tsroba: Bayesian estimation for the TSR model.

In OBASpatial: Objective Bayesian Analysis for Spatial Regression Models

Bayesian estimation for the TSR model.

Description

This function performs Bayesian estimation of θ=(\bold{β},σ^2,φ) for the TSR model using the based reference, Jeffreys' rule ,Jeffreys' independent and vague priors.

Usage

tsroba(formula, method="median",sdnu=1,
prior = "reference",coords.col = 1:2,kappa = 0.5,
cov.model = "matern", data,asigma=2.1, intphi = "default",
intnu="default",ini.pars,burn=500, iter=5000,thin=10,cprop = NULL)

Arguments

formula	A valid formula for a linear regression model.
method	Method to estimate (\bold{beta},σ,φ,ν). The methods availables are "mean","median" and "mode".
sdnu	Standard deviation logarithm for the lognormal proposal for ν
prior	Objective prior densities avaiable for the TSR model: ( reference: Reference based, jef.rul: Jeffreys' rule, jef.ind: Jeffreys' independent,vague: Vague).
coords.col	A vector with the column numbers corresponding to the spatial coordinates.
kappa	Shape parameter of the covariance function (fixed).
cov.model	Covariance functions available for the TSR model. matern: Matern, pow.exp: power exponential, exponential:exponential, cauchy: Cauchy, spherical: Spherical.
data	Data set with 2D spatial coordinates, the response and optional covariates.
asigma	Value of a for vague prior.
intphi	An interval for φ used for the uniform proposal. See DETAILS below.
intnu	An interval for ν used for the uniform proposal. See DETAILS below.
ini.pars	Initial values for (σ^2,φ,ν) in that order.
burn	Number of observations considered in burning process.
iter	Number of iterations for the sampling procedure.
thin	Number of observations considered in thin process.
cprop	A constant related to the acceptance probability (Default = NULL indicates that cprop is computed as the interval length of intphi). See DETAILS below.

Details

For the prior proposal, it was considered the structure π(φ,ν,λ)=φ(φ)π(ν|λ)π(λ). For the vague prior, φ follows an uniform distribution on the interval intphi, by default, this interval is computed using the empirical range of data as well as the constant cprop. On the other hand, ν|λ~ Texp(λ,A) with A the interval given by the argument intnu and λ~unif(0.02,0.5)

For the Jeffreys independent prior, the sampling procedure generates improper posterior distribution when intercept is considered for the mean function.

Value

dist	Joint sample (matrix object) obtaining for (\bold{beta},σ^2,φ).
betaF	Sample obtained for \bold{beta}.
sigmaF	Sample obtained for σ^2.
phiF	Sample obtained for φ.
nuF	Sample obtained for φ.
coords	Spatial data coordinates.
kappa	Shape parameter of the covariance function.
$X	Design matrix of the model.
$type	Covariance function of the model.
$theta	Bayesian estimator of (\bold{beta},σ,φ).
$y	Response variable.
$prior	Prior density considered.

Author(s)

Jose A. Ordonez, Marcos O. Prates, Larissa A. Matos, Victor H. Lachos.

References

Ordonez, J.A, M.O. Prattes, L.A. Matos, and V.H. Lachos (2020+). Objective Bayesian analysis for spatial Student-t regression models. (Submitted)

See Also

dnsrposoba,dtsrprioroba,dnsrprioroba,tsroba

Examples

set.seed(25)
data(dataca20)
d1=dataca20[1:158,]

xpred=model.matrix(calcont~altitude+area,data=dataca20[159:178,])
xobs=model.matrix(calcont~altitude+area,data=dataca20[1:158,])
coordspred=dataca20[159:178,1:2]

######covariance matern: kappa=0.3 prior:reference
res=tsroba(calcont~altitude+area, kappa = 0.3, data=d1,
           ini.pars=c(10,390,10),iter=11000,burn=1000,thin=10)

summary(res)

######covariance matern: kappa=0.3 prior:jef.rul
res1=tsroba(calcont~altitude+area, kappa = 0.3,
            data=d1,prior="jef.rul",ini.pars=c(10,390,10),
            iter=11000,burn=1000,thin=10)

summary(res1)

######covariance matern: kappa=0.3 prior:jef.ind
res2=tsroba(calcont~altitude+area, kappa = 0.3, data=d1,
            prior="jef.ind",ini.pars=c(10,390,10),iter=11000,
            burn=1000,thin=10)

summary(res2)

######covariance matern: kappa=0.3 prior:vague
res3=tsroba(calcont~altitude+area, kappa = 0.3,
     data=d1,prior="vague",ini.pars=c(10,390,10),,iter=11000,
     burn=1000,thin=10)

summary(res3)

####obtaining posterior probabilities
###(just comparing priors with kappa=0.3).
###the real aplication (see Ordonez et.al) consider kappa=0.3,0.5,0.7.

######### Using reference prior ###########
m1=intmT(prior="reference",formula=calcont~altitude+area,
kappa=0.3,cov.model="matern",data=dataca20,maxEval=1000)

######### Using Jeffreys' rule prior ###########
m1j=intmT(prior="jef.rul",formula=calcont~altitude+area,
kappa=0.3,cov.model="matern",data=dataca20,maxEval=1000)


######### Using Jeffreys' independent prior ###########
m1ji=intmT(prior="jef.ind",formula=calcont~altitude+area
,kappa=0.3,cov.model="matern",data=dataca20,maxEval=1000)

m1v=intmT(prior="vague",formula=calcont~altitude+area
,kappa=0.3,cov.model="matern",data=dataca20,maxEval=1000,intphi="default")


tot=m1+m1j+m1ji+m1v

####posterior probabilities#####
p1=m1/tot
pj=m1j/tot
pji=m1ji/tot
pv=m1v/tot


##########MSPE#######################################

pme=tsrobapred(res,xpred=xpred,coordspred=coordspred)
pme1=tsrobapred(res1,xpred=xpred,coordspred=coordspred)
pme2=tsrobapred(res2,xpred=xpred,coordspred=coordspred)
pme3=tsrobapred(res3,xpred=xpred,coordspred=coordspred)

mse=mean((pme-dataca20$calcont[159:178])^2)
mse1=mean((pme1-dataca20$calcont[159:178])^2)
mse2=mean((pme2-dataca20$calcont[159:178])^2)
mse3=mean((pme3-dataca20$calcont[159:178])^2)

OBASpatial documentation built on Sept. 11, 2022, 9:05 a.m.