 −80 −70

2030

4050

60

−0.13

−0.07

−0.01

0.05

GBIF

−120 −100 −80 −70

2030

4050

60

−25

−21

−17

−13

−9

BBS


Join the bandwagon!

Using continuous space - makes lifeeasierPulls together different ideas in oneframework


Not the final answer...

http://www.gocomics.com/nonsequitur/2014/06/24

More to do:

I Finish the paper (and R package)

I Get areas working

I Multi-species models