# Construct confidence regions for exceedance (excursion) sets. `confreg` constructs confidence regions for the exceedance (excursions) sets of geostatistical processes. These will actually be credible regions if `obj` contains samples from the joint posterior predictive distribution in a Bayesian setting. ```r confreg( obj, level, statistic = NULL, conf.level = 0.95, direction = ">", type = "o", method = "test", greedy = FALSE ) ``` ## Arguments - `obj`: An object of the appropriate type (`matrix`, `krigeConditionalSample`, or `jointPredictiveSample`. See Details. - `level`: The threshold level for the exceedance region. - `statistic`: The statistic used in constructing the confidence region. Should be a vector containing a value for each location - `conf.level`: The confidence level of the confidence region. Default is 0.95. - `direction`: The direction of the exceedance region. `">"` indicates the exceedance region is values above a threshold, while `"<"` indicates values below a threshold. - `type`: `"o"` indicates on outer confidence region while `"i"` indicates in inner confidence region. - `method`: `"test"` indicates a testing-based method, while `"direct"` indicates a direct method using joint probabilities. - `greedy`: Only applicable for the direct construction method. Default is `FALSE`. If `TRUE`, then grid cells are added to the confidence region using a greedy algorithm based on joint probability. ## Returns Returns an object of class `confreg` with the following components: - **confidence**: The sites included in the confidence region. - **complement**: The complement of the confidence region. ## Details `obj` can be an object of class `matrix`, `krigeConditionalSample`, or `jointPredictiveSample`. If `obj` is a `matrix`, then it should have `m` rows and `nsim` columns. In that case, each row of `obj` corresponds to a sample from the conditional distribution of the response conditional on the observed data. Each row represents a different location. Generally, these locations are assumed to be on a grid spanning the spatial domain of interest. A `krigeConditionalSample` object can be obtained using the `krige.sk`, `krige.ok`, or `krige.uk` functions in the `SpatialTools` package. In these functions, the `nsim` argument must be greater than 0, and indicates the number of samples used to construct the confidence region. A `jointPredictiveSample` object can be obtained using the `spLMPredictJoint` function in the `SpatialTools` package. Since this is in the context of Bayesian statistics, the function actually produces credible region. If `statistic` is supplied for the direct construction procedure, then the locations are ordered by marginal probability and then the statistic. `statistic` should be a vector of length `m`, where `m` is the number of prediction locations at which samples were drawn for in `obj`. If type == `"o"`, then an outer credible region is constructed. The outer credible region should entirely contain the true exceedanace region with the specified posterior probability. If type == `"i"`, then an inner credible region is constructed. The inner confidence region should be entirely contained within the true exceedanace region with specified posterior probability. ## Examples ```r # Set parameters n <- 100 mygrid = create.pgrid(0, 1, 0, 1, nx = 5, ny = 4) n.samples <- 10 burnin.start <- 1 sigmasq <- 1 tausq <- 0.0 phi <- 1 cov.model <- "exponential" n.report <- 5 # Generate coordinates coords <- matrix(runif(2 * n), ncol = 2) pcoords <- mygrid$pgrid # Construct design matrices X <- as.matrix(cbind(1, coords)) Xp <- cbind(1, pcoords) # Specify priors starting <- list("phi" = phi, "sigma.sq"= sigmasq, "tau.sq" = tausq) tuning <- list("phi"=0.1, "sigma.sq"=0.1, "tau.sq"=0.1) priors.1 <- list("beta.Norm"=list(c(1, 2, 1), diag(100, 3)), "phi.Unif"=c(0.00001, 10), "sigma.sq.IG"=c(1, 1)) # Generate data library(SpatialTools) B <- rnorm(3, c(1, 2, 1), sd = 10) phi <- runif(1, 0, 10) sigmasq <- 1/rgamma(1, 1, 1) V <- simple.cov.sp(D = dist1(coords), cov.model, c(sigmasq, 1/phi), error.var = tausq, smoothness = nu, finescale.var = 0) y <- X %*% B + rmvnorm(1, rep(0, n), V) + rnorm(n, 0, sqrt(tausq)) # Create spLM object library(spBayes) m1 <- spBayes::spLM(y ~ X - 1, coords = coords, starting = starting, tuning = tuning, priors = priors.1, cov.model = cov.model, n.samples = n.samples, verbose = FALSE, n.report = n.report) # Sample from joint posterior predictive distribution y1 <- spLMPredictJoint(m1, pred.coords = pcoords, pred.covars = Xp, start = burnin.start, verbose = FALSE, method = "chol") u = quantile(y, .5) myfun = function(x) { (mean(x) - u)/sd(x) } myfun2 = function(x) { mean(x > u) } stat1 = apply(y1, 1, myfun) stat2 = apply(y1, 1, myfun2) myconf = confreg(y1, level = u, statistic = NULL, direction = ">", type = "o", method = "direct") myconf2 = confreg(y1, level = u, statistic = stat1, direction = ">", type = "o") myconf3 = confreg(y1, level = u, statistic = stat2, direction = ">", type = "o") ``` ## References Joshua P. French and Stephan R. Sain (2013). Spatio-temporal exceedance locations and confidence regions. Annals of Applied Statistics. 7(3):1421-1449. French, J. P. (2014), Confidence regions for the level curves of spatial data, Environmetrics, 25, pages 498–512, DOI: 10.1002/env.2295 French, J. P., and Hoeting, J. A. (2016) Credible regions for exceedance sets of geostatistical data. Environmetrics, 27: 4–14. doi: 10.1002/env.2371. ## Author(s) Joshua French

confreg function