occuMS: Fit Single-Season and Dynamic Multi-State Occupancy Models

Description

This function fits single-season and dynamic multi-state occupancy models with both the multinomial and conditional binomial parameterizations.

Usage

occuMS(detformulas, psiformulas, phiformulas=NULL, data, 
    parameterization=c("multinomial","condbinom"),
    starts, method="BFGS", se=TRUE, engine=c("C","R"), silent=FALSE, ...)

Value

unmarkedFitOccuMS object describing the model fit.

Arguments

detformulas: Character vector of formulas for detection probabilities. See details for a description of how to order these formulas.
psiformulas: Character vector of formulas for occupancy probabilities. See details for a description of how to order these formulas.
phiformulas: Character vector of formulas for state transition probabilities. Only used if you are fitting a dynamic model. See details for a description of how to order these formulas.
data: An unmarkedFrameOccuMS object
parameterization: Either "multinomial" for the multinomial parameterization (MacKenzie et al. 2009) which allows an arbitrary number of occupancy states, or "condbinom" for the conditional binomial parameterization (Nichols et al. 2007) which requires exactly 3 occupancy states. See details.
starts: Vector of parameter starting values.
method: Optimization method used by optim.
se: Logical specifying whether or not to compute standard errors.
engine: Either "C" to use fast C++ code or "R" to use native R code during the optimization.
silent: Boolean; if TRUE, suppress warnings.
...: Additional arguments to optim, such as lower and upper bounds

Author

Ken Kellner contact@kenkellner.com

Details

Traditional occupancy models fit data with exactly two states: detection and non-detection (MacKenzie et al. 2002). The occuMS function fits models to occupancy data for which there are greater than 2 states (Nichols et al 2007, MacKenzie et al. 2009). For example, detections may be further divided into multiple biologically relevant categories, e.g. breeding vs. non-breeding, or some/many individuals present. As with detection status, classification of these additional occupancy states is likely to be imperfect.

Multiple parameterizations for multi-state occupancy models have been proposed. The occuMS function fits two at present: the "conditional binomial" parameterization of Nichols et al. (2007), and the more general "multinomial" parameterization of MacKenzie et al. (2009). Both single-season and dynamic models are possible with occuMS (MacKenzie et al. 2009).

The conditional binomial parameterization (parameterization = 'condbinom') models occupancy and the presence or absence of an additional biological state of interest given the species is present (typically breeding status). Thus, there should be exactly 3 occupancy states in the data: 0 (non-detection); 1 (detection, no evidence of breeding); or 2 (detection, evidence of breeding).

Two state parameters are estimated: $\psi$, the probability of occupancy, and $R$, the probability of successful reproduction given an occupied state (although this could be some other binary biological condition). Covariates (in siteCovs) can be supplied for either or both of these parameters with the stateformulas argument, which takes a character vector of R-style formulas with length = 2, with formulas in the order ($\psi$, $R$). For example, to fit a model where $\psi$ varies with a landcover covariate and $R$ is constant, stateformulas = c('~landcover','~1').

There are three detection parameters associated with the conditional binomial parameterization: $p_1$, the probability of detecting the species given true state 1; $p_2$, the probability of detecting the species given true state 2; and $\delta$, the probability of detecting state 2 (i.e., breeding), given that the species has been detected. See MacKenzie et al. (2009), pages 825-826 for more details. As with occupancy, covariates (in obsCovs) can be supplied for these detection probabilities with the detformulas argument, which takes a character vector of formulas with length = 3 in the order ($p_1$, $p_2$, $\delta$). So, to fit a model where $p_1$ varies with temperature and the other two parameters are constant, detformulas = c('~temp','~1','~1').

The multinomial parameterization (parameterization = "multinomial") is more general, allowing an arbitrary number of occupancy states $S$. $S$ - 1 occupancy probabilities $\psi$ are estimated. Thus, if there are $S$ = 4 occupancy states (0, 1, 2, 3), occuMS estimates $\psi_1$, $\psi_2$, and $\psi_3$ (the probability of state 0 can be obtained by subtracting the others from 1). Covariates can be supplied for each occupancy probability with a character vector with length $S-1$, e.g. stateformulas = c('~landcover','~1','~1') where $\psi_1$ varies with landcover and $\psi_2$ and $\psi_3$ are constant.

The number of detection probabilities estimated quickly expands as $S$ increases, equal to $S \times (S-1) / 2$. In the simplest case (when $S$ = 3), there are 3 detection probabilities: $p_{11}$, the probability of detecting state 1 given true state 1; $p_{12}$, the probability of detecting state 1 given true state 2; and $p_{22}$, the probability of detecting state 2 given true state 2. Covariates can be supplied for any or all of these detection probabilities with the detformulas argument, which takes a character vector of formulas with length = 3 in the order ($p_{11}$, $p_{12}$, $p_{22}$). So, to fit a model where $p_{11}$ varies with temperature and the other two detection probabilities are constant, detformulas = c('~temp','~1','~1'). If there were $S$ = 4 occupancy states, there are 6 estimated detection probabilities and the order is ($p_{11}$, $p_{12}$, $p_{13}$, $p_{22}$, $p_{23}$, $p_{33}$), and so on. See MacKenzie et al. (2009) for a more detailed explanation.

Dynamic (multi-season) models can be fit as well for both parameterizations (MacKenzie et al. 2009). In a standard dynamic occupancy model, additional parameters for probabilities of colonization (i.e., state 0 -> 1) and extinction (1 -> 0) are estimated. In a multi-state context, we must estimate a transition probability matrix ($\phi$) between all possible states. You can provide formulas for some of the probabilities in this matrix using the phiformulas argument. The approach differs depending on parameterization.

For the conditional binomial parameterization, phiformulas is a character vector of length 6. The first three elements are formulas for the probability a site is occupied at time $t$ given that it was previously in states 0, 1, or 2 at time $t-1$ (phi0, phi1, phi2). Elements 4-6 are formulas for the probability of reproduction (or other biological state) given state 0, 1, or 2 at time $t-1$ (R0, R1, R2). See umf@phiOrder$cond_binom for a reminder of the correct order, where umf is your unmarkedFrameOccuMS.

For the multinomial parameterization, phiformulas can be used to provide formulas for some transitions between different occupancy states. You can't give formulas for the probabilities of remaining in the same state between seasons to keep the model identifiable. Thus, if there are 3 possible states (0, 1, 2), phiformulas should contain 6 formulas for the following transitions: p(0->1), p(0->2), p(1->0), p(1->2), p(2->0), p(2->1), in that order (and similar for more than 3 states). The remaining probabilities of staying in the same state between seasons can be obtained via subtraction. See umf@phiOrder$multinomial for the correct order matching the number of states in your dataset.

See unmarkedFrame and unmarkedFrameOccuMS for a description of how to supply data to the data argument.

References

MacKenzie, D. I., J. D. Nichols, G. B. Lachman, S. Droege, J. Andrew Royle, and C. A. Langtimm. 2002. Estimating Site Occupancy Rates When Detection Probabilities Are Less Than One. Ecology 83: 2248-2255.

MacKenzie, D. I., Nichols, J. D., Seamans, M. E., and R. J. Gutierrez, 2009. Modeling species occurrence dynamics with multiple states and imperfect detection. Ecology 90: 823-835.

Nichols, J. D., Hines, J. E., Mackenzie, D. I., Seamans, M. E., and R. J. Gutierrez. 2007. Occupancy estimation and modeling with multiple states and state uncertainty. Ecology 88: 1395-1400.

Examples

Run this code


if (FALSE) {

#Simulate data

#Parameters
N <- 500; J <- 5; S <- 3
site_covs <- matrix(rnorm(N*2),ncol=2)
obs_covs <- matrix(rnorm(N*J*2),ncol=2)
a1 <- -0.5; b1 <- 1; a2 <- -0.6; b2 <- -0.7

##################################
## Multinomial parameterization ##
##################################

p11 <- -0.4; p12 <- -1.09; p22 <- -0.84
truth <- c(a1,b1,a2,b2,p11,0,p12,p22)

#State process
lp <- matrix(NA,ncol=S,nrow=N)
for (n in 1:N){
  lp[n,2] <- exp(a1+b1*site_covs[n,1])
  lp[n,3] <- exp(a2+b2*site_covs[n,2])
  lp[n,1] <- 1  
}
psi_mat <- lp/rowSums(lp)

z <- rep(NA,N)
for (n in 1:N){
  z[n] <- sample(0:2, 1, replace=T, prob=psi_mat[n,])
}

probs_raw <- matrix(c(1,0,0,1,exp(p11),0,1,exp(p12),exp(p22)),nrow=3,byrow=T)
probs_raw <- probs_raw/rowSums(probs_raw)
  
y <- matrix(0,nrow=N,ncol=J)
for (n in 1:N){

  probs <- switch(z[n]+1,
                  probs_raw[1,],
                  probs_raw[2,],
                  probs_raw[3,])
  if(z[n]>0){
    y[n,] <- sample(0:2, J, replace=T, probs)
  }
}

#Construct unmarkedFrame
umf <- unmarkedFrameOccuMS(y=y,siteCovs=as.data.frame(site_covs),
                           obsCovs=as.data.frame(obs_covs))

#Formulas

#3 states, so detformulas is a character vector of formulas of 
#length 3 in following order:
#1) p[11]: prob of detecting state 1 given true state 1
#2) p[12]: prob of detecting state 1 given true state 2
#3) p[22]: prob of detecting state 2 given true state 2
detformulas <- c('~V1','~1','~1')
#If you had 4 states, it would be p[11],p[12],p[13],p[22],p[23],p[33] and so on

#3 states, so stateformulas is a character vector of length 2 in following order:
#1) psi[1]: probability of state 1
#2) psi[2]: probability of state 2
#You can get probability of state 0 (unoccupied) as 1 - psi[1] - psi[2]
stateformulas <- c('~V1','~V2')

#Fit model
fit <- occuMS(detformulas, stateformulas, data=umf,
              parameterization="multinomial")

#Look at results
fit
#Compare with truth
cbind(truth=truth,estimate=coef(fit))

#Generate predicted values
lapply(predict(fit,type='psi'),head)
lapply(predict(fit,type='det'),head)

#Fit a null model
detformulas <- rep('~1',3)
stateformulas <- rep('~1',2)
fit_null <- occuMS(detformulas, stateformulas, data=umf,
                   parameterization="multinomial")

#Compare fits
modSel(fitList(fit,fit_null))

###########################################
## Conditional binomial parameterization ##
###########################################

p11 <- 0.4; p12 <- 0.6; p22 <- 0.8
truth_cb <- c(a1,b1,a2,b2,qlogis(p11),0,qlogis(c(p12,p22)))

#Simulate data

#State process
psi_mat <- matrix(NA,ncol=S,nrow=N)
for (n in 1:N){
  psi_mat[n,2] <- plogis(a1+b1*site_covs[n,1])
  psi_mat[n,3] <- plogis(a2+b2*site_covs[n,2])
}
psi_bin <- matrix(NA,nrow=nrow(psi_mat),ncol=ncol(psi_mat))
psi_bin[,1] <- 1-psi_mat[,2]
psi_bin[,2] <- (1-psi_mat[,3])*psi_mat[,2]
psi_bin[,3] <- psi_mat[,2]*psi_mat[,3]
z <- rep(NA,N)
for (n in 1:N){
  z[n] <- sample(0:2, 1, replace=T, prob=psi_bin[n,])
}

#Detection process
y_cb <- matrix(0,nrow=N,ncol=J)
for (n in 1:N){
  #p11 = p1; p12 = p2; p22 = delta
  probs <- switch(z[n]+1,
                  c(1,0,0),
                  c(1-p11,p11,0),
                  c(1-p12,p12*(1-p22),p12*p22)) 
  if(z[n]>0){
    y_cb[n,] <- sample(0:2, J, replace=T, probs)
  }
}

#Build unmarked frame
umf2 <- unmarkedFrameOccuMS(y=y_cb,siteCovs=as.data.frame(site_covs),
                           obsCovs=as.data.frame(obs_covs))

#Formulas

#detformulas is a character vector of formulas of length 3 in following order:
#1) p[1]: prob of detecting species given true state 1
#2) p[2]: prob of detecting species given true state 2
#3) delta: prob of detecting state 2 (eg breeding) given species was detected
detformulas <- c('~V1','~1','~1')

#stateformulas is a character vector of length 2 in following order:
#1) psi: probability of occupancy
#2) R: probability state 2 (eg breeding) given occupancyc
stateformulas <- c('~V1','~V2')

#Fit model
fit_cb <- occuMS(detformulas, stateformulas, data=umf2,
                 parameterization='condbinom')

#Look at results
fit_cb
#Compare with truth
cbind(truth=truth_cb,estimate=coef(fit_cb))

#Generate predicted values
lapply(predict(fit_cb,type='psi'),head)
lapply(predict(fit_cb,type='det'),head)


##################################
## Dynamic (multi-season) model ##
##################################

#Simulate data-----------------------------------------------
N <- 500 #Number of sites
T <- 3 #Number of primary periods
J <- 5 #Number of secondary periods
S <- 3 #Number of occupancy states (0,1,2)

#Generate covariates
site_covs <- as.data.frame(matrix(rnorm(N*2),ncol=2))
yearly_site_covs <- as.data.frame(matrix(rnorm(N*T*2),ncol=2))
obs_covs <- as.data.frame(matrix(rnorm(N*J*T*2),ncol=2))

#True parameter values
b <- c(
  #Occupancy parameters
  a1=-0.5, b1=1, a2=-0.6, b2=-0.7,
  #Transition prob (phi) parameters
  phi01=0.7, phi01_cov=-0.5, phi02=-0.5, phi10=1.2, 
  phi12=0.3, phi12_cov=1.1, phi20=-0.3, phi21=1.4, phi21_cov=0,
  #Detection prob parameters
  p11=-0.4, p11_cov=0, p12=-1.09, p22=-0.84
)

#Generate occupancy probs (multinomial parameterization)
lp <- matrix(1, ncol=S, nrow=N)
lp[,2] <- exp(b[1]+b[2]*site_covs[,1])
lp[,3] <- exp(b[3]+b[4]*site_covs[,2])
psi <- lp/rowSums(lp)

#True occupancy state matrix
z <- matrix(NA, nrow=N, ncol=T)

#Initial occupancy
for (n in 1:N){
  z[n,1] <- sample(0:(S-1), 1, prob=psi[n,])
}

#Raw phi probs
phi_raw <- matrix(NA, nrow=N*T, ncol=S^2-S)
phi_raw[,1] <- exp(b[5]+b[6]*yearly_site_covs[,1]) #p[0->1]
phi_raw[,2] <- exp(b[7]) #p[0->2]
phi_raw[,3] <- exp(b[8]) #p[1->0]
phi_raw[,4] <- exp(b[9]+b[10]*yearly_site_covs[,2]) #p[1->2]
phi_raw[,5] <- exp(b[11]) #p[2->0]
phi_raw[,6] <- exp(b[12]+b[13]*yearly_site_covs[,1])

#Generate states in times 2..T
px <- 1
for (n in 1:N){
  for (t in 2:T){
    phi_mat <- matrix(c(1, phi_raw[px,1], phi_raw[px,2],  # phi|z=0
                        phi_raw[px,3], 1, phi_raw[px,4],  # phi|z=1
                        phi_raw[px,5], phi_raw[px,6], 1), # phi|z=2
                      nrow=S, byrow=T)
    phi_mat <- phi_mat/rowSums(phi_mat)
    z[n, t] <- sample(0:(S-1), 1, prob=phi_mat[z[n,(t-1)]+1,])
    px <- px + 1
    if(t==T) px <- px + 1 #skip last datapoint for each site
  }
}

#Raw p probs
p_mat <- matrix(c(1, 0, 0, #p|z=0
                  1, exp(b[14]), 0, #p|z=1
                  1, exp(b[16]), exp(b[17])), #p|z=2 
                nrow=S, byrow=T)
p_mat <- p_mat/rowSums(p_mat)

#Simulate observation data
y <- matrix(0, nrow=N, ncol=J*T)
for (n in 1:N){
  yx <- 1
  for (t in 1:T){
    if(z[n,t]==0){
      yx <- yx + J
      next
    }
    for (j in 1:J){
      y[n, yx] <- sample(0:(S-1), 1, prob=p_mat[z[n,t]+1,])
      yx <- yx+1
    }
  }
}
#-----------------------------------------------------------------

#Model fitting

#Build UMF
umf <- unmarkedFrameOccuMS(y=y, siteCovs=site_covs,
                           obsCovs=obs_covs,
                           yearlySiteCovs=yearly_site_covs,
                           numPrimary=3)
summary(umf)

#Formulas
#Initial occupancy
psiformulas <- c('~V1','~V2') #on psi[1] and psi[2]

#Transition probs
#Guide to order:
umf@phiOrder$multinomial
phiformulas <- c('~V1','~1','~1','~V2','~1','~V1')

#Detection probability
detformulas <- c('~V1','~1','~1') #on p[1|1], p[1|2], p[2|2]

#Fit model
(fit <- occuMS(detformulas=detformulas, psiformulas=psiformulas,
              phiformulas=phiformulas, data=umf))

#Compare with truth
compare <- cbind(b,coef(fit),
                 coef(fit)-1.96*SE(fit),coef(fit)+1.96*SE(fit))
colnames(compare) <- c('truth','estimate','lower','upper')
round(compare,3)

#Estimated phi matrix for site 1
phi_est <- predict(fit, 'phi', se.fit=F)
phi_est <- sapply(phi_est, function(x) x$Predicted[1])
phi_est_mat <- matrix(NA, nrow=S, ncol=S)
phi_est_mat[c(4,7,2,8,3,6)] <- phi_est
diag(phi_est_mat) <- 1 - rowSums(phi_est_mat,na.rm=T)

#Actual phi matrix for site 1
phi_act_mat <- diag(S)
phi_act_mat[c(4,7,2,8,3,6)] <- phi_raw[1,]
phi_act_mat <- phi_act_mat/rowSums(phi_act_mat)

#Compare
cat('Estimated phi\n')
phi_est_mat
cat('Actual phi\n')
phi_act_mat

#Rough check of model fit
fit_sim <- simulate(fit, nsim=20)
hist(sapply(fit_sim,mean),col='gray')
abline(v=mean(umf@y),col='red',lwd=2)
#line should fall near middle of histogram

}

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