diveMove (version 0.8.2)

bouts2MLE: Maximum Likelihood Model of mixture of 2 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

bouts2.mleFUN(x, p, lambda1, lambda2)
bouts2.ll(x)
bouts2.LL(x)
bouts.mle(ll.fun, start, x, ...)
bouts2.mleBEC(fit)
plotBouts2.mle(fit, x, xlab="x", ylab="Log Frequency", bec.lty=2, ...)
plotBouts2.cdf(fit, x, draw.bec=FALSE, bec.lty=2, ...)

Arguments

x

Numeric vector with values to model.

p, lambda1, lambda2

Parameters of the mixture of Poisson processes.

ll.fun

function returning the negative of the maximum likelihood function that should be maximized. This should be a valid minuslogl argument to mle.

start, ...

Arguments passed to mle. For plotBouts2.cdf, arguments passed to plot.ecdf. For plotBouts2.mle, arguments passed to

fit

mle object.

xlab, ylab

Titles for the x and y axes.

bec.lty

Line type specification for drawing the BEC reference line.

draw.bec

Logical; do we draw the BEC?

Value

• bouts.mle returns an object of class mle.

  bouts2.mleBEC and bouts2.mleFUN return a numeric vector.

  bouts2.LL and bouts2.ll return a function.

  plotBouts2.mle and plotBouts2.cdf return nothing, but produce a plot as side effect.

Details

For now only a mixture of 2 Poisson processes is 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 corresponding to a 2-process model (Sibly et al. 1990). Second, calculate parameter p from the 2 alpha parameters obtained from the broken stick model, to get 3 tentative initial values for the 2-process model from Langton et al. (1995). Third, obtain MLE estimates for these 3 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 parameter p in step 2 is trivial from these estimates. Function bouts2.LL is 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 bouts2.ll is the -log L2 function corresponding to the un-transformed model, hence can be used for step 4.

bouts.mle 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. Therefore, it is almost always the rule to use method L-BFGS-B, again with 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.

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, in press.

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

See Also

mle, optim, logit, unLogit for transforming and fitting a reparameterized model.

Examples

data(divesSummary)
postdives <- divesSummary$postdive.dur[divesSummary$trip.no == 2]
postdives.diff <- abs(diff(postdives))

## Remove isolated dives
postdives.diff <- postdives.diff[postdives.diff < 2000]
lnfreq <- boutfreqs(postdives.diff, bw=0.1, plot=FALSE)
startval <- boutinit(lnfreq, 50)
p <- startval$a1 / (startval$a1 + startval$a2)

## Fit the reparameterized (transformed parameters) model
init.parms <- list(p=logit(p), lambda1=log(startval$lambda1),
                   lambda2=log(startval$lambda2))
bout.fit1 <- bouts.mle(bouts2.LL, start=init.parms, x=postdives.diff,
                       method="L-BFGS-B", lower=c(-2, -5, -10))
coefs <- as.vector(coef(bout.fit1))

## Un-transform and fit the original parameterization
init.parms <- list(p=unLogit(coefs[1]), lambda1=exp(coefs[2]),
                   lambda2=exp(coefs[3]))
bout.fit2 <- bouts.mle(bouts2.ll, x=postdives.diff, start=init.parms,
                       method="L-BFGS-B", lower=rep(1e-08, 3),
                       control=list(parscale=c(1, 0.1, 0.01)))
plotBouts(bout.fit2, postdives.diff)

## Plot cumulative frequency distribution
plotBouts2.cdf(bout.fit2, postdives.diff)

## Estimated BEC
bec2(bout.fit2)