sbsm: Semiparametric Bayesian spline model

Description

Monte Carlo sampling for Bayesian spline regression with an unknown (nonparametric) transformation. Cubic B-splines are used with a prior that penalizes roughness.

Usage

sbsm(
  y,
  x = NULL,
  x_test = x,
  psi = NULL,
  laplace_approx = TRUE,
  fixedX = (length(y) >= 500),
  approx_g = FALSE,
  nsave = 1000,
  ngrid = 100,
  verbose = TRUE
)

Value

a list with the following elements:

coefficients the posterior mean of the regression coefficients
fitted.values the posterior predictive mean at the test points x_test
post_theta: nsave x p samples from the posterior distribution of the regression coefficients
post_ypred: nsave x n_test samples from the posterior predictive distribution at x_test
post_g: nsave posterior samples of the transformation evaluated at the unique y values
model: the model fit (here, sbsm)

as well as the arguments passed in.

Arguments

y: n x 1 response vector
x: n x 1 vector of observation points; if NULL, assume equally-spaced on [0,1]
x_test: n_test x 1 vector of testing points; if NULL, assume equal to x
psi: prior variance (inverse smoothing parameter); if NULL, sample this parameter
laplace_approx: logical; if TRUE, use a normal approximation to the posterior in the definition of the transformation; otherwise the prior is used
fixedX: logical; if TRUE, treat the design as fixed (non-random) when sampling the transformation; otherwise treat covariates as random with an unknown distribution
approx_g: logical; if TRUE, apply large-sample approximation for the transformation
nsave: number of Monte Carlo simulations
ngrid: number of grid points for inverse approximations
verbose: logical; if TRUE, print time remaining

Details

This function provides fully Bayesian inference for a transformed spline regression 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. By default, fixedX is set to FALSE for smaller datasets (n < 500) and TRUE for larger datasets (n >= 500).

Examples

Run this code

# \donttest{
# Simulate some data:
n = 200 # sample size
x = sort(runif(n)) # observation points

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

# Fit the semiparametric Bayesian spline model:
fit = sbsm(y = y, x = x)
names(fit) # what is returned

# Note: this is Monte Carlo sampling...no need for MCMC diagnostics!

# 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') # observed points
lines(x, fitted(fit), lwd = 3) # fitted curve
# }