Simultaneous confidence regions for Gaussian models

simconf is used for calculating simultaneous confidence regions for Gaussian models \(x\). The function returns upper and lower bounds \(a\) and \(b\) such that \(P(a<x<b) = 1-\alpha\).

Usage

simconf(
  alpha,
  mu,
  Q,
  n.iter = 10000,
  Q.chol,
  vars,
  ind = NULL,
  verbose = 0,
  max.threads = 0,
  seed = NULL,
  tol = NULL
)

Arguments

alpha: Error probability for the region.
mu: Expectation vector for the Gaussian distribution.
Q: Precision matrix for the Gaussian distribution.
n.iter: Number or iterations in the MC sampler that is used for approximating probabilities. The default value is 10000. If tol is given, this is the maximal number of iterations.
Q.chol: The Cholesky factor of the precision matrix (optional).
vars: Precomputed marginal variances (optional).
ind: Indices of the nodes that should be analyzed (optional).
verbose: Set to TRUE for verbose mode (optional).
max.threads: The number of threads that the program can use. The default, 0, uses the default number of threads of OpenMP, which can be set with the environment variable OMP_NUM_THREADS. The number of threads is at most OMP_THREAD_LIMIT, and is one if the package was built without OpenMP.
seed: Random seed (optional).
tol: Target for the estimated errors of the probabilities that are computed to find the band (optional). If tol is given, the number of iterations is chosen adaptively, using at most n.iter iterations, see gaussint(). By default, n.iter iterations are always used.

Value

An object of class "excurobj" with elements

a: The lower bound.
b: The upper bound.
a.marginal: The lower bound for pointwise confidence bands.
b.marginal: The upper bound for pointwise confidence bands.

Details

The pointwise confidence bands are based on the marginal quantiles, meaning that a.marignal is a vector where the ith element equals \(\mu_i + q_{\alpha,i}\) and b.marginal is a vector where the ith element equals \(\mu_i + q_{1-\alpha,i}\), where \(\mu_i\) is the expected value of the \(x_i\) and \(q_{\alpha,i}\) is the \(\alpha\)-quantile of \(x_i-\mu_i\).

The simultaneous confidence band is defined by the lower limit vector a and the upper limit vector b, where \(a_i = \mu_i +c q_{\alpha}\) and \(b_i = \mu_i + c q_{1-\alpha}\), where \(c\) is a constant computed such that \(P(a < x < b) = 1-\alpha\).

References

Bolin et al. (2015) Statistical prediction of global sea level from global temperature, Statistica Sinica, vol 25, pp 351-367.

Bolin, D. and Lindgren, F. (2018), Calculating Probabilistic Excursion Sets and Related Quantities Using excursions, Journal of Statistical Software, vol 86, no 1, pp 1-20.

Author

David Bolin davidbolin@gmail.com and Finn Lindgren finn.lindgren@gmail.com

Examples

## Create mean and a tridiagonal precision matrix
n <- 11
mu.x <- seq(-5, 5, length = n)
Q.x <- Matrix(toeplitz(c(1, -0.1, rep(0, n - 2))))
## calculate the confidence region
conf <- simconf(0.05, mu.x, Q.x, max.threads = 2)
## Plot the region
plot(mu.x,
  type = "l", ylim = c(-10, 10),
  main = "Mean (black) and confidence region (red)"
)
lines(conf$a, col = 2)
lines(conf$b, col = 2)