Complex models in ecology: challenges and solutions

Motivation Bayesian tools Data cloning HPC dcmle PVAClone Next
Complex models in ecology: challenges and solutions
Péter Sólymos
with K. Nadeem and S. R. Lele
University of Alberta
41st Annual Meeting of the SSC
Recent developments in R packages
May 27, 2013

Complex models are everywhere
ˆ Ecology is the scientic study of interactions of organisms with one another and
with their environment.
ˆ Data is growing fast, models are becoming more complex.
ˆ We need complex models for dealing with:
ˆ non-independence (spatial, temporal, phylogenetic),ˆ missing data,ˆ observation and measurement error.

Hierarchical models
ˆ Inference:
ˆ (y | X = x) ∼ h(y ; X = x, θ1)
ˆ X ∼ g (x; θ2)
ˆ θ = (θ1, θ2)
ˆ L(θ; y ) = h(y | x; θ1)g (x; θ2)dx
ˆ Computation:
ˆ high dimensional integral hard to calculate,ˆ noisy likelihood surface numerical search is hard,ˆ second derivatives hard to calculate.ˆ life is hard if you are a frequentist.

The Bayesian toolkit
ˆ MCMC is easy because others did the heavy lifting:
ˆ WinBUGS, OpenBUGSˆ JAGSˆ STAN
ˆ Great interfaces with R:
ˆ R2WinBUGS, R2OpenBUGS, BRugsˆ coda, rjagsˆ rstan
ˆ Inference based on the posterior distribution:
ˆ π(θ | y ) =
L(θ;y)π(θ)
L(θ;y)π(θ)dθ
,
ˆ π(θ) is the prior distribution.

NormalNormal model
ˆ Yij | µij ∼ Normal(µij, σ2)
ˆ i = 1, . . . , n; j = 1, . . . , mn
ˆ µij = XT
ij θ + i
ˆ i ∼ Normal(0, τ2)
model {
for (ij in 1:nm) { #### - likelihood
Y[ij] ~ dnorm(mu[ij], 1/sigma^2)
mu[ij] - inprod(X[ij,], theta) + e[gr[ij]]
}
for (i in 1:n) {
e[i] ~ dnorm(0, 1/tau^2)
}
for (k in 1:np) { #### - priors
theta[k] ~ dnorm(0, 0.001)
}
sigma ~ dlnorm(0, 0.001)
tau ~ dlnorm(0, 0.001)
}

NormalNormal model
library(rjags)
library(dclone)
set.seed(1234)
theta - c(1, -1)
sigma - 0.6
tau - 0.3
n - 50 # number of clusters
m - 10 # number of repeats within each cluster
nm - n * m # total number of observations
gr - rep(1:n, each = m) # group membership defining clusters
x - rnorm(nm) # covariate
X - model.matrix(~x) # design matrix
e - rnorm(n, 0, tau) # random effect
mu - drop(X %*% theta) + e[gr] # mean
Y - rnorm(nm, mu, sigma) # outcome

JAGS
dat - list(Y = Y, X = X, nm = nm, n = n, np = ncol(X), gr = gr)
str(dat)
List of 6
$ Y : num [1:500] 1.669 0.34 -0.474 3.228 0.968 ...
$ X : num [1:500, 1:2] 1 1 1 1 1 1 1 1 1 1 ...
..- attr(*, dimnames)=List of 2
.. ..$ : chr [1:500] 1 2 3 4 ...
.. ..$ : chr [1:2] (Intercept) x
..- attr(*, assign)= int [1:2] 0 1
$ nm: num 500
$ n : num 50
$ np: int 2
$ gr: int [1:500] 1 1 1 1 1 1 1 1 1 1 ...
m - jags.fit(dat, c(theta, sigma, tau), model, n.update = 2000)

JAGS
summary(m)
Iterations = 3001:8000
Thinning interval = 1
Number of chains = 3
Sample size per chain = 5000
1. Empirical mean and standard deviation for each variable,
plus standard error of the mean:
Mean SD Naive SE Time-series SE
sigma 0.581 0.0193 0.000158 0.000225
tau 0.279 0.0410 0.000335 0.000739
theta[1] 0.959 0.0479 0.000391 0.000959
theta[2] -1.032 0.0260 0.000213 0.000230
2. Quantiles for each variable:
2.5% 25% 50% 75% 97.5%
sigma 0.546 0.568 0.581 0.594 0.621
tau 0.205 0.250 0.276 0.305 0.365
theta[1] 0.862 0.927 0.959 0.991 1.052
theta[2] -1.083 -1.050 -1.032 -1.014 -0.981

JAGS
sigma
0.15 0.40 −1.15 −0.95
0.55
0.150.40
20
40
tau
20 10
theta[1]
0.81.1
0.55
−1.15−0.95
50
20
0.8 1.1
10
theta[2]

Data cloning (DC)
ˆ Basic results1:
ˆ y
(K)
= (y , . . . , y )
ˆ L(θ; y
K) = L(θ; y )K
ˆ πK(θ | y ) =
[L(θ;y)]Kπ(θ)
[L(θ;y)]Kπ(θ)dθ
,
ˆ πK(θ | y ) ∼ MVN(ˆθ, 1
K I
−1(ˆθ))
ˆ Implications:
ˆ we can use Bayesian MCMC toolkit for frequentist inference,ˆ mean of the posterior is the MLE (ˆθ),ˆ K times the posterior variance is the variance of the MLE.ˆ High dimensional integral no need to calculate,ˆ noisy likelihood surface no numerical optimization involved,ˆ second derivatives no need to calculate.ˆ This is independent of the specication of the prior distribution.
1Lele et al. 2007 ELE; Lele et al. 2010 JASA

Iterative model tting
str(dclone(dat, n.clones = 2, unchanged = np, multiply = c(nm,
+ n)))
List of 6
$ Y : atomic [1:1000] 1.669 0.34 -0.474 3.228 0.968 ...
..- attr(*, n.clones)= atomic [1:1] 2
.. ..- attr(*, method)= chr rep
$ X : num [1:1000, 1:2] 1 1 1 1 1 1 1 1 1 1 ...
..- attr(*, dimnames)=List of 2
.. ..$ : chr [1:1000] 1_1 2_1 3_1 4_1 ...
.. ..$ : chr [1:2] (Intercept) x
$ nm: atomic [1:1] 1000
.. ..- attr(*, method)= chr multi
$ n : atomic [1:1] 100
.. ..- attr(*, method)= chr multi
$ np: int 2
$ gr: atomic [1:1000] 1 1 1 1 1 1 1 1 1 1 ...
mk - dc.fit(dat, c(theta, sigma, tau), model, n.update = 2000,
+ n.clones = c(1, 2, 4, 8), unchanged = np, multiply = c(nm,
+ n))

Marginal posterior summaries
sigma
Number of clones
1 2 4 8
0.540.60
x x x x
x
x
x
x
tau
Number of clones
Estimate
1 2 4 8
0.200.35
x
x x x
x
x x x
theta[1]
1 2 4 8
0.901.00
x x x x
x x x x
theta[2]Estimate
1 2 4 8
−1.08−1.00
x
x
x x
x
x
x x

DC diagnostics
dcdiag(mk)
n.clones lambda.max ms.error r.squared r.hat
1 1 0.002150 0.04172 0.003217 1.001
2 2 0.002234 0.08290 0.006286 1.008
3 4 0.002322 0.10398 0.010156 1.001
4 8 0.002117 0.08042 0.006313 1.012
ˆ It can help in dientifying the number of clones required.
ˆ Non-identiability can be spotted as a bonus.

Computational demands
1.0 2.0 3.0 4.0
04812
Number of chains
Processingtime(sec)
1 2 3 4 5 6 7 8
0204060
Number of clones
Processingtime(sec)

HPC to the rescue!
ˆ MCMC is labelled as embarassingly parallel problem.
ˆ Distributing independent chanins to workers:
ˆ Proper initialization (initial values, RNGs),ˆ run the chains on workers,ˆ collect results (large MCMC object might mean more communication overhead),ˆ repeat this for dierent number of clones.
ˆ The paradox of burn-in:
ˆ one long chain: burn-in happens only once,ˆ few-to-many chains: best trade-o w.r.t. burn-in,ˆ n.iter chains: burn-in happens n.iter times (even with ∞ chains).
ˆ Computing time should drop to 1/n.iter + overhead.
ˆ This works for Bayesian analysis and DC.
ˆ Learning can happen, i.e. results can be used to make priors more informative
(this improves mixing and reduces burn-in).

Workload optimization
ˆ Size balancing for DC:
ˆ start with largest problem chighest K ,ˆ run smaller problems on other workers.ˆ collect results (collect whole MCMC object for highest K , only summaries for
others),
ˆ Learning is not an option here, need to have good guesses or rely on
non-informative priors.
ˆ Can be combined with the parallel chains approach.
ˆ Computing time is dened by the highest K problem.

Size balancing
0 2 4 6 8 12
2
1
No Balancing
Max = 12
Approximate Processing Time
Workers
1 2
3 4 5
0 2 4 6 8 12
2
1
Size Balancing
Max = 8
Approximate Processing Time
Workers
5 2 1
4 3

Types of parallelism
## snow type clusters
cl - makeCluster(3)
m - jags.parfit(cl, ...)
mk - dc.parfit(cl, ...)
stopCluster(cl)

## muticore type forking
m - jags.parfit(3, ...)
mk - dc.parfit(3, ...)
ˆ Parallel chains approach not available for WinBUGS/OpenBUGS,
ˆ dc.parfit allows size balancing for WinBUGS/OpenBUGS.
ˆ Forking does not work on Windows.
ˆ (STAN: all works, see R-Forge2)
2http://dcr.r-forge.r-project.org/extras/stan.t.R

Random Number Generation
ˆ WinBUGS/OpenBUGS: seeds approach does not guarantee independence of very
long chains.
ˆ STAN uses L'Ecuyer RNGs.
ˆ JAGS: uses 4 RNGs in base module, but there is the lecuyer module which
allows high number of independent chains.
## 'base::BaseRNG' factory
str(parallel.inits(NULL, 2))
List of 2
$ :List of 2
..$ .RNG.name : chr base::Marsaglia-Multicarry
..$ .RNG.state: int [1:2] 2087563718 113920118
$ :List of 2
..$ .RNG.name : chr base::Super-Duper
..$ .RNG.state: int [1:2] 1968324500 1720729645
## 'lecuyer::RngStream' factory
load.module(lecuyer, quiet = TRUE)
str(parallel.inits(NULL, 2))
List of 2
$ :List of 2
..$ .RNG.name : chr lecuyer::RngStream
..$ .RNG.state: int [1:6] -1896643356 145063650 -1913397488 -341376786 297806844 1434416058
$ :List of 2
..$ .RNG.name : chr lecuyer::RngStream
..$ .RNG.state: int [1:6] -1118475959 1854133089 -159660578 -80247816 -567553258 -1472234812
unload.module(lecuyer, quiet = TRUE)

HMC in STAN
ˆ Another way to cut back on burn-in is to use Hamiltonian Monte Carlo and
related algorithms (No-U-Turn sampler, NUTS).
ˆ This is also helpful when parameters are correlated.
ˆ See http://mc-stan.org/.
ˆ DC support exists3: currently not through CRAN because rstan is not hosted on
CRAN (might never be).
3http://dcr.r-forge.r-project.org/extras/stan.t.R

The two mind-sets
1. Analytic mid-set
ˆ use a predened general model,ˆ possibly t it to many similar data sets,ˆ not that interested in algorithms (...),ˆ want something like this:
FIT - MODEL(y ~ x, DATA, ...)
2. Algorithmic mid-set
ˆ t a specic modelˆ to a specic data set,ˆ more focus on algorithmic settings:
DATA - list(y = y, x = x)
MODEL - y ~ x
FIT - WRAPPER(DATA, MODEL, ...)
How do we provide estimating procedures for folks with an analytic mind-set?

sharx: an example
sharx is a package to t hierarchical speciesare relationship models (a kind of
multivariate mixed model for mata-analysis).
library(sharx)
hsarx
function (formula, data, n.clones, cl = NULL, ...)
{
if (missing(n.clones))
stop('n.clones' argument missing)
if (missing(data))
data - parent.frame()
tmp - parse_hsarx(formula, data)
dcf - make_hsarx(tmp$Y, tmp$X, tmp$Z, tmp$G)
dcm - dcmle(dcf, n.clones = n.clones, cl = cl, nobs = length(tmp$Y),
...)
out - as(dcm, hsarx)
title - if (ncol(tmp$X) 2)
SARX
else SAR
if (!is.null(tmp$Z)) {
if (title != SARX NCOL(tmp$Z) 1)
title - paste(title, X, sep = )
title - paste(H, title, sep = )
}
out@title - paste(title, Model)
out@data - dcf
out
}
environment: namespace:sharx

dcmle
ˆ The dcmle package was motivated by stats4:::mle and the modeltools
package.
ˆ Wanted to provide:
ˆ a wrapper around wrappers around wrappers (another abstraction layer),ˆ unied S4 object classes for data and tted models for Bayesian analysis and DC,ˆ lots of methods for access, coercion, summaries, plots.
ˆ This is the engine for package development with DC.
ˆ Classic BUGS examples:
module glm loaded
library(dcmle)
as.character(listDcExamples()$topic)
[1] blocker bones dyes epil
[5] equiv leuk litters lsat
[9] mice oxford pump rats
[13] salm seeds air alli
[17] asia beetles biops birats
[21] cervix dugongs eyes hearts
[25] ice jaw orange pigs
[29] schools paramecium

seeds example
sourceDcExample(seeds)
seeds
Formal class 'dcFit' [package dcmle] with 10 slots
..@ multiply : chr N
..@ unchanged: NULL
..@ update : NULL
..@ updatefun: NULL
..@ initsfun : NULL
..@ flavour : chr jags
..@ data :List of 5
.. ..$ N : num 21
.. ..$ r : num [1:21] 10 23 23 26 17 5 53 55 32 46 ...
.. ..$ n : num [1:21] 39 62 81 51 39 6 74 72 51 79 ...
.. ..$ x1: num [1:21] 0 0 0 0 0 0 0 0 0 0 ...
.. ..$ x2: num [1:21] 0 0 0 0 0 1 1 1 1 1 ...
..@ model :function ()
..@ params : chr [1:5] alpha0 alpha1 alpha2 alpha12 ...
..@ inits :List of 5
.. ..$ tau : num 1
.. ..$ alpha0 : num 0
.. ..$ alpha1 : num 0
.. ..$ alpha2 : num 0
.. ..$ alpha12: num 0

seeds example
custommodel(seeds@model)
Object of class custommodel:
model
{
alpha0 ~ dnorm(0.00000E+00, 1.00000E-06)
alpha1 ~ dnorm(0.00000E+00, 1.00000E-06)
alpha2 ~ dnorm(0.00000E+00, 1.00000E-06)
alpha12 ~ dnorm(0.00000E+00, 1.00000E-06)
tau ~ dgamma(0.001, 0.001)
sigma - 1/sqrt(tau)
for (i in 1:N) {
b[i] ~ dnorm(0.00000E+00, tau)
logit(p[i]) - alpha0 + alpha1 * x1[i] + alpha2 * x2[i] +
alpha12 * x1[i] * x2[i] + b[i]
r[i] ~ dbin(p[i], n[i])
}
}

seeds example
dcm - dcmle(seeds, n.clones = 1:3, n.iter = 1000)
summary(dcm)
Maximum likelihood estimation with data cloning
Call:
dcmle(x = seeds, n.clones = 1:3, n.iter = 1000)
Settings:
start end thin n.iter n.chains n.clones
1001 2000 1 1000 3 3
Coefficients:
Estimate Std. Error z value Pr(|z|)
alpha0 -0.5556 0.1738 -3.20 0.0014 **
alpha1 0.0981 0.2909 0.34 0.7360
alpha12 -0.8319 0.3947 -2.11 0.0350 *
alpha2 1.3547 0.2542 5.33 9.8e-08 ***
sigma 0.2450 0.1244 1.97 0.0490 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Convergence:
n.clones lambda.max ms.error r.squared r.hat
1 0.494 NA NA 1.05
2 0.184 NA NA 1.04
3 0.130 NA NA 1.02

PVAClone
ˆ Likelihood based population viability analysis in the presence of observation error
and missing data4.
ˆ Builds upon JAGS, dclone, and dcmle.
ˆ Growth models: Xt | σ2 ∼ Normal(m(Xt−1, η), σ2)
ˆ gompertz: m(Xt−1) = Xt−1 + a + bXt−1,
ˆ ricker: m(Xt−1) = Xt−1 + a + be
Xt−1 ,ˆ bevertonholt,ˆ thetalogistic,ˆ thetalogistic_D.
ˆ Observation error: Yt ∼ f (yt; Xt, Ψ)
ˆ none,ˆ poisson,ˆ normal,
4Nadeem and Lele 2012 OIKOS

Growth model objects
library(PVAClone)
gm - ricker(normal, fixed = c(sigma = 0.5))
str(gm)
Formal class 'pvamodel' [package PVAClone] with 14 slots
..@ growth.model: chr ricker
..@ obs.error : chr normal
..@ model :Class 'custommodel' chr [1:20] model { for (i in 1:kk) { N[1,i] -
..@ genmodel :function ()
..@ p : int 4
..@ support : num [1:4, 1:2] -Inf -Inf 2.22e-16 2.22e-16 Inf ...
.. ..- attr(*, dimnames)=List of 2
.. .. ..$ : chr [1:4] a b sigma tau
.. .. ..$ : chr [1:2] Min Max
..@ params : chr [1:3] a b lntau
..@ varnames : chr [1:4] a b sigma tau
..@ fixed : Named num 0.5
.. ..- attr(*, names)= chr sigma
..@ fancy : chr [1:2] Ricker Normal
..@ transf :function (mcmc, obs.error)
..@ backtransf :function (mcmc, obs.error)
..@ logdensity :function (logx, mle, data, null_obserror = FALSE, alt_obserror = FALSE)
..@ neffective :function (obs)

Model with xed parameters
gm@model
Object of class custommodel:
model {
for (i in 1:kk) {
N[1,i] - exp(y[1,i])
x[1,i] - y[1,i]
for (j in 2:T) {
x[j,i] ~ dnorm(mu[j,i], prcx)
mu[j,i] - a + b * N[j-1,i] + x[j-1,i]
N[j,i] - min(exp(x[j,i]), 10000)
y[j,i] ~ dnorm(x[j,i], prcy)
}
}
sigma - 0.5
tau - exp(lntau)
lnsigma - log(sigma)
lntau ~ dnorm(0, 1)
a ~ dnorm(0, 0.01)
b ~ dnorm(0, 10)
prcx - 1/sigma^2
prcy - 1/tau^2
}

Likelihood ratio test (DCLR)
m1 - pva(redstart, gompertz(normal), 50, n.update = 2000,
+ n.iter = 1000)
m2 - pva(redstart, ricker(normal), 50, n.update = 2000, n.iter = 1000)
ms - model.select(m1, m2)
coef(m2)
a b sigma tau
0.07159 -0.01721 0.05096 0.58996
ms
PVA Model Selection:
Time series with 30 observations (missing: 0)
Null Model: m1
Gompertz growth model with Normal observation error
Alternative Model: m2
Ricker growth model with Normal observation error
log_LR delta_AIC delta_BIC delta_AICc
1 -249.6 499.3 499.4 499.3
Alternative Model is strongly supported over the Null Model

Prole likelihood
alt - pva(redstart, ricker(normal, fixed = c(sigma = 0.05)),
+ 50, n.update = 2000, n.iter = 1000)
p - generateLatent(alt, n.chains = 1, n.iter = 10000)
a - c(-0.1, -0.05, 0, 0.05, 0.1, 0.15, 0.2)
llr_res - numeric(length(a))
for (i in seq_len(length(a))) {
+ null - pva(redstart, ricker(normal, fixed = c(a = a[i],
+ sigma = 0.05)), 50, n.update = 2000, n.iter = 1000)
+ llr_res[i] - pva.llr(null, alt, pred = p)
+ }

Prole likelihood
−0.10 −0.05 0.00 0.05 0.10 0.15 0.20
−300−200−1000
a
Profilelog−likelihood

What's next?
ˆ Things done:
ˆ DC support for OpenBUGS, WinBUGS, JAGS, STAN.ˆ Support for parallel computing.ˆ dcmle engine for package development (sharx, PVAClone, and soon detect).
ˆ Things to do:
ˆ Full integration with STAN (dc.fit, dcmle).ˆ More examples.ˆ Prediction/forecasting features for PVAClone.
ˆ Find out more:
ˆ Sólymos 2010 R Journal [PDF]ˆ http://dcr.r-forge.r-project.org/

Complex models in ecology: challenges and solutions

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