Dynamic occupancy models [@mackenzie_estimating_2003] allow inference about
the occurrence of "things" at collections of "sites"
and about how changes in occurrence are driven by colonization and
local extinction. These models also account for imperfect detection
probability. Depending on how "thing" and "site" are defined,
occupancy may have vastly different biological meanings,
including the presence of a disease in an individual (disease
incidence) of a species at a site (occurrence, distribution), or of an
individual in a territory.
Dynamic occupancy models in unmarked
are fit using the
function colext
.
All parameters can be modeled as functions of covariates, i.e.,
first-year occupancy with covariates varying by site
(site-covariates),
colonization and survival with site- and yearly-site-covariates and
detection with site-, yearly-site- and sample-occasion-covariates.
We give two commented example analyses: one for a simulated data set
and another for a real data set on crossbills in the Swiss breeding
bird survey MHB.
We also give examples to show how predictions, along with standard
errors and confidence intervals, can be obtained.
Occurrence is a quantity of central importance in many branches of ecology and related sciences. The presence of a disease in an individual or of a species at a site are two common types of occurrence studies. The associated biological metrics are the incidence of the disease and species occurrence or species distribution. Thus, depending on how we define the thing we are looking for and the sample unit, very different biological quantities can be analyzed using statistical models for occupancy.
If we denote presence of the "thing" as $y=1$ and its absence as $y=0$, then it is natural to characterize all these metrics by the probability that a randomly chosen sample unit ("site") is occupied, i.e., has a "thing" present: $Pr(y=1) = \psi$. We call this the occupancy probability, or occupancy for short, and from now on will call the sample unit, where the presence or absence of a "thing" is assessed, generically a "site".
Naturally, we would like to explore factors that affect the likelihood that a site is occupied. A binomial generalized linear model, or logistic regression, is the customary statistical model for occurrence. In this model, we treat occurrence $y$ as a binomial random variable with trial size 1 and success probability $p$, or, equivalently, a Bernoulli trial with $p$. "Success" means occurrence, so $p$ is the occurrence probability. It can be modeled as a linear or other function of covariates via a suitable link function, e.g., the logit link. This simple model is described in many places, including @McCullagh_1989, Royle and Dorazio [-@royle_dorazio:2008, chapter 3], Kéry [-@Kery_2010, chapter 17] and Kéry and Schaub [-@Kery_2011, chapter 3].
A generalization of this model accounts for changes in the occupancy
state of sites by introducing parameters for survival
(or alternatively, extinction) and colonization probability.
Thus, when we have observations of occurrence for more than a single
point in time, we can model the transition of the occupancy
state at site $i$ between successive times as another Bernoulli trial.
To model the fate of an occupied site, we denote the probability that
a site occupied at $t$ is again occupied at $t+1$ as $Pr(y_{i,t+1} = 1
| y_{i,t} = 1 ) = \phi$.
This represents the survival probability of a site that is occupied.
Of course, we could also choose to express this component of occupancy
dynamics by the converse, extinction probability $\epsilon$ ---
the parameterization used in unmarked
.
To model the fate of an unoccupied site, we denote as $Pr(y_{i,t+1} =
1 | y_{i,t} = 0 ) = \gamma$ the probability that an unoccupied site at
$t$ becomes occupied at $t+1$.
This is the colonization probability of an empty site.
Such a dynamic model of occurrence has become famous in the ecological literature under the name "metapopulation model" [@Hanski_1998].
However, when using ecological data collected in the field to fit such models of occurrence, we face the usual challenge of imperfect detection [e.g. @Kery_2008]. For instance, a species can go unobserved at a surveyed site or an occupied territory can appear unoccupied during a particular survey, perhaps because both birds are away hunting. Not accounting for detection error may seriously bias all parameter estimators of a metapopulation model [@Moilanen_2002; @royle_dorazio:2008]. To account for this additional stochastic component in the generation of most ecological field data, the classical metapopulation model may be generalized to include a submodel for the observation process, which allows an occupied site to be recorded as unoccupied. This model has been developed by @mackenzie_estimating_2003. It is described as a hierarchical model by @Royle_2007, Royle and Dorazio [-@royle_dorazio:2008, chapter 9] and Kéry and Schaub [-@Kery_2011, chapter 13]. The model is usually called a multi-season, multi-year or a dynamic site-occupancy model. The former terms denote the fact that it is applied to multiple "seasons" or years and the latter emphasizes that the model allows for between-season occurrence dynamics.
This vignette describes the use of the unmarked
function
colext
to fit dynamic occupancy models. Note that we will use
italics for the names of functions.
Static occupancy models, i.e., for a single season without changes in
the occupancy state [@mackenzie_estimating_2002], can be fit with occu
,
for the model described by @mackenzie_estimating_2002 and @Tyre_2002, and with occuRN
, for the heterogeneity occupancy model
described by @royle_estimating_2003.
In the next section (section 2), we give a more technical description
of the dynamic occupancy model.
In section 3, we provide R code for generating data under a basic
dynamic occupancy model and illustrate use of colext
for fitting the
model.
In section 4, we use real data from the Swiss breeding bird survey MHB
[@schmid_etal:2004] to fit a few more elaborate models with
covariates for all parameters.
We also give examples illustrating how to compute predictions, with
standard errors and 95% confidence intervals, for the parameters.
To be able to estimate the parameters of the dynamic occupancy model (probabilities of occurrence, survival and colonization) separately from the parameters for the observation process (detection probability), replicate observations are required from a period of closure, during which the occupancy state of a site must remain constant, i.e., it is either occupied or unoccupied. The modeled data $y_{ijt}$ are indicators for whether a species is detected at site $i$ ($i = 1, 2, \ldots M$), during replicate survey $j$ ($j = 1, 2, \ldots J$) in season $t$ ($t = 1, 2, \ldots T$). That is, $y_{ijt}=1$ if at least one individual is detected and $y_{ijt}=0$ if none is detected.
The model makes the following assumptions: replicate surveys at a site during a single season are independent (or else dependency must be modeled) occurrence state $z_{it}$ (see below) does not change over replicate surveys at site $i$ during season $t$ * there are no false-positive errors, i.e., a species can only be overlooked where it occurs, but it cannot be detected where it does not in fact occur (i.e., there are no false-positives)
The complete model consists of one submodel to describe the ecological process, or state, and another submodel for the observation process, which is dependent on the result of the ecological process. The ecological process describes the latent occurrence dynamics for all sites in terms of parameters for the probability of initial occurrence and site survival and colonization. The observation process describes the probability of detecting a presence (i.e., $y = 1$) at a site that is occupied and takes account of false-negative observation errors.
This initial state is denoted $z_{i1}$ and represents occurrence at site $i$ during season 1. For this, the model assumes a Bernoulli trial governed by the occupancy probability in the first season $\psi_{i1}$:
$$ z_{i1} = Bernoulli(\psi_{i1}) $$
We must distinguish the sample quantity "occurrence" at a site, $z$, from the population quantity "occupancy probability", $\psi$. The former is the realization of a Bernoulli random variable with parameter $\psi$. This distinction becomes important when we want to compute the number of occupied sites among the sample of surveyed sites; see @Royle_2007 and @Weir_2009 for this distinction.
For all later seasons ($t = 2, 3, \ldots T$), occurrence is a function of occurrence at site $i$ at time $t-1$ and one of two parameters that describe the colonization-extinction dynamics of the system. These dynamic parameters are the probability of local survival $\phi_{it}$, also called probability of persistence (= 1 minus the probability of local extinction), and the probability of colonization $\gamma_{it}$.
$$ z_{it} \sim Bernoulli(z_{i,t-1} \phi_{it} + (1-z_{i,t-1}) \gamma_{it}) $$
Hence, if site $i$ is unoccupied at $t-1$ , $z_{i,t-1}=0$, and the success probability of the Bernoulli is $0\phi_{it} + (1-0) * \gamma_{it}$, so the site is occupied (=colonized) in season $t$ with probability $\gamma_{it}$ . Conversely, if site $i$ is occupied at $t-1$ , $z_{i,t-1}=1$, and the success probability of the Bernoulli is given by $1\phi_{it} + (1-1) * \gamma_{it}$, so the site is occupied in (=survives to) season $t$ with probability $\phi_{it}$.
Occupancy probability ($\psi_{it}$) and occurrence ($z_{it}$) at all later times $t$ can be computed recursively from $\psi_{i1}$, $z_{i1}$ , $\phi_{it}$ and $\gamma_{it}$. Variances of these derived estimates can be obtained via the delta method or the bootstrap.
To account for the observation error (specifically, false-negative observations), the conventional Bernoulli detection process is assumed, such that
$$ y_{ijt} \sim Bernoulli(z_{it} p_{ijt}) $$
Here, $y_{ijt}$ is the detection probability at site $i$ during survey $j$ and season $t$. Detection is conditional on occurrence, and multiplying $p_{ijt}$ with $z_{it}$ ensures that occurrence can only be detected where in fact a species occurs, i.e. where $z_{it}=1$.
The preceding, fully general model description allows for site-($i$) dependence of all parameters. In addition to that, survival and colonization probabilities may be season-($t$)dependent and detection probability season-($t$) and survey-($j$) dependent. All this complexity may be dropped, especially the dependence on sites. On the other hand, all parameters that are indexed in some way can be modeled, e.g., as functions of covariates that vary along the dimension denoted by an index. We will fit linear functions (on the logit link scale) of covariates into first-year occupancy, survival and colonization and into detection probability. That is, for probabilities of first-year occupancy, survival, colonization and detection, respectively, we will fit models of the form $logit(\psi_{i1}) = \alpha + \beta x_i$, where $x_i$ may be forest cover or elevation of site $i$ , $logit(\phi_{it}) = \alpha + \beta x_{it}$, where $x_{it}$ may be tree mast at site $i$ during season $t$, $logit(\gamma_{it}) = \alpha + \beta x_{it}$, for a similarly defined covariate $x_{it}$, or $logit(p_{ijt}) = \alpha + \beta x_{ijt}$ , where $x_{ijt}$ is the Julian date of the survey $j$ at site $i$ in season $t$.
We note that for first-year occupancy, only covariates that vary among
sites ("site covariates") can be fitted, while for survival and
colonization, covariates that vary by site and by season ("yearly
site covariates") may be fitted as well.
For detection, covariates of three formats may be fitted:
"site-covariates", "yearly-site-covariates" and
"observation-covariates", as
they are called in unmarked
.
We first generate a simple, simulated data set with specified, year-specific values for the parameters as well as design specifications, i.e., number of sites, years and surveys per year. Then, we show how to fit a dynamic occupancy model with year-dependence in the parameters for colonization, extinction and detection probability.
To simulate the data, we execute the following R code. The actual values for these parameters for each year are drawn randomly from a uniform distribution with the specified bounds.
M <- 250 # Number of sites J <- 3 # num secondary sample periods T <- 10 # num primary sample periods psi <- rep(NA, T) # Occupancy probability muZ <- z <- array(dim = c(M, T)) # Expected and realized occurrence y <- array(NA, dim = c(M, J, T)) # Detection histories set.seed(13973) psi[1] <- 0.4 # Initial occupancy probability p <- c(0.3,0.4,0.5,0.5,0.1,0.3,0.5,0.5,0.6,0.2) phi <- runif(n=T-1, min=0.6, max=0.8) # Survival probability (1-epsilon) gamma <- runif(n=T-1, min=0.1, max=0.2) # Colonization probability # Generate latent states of occurrence # First year z[,1] <- rbinom(M, 1, psi[1]) # Initial occupancy state # Later years for(i in 1:M){ # Loop over sites for(k in 2:T){ # Loop over years muZ[k] <- z[i, k-1]*phi[k-1] + (1-z[i, k-1])*gamma[k-1] z[i,k] <- rbinom(1, 1, muZ[k]) } } # Generate detection/non-detection data for(i in 1:M){ for(k in 1:T){ prob <- z[i,k] * p[k] for(j in 1:J){ y[i,j,k] <- rbinom(1, 1, prob) } } } # Compute annual population occupancy for (k in 2:T){ psi[k] <- psi[k-1]*phi[k-1] + (1-psi[k-1])*gamma[k-1] }
We have now generated a single realization from the stochastic system thus defined. Figure 1 illustrates the fundamental issue of imperfect detection --- the actual proportion of sites occupied differs greatly from the observed proportion of sites occupied, and because $p$ varies among years, the observed data cannot be used as a valid index of the parameter of interest $\psi_i$.
plot(1:T, colMeans(z), type = "b", xlab = "Year", ylab = "Proportion of sites occupied", col = "black", xlim=c(0.5, 10.5), xaxp=c(1,10,9), ylim = c(0,0.6), lwd = 2, lty = 1, frame.plot = FALSE, las = 1, pch=16) psi.app <- colMeans(apply(y, c(1,3), max)) lines(1:T, psi.app, type = "b", col = "blue", lty=3, lwd = 2) legend(1, 0.6, c("truth", "observed"), col=c("black", "blue"), lty=c(1,3), pch=c(16,1))
To analyze this data set with a dynamic occupancy model in
unmarked
, we first load the package.
library(unmarked)
Next, we reformat the detection/non-detection data from a 3-dimensional array (as generated) into a 2-dimensional matrix with M rows. That is, we put the annual tables of data (the slices of the former 3-D array) sideways to produce a "wide" layout of the data.
yy <- matrix(y, M, J*T)
Next, we create a matrix indicating the year each site was surveyed.
year <- matrix(c('01','02','03','04','05','06','07','08','09','10'), nrow(yy), T, byrow=TRUE)
To organize the data in the format required by colext
, we make
use of the function unmarkedMultFrame
. The only required
arguments are y
, the detection/non-detection data, and
numPrimary
, the number of seasons. The three types of
covariates described earlier can also be supplied using the arguments
siteCovs
, yearlySiteCovs
, and obsCovs
. In this case,
we only make use of the second type, which must have M rows and T
columns.
simUMF <- unmarkedMultFrame( y = yy, yearlySiteCovs = list(year = year), numPrimary=T) summary(simUMF)
## unmarkedFrame Object ## ## 250 sites ## Maximum number of observations per site: 30 ## Mean number of observations per site: 30 ## Number of primary survey periods: 10 ## Number of secondary survey periods: 3 ## Sites with at least one detection: 195 ## ## Tabulation of y observations: ## 0 1 ## 6430 1070 ## ## Yearly-site-level covariates: ## year ## 01 : 250 ## 02 : 250 ## 03 : 250 ## 04 : 250 ## 05 : 250 ## 06 : 250 ## (Other):1000
We are ready to fit a few dynamic occupancy models. We will fit a model with constant values for all parameters and another with full time-dependence for colonization, extinction and detection probability. We also time the calculations.
# Model with all constant parameters m0 <- colext(psiformula= ~1, gammaformula = ~ 1, epsilonformula = ~ 1, pformula = ~ 1, data = simUMF, method="BFGS") summary(m0)
## ## Call: ## colext(psiformula = ~1, gammaformula = ~1, epsilonformula = ~1, ## pformula = ~1, data = simUMF, method = "BFGS") ## ## Initial (logit-scale): ## Estimate SE z P(>|z|) ## -0.813 0.158 -5.16 2.46e-07 ## ## Colonization (logit-scale): ## Estimate SE z P(>|z|) ## -1.77 0.0807 -22 2.75e-107 ## ## Extinction (logit-scale): ## Estimate SE z P(>|z|) ## -0.59 0.102 -5.79 7.04e-09 ## ## Detection (logit-scale): ## Estimate SE z P(>|z|) ## -0.0837 0.0562 -1.49 0.137 ## ## AIC: 4972.597 ## Number of sites: 250 ## optim convergence code: 0 ## optim iterations: 27 ## Bootstrap iterations: 0
The computation time was only a few seconds.
Note that all parameters were estimated on the logit scale. To
back-transform to the original scale, we can simply use the
inverse-logit function, named plogis
in R.
plogis(-0.813)
## [1] 0.3072516
Alternatively, we can use backTransform
, which
computes standard errors using the delta method. Confidence intervals
are also easily obtained using the function confint
.
We first remind ourselves of the names of parameters, which can all be
used as arguments for these functions.
names(m0)
## [1] "psi" "col" "ext" "det"
backTransform(m0, type="psi")
## Backtransformed linear combination(s) of Initial estimate(s) ## ## Estimate SE LinComb (Intercept) ## 0.307 0.0335 -0.813 1 ## ## Transformation: logistic
confint(backTransform(m0, type="psi"))
## 0.025 0.975 ## 0.2457313 0.3765804
Next, we fit the dynamic occupancy model with full year-dependence in the parameters describing occupancy dynamics and also in detection. This is the same model under which we generated the data set, so we would expect accurate estimates.
By default in R, a factor such as year in this analysis, is a parameterized in terms of an intercept and effects representing differences. This would mean that the parameter for the first year is the intercept and the effects would denote the differences between the parameter values in all other years, relative to the parameter value in the first year, which serves as a reference level. This treatment or effects parameterization is useful for testing for differences. For simple presentation, a means parameterization is more practical. It can be specified by adding a -1 to the formula for the time-dependent parameters.
m1 <- colext(psiformula = ~1, # First-year occupancy gammaformula = ~ year-1, # Colonization epsilonformula = ~ year-1, # Extinction pformula = ~ year-1, # Detection data = simUMF) m1
## ## Call: ## colext(psiformula = ~1, gammaformula = ~year - 1, epsilonformula = ~year - ## 1, pformula = ~year - 1, data = simUMF) ## ## Initial: ## Estimate SE z P(>|z|) ## -0.273 0.302 -0.906 0.365 ## ## Colonization: ## Estimate SE z P(>|z|) ## year01 -2.08 0.951 -2.19 2.86e-02 ## year02 -2.18 0.365 -5.96 2.52e-09 ## year03 -1.98 0.274 -7.23 4.88e-13 ## year04 -2.32 0.678 -3.42 6.37e-04 ## year05 -1.89 0.478 -3.95 7.78e-05 ## year06 -1.76 0.294 -5.97 2.44e-09 ## year07 -1.55 0.230 -6.73 1.75e-11 ## year08 -1.43 0.228 -6.29 3.19e-10 ## year09 -2.35 0.470 -5.00 5.64e-07 ## ## Extinction: ## Estimate SE z P(>|z|) ## year01 -1.4209 0.418 -3.401 6.72e-04 ## year02 -0.4808 0.239 -2.009 4.45e-02 ## year03 -1.2606 0.366 -3.440 5.83e-04 ## year04 -0.0907 0.650 -0.139 8.89e-01 ## year05 -0.6456 0.599 -1.078 2.81e-01 ## year06 -0.9586 0.378 -2.539 1.11e-02 ## year07 -1.2279 0.365 -3.362 7.74e-04 ## year08 -1.1894 0.292 -4.076 4.58e-05 ## year09 -0.6292 0.635 -0.991 3.22e-01 ## ## Detection: ## Estimate SE z P(>|z|) ## year01 -1.0824 0.244 -4.434 9.26e-06 ## year02 -0.2232 0.148 -1.508 1.32e-01 ## year03 0.2951 0.154 1.918 5.52e-02 ## year04 0.0662 0.161 0.412 6.81e-01 ## year05 -2.0396 0.433 -4.706 2.52e-06 ## year06 -0.6982 0.232 -3.005 2.66e-03 ## year07 0.2413 0.165 1.466 1.43e-01 ## year08 0.0847 0.155 0.548 5.84e-01 ## year09 0.6052 0.140 4.338 1.44e-05 ## year10 -1.1699 0.306 -3.828 1.29e-04 ## ## AIC: 4779.172
Again, all estimates are shown on the logit-scale. Back-transforming
estimates when covariates, such as year, are present involves an
extra step. Specifically, we need to tell unmarked
the values
of our covariate
at which we want an estimate. This can be done using
backTransform
in combination with linearComb
, although
it can be easier to use predict
. predict
allows the user
to supply a data.frame in which each row represents a combination of
covariate values of interest. Below, we create data.frames called
nd
with each row representing a year.
Then we request yearly estimates of the probability of extinction,
colonization and detection,
and compare them to "truth", i.e., the values with which we
simulated the data set. Note that there are T-1 extinction and
colonization parameters in this case, so we do not need to include
year 10 in nd
.
nd <- data.frame(year=c('01','02','03','04','05','06','07','08','09')) E.ext <- predict(m1, type='ext', newdata=nd) E.col <- predict(m1, type='col', newdata=nd) nd <- data.frame(year=c('01','02','03','04','05','06','07','08','09','10')) E.det <- predict(m1, type='det', newdata=nd)
predict
returns the predictions along with standard errors and
confidence intervals. These can be used to create plots. The
with
function is used to simplify the process of requesting the
columns of data.frame
returned by predict
.
op <- par(mfrow=c(3,1), mai=c(0.6, 0.6, 0.1, 0.1)) with(E.ext, { # Plot for extinction probability plot(1:9, Predicted, pch=1, xaxt='n', xlab='Year', ylab=expression(paste('Extinction probability ( ', epsilon, ' )')), ylim=c(0,1), col=4) axis(1, at=1:9, labels=nd$year[1:9]) arrows(1:9, lower, 1:9, upper, code=3, angle=90, length=0.03, col=4) points((1:9)-0.1, 1-phi, col=1, lwd = 1, pch=16) legend(7, 1, c('Parameter', 'Estimate'), col=c(1,4), pch=c(16, 1), cex=0.8) }) with(E.col, { # Plot for colonization probability plot(1:9, Predicted, pch=1, xaxt='n', xlab='Year', ylab=expression(paste('Colonization probability ( ', gamma, ' )')), ylim=c(0,1), col=4) axis(1, at=1:9, labels=nd$year[1:9]) arrows(1:9, lower, 1:9, upper, code=3, angle=90, length=0.03, col=4) points((1:9)-0.1, gamma, col=1, lwd = 1, pch=16) legend(7, 1, c('Parameter', 'Estimate'), col=c(1,4), pch=c(16, 1), cex=0.8) }) with(E.det, { # Plot for detection probability: note 10 years plot(1:10, Predicted, pch=1, xaxt='n', xlab='Year', ylab=expression(paste('Detection probability ( ', p, ' )')), ylim=c(0,1), col=4) axis(1, at=1:10, labels=nd$year) arrows(1:10, lower, 1:10, upper, code=3, angle=90, length=0.03, col=4) points((1:10)-0.1, p, col=1, lwd = 1, pch=16) legend(7.5, 1, c('Parameter','Estimate'), col=c(1,4), pch=c(16, 1), cex=0.8) })
par(op)
Figure 2 shows that the 95% confidence intervals include the true parameter values, and the point estimates are not too far off.
Estimates of occupancy probability in years $T>1$ must be derived from the
estimates of first-year occupancy and the two parameters governing the
dynamics, extinction/survival and colonization.
unmarked
does this automatically in two ways. First, the
population-level estimates of occupancy probability
$\psi_t = \psi_{t-1}\phi_{t-1} + (1-\phi_{t-1})\gamma$ are calculated
and stored in the slot named \emph{projected}. Slots can be accessed
using the @
operator, e.g. fm@projected
.
In some cases, interest may lie in making
inference about the proportion of the sampled sites that are occupied,
rather than the entire population of sites. These estimates are
contained in the smoothed
slot of the fitted model. Thus, the
projected
values are estimates of population parameters, and
the smoothed
estimates are of the finite-sample
quantities. Discussions of the differences can be found in @Weir_2009.
Bootstrap methods can be used to compute standard errors of derived parameter estimates. Here we employ a non-parametric bootstrap to obtain standard errors of the smoothed estimates of occupancy probability during each year.
m1 <- nonparboot(m1, B = 10) cbind(psi=psi, smoothed=smoothed(m1)[2,], SE=m1@smoothed.mean.bsse[2,])
## psi smoothed SE ## 1 0.4000000 0.4320671 0.06783911 ## 2 0.3493746 0.4110124 0.03786402 ## 3 0.2977125 0.3139967 0.02780818 ## 4 0.3148447 0.3278179 0.04303542 ## 5 0.3192990 0.2316695 0.10858419 ## 6 0.2915934 0.2528485 0.04179036 ## 7 0.3114415 0.2928429 0.03113920 ## 8 0.3636580 0.3504885 0.04224678 ## 9 0.3654064 0.3936991 0.02103870 ## 10 0.3460641 0.3095786 0.06830698
In practice, B
should be much higher, possibly >1000 for complex
models .
Another derived parameters of interest is turnover probability
$$ \tau_t = \frac{\gamma_{t-1}(1-\psi_{t-1})}{\gamma_{t-1}(1-\psi_{t-1}) + \phi_{t-1}\psi_{t-1}} $$
The following function returns these estimates.
turnover <- function(fm) { psi.hat <- plogis(coef(fm, type="psi")) if(length(psi.hat) > 1) stop("this function only works if psi is scalar") T <- getData(fm)@numPrimary tau.hat <- numeric(T-1) gamma.hat <- plogis(coef(fm, type="col")) phi.hat <- 1 - plogis(coef(fm, type="ext")) if(length(gamma.hat) != T-1 | length(phi.hat) != T-1) stop("this function only works if gamma and phi T-1 vectors") for(t in 2:T) { psi.hat[t] <- psi.hat[t-1]*phi.hat[t-1] + (1-psi.hat[t-1])*gamma.hat[t-1] tau.hat[t-1] <- gamma.hat[t-1]*(1-psi.hat[t-1]) / psi.hat[t] } return(tau.hat) }
The bootstrap again offers a means of estimating variance. Here we show how to generate 95\% confidence intervals for the turnover estimates using the parametric bootstrap.
pb <- parboot(m1, statistic=turnover, nsim=2) turnCI <- cbind(pb@t0, t(apply(pb@t.star, 2, quantile, probs=c(0.025, 0.975)))) colnames(turnCI) <- c("tau", "lower", "upper") turnCI
## tau lower upper ## t*1 0.1532645 0.00536045 0.1974714 ## t*2 0.1911530 0.07881180 0.2119585 ## t*3 0.2537292 0.19777204 0.2785973 ## t*4 0.2604356 0.04063769 0.4197328 ## t*5 0.3989303 0.34078483 0.4720357 ## t*6 0.3758690 0.32703698 0.5370796 ## t*7 0.3537473 0.32696166 0.3564059 ## t*8 0.3174983 0.32925238 0.4139696 ## t*9 0.1704449 0.18946470 0.3186236
Which bootstrap method is most appropriate for variance estimation? For detailed distinctions between the non-parametric and the parametric bootstrap, see @Davison_1997. We note simply that the parametric bootstrap resamples from the fitted model, and thus the measures of uncertainty are purely functions of the distributions assumed by the model. Non-parametric bootstrap samples, in contrast, are obtained by resampling the data, not the model, and thus are not necessarily affected by the variance formulas of the model's distributions.
In addition to estimating the variance of an estimate, the parametric bootstrap can be used to assess goodness-of-fit. For this purpose, a fit-statistic, i.e. one that compares observed and expected values, is evaluated using the original fitted model, and numerous other models fitted to simulated datasets. The simulation yields an approximation of the distribution of the fit-statistic, and a \emph{P}-value can be computed as the proportion of simulated values greater than the observed value.
@Hosmer_1997 found that a $\chi^2$ statistic performed reasonably well in assessing lack of fit for logistic regression models. We know of no studies formally evaluating the performance of various fit-statistics for dynamic occupancy models, so this approach should be considered experimental. Fit-statistics applied to aggregated encounter histories offer an alternative approach [@MacKenzie_2004], but are difficult to implement when J*T is high and missing values or continuous covariates are present.
chisq <- function(fm) { umf <- getData(fm) y <- getY(umf) sr <- fm@sitesRemoved if(length(sr)>0) y <- y[-sr,,drop=FALSE] fv <- fitted(fm, na.rm=TRUE) y[is.na(fv)] <- NA sum((y-fv)^2/(fv*(1-fv))) } set.seed(344) pb.gof <- parboot(m0, statistic=chisq, nsim=100) plot(pb.gof, xlab=expression(chi^2), main="", col=gray(0.95), xlim=c(7300, 7700))
Figure 3 indicates that, as expected, the constant parameter model does not fit the data well.
The crossbill data are included with the unmarked
package.
The dataset contains the results of nine years of surveys (1999--2007)
for the European crossbill (Loxia curvirostra),
a pine-seed eating finch, in 267 1-km$^2$ sample quadrats in Switzerland.
Quadrats are surveyed annually as part of the Swiss breeding bird
survey MHB [@schmid_etal:2004].
They are laid out as a grid over Switzerland and surveyed 2 or 3 times
every breeding season (mid-April to late June)
by experienced field ornithologists along a haphazard survey route of
length 1-9 km (average 5 km).
High-elevation sites are only surveyed twice per breeding season.
The data can be loaded into an open R workspace using the data
command.
data(crossbill) colnames(crossbill)
## [1] "id" "ele" "forest" "surveys" "det991" "det992" "det993" ## [8] "det001" "det002" "det003" "det011" "det012" "det013" "det021" ## [15] "det022" "det023" "det031" "det032" "det033" "det041" "det042" ## [22] "det043" "det051" "det052" "det053" "det061" "det062" "det063" ## [29] "det071" "det072" "det073" "date991" "date992" "date993" "date001" ## [36] "date002" "date003" "date011" "date012" "date013" "date021" "date022" ## [43] "date023" "date031" "date032" "date033" "date041" "date042" "date043" ## [50] "date051" "date052" "date053" "date061" "date062" "date063" "date071" ## [57] "date072" "date073"
We have three covariates that vary by site: median elevation of the
quadrat (ele
, in metres), forest cover of the quadrat (forest
, in
percent) and the number of surveys per season (i.e., 2 or 3,
surveys).
These are called site covariates, because they vary by sites only.
The 27 columns entitled det991
- det073
contain the crossbill
detection/nondetection data during all surveys over the 9 years.
They contain a 1 when at least one crossbill was recorded during a
survey and a 0 otherwise.
NA
s indicate surveys that did not take place, either because a site is
high-elevation and has no third survey or because it failed to be
surveyed altogether in a year.
The final 27 columns entitled date991
- date073
give the Julian
date of each survey.
They represent a "survey-covariate" or "observation covariate".
We note that the paper by @Royle_2007 used a subset of this
data set.
AIC-based model selection (see section 5.4) requires
that all models are fit to the same data.
unmarked
removes missing data in a context specific way. For
missing siteCovs
, the entire row of data must be removed. However, for
missing yearlySiteCovs
or obsCovs
, only the
corresponding observation
are removed. Thus, if unmarked
removes different observations
from different models, the models cannot be compared using AIC. A way
around this is to remove the detection data corresponding to
missing covariates before fitting the models.
The crossbill data have missing dates and so we remove the associated
detection/non-detection data.
DATE <- as.matrix(crossbill[,32:58]) y.cross <- as.matrix(crossbill[,5:31]) y.cross[is.na(DATE) != is.na(y.cross)] <- NA
In addition, continuous covariates should be transformed in a way
that brings their values close to zero in order to improve
or even enable numerical convergence of the maximum-likelihood routine.
We do this "by hand" and note that we could also have used the R
function scale
. We subtract the mean and divide by the standard
deviation.
sd.DATE <- sd(c(DATE), na.rm=TRUE) mean.DATE <- mean(DATE, na.rm=TRUE) DATE <- (DATE - mean.DATE) / sd.DATE
Before we can fit occupancy models, we need to format this data set appropriately.
years <- as.character(1999:2007) years <- matrix(years, nrow(crossbill), 9, byrow=TRUE) umf <- unmarkedMultFrame(y=y.cross, siteCovs=crossbill[,2:3], yearlySiteCovs=list(year=years), obsCovs=list(date=DATE), numPrimary=9)
We fit a series of models that represent different hypotheses about
the colonization-extinction dynamics of Swiss crossbills
at a spatial scale of 1 km$^2$.
We fit year effects on colonization and extinction in the means
parameterization,
but for detection probability, we choose an effects parameterization.
The latter is more useful for getting predictions in the presence of
other explanatory variables for that parameter.
For model fm5
with more complex covariate relationships, we use as
starting values for the optimization routine
the solution from a "neighboring" model with slightly less
complexity, model fm4
.
Wise choice of starting values can be decisive for success or failure
of maximum likelihood estimation.
# A model with constant parameters fm0 <- colext(~1, ~1, ~1, ~1, umf) # Like fm0, but with year-dependent detection fm1 <- colext(~1, ~1, ~1, ~year, umf) # Like fm0, but with year-dependent colonization and extinction fm2 <- colext(~1, ~year-1, ~year-1, ~1, umf) # A fully time-dependent model fm3 <- colext(~1, ~year-1, ~year-1, ~year, umf) # Like fm3 with forest-dependence of 1st-year occupancy fm4 <- colext(~forest, ~year-1, ~year-1, ~year, umf) # Like fm4 with date- and year-dependence of detection fm5 <- colext(~forest, ~year-1, ~year-1, ~year + date + I(date^2), umf, starts=c(coef(fm4), 0, 0)) # Same as fm5, but with detection in addition depending on forest cover fm6 <- colext(~forest, ~year-1, ~year-1, ~year + date + I(date^2) + forest, umf)
We can compare models using the Akaike information criterion
($AIC$).
Note that unmarked
yields $AIC$, not $AIC_c$
because the latter would require the sample size,
which is not really known for
hierarchical models such as the dynamic occupancy model.
Model selection and model-averaged prediction in unmarked
require that we create a list of models using fitList
.
This function organizes models and conducts a series of tests to
ensure that the models were fit to the same data.
models <- fitList('psi(.)gam(.)eps(.)p(.)' = fm0, 'psi(.)gam(.)eps(.)p(Y)' = fm1, 'psi(.)gam(Y)eps(Y)p(.)' = fm2, 'psi(.)gam(Y)eps(Y)p(Y)' = fm3, 'psi(F)gam(Y)eps(Y)p(Y)' = fm4, 'psi(F)gam(Y)eps(Y)p(YD2)' = fm5, 'psi(F)gam(Y)eps(Y)p(YD2F)' = fm6) ms <- modSel(models) ms
## nPars AIC delta AICwt cumltvWt ## psi(F)gam(Y)eps(Y)p(YD2F) 30 4986.39 0.00 1.0e+00 1.00 ## psi(F)gam(Y)eps(Y)p(YD2) 29 5059.30 72.91 1.5e-16 1.00 ## psi(F)gam(Y)eps(Y)p(Y) 27 5095.38 108.99 2.2e-24 1.00 ## psi(.)gam(.)eps(.)p(Y) 12 5111.32 124.93 7.5e-28 1.00 ## psi(.)gam(Y)eps(Y)p(Y) 26 5127.63 141.24 2.1e-31 1.00 ## psi(.)gam(Y)eps(Y)p(.) 18 5170.54 184.15 1.0e-40 1.00 ## psi(.)gam(.)eps(.)p(.) 4 5193.50 207.11 1.1e-45 1.00
One model has overwhelming support, so we can base inference on that
one alone. Before doing so, we point out how to extract coefficients
from a fitList
object, and convert the results to a
data.frame
, which could be exported from R.
coef(ms) # Estimates only SE(ms) # Standard errors only toExport <- as(ms, "data.frame") # Everything
Fitted models can be used to predict expected outcomes when given new data. For example, one could ask "how many crossbills would you expect to find in a quadrat with 50% forest cover?" Prediction also offers a way of presenting the results of an analysis. We illustrate by plotting the predictions of $\psi$ and $p$ over the range of covariate values studied. Note that because we standardized date, we need to transform it back to its original scale after obtaining predictions on the standardized scale.
op <- par(mfrow=c(1,2), mai=c(0.8,0.8,0.1,0.1)) nd <- data.frame(forest=seq(0, 100, length=50)) E.psi <- predict(fm6, type="psi", newdata=nd, appendData=TRUE) with(E.psi, { plot(forest, Predicted, ylim=c(0,1), type="l", xlab="Percent cover of forest", ylab=expression(hat(psi)), cex.lab=0.8, cex.axis=0.8) lines(forest, Predicted+1.96*SE, col=gray(0.7)) lines(forest, Predicted-1.96*SE, col=gray(0.7)) }) nd <- data.frame(date=seq(-2, 2, length=50), year=factor("2005", levels=c(unique(years))), forest=50) E.p <- predict(fm6, type="det", newdata=nd, appendData=TRUE) E.p$dateOrig <- E.p$date*sd.DATE + mean.DATE with(E.p, { plot(dateOrig, Predicted, ylim=c(0,1), type="l", xlab="Julian date", ylab=expression( italic(p) ), cex.lab=0.8, cex.axis=0.8) lines(dateOrig, Predicted+1.96*SE, col=gray(0.7)) lines(dateOrig, Predicted-1.96*SE, col=gray(0.7)) })
par(op)
Acknowledgments
Special thanks goes to Ian Fiske, the author of colext
and the
original developer of unmarked
. Andy Royle provided the
initial funding and support for the package. The questions of many
people on the users' list motivated the writing of this document.
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