dCJS {nimbleEcology}

Cormack-Jolly-Seber distribution for use in nimble models

Description

dCJS_** and rCJS_** provide Cormack-Jolly-Seber capture-recapture distributions that can be used directly from R or in nimble models.

Usage

dCJS_ss(x, probSurvive, probCapture, len = 0, log = 0)

dCJS_sv(x, probSurvive, probCapture, len = 0, log = 0)

dCJS_vs(x, probSurvive, probCapture, len = 0, log = 0)

dCJS_vv(x, probSurvive, probCapture, len = 0, log = 0)

rCJS_ss(n, probSurvive, probCapture, len = 0)

rCJS_sv(n, probSurvive, probCapture, len = 0)

rCJS_vs(n, probSurvive, probCapture, len = 0)

rCJS_vv(n, probSurvive, probCapture, len = 0)

Arguments

x

capture history vector of 0s (not captured) and 1s (captured). Include the initial capture, so x[1] should equal 1.

probSurvive

survival probability, either a time-independent scalar (for dCJS_s*) or a time-dependent vector (for dCJS_v*) with length len - 1.

probCapture

capture probability, either a time-independent scalar (for dCJS_*s) or a time-dependent vector (for dCJS_*v) with length len. If a vector, first element is ignored, as the total probability is conditioned on the capture at t = 1.

len

length of capture history. Should equal length(x)

log

TRUE or 1 to return log probability. FALSE or 0 to return probability.

n

number of random draws, each returning a vector of length len. Currently only n = 1 is supported, but the argument exists for standardization of "r" functions.

Details

These nimbleFunctions provide distributions that can be used directly in R or in nimble hierarchical models (via nimbleCode and nimbleModel).

The letters following the 'dCJS_' indicate whether survival and/or capture probabilities, in that order, are scalar (s, meaning the probability applies to every x[t]) or vector (v, meaning the probability is a vector aligned with x). When probCapture and/or probSurvive is a vector, they must be the same length as x.

It is important to use the time indexing correctly for survival. probSurvive[t] is the survival probabilty from time t to time t + 1. When a vector, probSurvive may have length greater than length(x) - 1, but all values beyond that index are ignored.

Time indexing for detection is more obvious: probDetect[t] is the detection probability at time t.

When called from R, the len argument to dCJS_** is not necessary. It will default to the length of x. When used in nimble model code (via nimbleCode), len must be provided (even though it may seem redundant).

For more explanation, see package vignette (vignette("Introduction_to_nimbleEcology")).

Compared to writing nimble models with a discrete latent state for true alive/dead status at each time and a separate scalar datum for each observation, use of these distributions allows one to directly sum (marginalize) over the discrete latent states and calculate the probability of the detection history for one individual jointly.

These are nimbleFunctions written in the format of user-defined distributions for NIMBLE's extension of the BUGS model language. More information can be found in the NIMBLE User Manual at https://r-nimble.org.

When using these distributions in a nimble model, the left-hand side will be used as x, and the user should not provide the log argument.

For example, in nimble model code,

captures[i, 1:T] ~ dCSJ_ss(survive, capture, T)

declares a vector node, captures[i, 1:T], (detection history for individual i, for example) that follows a CJS distribution with scalar survival probability survive and scalar capture probability capture (assuming survive and capture are defined elsewhere in the model).

This will invoke (something like) the following call to dCJS_ss when nimble uses the model such as for MCMC:

dCJS_ss(captures[i, 1:T], survive, capture, len = T, log = TRUE)

If an algorithm using a nimble model with this declaration needs to generate a random draw for captures[i, 1:T], it will make a similar invocation of rCJS_ss, with n = 1.

If both survival and capture probabilities are time-dependent, use

captures[i,1:T] ~ dCSJ_vv(survive[1:(T-1)], capture[1:T], T)

and so on for each combination of time-dependent and time-independent parameters.

Value

For dCJS_**: the probability (or likelihood) or log probability of observation vector x.

For rCJS_**: a simulated capture history, x.

Notes for use with automatic differentiation

The dCJS_** distributions should all work for models and algorithms that use nimble's automatic differentiation (AD) system. In that system, some kinds of values are "baked in" (cannot be changed) to the AD calculations from the first call, unless and until the AD calculations are reset. For the dCJS_** distributions, the lengths of vector inputs and the data (x) values themselves are baked in. These can be different for different iterations through a for loop (or nimble model declarations with different indices, for example), but the lengths and data values for each specific iteration will be "baked in" after the first call. In other words, it is assumed that x are data and are not going to change.

Author(s)

Ben Goldstein, Perry de Valpine, and Daniel Turek

References

D. Turek, P. de Valpine and C. J. Paciorek. 2016. Efficient Markov chain Monte Carlo sampling for hierarchical hidden Markov models. Environmental and Ecological Statistics 23:549–564. DOI 10.1007/s10651-016-0353-z

See Also

For multi-state or multi-event capture-recapture models, see dHMM or dDHMM.

Examples

# Set up constants and initial values for defining the model
dat <- c(1,1,0,0,0) # A vector of observations
probSurvive <- c(0.6, 0.3, 0.3, 0.1)
probCapture <- 0.4


# Define code for a nimbleModel
nc <- nimbleCode({
  x[1:4] ~ dCJS_vs(probSurvive[1:4], probCapture, len = 4)
  probCapture ~ dunif(0,1)
  for (i in 1:4) probSurvive[i] ~ dunif(0, 1)
})

# Build the model, providing data and initial values
CJS_model <- nimbleModel(nc, data = list(x = dat),
                         inits = list(probSurvive = probSurvive,
                                      probCapture = probCapture))

# Calculate log probability of data from the model
CJS_model$calculate()
# Use the model for a variety of other purposes...