covarTx: Variance matrix approximation

Description Usage Arguments Details Value Author(s) Examples

Description

Approximating the variance-covariance matrix of statistics

Usage

1
covarTx(qsd, W = NULL, theta = NULL, cvm = NULL, doInvert = FALSE)

Arguments

qsd

object of class QLmodel

W

weight matrix for weighted average approximation

theta

parameter vector for weighted average approximation

cvm

list of fitted cross-validation models, see prefitCV

doInvert

if TRUE, return the inverse of the variance matrix approximation

Details

The function estimates the variance matrix of statistics at some (unsampled) point 'theta' by either averaging (the Cholesky decomposed terms or matrix logarithms) over all simulated variance matrices of statistics at previously evaluated points of the parameter space or by a kriging approach which treats the Cholesky decomposed terms of each variance matrix as a data vector for kriging.

In addition, a Nadaraya-Watson kernel-weighted average approximation can also be applied in order to bias the variance estimation towards a more locally weighted estimation, where smaller weights are assigned to points being more distant to an estimate of the model parameter 'theta'. A reasonable symmetric weighting matrix 'W' of size equal to the problem dimension, say q, can be freely chosen by the user. In addition, the user can select different types of variance averaging methods such as "cholMean", "wcholMean", "logMean", "wlogMean", "kriging" (kriging variance matrix adding three times the prediction standard errors for each Cholesky entry) defined by 'qsd$var.type', where the prefix "w" indicats its corresponding weighted version of the approximation type. If 'theta' is not given, then no prediction variances are included and thus the variance matrix estimate of the statistics only refers to the variances due to simulation replications and not the ones due to the use of kriging approximations of the statistics. Otherwise, the mean variance matrix estimate is given by

\hat{V}+\textrm{diag}(σ(θ)),

where \hat{V} denotes one of the above variance approximation types.

The prediction variances σ are either derived from the kriging results of statistics or based on a (possibly more robust) cross-validation (CV) approach, see vignette for details.

Value

List of variance matrices with the following structure:

VTX

variance matrix approximation

sig2

kriging prediction variances of statistics at 'theta'

var

matrix 'VTX' with added variances 'sig2' to the diagonal terms

inv

if applicable, the inverse of either 'VTX' or 'var'

Author(s)

M. Baaske

Examples

1
2
3
 data(normal)
 # average approximation of variance matrices
 covarTx(qsd,theta=c("mu"=2,"sd"=1))

qle documentation built on May 2, 2019, 5:26 p.m.