OBASpatial (version 1.9)

dtsrposoba: Objective posterior density for the TSR model

Description

It calculates the density function \(\pi(\phi,\nu)\) (up to a proportionality constant) for the TSR model using the based reference, Jeffreys' rule, Jeffreys' independent and vague priors. In this context \(\phi\) corresponds to the range parameter and \(\nu\) to the degrees of freedom.

Usage

dtsrposoba(x,formula,prior="reference",coords.col=1:2,
kappa=0.5,cov.model="exponential",data,asigma=2.1,intphi,intnu)

Value

Posterior density of x=(\(\phi,\nu\)) for the reference based, Jeffreys' rule and Jeffreys' independent priors. For the vague the result is the posterior density of x=(\(\phi,\nu,\lambda\))

Arguments

x

A vector with the quanties \((\phi,\nu)\). For the vague prior x must be a three dimension vector \((\phi,\nu,\lambda)\) with \(\lambda\) a number in the interval \((0.02,0.5)\). See DETAILS below.

formula

A valid formula for a linear regression model.

prior

Objective prior densities avaiable for the TSR model: ( reference: Reference based, jef.rul: Jeffreys' rule, jef.ind: Jeffreys' independent).

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 \(\phi\) used for vague prior.

intnu

An interval for \(\nu\) used for vague prior.

Author

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

Details

The posterior distribution is computed for this priors under the improper family \(\frac{\pi(\phi,\nu)}{(\sigma^2)^a}\). For the vague prior, it was considered the structure \(\pi(\phi,\nu,\lambda)=\phi(\phi)\pi(\nu|\lambda)\pi(\lambda)\) where a priori, \(\phi\) follows an uniform distribution on the interval intphi, \(\nu|\lambda~ Texp(\lambda,A)\) with A the interval given by the argument intnu and \(\lambda~unif(0.02,0.5)\).

For the Jeffreys independent prior, this family of priors generates improper posterior distribution when intercept is considered for the mean function.

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

Examples

data(dataca20)

######### Using reference prior ###########
dtsrposoba(x=c(5,11),prior="reference",formula=calcont~altitude+area,
kappa=0.3,cov.model="matern",data=dataca20)

######### Using Jeffreys' rule prior ###########
dtsrposoba(x=c(5,11),prior="jef.rul",formula=calcont~altitude+area,
kappa=0.3,cov.model="matern",data=dataca20)


######### Using Jeffreys' independent prior ###########
dtsrposoba(x=c(5,11),prior="jef.ind",formula=calcont~altitude+area
,kappa=0.3,cov.model="matern",data=dataca20)

######### Using vague independent prior ###########
dtsrposoba(x=c(5,11,.3),prior="vague",formula=calcont~altitude+area,
kappa=0.3,cov.model="matern",data=dataca20,intphi=c(0.1,10),
intnu=c(4.1,30))