Help for package anMC

Type: Package

Title: Compute High Dimensional Orthant Probabilities

Version: 0.2.5

Date: 2023年08月22日

Maintainer: Dario Azzimonti <dario.azzimonti@gmail.com>

Description: Computationally efficient method to estimate orthant probabilities of high-dimensional Gaussian vectors. Further implements a function to compute conservative estimates of excursion sets under Gaussian random field priors.

URL: https://doi.org/10.1080/10618600.2017.1360781

License: GPL-3

Encoding: UTF-8

Depends: mvtnorm

Imports: Rcpp (≥ 0.11.1)

Suggests: DiceKriging, TruncatedNormal

LinkingTo: Rcpp, RcppArmadillo

RoxygenNote: 7.2.3

NeedsCompilation: yes

Packaged: 2023年08月22日 07:47:20 UTC; dario.azzimonti

Author: Dario Azzimonti ORCID iD [aut, cre, cph]

Repository: CRAN

Date/Publication: 2023年08月22日 10:30:02 UTC

anMC: Compute High Dimensional Orthant Probabilities

Description

Computationally efficient method to estimate orthant probabilities of high-dimensional Gaussian vectors. Further implements a function to compute conservative estimates of excursion sets under Gaussian random field priors.

Details

Efficient estimation of high dimensional orthant probabilities. The package main functions are:

ProbaMax : the main function for high dimensional othant probabilities. Computes P(max X > t), where X is a Gaussian vector and t is the selected threshold. It implements the GANMC algorithm and allows for user-defined sampler and core probability estimates.
ProbaMin : analogous of ProbaMax for the problem P(min X < t), where X is a Gaussian vector and t is the selected threshold. It implements the GANMC algorithm and allows for user-defined sampler and core probability estimates.
conservativeEstimate : the main function for conservative estimates computation. Requires the mean and covariance of the posterior field at a discretization design.

Note

This work was supported in part by the Swiss National Science Foundation, grant number 146354 and the Hasler Foundation, grant number 16065. Thanks to David Ginsbourger for the fruitful discussions and his continuous help in testing and improving the package.

Author(s)

Maintainer: Dario Azzimonti dario.azzimonti@gmail.com (ORCID) [copyright holder]

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. doi:10.1080/10618600.2017.1360781

Azzimonti, D. (2016). Contributions to Bayesian set estimation relying on random field priors. PhD thesis, University of Bern.

Bolin, D. and Lindgren, F. (2015). Excursion and contour uncertainty regions for latent Gaussian models. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 77(1):85–106.

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

Dickmann, F. and Schweizer, N. (2014). Faster comparison of stopping times by nested conditional Monte Carlo. arXiv preprint arXiv:1402.0243.

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

Genz, A. and Bretz, F. (2009). Computation of Multivariate Normal and t Probabilities. Lecture Notes in Statistics 195. Springer-Verlag.

Horrace, W. C. (2005). Some results on the multivariate truncated normal distribution. Journal of Multivariate Analysis, 94(1):209–221.

Robert, C. P. (1995). Simulation of truncated normal variables. Statistics and Computing, 5(2):121–125.

ANMC estimate for the remainder

Description

Asymmetric nested Monte Carlo estimation of P(max X^{-q} > threshold | max X^{q} \le threshold) where X is a normal vector. It is used for the bias correction in ProbaMax and ProbaMin .

Usage

ANMC_Gauss(
 compBdg,
 problem,
 delta = 0.4,
 type = "M",
 trmvrnorm = trmvrnorm_rej_cpp,
 typeReturn = 0,
 verb = 0
)

Arguments

compBdg

total computational budget in seconds.

problem

list defining the problem with mandatory fields

muEq = mean vector of X^{q};
sigmaEq = covariance matrix of X^q;
threshold = fixed threshold t;
muEmq = mean vector of X^{-q};
wwCondQ = “weights” for X^{-q} | X^q [the vector \Sigma^{-q,q}(\Sigma^q)^{-1}];
sigmaCondQChol = Cholesky factorization of the conditional covariance matrix \Sigma^{-q | q};

delta

total proportion of budget assigned to initial estimate (default 0.4), the actual proportion used might be smaller.

type

type of excursion: "m", for minimum below threshold or "M", for maximum above threshold.

trmvrnorm

function to generate truncated multivariate normal samples, it must have the following signature trmvrnorm(n,mu,sigma,upper,lower,verb), where

n: number of simulations;
mu: mean vector of the Normal variable of dimension d;
sigma: covariance matrix of dimension d x d;
upper: vector of upper limits of length d;
lower: vector of lower limits of length d;
verb: the level of verbosity 3 basic, 4 extended.

It must return a matrix d x n of realizations. If not specified, the rejection sampler trmvrnorm_rej_cpp is used.

typeReturn

integer chosen between

0 a number with only the probability estimation;
1 light return: a list with the probability estimator, the variance of the estimator, the vectors of conditional quantities used to obtain m^* and the system dependent parameters;
2 heavy return: the same list as light return with also the computational times and additional intermediate parameters.

verb

level of verbosity (0,1 for this function), also sets the verbosity of trmvrnorm (to verb-1).

Value

A list containing the estimated probability of excursion, see typeReturn for details.

References

Azzimonti, D. (2016). Contributions to Bayesian set estimation relying on random field priors. PhD thesis, University of Bern.

Dickmann, F. and Schweizer, N. (2014). Faster comparison of stopping times by nested conditional Monte Carlo. arXiv preprint arXiv:1402.0243.

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

MC estimate for the remainder

Description

Standard Monte Carlo estimate for P(max X^{-q} >threshold | max X^{q}\le threshold) or P(min X^{-q} <threshold | min X^{q}\ge threshold) where X is a normal vector. It is used for the bias correction in ProbaMax and ProbaMin .

Usage

MC_Gauss(
 compBdg,
 problem,
 delta = 0.1,
 type = "M",
 trmvrnorm = trmvrnorm_rej_cpp,
 typeReturn = 0,
 verb = 0,
 params = NULL
)

Arguments

compBdg

total computational budget in seconds.

problem

list defining the problem with mandatory fields:

muEq = mean vector of X^{q};
sigmaEq = covariance matrix of X^q;
threshold = threshold;
muEmq = mean vector of X^{-q};
wwCondQ = “weights” for X^{-q} | X^q [ the vector \Sigma^{-q,q}(\Sigma^q)^{-1}];
sigmaCondQChol = Cholesky factorization of the conditional covariance matrix \Sigma^{-q | q}.

delta

total proportion of budget assigned to initial estimate (default 0.1), the actual proportion used might be smaller.

type

type of excursion: "m", for minimum below threshold or "M", for maximum above threshold.

trmvrnorm

function to generate truncated multivariate normal samples, it must have the following signature trmvrnorm(n,mu,sigma,upper,lower,verb), where

n: number of simulations;
mu: mean vector of the Normal variable of dimension d;
sigma: covariance matrix of dimension d x d;
upper: vector of upper limits of length d;
lower: vector of lower limits of length d;
verb: the level of verbosity 3 basic, 4 extended.

It must return a matrix d x n of realizations. If not specified, the rejection sampler trmvrnorm_rej_cpp is used.

typeReturn

integer: 0 (only the estimate) or 1 (heavy return with variance of the estimate, parameters of the estimator and computational times).

verb

the level of verbosity, also sets the verbosity of trmvrnorm (to verb-1).

params

system dependent parameters (if NULL they are estimated).

Value

A list containing the estimated probability of excursion, see typeReturn for details.

References

Azzimonti, D. (2016). Contributions to Bayesian set estimation relying on random field priors. PhD thesis, University of Bern.

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

Probability of exceedance of maximum of Gaussian vector

Description

Computes P(max X > threshold) with choice of algorithm between ANMC_Gauss and MC_Gauss. By default, the computationally expensive sampling parts are computed with the Rcpp functions.

Usage

ProbaMax(
 cBdg,
 threshold,
 mu,
 Sigma,
 E = NULL,
 q = NULL,
 pn = NULL,
 lightReturn = T,
 method = 4,
 verb = 0,
 Algo = "ANMC",
 trmvrnorm = trmvrnorm_rej_cpp,
 pmvnorm_usr = pmvnorm
)

Arguments

cBdg

computational budget.

threshold

threshold.

mu

mean vector.

Sigma

covariance matrix.

E

discretization design for the field. If NULL, a simplex-lattice design n,n is used, with n=length(mu). In this case the choice of method=4,5 are not advised.

q

number of active dimensions, it can be either

an integer: in this case the optimal q active dimension are chosen;
a numeric vector of length 2: this is the range where to search for the best number of active dimensions;
NULL: q is selected as the best number of active dimensions in the feasible range.

pn

coverage probability function evaluated with mu, Sigma. If NULL it is computed automatically.

lightReturn

boolean, if TRUE light return.

method

method chosen to select the active dimensions. See selectActiveDims for details.

verb

level of verbosity (0-5), selects verbosity also for ANMC_Gauss (verb-1) and MC_Gauss (verb-1).

Algo

choice of algorithm to compute the remainder Rq ("ANMC" or "MC").

trmvrnorm

function to generate truncated multivariate normal samples, it must have the following signature trmvrnorm(n,mu,sigma,upper,lower,verb), where

n: number of simulations;
mu: mean vector of the Normal variable of dimension d;
sigma: covariance matrix of dimension d x d;
upper: vector of upper limits of length d;
lower: vector of lower limits of length d;
verb: the level of verbosity 3 basic, 4 extended.

It must return a matrix d x n of realizations. If not specified, the rejection sampler trmvrnorm_rej_cpp is used.

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

A list containing

probability: The probability estimate
variance: the variance of the probability estimate
q:the number of selected active dimensions

If lightReturn=F then the list also contains:

aux_probabilities: a list with the probability estimates: probability the actual probability, pq the biased estimator p_q, Rq the conditional probability R_q
Eq: the points of the design E selected for p_q
indQ: the indices of the active dimensions chosen for p_q
resRq: The list returned by the MC method used for R_q

References

Azzimonti, D. (2016). Contributions to Bayesian set estimation relying on random field priors. PhD thesis, University of Bern.

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

Dickmann, F. and Schweizer, N. (2014). Faster comparison of stopping times by nested conditional Monte Carlo. arXiv preprint arXiv:1402.0243.

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

Examples

## Not run: 
# Compute probability P(X \in (-\infty,0]) with X~N(0,Sigma)
d<-200 # example dimension
mu<-rep(0,d) # mean of the normal vector
# correlation structure (Miwa et al. 2003, Craig 2008, Botev 2016)
Sigma<-0.5*diag(d)+ 0.5*rep(1,d)%*%t(rep(1,d))
pANMC<-ProbaMax(cBdg=20, q=min(50,d/2), E=seq(0,1,,d), threshold=0, mu=mu, Sigma=Sigma,
 pn = NULL, lightReturn = TRUE, method = 3, verb = 2, Algo = "ANMC")
proba<-1-pANMC$probability
# Percentage error
abs(1-pANMC$probability-1/(d+1))/(1/(d+1))
# Implement ProbaMax with user defined function for active dimension probability estimate
if(!requireNamespace("TruncatedNormal", quietly = TRUE)) {
stop("Package TruncatedNormal needed for this example to work. Please install it.",
 call. = FALSE)
}
# define pmvnorm_usr with the function mvNcdf from the package TruncatedNormal
pmvnorm_usr<-function(lower,upper,mean,sigma){
 pMET<-TruncatedNormal::mvNcdf(l = lower-mean,u = upper-mean,Sig = sigma,n = 5e4)
 res<-pMET$prob
 attr(res,"error")<-pMET$relErr
 return(res)
}
pANMC<-ProbaMax(cBdg=20, q=min(50,d/2), E=seq(0,1,,d), threshold=0, mu=mu, Sigma=Sigma,
 pn = NULL, lightReturn = TRUE, method = 3, verb = 2, Algo = "ANMC",pmvnorm_usr=pmvnorm_usr)
proba<-1-pANMC$probability
# Percentage error
abs(1-pANMC$probability-1/(d+1))/(1/(d+1))
# Implement ProbaMax with user defined function for truncated normal sampling
if(!requireNamespace("tmg", quietly = TRUE)) {
stop("Package tmg needed for this example to work. Please install it.",
 call. = FALSE)
}
trmvrnorm_usr<-function(n,mu,sigma,upper,lower,verb){
 M<-chol2inv(chol(sigma))
 r=as.vector(M%*%mu)
 if(all(lower==-Inf) && all(upper==Inf)){
 f<- NULL
 g<- NULL
 }else{
 if(all(lower==-Inf)){
 f<--diag(length(mu))
 g<-upper
 initial<-(upper-1)/2
 }else if(all(upper==Inf)){
 f<-diag(length(mu))
 g<- -lower
 initial<-2*(lower+1)
 }else{
 f<-rbind(-diag(length(mu)),diag(length(mu)))
 g<-c(upper,-lower)
 initial<-(upper-lower)/2
 }
 }
 reals_tmg<-tmg::rtmg(n=n,M=M,r=r,initial = initial,f=f,g=g)
 return(t(reals_tmg))
}
pANMC<-ProbaMax(cBdg=20, q=min(50,d/2), E=seq(0,1,,d), threshold=0, mu=mu, Sigma=Sigma,
 pn = NULL, lightReturn = TRUE, method = 3, verb = 2, Algo = "ANMC",trmvrnorm=trmvrnorm_usr)
proba<-1-pANMC$probability
# Percentage error
abs(1-pANMC$probability-1/(d+1))/(1/(d+1))
## End(Not run)

Probability of exceedance of minimum of Gaussian vector

Description

Computes P(min X \le threshold) with choice of algorithm between ANMC_Gauss and MC_Gauss. By default, the computationally expensive sampling parts are computed with the Rcpp functions.

Usage

ProbaMin(
 cBdg,
 threshold,
 mu,
 Sigma,
 E = NULL,
 q = NULL,
 pn = NULL,
 lightReturn = T,
 method = 4,
 verb = 0,
 Algo = "ANMC",
 trmvrnorm = trmvrnorm_rej_cpp,
 pmvnorm_usr = pmvnorm
)

Arguments

cBdg

computational budget.

threshold

threshold.

mu

mean vector.

Sigma

covariance matrix.

E

discretization design for the field. If NULL, a simplex-lattice design n,n is used, with n=length(mu). In this case the choice of method=4,5 are not advised.

q

number of active dimensions, it can be either

an integer: in this case the optimal q active dimension are chosen;
a numeric vector of length 2: this is the range where to search for the best number of active dimensions;
NULL: q is selected as the best number of active dimensions in the feasible range.

pn

coverage probability function evaluated with mu, Sigma. If NULL it is computed automatically.

lightReturn

boolean, if TRUE light return.

method

method chosen to select the active dimensions. See selectActiveDims for details.

verb

level of verbosity (0-5), selects verbosity also for ANMC_Gauss (verb-1) and MC_Gauss (verb-1).

Algo

choice of algorithm to compute the remainder Rq ("ANMC" or "MC").

trmvrnorm

function to generate truncated multivariate normal samples, it must have the following signature trmvrnorm(n,mu,sigma,upper,lower,verb), where

n: number of simulations;
mu: mean vector of the Normal variable of dimension d;
sigma: covariance matrix of dimension d x d;
upper: vector of upper limits of length d;
lower: vector of lower limits of length d;
verb: the level of verbosity 3 basic, 4 extended.

It must return a matrix d x n of realizations. If not specified, the rejection sampler trmvrnorm_rej_cpp is used.

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

A list containing

probability: The probability estimate
variance: the variance of the probability estimate
q:the number of selected active dimensions

If lightReturn=F then the list also contains:

aux_probabilities: a list with the probability estimates: probability the actual probability, pq the biased estimator p_q, Rq the conditional probability R_q
Eq: the points of the design E selected for p_q
indQ: the indices of the active dimensions chosen for p_q
resRq: The list returned by the MC method used for R_q

References

Azzimonti, D. (2016). Contributions to Bayesian set estimation relying on random field priors. PhD thesis, University of Bern.

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

Dickmann, F. and Schweizer, N. (2014). Faster comparison of stopping times by nested conditional Monte Carlo. arXiv preprint arXiv:1402.0243.

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

Examples

## Not run: 
# Compute probability P(X \in [0,\infty]) with X~N(0,Sigma)
d<-200 # example dimension
mu<-rep(0,d) # mean of the normal vector
# correlation structure (Miwa et al. 2003, Craig 2008, Botev 2016)
Sigma<-0.5*diag(d)+ 0.5*rep(1,d)%*%t(rep(1,d))
pANMC<-ProbaMin(cBdg=20, q=min(50,d/2), E=seq(0,1,,d), threshold=0, mu=mu, Sigma=Sigma,
 pn = NULL, lightReturn = TRUE, method = 3, verb = 2, Algo = "ANMC")
proba<-1-pANMC$probability
# Percentage error
abs(1-pANMC$probability-1/(d+1))/(1/(d+1))
# Implement ProbaMin with user defined function for active dimension probability estimate
if(!requireNamespace("TruncatedNormal", quietly = TRUE)) {
stop("TruncatedNormal needed for this example to work. Please install it.",
 call. = FALSE)
}
# define pmvnorm_usr with the function mvNcdf from the package TruncatedNormal
pmvnorm_usr<-function(lower,upper,mean,sigma){
 pMET<-TruncatedNormal::mvNcdf(l = lower-mean,u = upper-mean,Sig = sigma,n = 5e4)
 res<-pMET$prob
 attr(res,"error")<-pMET$relErr
 return(res)
}
pANMC<-ProbaMin(cBdg=20, q=min(50,d/2), E=seq(0,1,,d), threshold=0, mu=mu, Sigma=Sigma,
 pn = NULL, lightReturn = TRUE, method = 3, verb = 2, Algo = "ANMC",pmvnorm_usr=pmvnorm_usr)
proba<-1-pANMC$probability
# Percentage error
abs(1-pANMC$probability-1/(d+1))/(1/(d+1))
# Implement ProbaMin with user defined function for truncated normal sampling
if(!requireNamespace("tmg", quietly = TRUE)) {
stop("Package tmg needed for this example to work. Please install it.",
 call. = FALSE)
}
trmvrnorm_usr<-function(n,mu,sigma,upper,lower,verb){
 M<-chol2inv(chol(sigma))
 r=as.vector(M%*%mu)
 if(all(lower==-Inf) && all(upper==Inf)){
 f<- NULL
 g<- NULL
 }else{
 if(all(lower==-Inf)){
 f<--diag(length(mu))
 g<-upper
 initial<-(upper-1)/2
 }else if(all(upper==Inf)){
 f<-diag(length(mu))
 g<- -lower
 initial<-2*(lower+1)
 }else{
 f<-rbind(-diag(length(mu)),diag(length(mu)))
 g<-c(upper,-lower)
 initial<-(upper-lower)/2
 }
 }
 reals_tmg<-tmg::rtmg(n=n,M=M,r=r,initial = initial,f=f,g=g)
 return(t(reals_tmg))
}
pANMC<-ProbaMin(cBdg=20, q=min(50,d/2), E=seq(0,1,,d), threshold=0, mu=mu, Sigma=Sigma,
 pn = NULL, lightReturn = TRUE, method = 3, verb = 2, Algo = "ANMC",trmvrnorm=trmvrnorm_usr)
proba<-1-pANMC$probability
# Percentage error
abs(1-pANMC$probability-1/(d+1))/(1/(d+1))
## End(Not run)

Computationally efficient conservative estimate

Description

Computes conservative estimates with two step GANMC procedure for a Gaussian vector. The probability is approximated with a biased low dimensional estimator and the bias is corrected with a MC estimator.

Usage

conservativeEstimate(
 alpha = 0.95,
 pred,
 design,
 threshold,
 pn = NULL,
 type = ">",
 verb = 1,
 lightReturn = T,
 algo = "GANMC"
)

Arguments

alpha

probability of conservative estimate.

pred

list containing mean vector (pred$mean) and covariance matrix (pred$cov).

design

a matrix of size length(pred$mean)x(input space dimension) that contains the design where pred$mean was computed.

threshold

threshold, real number.

pn

coverage probability function, vector of the same length as pred$mean (if not specified it is computed).

type

type of excursion: ">" for excursion above threshold or "<" for below.

verb

level of verbosity, integer from 1–7.

lightReturn

boolean for light return.

algo

choice of algorithm for computing probabilities ("GANMC", "GMC").

Value

A list containing the conservative estimate (set), the Vorob'ev level (lvs). If lightReturn=FALSE, it also returns the actual probability of the set (proba) and the variance of this estimate (vars).

References

Azzimonti, D. (2016). Contributions to Bayesian set estimation relying on random field priors. PhD thesis, University of Bern.

Bolin, D. and Lindgren, F. (2015). Excursion and contour uncertainty regions for latent Gaussian models. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 77(1):85–106.

Examples

if (!requireNamespace("DiceKriging", quietly = TRUE)) {
stop("DiceKriging needed for this example to work. Please install it.",
 call. = FALSE)
}
# Compute conservative estimate of excursion set of testfun below threshold
# Initialize
testfun<-function(x){return(((3*x^2+7*x-3)*exp(-1*(x)^2)*cos(5*pi*x^2)-1.2*x^2))}
mDet<- 1500
# Uniform design points
set.seed(42)
doe<-runif(n = 8)
res<-testfun(doe)
threshold<-0
# create km
smallKm <- DiceKriging::km(design = matrix(doe,ncol=1),
response = res,covtype = "matern5_2",coef.trend = -1,coef.cov = c(0.05),coef.var = 1.1)
# prediction at newdata
newdata<-data.frame(matrix(seq(0,1,,mDet),ncol=1)); colnames(newdata)<-colnames(smallKm@X)
pred<-DiceKriging::predict.km(object = smallKm,newdata = newdata,type = "UK",cov.compute = TRUE)
## Not run: 
# Plot (optional)
plot(seq(0,1,,mDet),pred$mean,type='l')
points(doe,res,pch=10)
abline(h = threshold)
lines(seq(0,1,,mDet),pred$mean+pred$sd,lty=2,col=1)
lines(seq(0,1,,mDet),pred$mean-pred$sd,lty=2,col=1)
## End(Not run)
# Compute the coverage function
pn<-pnorm((threshold-pred$mean)/pred$sd)
## Not run: 
pred$cov <- pred$cov + 1e-7*diag(nrow = nrow(pred$cov),ncol = ncol(pred$cov))
CE<-conservativeEstimate(alpha = 0.95,pred = pred,design = as.matrix(newdata),
threshold = threshold,type = "<",verb=1, pn=pn,algo = "ANMC")
points(newdata[CE$set,],rep(-0.1,mDet)[CE$set],col=4,pch="-",cex=2)
## End(Not run)

Measure elapsed time with C++11 chrono library

Description

Returns a time indicator that can be used to accurately measure elapsed time. The C++11 clock used is chrono::high_resolution_clock.

Usage

get_chronotime()

Value

A double with the number of nanoseconds elapsed since a fixed epoch.

Examples

# Measure 1 second sleep
initT<-get_chronotime()
Sys.sleep(1)
measT<-(get_chronotime()-initT)*1e-9
cat("1 second passed in ",measT," seconds.\n")

Sample from multivariate normal distribution with C++

Description

Simulates realizations from a multivariate normal with mean mu and covariance matrix sigma.

Usage

mvrnormArma(n, mu, sigma, chol)

Arguments

n

number of simulations.

mu

mean vector.

sigma

covariance matrix or Cholesky decomposition of the matrix (see chol).

chol

integer, if 0 sigma is a covariance matrix, otherwise it is the Cholesky decomposition of the matrix.

Value

A matrix of size d x n containing the samples.

Examples

# Simulate 1000 realizations from a multivariate normal vector
mu <- rep(0,200)
Sigma <- diag(rep(1,200))
realizations<-mvrnormArma(n=1000,mu = mu,sigma=Sigma, chol=0)
empMean<-rowMeans(realizations)
empCov<-cov(t(realizations))
# check if the sample mean is close to the actual mean
maxErrorOnMean<-max(abs(mu-empMean))
# check if we can estimate correctly the covariance matrix
maxErrorOnVar<-max(abs(rep(1,200)-diag(empCov)))
maxErrorOnCov<-max(abs(empCov[lower.tri(empCov)]))
## Not run: 
plot(density(realizations[2,]))
## End(Not run)

Select active dimensions for small dimensional estimate

Description

The function selectActiveDims selects the active dimensions for the computation of p_q with an heuristic method.

Usage

selectActiveDims(
 q = NULL,
 E,
 threshold,
 mu,
 Sigma,
 pn = NULL,
 method = 1,
 verb = 0,
 pmvnorm_usr = pmvnorm
)

Arguments

q

either the fixed number of active dimensions or the range where the number of active dimensions is chosen with selectQdims. If NULL the function selectQdims is called.

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.

verb

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

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

A vector of integers denoting the chosen active dimensions of the vector mu.

References

Azzimonti, D. (2016). Contributions to Bayesian set estimation relying on random field priors. PhD thesis, University of Bern.

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.

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

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.

Sample from truncated multivariate normal distribution with C++

Description

Simulates realizations from a truncated multivariate normal with mean mu, covariance matrix sigma in the bounds lower upper.

Usage

trmvrnorm_rej_cpp(n, mu, sigma, lower, upper, verb)

Arguments

n

number of simulations.

mu

mean vector.

sigma

covariance matrix.

lower

vector of lower bounds.

upper

vector of upper bounds.

verb

level of verbosity: if lower than 3 nothing, 3 minimal, 4 extended.

Value

A matrix of size d x n containing the samples.

References

Horrace, W. C. (2005). Some results on the multivariate truncated normal distribution. Journal of Multivariate Analysis, 94(1):209–221.

Robert, C. P. (1995). Simulation of truncated normal variables. Statistics and Computing, 5(2):121–125.

Examples

# Simulate 1000 realizations from a truncated multivariate normal vector
mu <- rep(0,10)
Sigma <- diag(rep(1,10))
upper <- rep(3,10)
lower <- rep(-0.5,10)
realizations<-trmvrnorm_rej_cpp(n=1000,mu = mu,sigma=Sigma, lower =lower, upper= upper,verb=3)
empMean<-rowMeans(realizations)
empCov<-cov(t(realizations))
# check if the sample mean is close to the actual mean
maxErrorOnMean<-max(abs(mu-empMean))
# check if we can estimate correctly the covariance matrix
maxErrorOnVar<-max(abs(rep(1,200)-diag(empCov)))
maxErrorOnCov<-max(abs(empCov[lower.tri(empCov)]))
## Not run: 
plot(density(realizations[1,]))
hist(realizations[1,],breaks="FD")
## End(Not run)

anMC: Compute High Dimensional Orthant Probabilities

Description

Details

Note

Author(s)

References

See Also

ANMC estimate for the remainder

Description

Usage

Arguments

Value

References

MC estimate for the remainder

Description

Usage

Arguments

Value

References

Probability of exceedance of maximum of Gaussian vector

Description

Usage

Arguments

Value

References

Examples

Probability of exceedance of minimum of Gaussian vector

Description

Usage

Arguments

Value

References

Examples

Computationally efficient conservative estimate

Description

Usage

Arguments

Value

References

Examples

Measure elapsed time with C++11 chrono library

Description

Usage

Value

Examples

Sample from multivariate normal distribution with C++

Description

Usage

Arguments

Value

Examples

Select active dimensions for small dimensional estimate

Description

Usage

Arguments

Value

References

Iteratively select active dimensions

Description

Usage

Arguments

Value

References

Sample from truncated multivariate normal distribution with C++

Description

Usage

Arguments

Value

References

Examples