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Bayesian linear model with a Box-Cox transformation

Source: R/source_compete.R
blm_bc.Rd

MCMC sampling for Bayesian linear regression with a (known or unknown) Box-Cox transformation. A g-prior is assumed for the regression coefficients.

Usage

blm_bc(
  y,
  X,
  X_test = X,
  psi = length(y),
  lambda = NULL,
  sample_lambda = TRUE,
  nsave = 1000,
  nburn = 1000,
  nskip = 0,
  verbose = TRUE
)

Arguments

y

n x 1 vector of observed counts

X

n x p matrix of predictors (no intercept)

X_test

n_test x p matrix of predictors for test data; default is the observed covariates X

psi

prior variance (g-prior)

lambda

Box-Cox transformation; if NULL, estimate this parameter

sample_lambda

logical; if TRUE, sample lambda, otherwise use the fixed value of lambda above or the MLE (if lambda unspecified)

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

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 test points X_test

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

  • post_lambda nsave posterior samples of lambda

  • post_sigma nsave posterior samples of sigma

  • model: the model fit (here, blm_bc)

as well as the arguments passed in.

Details

This function provides fully Bayesian inference for a transformed linear model via MCMC sampling. The transformation is parametric from the Box-Cox family, which has one parameter lambda. That parameter may be fixed in advanced or learned from the data.

Note

Box-Cox transformations may be useful in some cases, but in general we recommend the nonparametric transformation (with Monte Carlo, not MCMC sampling) in sblm.

An intercept is automatically added to X and X_test. The coefficients reported do *not* include this intercept parameter, since it is not identified under more general transformation models (e.g., sblm).

Examples

# Simulate some data:
dat = simulate_tlm(n = 100, p = 5, g_type = 'step')
y = dat$y; X = dat$X # training data
y_test = dat$y_test; X_test = dat$X_test # testing data

hist(y, breaks = 25) # marginal distribution


# Fit the Bayesian linear model with a Box-Cox transformation:
fit = blm_bc(y = y, X = X, X_test = X_test)
#> [1] "1 sec remaining"
#> [1] "1 sec remaining"
#> [1] "Total time: 0 seconds"
names(fit) # what is returned
#>  [1] "coefficients"  "fitted.values" "post_theta"    "post_ypred"   
#>  [5] "post_g"        "post_lambda"   "post_sigma"    "model"        
#>  [9] "y"             "X"             "X_test"        "psi"          
round(quantile(fit$post_lambda), 3) # summary of unknown Box-Cox parameter
#>    0%   25%   50%   75%  100% 
#> 0.000 0.004 0.010 0.019 0.072