bisoreg (version 1.1)

bisoreg: Bayesian Isotonic Regression

Description

Computes MCMC simulations from the poseterior distribution of the monotonic Bernstein polynomial regression function.

Usage

bisoreg(x, y, m = floor(n/2), priors = list(), inits = list(), n.sim = 5000, n.burn = 1000, n.thin = 10, seed)

Arguments

x

vector of covariate values

y

vector of response values

m

positive integer specifying the order of the Bernstein polynomial basis expansion. Default is half the number of observations in the data set. This value is reset with m <- floor(m) to prevent noninteger values from being used.

priors

list of values of prior parameters. Values in the list must be named one of (with default values in parentheses) a.sig (0.1), b.sig (0.1), a.tau (1), b.tau (1), a.p (1), b.p (1)

inits

list of initial values for model parameters. Values in the list must be named one of u0, u (must be a vector of length equal to m), sigsq, tausq, p. If not specified, initial

n.sim

number of MCMC simulations after burn-in and thinning. The total number of MCMC iterations run is n.sim*n.thin + n.burn. Thus, the final number MCMC simulations in the output of the function will always be n.sim.

n.burn

number of simulations to burn

n.thin

keep every thin draw from the MCMC simulations

seed

random number seed for the MCMC simulations

Value

• An S3 object with class "biso" and the following components:
• mcmctime
  the time it took to run the sampling algorithm. mcmctime is the output from a call to function system.time.
• postdraws
  a data frame with posterior draws of all parameters
• W
  the "W" matrix
• x
  original x values
• y
  original y values
• m
  order of the Bernstein polynomial basis expansion
• DIC
  "original" DIC value as in Spiegelhalter et. al. (2002)
• DIC2
  "alternative" DIC value as in Gelman et. al. (2004)
• pD
  effective number of parameters as in Spiegelhalter et. al. (2002)
• pD2
  alternative effective number of parameters as in Gelman et. al. (2004)

Details

Generates MCMC draws from the posterior distribution of the nonparametric Bernstein regression function as described in Curtis and Ghosh (2009).

References

Curtis, S. M. and Ghosh, S. K. (2009) "A variable selection approach to monotonic regression with Bernstein polynomials." Gelman, A., et. al. (2003). Bayesian Data Analysis. Chapman and Hall / CRC. Speigelhalter, D. J., et. al. (2002). "Bayesian measures of model complexity and fit." JRSSB 64(4): 583--616.

See Also

pflat.biso,fitted.biso,plot.biso

Examples

data(childgrowth)

## Run three different chains ##
out1 <- bisoreg(childgrowth$day, childgrowth$height, m=80, n.sim=15000, n.thin=10)
out2 <- bisoreg(childgrowth$day, childgrowth$height, m=80, n.sim=15000, n.thin=10)
out3 <- bisoreg(childgrowth$day, childgrowth$height, m=80, n.sim=15000, n.thin=10)

## Convert to MCMC object ##
out <- list()
out[[1]] <- mcmc(out1$postdraws)
out[[2]] <- mcmc(out2$postdraws)
out[[3]] <- mcmc(out3$postdraws)
mcmcout <- mcmc.list(out)

## Gelman Rubin diagnostics ##
gelman.diag(mcmcout)

## Posterior probability of a flat function ##
pflat(out1)

## Create scatterplot and plot Bayesian regression fit ##
plot(out1, color="black", cl=F, lwd=2, xlab="day", ylab="height")
title("Child Growth Data")

## Compute isoreg fit ##
iout = isoreg(childgrowth$day, childgrowth$height)
lines(as.stepfun(iout), col.h="red", col.v="red", lwd=2)

## Compute monreg estimate of Dette et. al. ##
mout = monreg.wrapper(childgrowth$day, childgrowth$height)
lines(childgrowth$day, mout$est, lwd=2, col="blue")

## Compute local polynomial regression fit ##
lines(childgrowth$day, predict(loess.wrapper(childgrowth$day, childgrowth$height)), lwd=2, col="green")

## Add a legend ##
gplots:::smartlegend("left", "top",
  legend=c("bayes", "isoreg", "monreg", "loess"),
  col=c("black", "red", "blue", "green"), lwd=2)

## Check sensitivity to the prior on p ##
## p ~ Beta(a=1,b=1) ##
outa1b1 <- bisoreg(childgrowth$day, childgrowth$height, 40, priors=list(a.p=1,b.p=1), n.sim=15000, n.thin=10)
## p ~ Beta(a=1,b=3) ##
outa1b3 <- bisoreg(childgrowth$day, childgrowth$height, 40, priors=list(a.p=1,b.p=3), n.sim=15000, n.thin=10)
## p ~ Beta(a=3,b=1) ##
outa3b1 <- bisoreg(childgrowth$day, childgrowth$height, 40, priors=list(a.p=3, b.p=1), n.sim=15000, n.thin=10)

## Create plot to compare model fits ##
plot(outa1b1,cl=F,color="black")
plot(outa1b3,cl=F,add=T,col="red")
plot(outa3b1,cl=F,add=T,col="green")
gplots:::smartlegend("left","top",
                    legend=c("a=1 b=1","a=1 b=3","a=3 b=1"),
                    col=c("black","red","green"),
                    lwd=1)

## Check sensitivity to the choice of M ##
outm20 <- bisoreg(childgrowth$day, childgrowth$height, m=20, n.sim=15000, n.thin=10)
outm30 <- bisoreg(childgrowth$day, childgrowth$height, m=30, n.sim=15000, n.thin=10)
outm40 <- bisoreg(childgrowth$day, childgrowth$height, m=40, n.sim=15000, n.thin=10)

## Create plot to compare model fits ##
plot(outm20, cl=F, col="blue")
plot(outm30, cl=F, add=T, col="red")
plot(outm40, cl=F, add=T, col="green")
plot(out1, cl=F, add=T, col="black")
gplots:::smartlegend("left", "top",
                    legend=c(20,30,40,80),
                    col=c("blue", "red", "green", "black"),
                    lwd=1)


## A method for choosing 'm' ##
## (from Sujit K. Ghosh) ##

## Piece-wise linear 'true' regression function ##
mu <- function(x){
  ifelse(-3<=x & x<=-1, x, 0) + ifelse(-1<x & x<1, -1, 0) + ifelse(1<=x & x<3, x - 2, 0)
}
par(mfrow=c(2,2))
curve(mu, -2.9, 2.9)

## Simulate the data ##
n <- 100
x <- runif(n, -2.9, 2.9)
y <- mu(x) + rnorm(n, 0, 0.1)

## Fit initially with default 'm=n/2' ##
fit0 <- bisoreg(x, y, m=floor(n/2))
plot(fit0)
lines(sort(x), mu(sort(x)), col="blue")
fit0$DIC2
fit0$pD2

## Use effective number of parameters ##
## as new value of 'm' ##
fit <- bisoreg(x, y, m=ceiling(fit0$pD2))
plot(fit)
lines(sort(x), mu(sort(x)), col="blue")
fit$DIC2
fit$pD2

## Compare fits ##
plot(fitted(fit0), fitted(fit))
abline(0, 1)
mean(abs(y - fitted(fit0)))
mean(abs(y - fitted(fit)))