mahalDist: Mahalanobis distance of statistics

In qle: Simulation-Based Quasi-Likelihood Estimation

Description

Compute the Mahalanobis distance (MD) based on the kriging models of statistics

Usage

mahalDist(points, qsd, Sigma = NULL, ..., cvm = NULL, obs = NULL,
  inverted = FALSE, check = TRUE, value.only = FALSE, na.rm = TRUE,
  cl = NULL, verbose = FALSE)

Arguments

points	either matrix or list of points or a vector of parameters (but then considered as a single (multidimensional) point)
qsd	object of class QLmodel
Sigma	either a constant variance matrix estimate or an prespecified value
...	further arguments passed to covarTx for variance average approximation
cvm	list of fitted cross-validation models (see prefitCV)
obs	numeric vector of observed statistics (overwrites 'qsd$obs' if given)
inverted	logical, FALSE (default), whether 'Sigma' is already inverted when used as constant variance matrix only
check	logical, TRUE (default), whether to check all input arguments
value.only	only return the value of the MD
na.rm	logical, if TRUE (default) remove 'Na' values from the results
cl	cluster object, NULL (default), of class MPIcluster, SOCKcluster, cluster
verbose	if TRUE, then print intermediate output

Details

The function computes the Mahalanobis distance of the given statistics T(X)\in R^p with different options for the approximation type of the variance matrix. The Mahalanobis distance can be used as an alternative criterion function for estimating the unknown model parameter during the main estimation function qle.

There are several options how to estimate or choose the variance matrix of the statistics Σ. First, in case of a given constant variance matrix estimate 'Sigma', the Mahalanobis distance reads

(T(x)-E_{θ}[T(X)])^tΣ^{-1}(T(x)-E_{θ}[T(X)])

and 'Sigma' is used as given.

As a second option, the variance matrix Σ can be estimated by the average approximation

\bar{V}=\frac{1}{n}∑_{i=1}^n V_i

based on the simulated variance matrices V_i=V(θ_i) of statistics over all sample points θ_1,...,θ_n (see vignette). Unless 'qsd$var.type' equals "const" additional prediction variances are added as diagonal terms to account for the kriging approximation error of the statistics using kriging with calculation of kriging variances if 'qsd$krig.type' equal to "var". Otherwise no additional variances are added. A weighted version of these average approximation types is also available (see covarTx).

As a continuous version of variance approximation we use a kriging approach (see [1]). Then

Σ(θ) = Var_{θ}(T(X))

denotes the variance matrix which depends on the parameter θ\in R^q and corresponds to the formal function argument 'points'. Each time a value of the criterion function is calculated at any parameter 'points' the variance matrix is estimated by the correpsonding approach either with or without using prediction variances as explained above. Note that in this case the argument 'Sigma' is ignored.

Value

Either a vector of MD values or a list of lists, where each contains the following elements:

value	Mahalanobis distance value
par	parameter estimate
I	approximate variance matrix of the parameter estimate
score	gradient of MD (for constant 'Sigma')
jac	Jacobian of sample mean values of statistics
varS	estimated variance matrix of 'score'

and, if applicable, the following attributes:

Sigma	estimate of variance matrix (if 'Sigma' is computed or was set as a constant matrix)
inverted	whether 'Sigma' was inverted

Author(s)

M. Baaske

Examples

 data(normal)
 # (weighted) least squares
 mahalDist(c(2,1), qsd, Sigma=diag(2))
 
 # generalized LS with variance average approximation 
	# here: same as quasi-deviance
 mahalDist(c(2,1), qsd)  
 

qle documentation built on May 2, 2019, 9:55 a.m.