selectQdims {anMC}

Iteratively select active dimensions

Description

The function selectQdims iteratively selects the number of active dimensions and the dimensions themselves for the computation of p_q. The number of dimensions is increased until p_{q}-p_{q-1} is smaller than the error of the procedure.

Usage

selectQdims(
  E,
  threshold,
  mu,
  Sigma,
  pn = NULL,
  method = 1,
  reducedReturn = T,
  verb = 0,
  limits = NULL,
  pmvnorm_usr = pmvnorm
)

Arguments

E

discretization design for the field.

threshold

threshold.

mu

mean vector.

Sigma

covariance matrix.

pn

coverage probability function based on threshold, mu and Sigma. If NULL it is computed.

method

integer chosen between

• 0 selects by taking equally spaced indexes in mu;

• 1 samples from pn;

• 2 samples from pn*(1-pn);

• 3 samples from pn adjusting for the distance (tries to explore all modes);

• 4 samples from pn*(1-pn) adjusting for the distance (tries to explore all modes);

• 5 samples with uniform probabilities.

reducedReturn

boolean to select the type of return. See Value for further details.

verb

level of verbosity: 0 returns nothing, 1 returns minimal info.

limits

numeric vector of length 2 with q_min and q_max. If NULL initialized at c(10,300)

pmvnorm_usr

function to compute core probability on active dimensions. Inputs:

• lower: the vector of lower limits of length d.

• upper: the vector of upper limits of length d.

• mean: the mean vector of length d.

• sigma: the covariance matrix of dimension d.

returns a the probability value with attribute "error", the absolute error. Default is the function pmvnorm from the package mvtnorm.

Value

If reducedReturn=F returns a list containing

• indQ: the indices of the active dimensions chosen for p_q;

• pq: the biased estimator p_q with attribute error, the estimated absolute error;

• Eq: the points of the design E selected for p_q;

• muEq: the subvector of mu selected for p_q;

• KEq: the submatrix of Sigma composed by the indexes selected for p_q.

Otherwise it returns only indQ.

References

Azzimonti, D. and Ginsbourger, D. (2018). Estimating orthant probabilities of high dimensional Gaussian vectors with an application to set estimation. Journal of Computational and Graphical Statistics, 27(2), 255-267. Preprint at hal-01289126

Chevalier, C. (2013). Fast uncertainty reduction strategies relying on Gaussian process models. PhD thesis, University of Bern.

Genz, A. (1992). Numerical computation of multivariate normal probabilities. Journal of Computational and Graphical Statistics, 1(2):141–149.