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Monte Carlo sampling for Bayesian Gaussian process regression with an unknown (nonparametric) transformation.


  X = NULL,
  covfun_name = "matern_isotropic",
  locs_test = locs,
  X_test = NULL,
  nn = 30,
  emp_bayes = TRUE,
  approx_g = FALSE,
  nsave = 1000,
  ngrid = 100



n x 1 response vector


n x d matrix of locations


n x p design matrix; if unspecified, use intercept only


string name of a covariance function; see ?GpGp


n_test x d matrix of locations at which predictions are needed; default is locs


n_test x p design matrix for test data; default is X


number of nearest neighbors to use; default is 30 (larger values improve the approximation but increase computing cost)


logical; if TRUE, use a (faster!) empirical Bayes approach for estimating the mean function


logical; if TRUE, apply large-sample approximation for the transformation


number of Monte Carlo simulations


number of grid points for inverse approximations


a list with the following elements:

  • coefficients the estimated regression coefficients

  • fitted.values the posterior predictive mean at the test points locs_test

  • fit_gp the fitted GpGp_fit object, which includes covariance parameter estimates and other model information

  • post_ypred: nsave x ntest samples from the posterior predictive distribution at locs_test

  • post_g: nsave posterior samples of the transformation evaluated at the unique y values

  • model: the model fit (here, sbgp)

as well as the arguments passed in.


This function provides Bayesian inference for a transformed Gaussian process model using Monte Carlo (not MCMC) sampling. The transformation is modeled as unknown and learned jointly with the regression function (unless approx_g = TRUE, which then uses a point approximation). This model applies for real-valued data, positive data, and compactly-supported data (the support is automatically deduced from the observed y values). The results are typically unchanged whether laplace_approx is TRUE/FALSE; setting it to TRUE may reduce sensitivity to the prior, while setting it to FALSE may speed up computations for very large datasets. For computational efficiency, the Gaussian process parameters are fixed at point estimates, and the latent Gaussian process is only sampled when emp_bayes = FALSE. However, the uncertainty from this term is often negligible compared to the observation errors, and the transformation serves as an additional layer of robustness.


# \donttest{
# Simulate some data:
n = 200 # sample size
x = seq(0, 1, length = n) # observation points

# Transform a noisy, periodic function:
y = g_inv_bc(
  sin(2*pi*x) + sin(4*pi*x) + rnorm(n, sd = .5),
             lambda = .5) # Signed square-root transformation

# Fit the semiparametric Bayesian Gaussian process:
fit = sbgp(y = y, locs = x)
#> [1] "Initial GP fit..."
#> [1] "Updated GP fit..."
#> [1] "Sampling..."
#> [1] "Done!"
names(fit) # what is returned
#>  [1] "coefficients"  "fitted.values" "fit_gp"        "post_ypred"   
#>  [5] "post_g"        "model"         "y"             "X"            
#>  [9] "approx_g"      "sigma_epsilon"
coef(fit) # estimated regression coefficients (here, just an intercept)
#> [1] 0.0152005
class(fit$fit_gp) # the GpGp object is also returned
#> [1] "GpGp_fit"

# Plot the model predictions (point and interval estimates):
pi_y = t(apply(fit$post_ypred, 2, quantile, c(0.05, .95))) # 90% PI
plot(x, y, type='n', ylim = range(pi_y,y),
     xlab = 'x', ylab = 'y', main = paste('Fitted values and prediction intervals'))
polygon(c(x, rev(x)),c(pi_y[,2], rev(pi_y[,1])),col='gray', border=NA)
lines(x, y, type='p')
lines(x, fitted(fit), lwd = 3)

# }