lnre.bootstrap {zipfR}

Parametric bootstrapping for LNRE models (zipfR)

Description

This function implements parametric bootstrapping for LNRE models, i.e. it draws a specified number of random samples from the population described by a given lnre object. For each sample, two callback functions are applied to perform transformations and/or extract statistics. In an important application (bootstrapped confidence intervals for model parameters), the first callback estimates a new LNRE model and the second callback extracts the relevant parameters from this model. See ‘Use Cases’ and ‘Examples’ below for other use cases.

Usage

lnre.bootstrap(model, N, ESTIMATOR, STATISTIC, 
               replicates=100, sample=c("spc", "tfl", "tokens"),
               simplify=TRUE, verbose=TRUE, parallel=1L, seed=NULL, ...)

Arguments

model

a trained LNRE model, i.e. an object belonging to a subclass of lnre. The model must provide a rlnre method to generate random samples from the underlying frequency distribution.

N

a single positive integer, specifying the size N (i.e. token count) of the individual bootstrap samples

ESTIMATOR

a callback function, normally used for estimating LNRE models in the bootstrap procedure. It is called once for each bootstrap sample with the sample as first argument (in the form determined by sample). Additional arguments (...) are passed on to the callback, so it is possible to use ESTIMATOR=lnre with appropriate settings. If this step is not needed, set ESTIMATOR=identity to pass samples through to the STATISTIC callback.

STATISTIC

a callback function, normally used to extract model parameters and other relevant statistics from the bootstrapped LNRE models. It is called once for each bootstrap sample, with the value returned by ESTIMATOR as its single argument. The return values are automatically aggregated across all bootstrap samples (see ‘Value’ below). If this step is not needed, set STATISTIC=identity in order to pass through the results of the ESTIMATOR callback. Note that STATISTIC must not return NULL, which is used internally to signal errors.

replicates

a single positive integer, specifying the number of bootstrap samples to be generated

sample

the form in which each sample is passed to ESTIMATOR: as a frequency spectrum (spc, the default), as a type-frequency list (tfl) or as a factor vector representing the token sequence (tokens). Warning: The latter can be computationally expensive for large N.

Alternatively, a callback function that will be invoked with arguments model and replicates and must return a random sample in the format expected by ESTIMATOR. See ‘Use Cases’ below for typical applications.

simplify

if TRUE, use rbind() to combine list of results into a single data structure. In this case, the estimator should return either a vector of fixed length or a single-row data frame or matrix. No validation is carried out before attempting the simplification.

verbose

if TRUE, show progress bar in R console during the bootstrapping process (which can take a long time). The progress bar may be updated quite infrequently if parallel processing is enabled.

parallel

whether to enable parallel processing. Either an integer specifying the number of worker processes to be forked, or a pre-initialised snow cluster created with makeCluster; see ‘Details’ below.

seed

a single integer value used to initialize the RNG in order to generate reproducible results

...

any further arguments are passed through to the ESTIMATOR callback function

Details

The parametric bootstrapping procedure works as follows:

1. replicates random samples of N tokens each are drawn from the population described by the LNRE model model (possibly using a callback function provided in argument sample)

2. Each sample is passed to the callback function ESTIMATOR in the form determined by sample (a frequency spectrum, type-frequency list, or factor vector of tokens). If ESTIMATOR fails, it is re-run with a different sample, otherwise the return value is passed on to STATISTIC. Use ESTIMATOR=identity to pass the original sample through to STATISTIC.

3. The callback function STATISTIC is used to extract relevant information for each sample. If STATISTIC fails, the procedure is repeated from step 2 with a different sample. The callback will typically return a vector of fixed length or a single-row data frame, and the results for all bootstrap samples are combined into a matrix or data frame if simplify=TRUE.

Warning: Keep in mind that sampling a token vector can be slow and consume large amounts of memory for very large N (several million tokens). If possible, use sample="spc" or sample="tfl", which can be generated more efficiently.

Parallelisation

Since bootstrapping is a computationally expensive procedure, it is usually desirable to use parallel processing. lnre.bootstrap supports two types of parallelisation, based on the parallel package:

• On Unix platforms, you can set parallel to an integer number in order to fork the specified number of worker processes, utilising multiple cores on the same machine. The detectCores function shows how many cores are available, but due to hyperthreading and memory contention, it is often better to set parallel to a smaller value. Note that forking may be unstable especially in a GUI environment, as explained on the mcfork manpage.

• On all platforms, you can pass a pre-initialised snow cluster in the argument, which consists of worker processes on the same machine or on different machines. A suitable cluster can be created with makeCluster; see the parallel package documentation for further information. It is your responsibility to set up the cluster so that all required data sets, packages and custom functions are available on the worker processes; lnre.bootstrap will only ensure that the zipfR package itself is loaded.

Note that parallel processing is not enabled by default and will only be used if parallel is set accordingly.

Value

If simplify=FALSE, a list of length replicates containing the statistics obtained from each individual bootstrap sample. In addition, the following attributes are set:

• N = sample size of the bootstrap replicates

• model = the LNRE model from which samples were generated

• errors = number of samples for which either the ESTIMATOR or the STATISTIC callback produced an error

If simplify=TRUE, the statistics are combined with rbind(). This is performed unconditionally, so make sure that STATISTIC returns a suitable value for all samples, typically vectors of the same length or single-row data frames with the same columns. The return value is usually a matrix or data frame with replicates rows. No additional attributes are set.

Use cases

Bootstrapped confidence intervals for model parameters:

The confint method for LNRE models uses bootstrapping to estimate confidence intervals for the model parameters.

For this application, ESTIMATOR=lnre re-estimates the LNRE model from each bootstrap sample. Configuration options such as the model type, cost function, etc. are passed as additional arguments in ..., and the sample must be provided in the form of a frequency spectrum. The return values are successfully estimated LNRE models.

STATISTIC extracts the model parameters and other coefficients of interest (such as the population diversity S) from each model and returns them as a named vector or single-row data frame. The results are combined with simplify=TRUE, then empirical confidence intervals are determined for each column.

Empirical sampling distribution of productivity measures:

For some of the more complex measures of productivity and lexical richness (see productivity.measures), it is difficult to estimate the sampling distribution mathematically. In these cases, an empirical approximation can be obtained by parametric bootstrapping.

The most convenient approach is to set ESTIMATOR=productivity.measures, so the desired measures can be passed as an additional argument measures= to lnre.bootstrap. The default sample="spc" is appropriate for most measures and is efficient enough to carry out the procedure for multiple sample sizes.

Since the estimator already returns the required statistics for each sample in a suitable format, set STATISTIC=identity and simplify=TRUE.

Empirical prediction intervals for vocabulary growth curves:

Vocabulary growth curves can only be generated from token vectors, so set sample="tokens" and keep N reasonably small.

ESTIMATOR=vec2vgc compiles vgc objects for the samples. Pass steps or stepsize as desired and set m.max if growth curves for V_1, V_2, \ldots are desired.

Either use STATISTIC=identity and simplify=FALSE to return a list of vgc objects, which can be plotted or processed further with sapply(). This strategy is particulary useful if one or more V_m are desired in addition to V.

Or use STATISTIC=function (x) x$V to extract y-coordinates for the growth curve and combine them into a matrix with simplify=TRUE, so that prediction intervals can be computed directly. Note that the corresponding x-coordinates are not returned and have to be inferred from N and stepsize.

Simulating non-randomness and mixture distributions:

More complex populations and non-random samples can be simulated by providing a user callback function in the sample argument. This callback is invoked with parameters model and n and has to return a sample of size n in the format expected by ESTIMATOR.

For simulating non-randomness, the callback will typically use rlnre to generate a random sample and then apply some transformation.

For simulating mixture distributions, it will typically generate multiple samples from different populations and merge them; the proportion of tokens from each population should be determined by a multinomial random variable. Individual populations might consist of LNRE models, or a finite number of “lexicalised” types. Note that only a single LNRE model will be passed to the callback; any other parameters have to be injected as bound variables in a local function definition.

See Also

lnre for more information about LNRE models. The high-level estimator function lnre uses lnre.bootstrap to collect data for approximate confidence intervals; lnre.productivity.measures uses it to approximate the sampling distributions of productivity measures.

Examples

## parametric bootstrapping from realistic LNRE model
model <- lnre("zm", spc=ItaRi.spc) # has quite a good fit

## estimate distribution of V, V1, V2 for sample size N=1000
res <- lnre.bootstrap(model, N=1000, replicates=200,
                      ESTIMATOR=identity,
                      STATISTIC=function (x) c(V=V(x), V1=Vm(x,1), V2=Vm(x,2)))
bootstrap.confint(res, method="normal")
## compare with theoretical expectations (EV/EVm = center, VV/VVm = spread^2)
lnre.spc(model, 1000, m.max=2, variances=TRUE)

## lnre.bootstrap() also captures and ignores occasional failures
res <- lnre.bootstrap(model, N=1000, replicates=200,
                      ESTIMATOR=function (x) if (runif(1) < .2) stop() else x,
                      STATISTIC=function (x) c(V=V(x), V1=Vm(x,1), V2=Vm(x,2)))

## empirical confidence intervals for vocabulary growth curve
## (this may become expensive because token-level samples have to be generated)
res <- lnre.bootstrap(model, N=1000, replicates=200, sample="tokens",
                      ESTIMATOR=vec2vgc, stepsize=100, # extra args passed to ESTIMATOR
                      STATISTIC=V) # extract vocabulary sizes at equidistant N
bootstrap.confint(res, method="normal")

## parallel processing is highly recommended for expensive bootstrapping
library(parallel)
## adjust number of processes according to available cores on your machine
cl <- makeCluster(2) # PSOCK cluster, should work on all platforms
res <- lnre.bootstrap(model, N=1e4, replicates=200, sample="tokens",
                      ESTIMATOR=vec2vgc, stepsize=1000, STATISTIC=V,
                      parallel=cl) # use cluster for parallelisation
bootstrap.confint(res, method="normal")
stopCluster(cl)

## on MacOS / Linux, simpler fork-based parallelisation also works well
## Not run: 
res <- lnre.bootstrap(model, N=1e5, replicates=400, sample="tokens",
                      ESTIMATOR=vec2vgc, stepsize=1e4, STATISTIC=V,
                      parallel=8) # if you have enough cores ...
bootstrap.confint(res, method="normal")

## End(Not run)