qle: Simulated quasi-likelihood parameter estimation
In mbaaske/qle: Simulation-Based Quasi-Likelihood Estimation

Description Usage Arguments Details Value Author(s) See Also Examples

This is the main function of the simulated quasi-likelihood estimation (QLE) approach.

qle(qsd, sim, ..., nsim, x0 = NULL, obs = NULL, Sigma = NULL,
  global.opts = list(), local.opts = list(), method = c("qscoring",
  "bobyqa", "direct"), qscore.opts = list(), control = list(),
  errType = "kv", multistart = FALSE, pl = 0, cl = NULL, iseed = NULL,
  plot = FALSE)

`qsd`	object of class `QLmodel`
`sim`	simulation function, see details
`...`	further arguments passed to the simulation function '`sim`'
`nsim`	optional, number of simulation replications at each new sample point, '`qsd$nsim`' (default)
`x0`	optional, numeric vector of starting parameters
`obs`	optional, numeric vector of observed statistics, overwrites '`qsd$obs`'
`Sigma`	optional, constant variance matrix estimate of statistics (see details)
`global.opts`	options for global search phase
`local.opts`	options for local search phase
`method`	vector of names of local search methods
`qscore.opts`	list of control arguments passed to `qscoring`
`control`	list of control arguments passed to any of the routines defined in '`method`'
`errType`	type of prediction variances, choose one of "`kv,cv,max`" (see details)
`multistart`	logical, `FALSE` (default), whether to search for local minimia or roots from multiple starting points at global phase
`pl`	print level, use `pl`>0 to print intermediate results
`cl`	cluster object, `NULL` (default), of class "`MPIcluster`", "`SOCKcluster`", "`cluster`"
`iseed`	integer seed, `NULL` (default) for default seeding of the random number generator (RNG) stream for each worker in the cluster
`plot`	if `TRUE`, plot newly sampled points (for 2D-parameter estimation problems only)

The function sequentially estimates the unknown model parameter. Basically, the user supplies a simulation function 'sim' which must return a vector of summary statistics (as the outcome of model simulations) and expects a vector of parameters as its first argument. Further arguments can be passed by the '...' argument. The object 'qsd' aggregates the type of variance matrix approximation, the data frame of observed and simulated data, the initial sample points and the covariance models of the involved statistics (see QLmodel). In addition, it defines the criterion function by 'qsd$criterion', which is either used to monitor the sampling process or minimized itself. The user also has the option to choose among different types of prediction variances: either "kv" (kriging variances), "cv" (CV variances) or the maximum of both, by "max", are available.

Criterion functions

The QD criterion function follows the quasi-likelihood estimation principle (see vignette) and seeks a solution to the quasi-score equation. Besides, the Mahalanobis distance (MD) as an alternative (simulation-based) criterion function has a more direct interpretation. It can be seen as a (weighted or generalized) least squares criterion depending on the employed type of variance matrix approximation. For this reason, we support several types of variance matrix approximations. In particular, given 'Sigma' and setting 'qsd$var.type' equal to "const" treats 'Sigma' as a constant estimate throughout the whole estimation procedure. Secondly, if 'Sigma' is supplied and used as an average variance approximation (see covarTx), it is considered an initial variance matrix approximation and recomputed each time an approximate (local) minimizer of the criterion function is found. This is commonly known as an iterative update strategy of variance matrices in the context of GMM estimation. Opposed to this, setting 'qsd$var.type' equal to "kriging" corresponds to continuously updating the variance matrix each time a new criterion function value is required at any point of the parameter space. In this way the algorithm can also be seen as a simulated version of a least squares method or even as a special case of a simulated method of moments (see, e.g. [3]). Note that some input combinations concerning the variance approximation types are not applicable since the criterion "qle", which uses the QD criterion function, does not support a constant variance matrix at all.

Monte Carlo (MC) hypothesis testing

The algorithm sequentially evaluates promising local minimizers of the criterion function during the local phase in order to assess the plausibility of being an approximate root of the corresponding quasi-score vector. We use essentially the same MC test procedure as in qleTest. First, having found a local minimum of the test statistic, i.e. the criterion function, given the data, new observations are simulated w.r.t. to the local minimizer and the algorithm re-estimates the approximate roots for each observation independently. If the current minimizer is accepted as an approximate root at the significance level 'local.opts$alpha', then the algorithm stays in its local phase and continues sampling around the current minimizer accoring to its asymptotic variance (measured by the inverse of the predicted quasi-information) and uses the additional simulations to improve the current kriging approximations. Otherwise we switch to the global phase and do not consider the current minimizer as an approximate root.

This procedure also allows for a stopping condition derived from the reults of the MC test. We can compare the estimated mean squared error (MSE) with the predicted error of the approximate root by its relative difference and terminate in case this value drops below a user-defined bound 'perr_tol' (either as a scalar value or numeric vector of length equal to the dimension of the unknown parameter). A value close to zero suggests a good match of both error measures. The testing procedure is disabled by default. Use 'local.opts$test=TRUE' for testing approximate roots. A value of the criterion function smaller than 'local.opts$ftol_abs' indicates that the corresponding minimizer could be an approximate root. Otherwise the last evaluation point is used as a starting point for next local searches which mimics a random multistart type minimization over the next iterations of the algorithm. This behaviour is also implemented for results of the above MC test when the local minimizer is not accepted as an approximate root. Note that this approach has the potential to escape regions where the criterion function value is quite low but, however, is not considered trustworthy in relation to the upper bound 'local.opts$ftol_abs' or the results of the MC test procedure.

If one of the other termination criteria is met in conjunction with a neglectable value of the criterion function, we say that the algorithm successfully terminated and converged to a local minimizer of the criterion function which could be an approximate root of the quasi-score vector. We then can perform a goodness-of-fit test in order to assess its plausibility (see qleTest) and quantify the empirical and predicted estimation error. If we wish to improve the final estimate the algorithm allows for a simple warm start strategy though not yet as an fully automated procedure. The algorithm can be easily restarted based on the final result of the preceeding run. We only need to extract the object 'OPT$qsd' as an input argument to function qle again.

Sampling new points

Our QLE approach dynamically switches from a local to a global search phase and vise versa for sampling new promising candidates for evaluation, that is, performing new simulations of the statistical model. Depending on the current value of the criterion function three different sampling criteria are used to select next evaluation points which aim on potentially improving the quasi-score or criterion function approximation. If a local minimizer of the criterion function has been accepted as an approximate root, then a local search tries to improve its accuracy. The next evaluation point is either selected according to a weighted minimum-distance criterion (see [2] and vignette), for the choice 'nextSample' equal to "score", or by maximizing the weighted variance of the quasi-score vector in case 'nextSample' is equal to "var". In all other cases, for example, if identifiable roots of the QS could not be found or the (numerical) convergence of the local solvers failed, the global phase of the algorithm is invoked and selects new potential candidates accross the whole search space based on a weighted selection criterion. This assigns large weights to candidates with low criterion function values and vise versa. During both phases the cycling between local and global candidates is controlled by the weights 'global.opts$weights' and 'locals.opts$weights', respectively. Besides this, the smaller the weights the more the candidates tend to be globally selected and vice versa during the global phase. Within the local phase, weights approaching one result in selecting candidates close to the current minimizer of the criterion function. Weights approximately zero maximize the minimum distance between candidates and previously sampled points and thus densely sample the search space almost everywhere if the algorithm is allowed to run infinitely. The choice of weights is somewhat ad hoc but may reflect the users preference on guiding the whole estimation more biased towards either a local or global search. In addition the local weights can be dynamically adjusted if 'useWeights' is FALSE depending on the current progress of estimation. In this case the first weight given by 'locals.opts$weights' is initially used for this kind of adjustment.

Some notes: For a 2D parameter estimation problem the function can visualize the sampling and selection process, which requires an active 2D graphical device in advance. The function can also be run in an cluster environment using the 'parallel' package. Make sure to export all functions to the cluster environment 'cl' beforehand, loading required packages on each cluster node, which are used in the simulation function (see clusterExport and clusterApply). If no cluster object is supplied, a local cluster is set up based on forking (under Linux) or as a socket connection for other OSs. One can also set an integer seed value 'iseed' to initialize each worker, see clusterSetRNGStream, for reproducible results of estimation in case a local cluster is used, i.e. cl=NULL and option mc.cores>1. If using a prespecified cluster object cl, then the user is responsible for seeding whereas the seed can be stored in the return value as well, see attribute 'optInfo$iseed'.

The following controls 'local.opts' for the local search phase are available:

ftol_rel: upper bound on relative change in criterion function values
lam_max: upper bound on the maximum eigenvalue of the generalized eigenvalue decomposition of the quasi-information matrix and estimated interpolation error (variance) of quasi-score. This stops the main iteration sampling new locations following the idea that in this case the quasi-score interpolation error has dropped below the estimated precision at best measured by quasi-information matrix for 'global.opts$NmaxLam' consecutive iterates.
pmin: minimum required probability that a new random candidate sample falls inside the parameter space. Dropping below this value triggers a global phase sampling step. This might indicate that the inverse of the quasi-information matrix does not sufficiently reflect the variance of the current parameter estimate due to a sparse sample or the (hyper)box constraints of the parameter space could be too restrictive.
nsample: sampling size of candidate locations at the local phase
weights: vector of weights, 0≤q\code{weights}≤q 1, for local sampling
useWeights: logical, if FALSE (default), dynamically adjust the weights, see vignette
ftol_abs: upper bound on the function criterion: values smaller trigger the local phase treating the current minimzer as an approximate root otherwise forces the algorithm to switch to the global phase and vice versa.
eta: values for decrease and increase of the local weights, which is intended to faciliate convergence while sampling new points more and more around the current best parameter estimate.
alpha: significance level for computation of empirical quantiles of one of the test statistics, that is, testing a parameter to be a root of the quasi-score vector in probability.
perr_tol upper bound on the relative difference of the empirical and predicted error of an approximate root
nfail: maximum number of consecutive failed iterations
nsucc: maximum number of consecutive successful iterations
nextSample: either "score" (default) or "var" (see details)

The following controls 'global.opts' for the global search phase are available:

stopval: stopping value related to the criterion function value, the main iteration terminates as soon as the criterion function value drops below this value. This might be preferable to a time consuming sampling procedure if one whishes to simply minimize the criterion function or find a first approximation to the unknown model parameter.
C_max: upper bound on the relative maximum quasi-score interpolation error. The algorithm terminates its value drops below after a number of 'global.opts$NmaxCV' consecutive iterations.
xtol_rel: relative change of found minimizer of the criterion function or root of quasi-score.
maxiter: maximum allowed global phase iterations
maxeval: maximum allowed global and local iterations
sampleTol: minimum allowed distance between sampled locations at global phase
weights: vector of \code{weights}>0 for global sampling
nsample: sampling size of candidate locations at the global phase
NmaxRel: maximum number of consecutive iterates until stopping according to 'xtol_rel'
NmaxCV: maximum number of consecutive iterates until stopping according to 'C_max'
NmaxSample: maximum number of consecutive iterations until stopping according to 'sampleTol'
NmaxLam: maximum number of consecutive iterations until stopping for which the generalized eigenvalue of the variance of the quasi-score vector within the kriging approximation model and its total variance measured by the quasi-information matrix at some estimated parameter drops below the upper bound 'local.opts$lam_max'
NmaxQI: maximum number of consecutive iterations until stopping for which the relative difference of the empirical error and predicted error of an approximate root drops below 'perr_tol'
Nmaxftol: maximum number of consecutive iterations until stopping for which the relative change in the values of the criterion function drops below 'local.opts$ftol_rel'

List of the following objects:

`par`	final parameter estimate
`value`	value of criterion function
`ctls`	a data frame with values of stopping conditions
`qsd`	final `QLmodel` object, including all sample points and covariance models
`cvm`	CV fitted covariance models
`why`	names of stopping conditions matched
`final`	final local minimization results of the criterion function, see `searchMinimizer`
`score`	quasi-score vector or gradient of the Mahalanobis distance
`convergence`	logical, whether the iterates converged, see details

Attributes:

`tracklist`	an object (list) of class `QDtrack` containing the local minimization results, evaluated sample points and the status of the corresponding iteration
`optInfo`	a list of arguments related to the estimation procedure:

x0: starting parameter vector
W: final weighting matrix (equal to quasi-information matrix at theta) used for both variance average approximation, if applicable, and as the predicted variance for (local) sampling of new candidate points according to a multivariate normal distribution with this variance and the current root as the mean parameter.
theta: the parameter corresponding to W, typically an approximate root or local minimzer
last.global: logical, whether last iteration sampled a point globally
minimized: whether last local minimization was successful
useCV: logical, whether the CV approach was applied
method: name of final search method applied
nsim: number of simulation replications at each evaluation point
iseed the seed to initialize the RNG

M. Baaske

mahalDist, quasiDeviance, qleTest

data(normal)
 
# main estimation with new evaluations
# (simulations of the statistical model)
OPT <- qle(qsd,qsd$simfn,nsim=10,
		    global.opts=list("maxeval"=1),
 		local.opts=list("test"=FALSE))