STAR_spline_gibbs: Gibbs sampler (data augmentation) for spline regression

In drkowal/rSTAR: MCMC and EM algorithms for Simultaneous Transformation and Rounding (STAR) Models

Gibbs sampler (data augmentation) for spline regression

Description

Compute MCMC samples from the predictive distributions of a STAR spline regression model. The Monte Carlo sampler STAR_spline is preferred unless n is large (> 500).

Usage

STAR_spline_gibbs(
  y,
  tau = NULL,
  transformation = "np",
  y_max = Inf,
  psi = NULL,
  approx_Fz = FALSE,
  approx_Fy = FALSE,
  nsave = 1000,
  nburn = 1000,
  nskip = 0,
  verbose = TRUE
)

Arguments

y	n x 1 vector of observed counts
tau	n x 1 vector of observation points; if NULL, assume equally-spaced on [0,1]
transformation	transformation to use for the latent data; must be one of "identity" (identity transformation) "log" (log transformation) "sqrt" (square root transformation) "bnp" (Bayesian nonparametric transformation using the Bayesian bootstrap) "np" (nonparametric transformation estimated from empirical CDF) "pois" (transformation for moment-matched marginal Poisson CDF) "neg-bin" (transformation for moment-matched marginal Negative Binomial CDF)
y_max	a fixed and known upper bound for all observations; default is Inf
psi	prior variance (1/smoothing parameter); if NULL, update in MCMC
approx_Fz	logical; in BNP transformation, apply a (fast and stable) normal approximation for the marginal CDF of the latent data
approx_Fy	logical; in BNP transformation, approximate the marginal CDF of y using the empirical CDF
nsave	number of MCMC iterations to save
nburn	number of MCMC iterations to discard
nskip	number of MCMC iterations to skip between saving iterations, i.e., save every (nskip + 1)th draw
verbose	logical; if TRUE, print time remaining

Details

STAR defines a count-valued probability model by (1) specifying a Gaussian model for continuous *latent* data and (2) connecting the latent data to the observed data via a *transformation and rounding* operation. Here, the continuous latent data model is a spline regression.

There are several options for the transformation. First, the transformation can belong to the *Box-Cox* family, which includes the known transformations 'identity', 'log', and 'sqrt'. Second, the transformation can be estimated (before model fitting) using the empirical distribution of the data y. Options in this case include the empirical cumulative distribution function (CDF), which is fully nonparametric ('np'), or the parametric alternatives based on Poisson ('pois') or Negative-Binomial ('neg-bin') distributions. For the parametric distributions, the parameters of the distribution are estimated using moments (means and variances) of y. The distribution-based transformations approximately preserve the mean and variance of the count data y on the latent data scale, which lends interpretability to the model parameters. Lastly, the transformation can be modeled using the Bayesian bootstrap ('bnp'), which is a Bayesian nonparametric model and incorporates the uncertainty about the transformation into posterior and predictive inference.

Value

post_ytilde: nsave x n samples from the posterior predictive distribution at the observation points tau

Note

For the 'bnp' transformation (without the Fy approximation), there are numerical stability issues when psi is modeled as unknown. In this case, it is better to fix psi at some positive number.

Examples

# Simulate some data:
n = 100
tau = seq(0,1, length.out = n)
y = round_floor(exp(1 + rnorm(n)/4 + poly(tau, 4)%*%rnorm(n=4, sd = 4:1)))

# Sample from the predictive distribution of a STAR spline model:
post_ytilde = STAR_spline_gibbs(y = y, tau = tau)

# Compute 90% prediction intervals:
pi_y = t(apply(post_ytilde, 2, quantile, c(0.05, .95)))

# Plot the results: intervals, median, and smoothed mean
plot(tau, y, ylim = range(pi_y, y))
polygon(c(tau, rev(tau)),c(pi_y[,2], rev(pi_y[,1])),col='gray', border=NA)
lines(tau, apply(post_ytilde, 2, median), lwd=5, col ='black')
lines(tau, smooth.spline(tau, apply(post_ytilde, 2, mean))$y, lwd=5, col='blue')
lines(tau, y, type='p')

drkowal/rSTAR documentation built on July 5, 2023, 2:18 p.m.