bayMCMC_np_local: Bayesian bandwidth estimation for a functional nonparametric regression with homoscedastic errors

Description

Estimate the bandwidths in the regression function approximated by the functional Nadaraya-Watson estimator and kernel-form error density with localised bandwidths, in a functional nonparametric regression

Usage

bayMCMC_np_local(data_x, data_y, data_xnew, warm = 1000, M = 1000, 
	mutprob = 0.44, errorprob = 0.44, epsilonprob = 0.44, mutsizp = 1, 
		errorsizp = 1, epsilonsizp = 1, prior_alpha = 1, prior_beta = 0.05, 
			err_int = c(-10, 10), err_ngrid = 10001, num_batch = 20, 
				step = 10, alpha = 0.95, ...)

Arguments

data_x

An (n by p) matrix of discretised data points of functional curves

data_y

A scalar-valued response of length n

data_xnew

A matrix of discretised data points of new functional curve(s)

warm

Number of iterations for the burn-in period

Number of iterations for the Markov chain Monte Carlo (MCMC)

mutprob

Optimal acceptance rate of the random-walk Metropolis algorithm for sampling the bandwidth in the regression function

errorprob

Optimal acceptance rate of the random-walk Metropolis algorithm for sampling the bandwidth in the kernel-form error density

epsilonprob

Optimal acceptance rate of the random-walk Metropolis algorithm for sampling the bandwidth adjust factor in the kernel-form error density

mutsizp

Initial step length of the random-walk Metropolis algorithm for sampling the bandwidth in the regression function. Its value will be updated at each iteration to achieve the optimal acceptance rate, given the MCMC converges to its target distribution

errorsizp

Initial step length of the random-walk Metropolis algorithm for sampling the bandwidth in the kernel-form error density. Its value will be updated at each iteration to achieve the optimal acceptance rate, given the MCMC converges to its target distribution

epsilonsizp

Initial step length of the random-walk Metropolis algorithm for sampling the bandwidth adjustment factor in the kernel-form error density. Its value will be updated at each iteration to achieve the optimal acceptance rate

prior_alpha

Hyperparameter of the inverse gamma prior distribution for the squared bandwidths

prior_beta

Hyperparameter of the inverse gamma prior distribution for the squared bandwidths

err_int

Range of the error-density grid for computing the probability density function and cumulative probability density function

err_ngrid

Number of the error-density grid points

num_batch

Number of batches to assess the convergence of the MCMC

step

Thinning parameter. For example, when step=10, it keeps every 10th iteration of the MCMC output

alpha

The nominal coverage probability of the prediction interval, customarily 95 percent

...

Other arguments used to define semi-metric. For a set of smoothed functional data, the semi-metric based on derivative is suggested. For a set of rough functional data, the semi-metric based on the functional principal component analysis is suggested

Value

xpfinalres: Estimated bandwidths
mhat: Estimated regression function
sif_value: Simulation inefficiency factor
mlikeres: Marginal likelihood calculated using the Chib's (1995) method
acceptnwMCMC: Acceptance rate for sampling bandwidth in the regression function
accepterroMCMC: Acceptance rate for sampling bandwidth in the kernel-form error density
acceptepsilonMCMC: Acceptance rate for sampling bandwidth adjustment factor in the kernel-form error density
fore.den.mkr: Estimated probability density function of the error
fore.cdf.mkr: Estimated cumulative density function of the error
point forecast: Predicted response
PI: Prediction interval of response

Details

The Bayesian method estimates the bandwidths in the regression function and kernel-form error density. It performs better than the functional cross validation in terms of estimation accuracy, since the latter one does not utilise the information about the unknown error density. Furthermore, it can estimate error density more accurate than the Bayesian method with a global bandwidth

References

H. L. Shang (2013) Bayesian bandwidth estimation for a semi-functional partial linear regression model with unknown error density, Computational Statistics, in press.

H. L. Shang (2013) Bayesian bandwidth estimation for a functional nonparametric regression model with unknown error density, Computational Statistics and Data Analysis, 67, 185-198.

X. Zhang and R. D. Brooks and M. L. King (2009) A Bayesian approach to bandwidth selection for multivariate kernel regression with an application to state-price density estimation, Journal of Econometrics, 153, 21-32.

F. Ferraty, I. Van Keilegom and P. Vieu (2010) On the validity of the bootstrap in non-parametric functional regression, Scandinavian Journal of Statistics, 37, 286-306.

R. Meyer and J. Yu (2000) BUGS for a Bayesian analysis of stochastic volatility models, Econometircs Journal, 3(2), 198-215.