gibbs_npc: Gibbs sampler for Bayesian semiparametric inference with the...

In beyondWhittle: Bayesian Spectral Inference for Time Series

Gibbs sampler for Bayesian semiparametric inference with the corrected AR likelihood

Description

Obtain samples of the posterior of the corrected autoregressive likelihood in conjunction with a Bernstein-Dirichlet prior on the correction.

Usage

gibbs_npc(
  data,
  ar.order,
  Ntotal,
  burnin,
  thin = 1,
  print_interval = 100,
  numerical_thresh = 1e-07,
  adaption.N = burnin,
  adaption.batchSize = 50,
  adaption.tar = 0.44,
  full_lik = F,
  rho.alpha = rep(1, ar.order),
  rho.beta = rep(1, ar.order),
  eta = T,
  M = 1,
  g0.alpha = 1,
  g0.beta = 1,
  k.theta = 0.01,
  tau.alpha = 0.001,
  tau.beta = 0.001,
  trunc_l = 0.1,
  trunc_r = 0.9,
  coars = F,
  kmax = 100 * coars + 500 * (!coars),
  L = max(20, length(data)^(1/3))
)

Arguments

data	numeric vector; NA values are interpreted as missing values and treated as random
ar.order	order of the autoregressive model (integer > 0)
Ntotal	total number of iterations to run the Markov chain
burnin	number of initial iterations to be discarded
thin	thinning number (postprocessing)
print_interval	Number of iterations, after which a status is printed to console
numerical_thresh	Lower (numerical pointwise) bound for the spectral density
adaption.N	total number of iterations, in which the proposal variances (of rho) are adapted
adaption.batchSize	batch size of proposal adaption for the rho_i's (PACF)
adaption.tar	target acceptance rate for the rho_i's (PACF)
full_lik	logical; if TRUE, the full likelihood for all observations is used; if FALSE, the partial likelihood for the last n-p observations
rho.alpha, rho.beta	prior parameters for the rho_i's: 2*(rho-0.5)~Beta(rho.alpha,rho.beta), default is Uniform(-1,1)
eta	logical variable indicating whether the model confidence eta should be included in the inference (eta=T) or fixed to 1 (eta=F)
M	DP base measure constant (> 0)
g0.alpha, g0.beta	parameters of Beta base measure of DP
k.theta	prior parameter for polynomial degree k (propto exp(-k.theta*k*log(k)))
tau.alpha, tau.beta	prior parameters for tau (inverse gamma)
trunc_l, trunc_r	left and right truncation of Bernstein polynomial basis functions, 0<=trunc_l<trunc_r<=1
coars	flag indicating whether coarsened or default bernstein polynomials are used (see Appendix E.1 in Ghosal and van der Vaart 2017)
kmax	upper bound for polynomial degree of Bernstein-Dirichlet mixture (can be set to Inf, algorithm is faster with kmax<Inf due to pre-computation of basis functions, but values 500<kmax<Inf are very memory intensive)
L	truncation parameter of DP in stick breaking representation

Details

Partial Autocorrelation Structure (PACF, uniform prior) and the residual variance sigma2 (inverse gamma prior) is used as model parametrization. A Bernstein-Dirichlet prior for c_eta with base measure Beta(g0.alpha, g0.beta) is used. Further details can be found in the simulation study section in the referenced paper by Kirch et al. For more information on the PACF parametrization, see the referenced paper by Barndorff-Nielsen and Schou.

Value

list containing the following fields:

psd.median, psd.mean	psd estimates: (pointwise) posterior median and mean
psd.p05, psd.p95	pointwise credibility interval
psd.u05, psd.u95	uniform credibility interval
k, tau, V, W	posterior traces of nonparametric correction
rho	posterior trace of model AR parameters (PACF parametrization)
eta	posterior trace of model confidence eta
lpost	trace of log posterior

References

C. Kirch et al. (2018) Beyond Whittle: Nonparametric Correction of a Parametric Likelihood With a Focus on Bayesian Time Series Analysis Bayesian Analysis <doi:10.1214/18-BA1126>

S. Ghosal and A. van der Vaart (2017) Fundamentals of Nonparametric Bayesian Inference <doi:10.1017/9781139029834>

O. Barndorff-Nielsen and G. Schou On the parametrization of autoregressive models by partial autocorrelations Journal of Multivariate Analysis (3),408-419 <doi:10.1016/0047-259X(73)90030-4>

Examples

## Not run: 

##
## Example 1: Fit a nonparametrically corrected AR(p) model to sunspot data:
##

# Use this variable to set the AR model order
p <- 2

data <- sqrt(as.numeric(sunspot.year))
data <- data - mean(data)

# If you run the example be aware that this may take several minutes
print("example may take some time to run")
mcmc <- gibbs_npc(data=data, ar.order=p, Ntotal=10000, burnin=4000, thin=2)

# Plot spectral estimate, credible regions and periodogram on log-scale
plot(mcmc, log=T)


##
## Example 2: Fit a nonparametrically corrected AR(p) model to high-peaked AR(1) data
##

# Use this variable to set the autoregressive model order
p <- 1

n <- 256
data <- arima.sim(n=n, model=list(ar=0.95)) 
data <- data - mean(data)
omega <- fourier_freq(n)
psd_true <- psd_arma(omega, ar=0.95, ma=numeric(0), sigma2=1)

# If you run the example be aware that this may take several minutes
print("example may take some time to run")
mcmc <- gibbs_npc(data=data, ar.order=p, Ntotal=10000, burnin=4000, thin=2)

# Compare estimate with true function (green)
plot(mcmc, log=F, pdgrm=F, credib="uniform")
lines(x=omega, y=psd_true, col=3, lwd=2)

# Compute the Integrated Absolute Error (IAE) of posterior median
cat("IAE=", mean(abs(mcmc$psd.median-psd_true)[-1]) , sep="")

## End(Not run)

beyondWhittle documentation built on June 22, 2024, 11:35 a.m.