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Semiparametric Bayesian spline model

Source: R/source_sba.R
sbsm.Rd

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
)

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

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.

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

# \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)
#> [1] "3 sec remaining"
#> [1] "2 sec remaining"
#> [1] "Total time: 4 seconds"
names(fit) # what is returned
#>  [1] "coefficients"   "fitted.values"  "post_theta"     "post_ypred"    
#>  [5] "post_g"         "post_psi"       "model"          "y"             
#>  [9] "X"              "sample_psi"     "laplace_approx" "fixedX"        
#> [13] "approx_g"      

# 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

# }