fitMLEbouts,numeric-method {diveMove}

Maximum Likelihood Model of mixtures of 2 or 3 Poisson Processes

Description

Functions to model a mixture of 2 random Poisson processes to identify bouts of behaviour. This follows Langton et al. (1995).

Usage

## S4 method for signature 'numeric'
fitMLEbouts(obj, start, optim_opts0 = NULL, optim_opts1 = NULL)

## S4 method for signature 'Bouts'
fitMLEbouts(obj, start, optim_opts0 = NULL, optim_opts1 = NULL)

Arguments

obj

Object of class Bouts.

start

passed to mle. A row- and column-named (2,N) matrix, as returned by boutinit.

optim_opts0

named list of optional arguments passed to mle for fitting the first model with transformed parameters.

optim_opts1

named list of optional arguments passed to mle for fitting the second model with parameters retrieved from the first model, untransformed to original scale.

Details

Mixtures of 2 or 3 Poisson processes are supported. Even in this relatively simple case, it is very important to provide good starting values for the parameters.

One useful strategy to get good starting parameter values is to proceed in 4 steps. First, fit a broken stick model to the log frequencies of binned data (see boutinit), to obtain estimates of 4 parameters in a 2-process model (Sibly et al. 1990), or 6 in a 3-process model. Second, calculate parameter(s) p from the alpha parameters obtained from the broken stick model, to get tentative initial values as in Langton et al. (1995). Third, obtain MLE estimates for these parameters, but using a reparameterized version of the -log L2 function. Lastly, obtain the final MLE estimates for the 3 parameters by using the estimates from step 3, un-transformed back to their original scales, maximizing the original parameterization of the -log L2 function.

boutinit can be used to perform step 1. Calculation of the mixing parameters p in step 2 is trivial from these estimates. Function boutsMLEll.chooser defines a reparameterized version of the -log L2 function given by Langton et al. (1995), so can be used for step 3. This uses a logit (see logit) transformation of the mixing parameter p, and log transformations for both density parameters lambda1 and lambda2. Function boutsMLEll.chooser can be used again to define the -log L2 function corresponding to the un-transformed model for step 4.

fitMLEbouts is the function performing the main job of maximizing the -log L2 functions, and is essentially a wrapper around mle. It only takes the -log L2 function, a list of starting values, and the variable to be modelled, all of which are passed to mle for optimization. Additionally, any other arguments are also passed to mle, hence great control is provided for fitting any of the -log L2 functions.

In practice, step 3 does not pose major problems using the reparameterized -log L2 function, but it might be useful to use method “L-BFGS-B” with appropriate lower and upper bounds. Step 4 can be a bit more problematic, because the parameters are usually on very different scales and there can be multiple minima. Therefore, it is almost always the rule to use method “L-BFGS-B”, again bounding the parameter search, as well as passing a control list with proper parscale for controlling the optimization. See Note below for useful constraints which can be tried.

Value

An object of class mle.

Methods (by class)

• numeric: Fit model via MLE on numeric vector.

• Bouts: Fit model via MLE on Bouts object.

Note

In the case of a mixture of 2 Poisson processes, useful values for lower bounds for the transformed negative log likelihood reparameterization are c(-2, -5, -10). For the un-transformed parameterization, useful lower bounds are rep(1e-08, 3). A useful parscale argument for the latter is c(1, 0.1, 0.01). However, I have only tested this for cases of diving behaviour in pinnipeds, so these suggested values may not be useful in other cases.

The lambdas can be very small for some data, particularly lambda2, so the default ndeps in optim can be so large as to push the search outside the bounds given. To avoid this problem, provide a smaller ndeps value.

Author(s)

Sebastian P. Luque [e-mail]

References

Langton, S.; Collett, D. and Sibly, R. (1995) Splitting behaviour into bouts; a maximum likelihood approach. Behaviour 132, 9-10.

Luque, S.P. and Guinet, C. (2007) A maximum likelihood approach for identifying dive bouts improves accuracy, precision, and objectivity. Behaviour, 144, 1315-1332.

Sibly, R.; Nott, H. and Fletcher, D. (1990) Splitting behaviour into bouts. Animal Behaviour 39, 63-69.

Examples

## Run example to retrieve random samples for two- and three-process
## Poisson mixtures with known parameters as 'Bouts' objects
## ('xbouts2', and 'xbouts3'), as well as starting values from
## broken-stick model ('startval2' and 'startval3')
utils::example("boutinit", package="diveMove", ask=FALSE)

## 2-process
opts0 <- list(method="L-BFGS-B", lower=c(-2, -5, -10))
opts1 <- list(method="L-BFGS-B", lower=c(1e-1, 1e-3, 1e-6))
bouts2.fit <- fitMLEbouts(xbouts2, start=startval2, optim_opts0=opts0,
                          optim_opts1=opts1)
plotBouts(bouts2.fit, xbouts2)

## 3-process
opts0 <- list(method="L-BFGS-B", lower=c(-5, -5, -6, -8, -12))
## We know 0 < p < 1, and can provide bounds for lambdas within an
## order of magnitude for a rough box constraint.
lo <- c(9e-2, 9e-2, 2e-3, 1e-3, 1e-5)
hi <- c(9e-1, 9.9e-1, 2e-1, 9e-2, 5e-3)
## Important to set the step size to avoid running below zero for
## the last lambda.
ndeps <- c(1e-3, 1e-3, 1e-3, 1e-3, 1e-5)
opts1 <- list(method="L-BFGS-B", lower=lo, upper=hi,
              control=list(ndeps=ndeps))
bout3.fit <- fitMLEbouts(xbouts3, start=startval3, optim_opts0=opts0,
                         optim_opts1=opts1)
bec(bout3.fit)
plotBoutsCDF(bout3.fit, xbouts3)