occuCOP: Fit the occupancy model using count dta

Description

This function fits a single season occupancy model using count data.

Usage

occuCOP(data, 
        psiformula = ~1, lambdaformula = ~1, 
        psistarts, lambdastarts, starts,
        method = "BFGS", se = TRUE, 
        engine = c("C", "R"), na.rm = TRUE, 
        return.negloglik = NULL, L1 = FALSE, ...)

Value

unmarkedFitOccuCOP object describing the model fit. See the unmarkedFit classes.

Arguments

data: An unmarkedFrameOccuCOP object created with the unmarkedFrameOccuCOP function.
psiformula: Formula describing the occupancy covariates.
lambdaformula: Formula describing the detection covariates.
psistarts: Vector of starting values for likelihood maximisation with optim for occupancy probability $\psi$. These values must be logit-transformed (with qlogis) (see details). By default, optimisation will start at 0, corresponding to an occupancy probability of 0.5 (plogis(0) is 0.5).
lambdastarts: Vector of starting values for likelihood maximisation with optim for detection rate $\lambda$. These values must be log-transformed (with log) (see details). By default, optimisation will start at 0, corresponding to detection rate of 1 (exp(0) is 1).
starts: Vector of starting values for likelihood maximisation with optim. If psistarts and lambdastarts are provided, starts = c(psistarts, lambdastarts).
method: Optimisation method used by optim.
se: Logical specifying whether to compute (se=TRUE) standard errors or not (se=FALSE).
engine: Code to use for optimisation. Either "C" for fast C++ code, or "R" for native R code.
na.rm: Logical specifying whether to fit the model (na.rm=TRUE) or not (na.rm=FALSE) if there are NAs in the unmarkedFrameOccuCOP object.
return.negloglik: A list of vectors of parameters (c(psiparams, lambdaparams)). If specified, the function will not maximise likelihood but return the negative log-likelihood for the those parameters in the nll column of a dataframe. See an example below.
L1: Logical specifying whether the length of observations (L) are purposefully set to 1 (L1=TRUE) or not (L1=FALSE).
...: Additional arguments to pass to optim, such as lower and upper bounds or a list of control parameters.

Author

Léa Pautrel

Details

See unmarkedFrameOccuCOP for a description of how to supply data to the data argument. See unmarkedFrame for a more general documentation of unmarkedFrame objects for the different models implemented in unmarked.

The COP occupancy model

occuCOP fits a single season occupancy model using count data, as described in Pautrel et al. (2023).

The occupancy sub-model is:

$$z_i \sim \text{Bernoulli}(\psi_i)$$

With $z_i$ the occupany state of site $i$. $z_i=1$ if site $i$ is occupied by the species, i.e. if the species is present in site $i$. $z_i=0$ if site $i$ is not occupied.
With $\psi_i$ the occupancy probability of site $i$.

The observation sub-model is:

$$ N_{ij} | z_i = 1 \sim \text{Poisson}(\lambda_{ij} L_{ij}) \\ N_{ij} | z_i = 0 \sim 0 $$

With $N_{ij}$ the count of detection events in site $i$ during observation $j$.
With $\lambda_{ij}$ the detection rate in site $i$ during observation $j$ (for example, 1 detection per day.).
With $L_{ij}$ the length of observation $j$ in site $i$ (for example, 7 days.).

What we call "observation" ($j$) here can be a sampling occasion, a transect, a discretised session. Consequently, the unit of $\lambda_{ij}$ and $L_{ij}$ can be either a time-unit (day, hour, ...) or a space-unit (kilometer, meter, ...).

The transformation of parameters $\psi$ and $\lambda$

In order to perform unconstrained optimisation, parameters are transformed.

The occupancy probability ($\psi$) is transformed with the logit function (psi_transformed = qlogis(psi)). It can be back-transformed with the "inverse logit" function (psi = plogis(psi_transformed)).

The detection rate ($\lambda$) is transformed with the log function (lambda_transformed = log(lambda)). It can be back-transformed with the exponential function (lambda = exp(lambda_transformed)).

References

Pautrel, L., Moulherat, S., Gimenez, O. & Etienne, M.-P. Submitted. Analysing biodiversity observation data collected in continuous time: Should we use discrete or continuous-time occupancy models? Preprint at tools:::Rd_expr_doi("10.1101/2023.11.17.567350").

Examples

Run this code

set.seed(123)
options(max.print = 50)

# We simulate data in 100 sites with 3 observations of 7 days per site.
nSites <- 100
nObs <- 3

# For an occupancy covariate, we associate each site to a land-use category.
landuse <- sample(factor(c("Forest", "Grassland", "City"), ordered = TRUE), 
                  size = nSites, replace = TRUE)
simul_psi <- ifelse(landuse == "Forest", 0.8, 
                    ifelse(landuse == "Grassland", 0.4, 0.1))
z <- rbinom(n = nSites, size = 1, prob = simul_psi)

# For a detection covariate, we create a fake wind variable.
wind <- matrix(rexp(n = nSites * nObs), nrow = nSites, ncol = nObs)
simul_lambda <- wind / 5
L = matrix(7, nrow = nSites, ncol = nObs)

# We now simulate count detection data
y <- matrix(rpois(n = nSites * nObs, lambda = simul_lambda * L), 
            nrow = nSites, ncol = nObs) * z

# We create our unmarkedFrameOccuCOP object
umf <- unmarkedFrameOccuCOP(
  y = y,
  L = L,
  siteCovs = data.frame("landuse" = landuse),
  obsCovs = list("wind" = wind)
)
print(umf)

# We fit our model without covariates
fitNull <- occuCOP(data = umf)
print(fitNull)

# We fit our model with covariates
fitCov <- occuCOP(data = umf, psiformula = ~ landuse, lambdaformula = ~ wind)
print(fitCov)

# We back-transform the parameter's estimates
## Back-transformed occupancy probability with no covariates
backTransform(fitNull, "psi")

## Back-transformed occupancy probability depending on habitat use
predict(fitCov,
        "psi",
        newdata = data.frame("landuse" = c("Forest", "Grassland", "City")),
        appendData = TRUE)

## Back-transformed detection rate with no covariates
backTransform(fitNull, "lambda")

## Back-transformed detection rate depending on wind
predict(fitCov,
        "lambda",
        appendData = TRUE)

## This is not easily readable. We can show the results in a clearer way, by:
##  - adding the site and observation
##  - printing only the wind covariate used to get the predicted lambda
cbind(
  data.frame(
    "site" = rep(1:nSites, each = nObs),
    "observation" = rep(1:nObs, times = nSites),
    "wind" = getData(fitCov)@obsCovs
  ),
  predict(fitCov, "lambda", appendData = FALSE)
)

# We can choose the initial parameters when fitting our model.
# For psi, intituively, the initial value can be the proportion of sites 
#          in which we have observations.
(psi_init <- mean(rowSums(y) > 0))

# For lambda, the initial value can be the mean count of detection events 
#             in sites in which there was at least one observation.
(lambda_init <- mean(y[rowSums(y) > 0, ]))

# We have to transform them.
occuCOP(
  data = umf,
  psiformula = ~ 1,
  lambdaformula = ~ 1,
  psistarts = qlogis(psi_init),
  lambdastarts = log(lambda_init)
)

# If we have covariates, we need to have the right length for the start vectors.
# psi ~ landuse --> 3 param to estimate: Intercept, landuseForest, landuseGrassland
# lambda ~ wind --> 2 param to estimate: Intercept, wind
occuCOP(
  data = umf,
  psiformula = ~ landuse,
  lambdaformula = ~ wind,
  psistarts = rep(qlogis(psi_init), 3),
  lambdastarts = rep(log(lambda_init), 2)
)

# And with covariates, we could have chosen better initial values, such as the
# proportion of sites in which we have observations per land-use category.
(psi_init_covs <- c(
  "City" = mean(rowSums(y[landuse == "City", ]) > 0),
  "Forest" = mean(rowSums(y[landuse == "Forest", ]) > 0),
  "Grassland" = mean(rowSums(y[landuse == "Grassland", ]) > 0)
))
occuCOP(
  data = umf,
  psiformula = ~ landuse,
  lambdaformula = ~ wind,
  psistarts = qlogis(psi_init_covs))

# We can fit our model with a different optimisation algorithm.
occuCOP(data = umf, method = "Nelder-Mead")

# We can run our model with a C++ or with a R likelihood function.
## They give the same result. 
occuCOP(data = umf, engine = "C", psistarts = 0, lambdastarts = 0)
occuCOP(data = umf, engine = "R", psistarts = 0, lambdastarts = 0)

## The C++ (the default) is faster.
system.time(occuCOP(data = umf, engine = "C", psistarts = 0, lambdastarts = 0))
system.time(occuCOP(data = umf, engine = "R", psistarts = 0, lambdastarts = 0))

## However, if you want to understand how the likelihood is calculated,
## you can easily access the R likelihood function.
print(occuCOP(data = umf, engine = "R", psistarts = 0, lambdastarts = 0)@nllFun)

# Finally, if you do not want to fit your model but only get the likelihood,
# you can get the negative log-likelihood for a given set of parameters.
occuCOP(data = umf, return.negloglik = list(
  c("psi" = qlogis(0.25), "lambda" = log(2)),
  c("psi" = qlogis(0.5), "lambda" = log(1)),
  c("psi" = qlogis(0.75), "lambda" = log(0.5))
))

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