Introduction to Species Distribution Models

By: Qing Zhao

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What are species distribution models (SDMs)?

“Species distribution models (SDMs) are numerical tools that combine observations of species occurrence or abundance with environmental estimates. They are used to gain ecological and evolutionary insights and to predict distributions across landscapes, sometimes requiring extrapolation in space and time.” – Elith & Leathwick 2009

Elith, J. and Leathwick, J.R., 2009. Species distribution models: ecological explanation and prediction across space and time. Annual review of ecology, evolution, and systematics, 40, pp.677-697.

Types of SDMs

  • Type of statistical models
    • Frequentist
    • Bayesian
    • Machine learning

Types of SDMs, cont’d

  • Type of response variables
    • Presence/absence (occurrence)
    • Count (abundance)
  • Number of response variables
    • One
      • single species
      • Uni-directional interspecific interactions (B -> A)
    • Two or more: multiple species
      • Interspecific similarity
      • Undirectional interspecific interactions (A - B)
      • Bi-directional interspecific interactions (A -> B & B -> A)

Types of SDMs, cont’d

  • Temporal processes represented
    • No: static
    • Yes: dynamic
  • Spatial processes represented
    • No
    • Implicitly (spatial autocorrelation)
    • Explicitly (dynamic)

Basic frequetist SDMs (that no one uses)

  • Occurrence: logistic regression \[y \sim Bernoulli(\pi)\] \[logit(\pi) = X\beta\]

  • Abundance: Poisson regression \[N \sim Poisson(\lambda)\] \[log(\lambda)=X\beta\]

Two directions of extension

Machine learning (non-parametric) Bayesian (parametric)
Non-linear relationship Easy to detect Hard to detect
Observation error Hard to account Easy to account

Today’s focus

  • Occurrence
  • Single species
  • Static, non-spatial

Two issues to consider

  • Non-linear relationships
  • Observation error

Two approaches for one goal

  • Multivariate adaptive regression spline (MARS)
    • Detect non-linear relationship
  • Occupancy model
    • Account for detection error

What is MARS

  • Developed by Friedman (Friedman, J.H., 1991. Multivariate adaptive regression splines. The annals of statistics, pp.1-67.)
  • Connect a bunch of straight segment to represent non-linear relationships

What are occupancy models

  • Developed by MacKenzie and colleagues (MacKenzie, D.I., Nichols, J.D., Lachman, G.B., Droege, S., Andrew Royle, J. and Langtimm, C.A., 2002. Estimating site occupancy rates when detection probabilities are less than one. Ecology, 83(8), pp.2248-2255.)
  • It is a hierarchical Bayesian model that includes an observation sub-model and a process sub-model

\[y_{i,j} \sim Binomial(z_{i} \times p, J_{i})\] Process: \[z_{i} \sim Bernoulli(\pi_{i})\] \[logit(\pi_{i}) = x^T \beta\]

Case study

  • Lion beetle in Alberta, Canada
  • Relate lion beetle occurrence with temperature and forest cover
  • Predict lion beetle distribution

Study area and survey sites

Look at the data

# Slide found here 
load('c:/A. UFL/rmeetup/rmeetup data.RData')
head(data, n=3)

Things can get complicated quickly

Distribution-environment relationship Model
Linear, additive Temperature
Temperature + Forest
Non-linear, additive Temperature ^ 2
Forest ^ 2
Temperature ^ 2 + Forest
Temperature + Forest ^ 2
Temperature ^ 2 + Forest ^ 2

And more

Distribution-environment relationship Model
Interaction Temperature * Forest
(Temperature ^ 2) * Forest
Temperature * (Forest ^ 2)
(Temperature ^ 2) * (Forest ^ 2)

Use MARS to make things easier

library(earth) # This is the library for MARS
# Slide found here 
# Create "occupancy" data for MARS
data$y01 <- ifelse(rowSums(data[,1:4])==0, 0, 1)
head(data, n=3)

Use MARS to analyze the data

Slide for these chunks found here

# Model one, only additive relationships are considered
fit1 <- earth(y01 ~ temperature + forest, data=data, 
              glm=list(family='binomial'), # for occurrence data
              degree=1, # for additive model
              pmethod='backward', nfold=20, keepxy=TRUE) # for k-fold cross validation
# Model two, consider interactions between covariates
fit2 <- earth(y01 ~ temperature + forest, data=data, 
              degree=2, # include level-one interactions
              pmethod='backward', nfold=20, keepxy=TRUE)

Model comparison

Look at the AUC values of cross-validation - slide found here

auc <- cbind(fit1$[1:nfold,1], fit2$[1:nfold,1], 
boxplot(auc, axes=F, ylim=c(0,1), xlab='Model', ylab='AUC')
axis(side=1, at=1:2, labels=c('Temp + Forest','Temp x Forest'))
axis(side=2, at=seq(0,1,.5))

The importance of covariates

# Look at slideshow

Look at the response curves

# Slide found here
npred <- 100
forest.highlow <- c(0, 1)
ypred <- matrix(0, npred, 2)
for (i in 1:2) {
  xpred <- cbind(seq(-3, 3, length.out=npred), 
                 rep(forest.highlow[i], npred))
  ypred[,i] <- inv.logit(predict(object=fit2, newdata=xpred))
xseq <- seq(-3, 3, length.out=npred)
plot(ypred[,1] ~ xseq, type='n', ylim=c(0,1), 
     xlab='Temperature', ylab='Prob. of Occu.')
for (i in 1:2) {lines(ypred[,i] ~ xseq, col=2^i)}
legend('topright', bty='n', lty=1, col=2^c(1:2), 
       legend=c('low','high'), title='Forest')

# Look at presentation for figure 

load('c:/A. UFL/rmeetup/ypred.RData')
npred <- 100
xseq <- seq(-3, 3, length.out=npred)
plot(ypred[,1] ~ xseq, type='n', ylim=c(0,1), 
     xlab='Temperature', ylab='Prob. of Occu.')
for (i in 1:2) {lines(ypred[,i] ~ xseq, col=2^i)}
legend('topright', bty='n', lty=1, col=2^c(1:2), 
       legend=c('low','high'), title='Forest')

What we learned from MARS

  • An interaction between temperature and forest is needed
  • A quandratic form of temperature is needed

Issues yet to solve

  • Observation error

Conduct occupancy modeling

library(unmarked) # This is the library for occupancy models
# Prepare data for "unmarked" <- unmarkedFrameOccu(y=data[,1:4], 

Run “unmarked” to fit the MacKenzie et al. (2002) Occupancy Model

fit <- occu(~1 ~temperature*forest + I(temperature^2)*forest, 
  , knownOcc=which(rowSums(y)==1))

Look at the parameter estimates

# slide found here

Overall detection probability

# slide found here
pobs <- inv.logit(0.007790955)
pobs.grand <- 1 - (1 - pobs) ^ 4

Check the response curve

load('c:/A. UFL/rmeetup/est.RData')
n <- 100
xseq <- seq(-3, 3, length.out=n)
pi0 <- inv.logit(est[1,1] + est[2,1] * xseq + est[4,1] * xseq ^ 2)
pi1 <- inv.logit(est[1,1] + est[2,1] * xseq + est[4,1] * xseq ^ 2 + 
                est[3,1] + est[5,1] * xseq + est[6,1] * xseq ^ 2)
plot(pi0 ~ xseq, type='l', ylim=c(0,1), col=2, 
     xlab='Temperature', ylab='Prob. of Occu.')
lines(pi1 ~ xseq, col=4)
legend('topleft', bty='n', lty=1, col=2^c(1:2), 
       legend=c('low','high'), title='Forest')

Predict the distribution of lion beetles


load('c:/A. UFL/rmeetup/est.RData')
load('c:/A. UFL/2. Alberta/data/original/EnvData.km2.RData')
pi <- inv.logit(est[1,1] + est[2,1] * cov$MAT + est[4,1] * cov$MAT ^ 2 + 
              est[3,1] * cov$forest + est[5,1] * cov$MAT * cov$forest + 
              est[6,1] * cov$MAT ^ 2 * cov$forest)
pi.scale <- round((pi - min(pi)) / (max(pi) - min(pi)) * 9) + 1


plot(cov$lat ~ cov$lon, pch=19, cex=.01, axes=F, 
     xlab='', ylab='', 

What did occupancy do for us?

  • Estimate detection probability
  • Explain beetle-environment relationships
  • Predict beetle distribution

Further reading

  • Stacked Species Distribution Modelling
    • You can use this package to ensemble a number of machine learning approaches such as Generalized additive model (GAM), Multivariate adaptive regression splines (MARS), Generalized boosted regressions model (GBM), Classification tree analysis (CTA), Random forest (RF), Maximum entropy (MAXENT), Artificial neural network (ANN), and Support vector machines (SVM)

Further reading

  • Multi-species modeling
    • Interspecific similarity & undirectional interspecific interactions
      • Ovaskainen, O., Tikhonov, G., Norberg, A., Guillaume Blanchet, F., Duan, L., Dunson, D., Roslin, T. and Abrego, N., 2017. How to make more out of community data? A conceptual framework and its implementation as models and software. Ecology Letters, 20(5), pp.561-576.
    • Uni-directional interspecific interactions
      • Waddle, J.H., Dorazio, R.M., Walls, S.C., Rice, K.G., Beauchamp, J., Schuman, M.J. and Mazzotti, F.J., 2010. A new parameterization for estimating co-occurrence of interacting species. Ecological Applications, 20(5), pp.1467-1475.

Further reading

  • Spatial autocorrelation
    • Reviews
      • Dormann et al., 2007. Methods to account for spatial autocorrelation in the analysis of species distributional data: a review. Ecography, 30(5), pp.609-628.
      • Beale et al., 2010. Regression analysis of spatial data. Ecology letters, 13(2), pp.246-264.
    • Recent advances
      • Crase, B., Liedloff, A.C. and Wintle, B.A., 2012. A new method for dealing with residual spatial autocorrelation in species distribution models. Ecography, 35(10), pp.879-888.
      • Hodges, J.S. and Reich, B.J., 2010. Adding spatially-correlated errors can mess up the fixed effect you love. The American Statistician, 64(4), pp.325-334.

Further reading

  • Dynamic models
    • MacKenzie, D.I., Nichols, J.D., Hines, J.E., Knutson, M.G. and Franklin, A.B., 2003. Estimating site occupancy, colonization, and local extinction when a species is detected imperfectly. Ecology, 84(8), pp.2200-2207.
    • Dail, D. and Madsen, L., 2011. Models for estimating abundance from repeated counts of an open metapopulation. Biometrics, 67(2), pp.577-587.

Further reading

  • Spatially explicit dynamic models
    • Bled, F., Royle, J.A. and Cam, E., 2011. Hierarchical modeling of an invasive spread: the Eurasian Collared-Dove Streptopelia decaocto in the United States. Ecological Applications, 21(1), pp.290-302.
    • Hefley, T.J., Hooten, M.B., Russell, R.E., Walsh, D.P. and Powell, J.A., 2017. When mechanism matters: Bayesian forecasting using models of ecological diffusion. Ecology Letters, 20(5), pp.640-650.
    • Zhao, Q., Royle, J.A. and Boomer, G.S., 2017. Spatially explicit dynamic N-mixture models. Population Ecology, 59(4), pp.293-300.

Thank you

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