bsaq {bsamGP}  R Documentation 
Bayesian ShapeRestricted Spectral Analysis Quantile Regression
Description
This function fits a Bayesian semiparametric quantile regression model to estimate shaperestricted functions using a spectral analysis of Gaussian process priors.
Usage
bsaq(formula, xmin, xmax, p, nbasis, nint, mcmc = list(), prior = list(),
shape = c('Free', 'Increasing', 'Decreasing', 'IncreasingConvex', 'DecreasingConcave',
'IncreasingConcave', 'DecreasingConvex', 'IncreasingS', 'DecreasingS',
'IncreasingRotatedS','DecreasingRotatedS','InvertedU','Ushape',
'IncMultExtreme','DecMultExtreme'), nExtreme = NULL,
marginal.likelihood = TRUE, spm.adequacy = FALSE, verbose = FALSE)
Arguments
formula 
an object of class “ 
xmin 
a vector or scalar giving userspecific minimum values of x. The default values are minimum values of x. 
xmax 
a vector or scalar giving userspecific maximum values of x. The default values are maximum values of x. 
p 
quantile of interest (default=0.5). 
nbasis 
number of cosine basis functions. 
nint 
number of grid points where the unknown function is evaluated for plotting. The default is 200. 
mcmc 
a list giving the MCMC parameters.
The list includes the following integers (with default values in parentheses):

prior 
a list giving the prior information. The list includes the following parameters
(default values specify the noninformative prior):

shape 
a vector giving types of shape restriction. 
nExtreme 
a vector of extreme points for 'IncMultExtreme', 'DecMultExtreme' shape restrictions. 
marginal.likelihood 
a logical variable indicating whether the log marginal likelihood is calculated. The methods of Gelfand and Dey (1994) and Newton and Raftery (1994) are used. 
spm.adequacy 
a logical variable indicating whether the log marginal likelihood of linear model is calculated. The marginal likelihood gives the values of the linear regression model excluding the nonlinear parts. 
verbose 
a logical variable. If 
Details
This generic function fits a Bayesian spectral analysis quantile regression model for estimating shaperestricted functions using Gaussian process priors. For enforcing shaperestrictions, the model assumed that the derivatives of the functions are squares of Gaussian processes.
Let y_i
and w_i
be the response and the vector of parametric predictors, respectively.
Further, let x_{i,k}
be the covariate related to the response through an unknown shaperestricted function.
The model for estimating shaperestricted functions is as follows.
y_i = w_i^T\beta + \sum_{k=1}^K f_k(x_{i,k}) + \epsilon_i, ~ i=1,\ldots,n,
where f_k
is an unknown shaperestricted function of the scalar x_{i,k} \in [0,1]
and
the error terms \{\epsilon_i\}
are a random sample from an asymmetric Laplace distribution, ALD_p(0,\sigma^2)
,
which has the following probability density function:
ALD_p(\epsilon; \mu, \sigma^2) = \frac{p(1p)}{\sigma^2}\exp\Big(\frac{(x\mu)[p  I(x \le \mu)]}{\sigma^2}\Big),
where 0 < p < 1
is the skew parameter, \sigma^2 > 0
is the scale parameter, \infty < \mu < \infty
is
the location parameter, and I(\cdot)
is the indication function.
The prior of function without shape restriction is:
f(x) = Z(x),
where Z
is a secondorder Gaussian process with mean function equal to zero and covariance function
\nu(s,t) = E[Z(s)Z(t)]
for s, t \in [0, 1]
. The Gaussian process is expressed with
the spectral representation based on cosine basis functions:
Z(x) = \sum_{j=0}^\infty \theta_j\varphi_j(x)
\varphi_0(x) = 1 ~~ \code{and} ~~ \varphi_j(x) = \sqrt{2}\cos(\pi j x), ~ j \ge 1, ~ 0 \le x \le 1
The shaperestricted functions are modeled by assuming the q
th derivatives of f
are squares of Gaussian processes:
f^{(q)}(x) = \delta Z^2(x)h(x), ~~ \delta \in \{1, 1\}, ~~ q \in \{1, 2\},
where h
is the squish function. For monotonic, monotonic convex, and concave functions, h(x)=1
, while
for S
and U
shaped functions, h
is defined by
h(x) = \frac{1  \exp[\psi(x  \omega)]}{1 + \exp[\psi(x  \omega)]}, ~~ \psi > 0, ~~ 0 < \omega < 1
For the spectral coefficients of functions without shape constraints, the scaleinvariant prior is used
(The intercept is included in \beta
):
\theta_j  \sigma, \tau, \gamma \sim N(0, \sigma^2\tau^2\exp[j\gamma]), ~ j \ge 1
The priors for the spectral coefficients of shape restricted functions are:
\theta_0  \sigma \sim N(m_{\theta_0}, \sigma v^2_{\theta_0}), \quad
\theta_j  \sigma, \tau, \gamma \sim N(m_{\theta_j}, \sigma\tau^2\exp[j\gamma]), ~ j \ge 1
To complete the model specification, the conjugate priors are assumed for \beta
and \sigma
:
\beta  \sigma \sim N(m_{0,\beta}, \sigma^2V_{0,\beta}), \quad \sigma^2 \sim IG\left(\frac{r_{0,\sigma}}{2}, \frac{s_{0,\sigma}}{2}\right)
Value
An object of class bsam
representing the Bayesian spectral analysis model fit.
Generic functions such as print
, fitted
and plot
have methods to show the results of the fit.
The MCMC samples of the parameters in the model are stored in the list mcmc.draws
,
the posterior samples of the fitted values are stored in the list fit.draws
, and
the MCMC samples for the log marginal likelihood are saved in the list loglik.draws
.
The output list also includes the following objects:
post.est 
posterior estimates for all parameters in the model. 
lmarg.lm 
log marginal likelihood for linear quantile regression model. 
lmarg.gd 
log marginal likelihood using GelfandDey method. 
lmarg.nr 
log marginal likelihood using NetwonRaftery method, which is biased. 
rsquarey 
correlation between 
call 
the matched call. 
mcmctime 
running time of Markov chain from 
References
Jo, S., Choi, T., Park, B. and Lenk, P. (2019). bsamGP: An R Package for Bayesian Spectral Analysis Models Using Gaussian Process Priors. Journal of Statistical Software, 90, 310320.
Lenk, P. and Choi, T. (2017) Bayesian Analysis of ShapeRestricted Functions using Gaussian Process Priors. Statistica Sinica, 27: 4369.
Gelfand, A. E. and Dey, K. K. (1994) Bayesian model choice: asymptotics and exact calculations. Journal of the Royal Statistical Society. Series B  Statistical Methodology, 56, 501514.
Kozumi, H. and Kobayashi, G. (2011) Gibbs sampling methods for Bayesian quantile regression. Journal of Statistical Computation and Simulation, 81(11), 15651578.
Newton, M. A. and Raftery, A. E. (1994) Approximate Bayesian inference with the weighted likelihood bootstrap (with discussion). Journal of the Royal Statistical Society. Series B  Statistical Methodology, 56, 348.
See Also
Examples
## Not run:
######################
# Increasingconcave #
######################
# Simulate data
set.seed(1)
n < 200
x < runif(n)
y < log(1 + 10*x) + rald(n, scale = 0.5, p = 0.5)
# Number of cosine basis functions
nbasis < 50
# Fit the model with default priors and mcmc parameters
fout1 < bsaq(y ~ fs(x), p = 0.25, nbasis = nbasis,
shape = 'IncreasingConcave')
fout2 < bsaq(y ~ fs(x), p = 0.5, nbasis = nbasis,
shape = 'IncreasingConcave')
fout3 < bsaq(y ~ fs(x), p = 0.75, nbasis = nbasis,
shape = 'IncreasingConcave')
# fitted values
fit1 < fitted(fout1)
fit2 < fitted(fout2)
fit3 < fitted(fout3)
# plots
plot(x, y, lwd = 2, xlab = 'x', ylab = 'y')
lines(fit1$xgrid, fit1$wbeta$mean[1] + fit1$fxgrid$mean, lwd=2, col=2)
lines(fit2$xgrid, fit2$wbeta$mean[1] + fit2$fxgrid$mean, lwd=2, col=3)
lines(fit3$xgrid, fit3$wbeta$mean[1] + fit3$fxgrid$mean, lwd=2, col=4)
legend('topleft', legend = c('1st Quartile', '2nd Quartile', '3rd Quartile'),
lwd = 2, col = 2:4, lty = 1)
## End(Not run)