Calcium Content In Soil Samples.

Description

This data set contains the calcium content measured in soil samples taken from the 0-20cm layer at 178 locations within a certain study area divided in three sub-areas. The elevation at each location was also recorded.See geoR package for details.

Usage

data("dataca20")

Format

A data frame with 178 observations on the following 3 variables.

east

X Coordinate.

north

Y coordinate.

calcont

Calcium content measured in mmol_c/dm^3.

altitude

A vector with the elevation of each sampling location,in meters.

area

A factor indicating the sub area to which the locations belongs.

References

Oliveira, M. C. N. (2003). Metodos de estimacao de parametros em modelos geoestatisticos com diferentes estruturas de covariancias: uma aplicacao ao teor de calcio no solo. Ph.D. thesis, ESALQ/USP/Brasil.

Surface elevations

Description

Surface elevation data taken from Davis (1972). An onject of the class geodata with elevation values at 52 locations.

Usage

data("dataelev")

Format

A data frame with 52 observations on the following 3 variables.

x

X coordinate (multiple of 50 feet).

y

Y coordinate (multiple of 50 feet).

elevation

elevations (multiples of 10 feet).

References

Davis, J.C. (1973) Statistics and Data Analysis in Geology. Wiley.

Objective posterior density for the NSR model

Description

It calculates the density function \pi(\phi) (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.

Usage

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

Arguments

Details

The posterior distribution is computed for this priors under the improper family \frac{\pi(\phi)}{(\sigma^2)^a}. For the vague prior, it was considered the structure where a priori, \phi folows an uniform distribution on the interval intphi.

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

Value

Posterior density of x=\phi.

Author(s)

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

References

Berger, J.O, De Oliveira, V. and Sanso, B. (2001). Objective Bayesian Analysis of Spatially Correlated Data. Journal of the American Statistical Association., 96, 1361 – 1374.

See Also

dtsrposoba,dtsrprioroba,dnsrprioroba

Examples

data(dataelev)

######### Using reference prior ###########
dnsrposoba(x=5,prior="reference",formula=elevation~1,
kappa=1,cov.model="matern",data=dataelev)

######### Using Jeffreys' rule prior ###########
dnsrposoba(x=5,prior="jef.rul",formula=elevation~1,
kappa=1,cov.model="matern",data=dataelev)

######### Using vague independent prior ###########
dnsrposoba(x=5,prior="vague",formula=elevation~1,
kappa=0.3,cov.model="matern",data=dataelev,intphi=c(0.1,10))

Objective prior density for the NSR model

Description

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

Usage

dnsrprioroba(x,trend="cte",prior="reference",coords.col=1:2,
kappa=0.5,cov.model="exponential",data)

Arguments

Details

Denote as \bold{c}=(c_{1},c_{2}) the coordinates of a spatial location. trend defines the design matrix as:

• 0 (zero,without design matrix) Only valid for the Independent Jeffreys' prior

• "cte", the design matrix is such that mean function \mu(\bold{c})=\mu is constant over the region.

• "1st", the design matrix is such that mean function becames a first order polynomial on the coordinates:

  \mu(\bold(c))=\beta_0+ \beta_1c_1+\beta_2c_2

• "2nd", the design matrix is such that mean function \mu(\bold{c})=\mu becames a second order polynomial on the coordinates:

  \mu(\bold(c))=\beta_0+ \beta_1c_1+\beta_2c_2 + \beta_3c_{1}^2+ \beta_4c_{2}^2+ \beta_5c_1c_2

• ~model a model specification to include covariates (external trend) in the model.

Value

Prior density of x=\phi

Author(s)

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

References

Berger, J.O, De Oliveira, V. and Sanso, B. (2001). Objective Bayesian Analysis of Spatially Correlated Data. Journal of the American Statistical Association., 96, 1361 – 1374.

See Also

dtsrposoba,dtsrprioroba,dnsrposoba

Examples

data(dataelev)## data using by Berger et. al (2001)

######### Using reference prior ###########
dnsrprioroba(x=20,kappa=0.3,cov.model="matern",data=dataelev)


######### Using jef.rule prior###########
dnsrprioroba(x=20,prior="jef.rul",kappa=0.3,cov.model="matern",
data=dataelev)

######### Using  jef.ind prior ###########
dnsrprioroba(x=20,prior="jef.ind",trend=0,
kappa=0.3,cov.model="matern",data=dataelev)

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)

Arguments

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.

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)

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

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))

Objective prior 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 and Jeffreys' independent priors. In this context \phi corresponds to the range parameter and \nu to the degrees of freedom.

Usage

dtsrprioroba(x,trend="cte",prior="reference",coords.col=1:2,
kappa=0.5,cov.model="exponential",data)

Arguments

Details

Denote as \bold{c}=(c_{1},c_{2}) the coordinates of a spatial location. trend defines the design matrix as:

• 0 (zero,without design matrix) Only valid for the Independent Jeffreys' prior

• "cte", the design matrix is such that mean function \mu(\bold{c})=\mu is constant over the region.

• "1st", the design matrix is such that mean function becames a first order polynomial on the coordinates:

  \mu(\bold(c))=\beta_0+ \beta_1c_1+\beta_2c_2

• "2nd", the design matrix is such that mean function \mu(\bold{c})=\mu becames a second order polynomial on the coordinates:

  \mu(\bold(c))=\beta_0+ \beta_1c_1+\beta_2c_2 + \beta_3c_{1}^2+ \beta_4c_{2}^2+ \beta_5c_1c_2

• ~model a model specification to include covariates (external trend) in the model.

Value

Density of x=(\phi,\nu)

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

dtsrposoba,dnsrprioroba,dnsrposoba

Examples

data(dataca20)

######### Using reference prior and a constant trend###########
dtsrprioroba(x=c(6,100),kappa=0.3,cov.model="matern",data=dataca20)


######### Using jef.rule prior and 1st trend###########
dtsrprioroba(x=c(6,100),prior="jef.rul",trend=~altitude+area,
kappa=0.3,cov.model="matern",data=dataca20)

######### Using  jef.ind prior ###########
dtsrprioroba(x=c(6,100),prior="jef.ind",trend=0,
kappa=0.3,cov.model="matern",data=dataca20)

Marginal posterior density for a model.

Description

It calculates the marginal density density for a model M (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

intmT(formula,prior="reference",coords.col=1:2,kappa=0.5,
cov.model="exponential",data,asigma,intphi="default",intnu=c(4.1,Inf),maxEval)

Arguments

Details

Let m_k a parametric model with parameter vector \theta_k. Under the TSR model and the prior density proposal:

\frac{\pi(\phi,\nu)}{(\sigma^2)^a}

we have that the marginal density is given by:

\int L(\theta_{m_k})\pi(m_k)dm_k

This quantity can be useful as a criteria for model selection. The computation of m_k could be compute demanding depending on the number of iterations in maxEval.

Value

Marginal density of the model m_k for the reference based, Jeffreys' rule, Jeffreys' independent and vague priors.

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

Examples

set.seed(25)
data(dataca20)



######### 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: higher probability:
#########prior="reference", kappa=0.3


p1=m1/tot
pj=m1j/tot
pji=m1ji/tot
pv=m1v/tot

Marginal posterior density for a model.

Description

It calculates the marginal density density for a model M (up to a proportionality constant) for the NSR model using the based reference, Jeffreys' rule, Jeffreys' independent and vague priors. In this context \phi corresponds to the range parameter.

Usage

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

Arguments

Details

Let m_k a parametric model with parameter vector \theta_k. Under the TSR model and the prior density proposal:

\frac{\pi(\phi)}{(\sigma^2)^a}

we have that the marginal density is given by:

\int L(\theta_{m_k})\pi(m_k)dm_k

This quantity can be useful as a criteria for model selection. The computation of m_k could be compute demanding depending on the number of iterations in maxEval.

Value

Marginal density of the model m_k for the reference based, Jeffreys' rule, Jeffreys' independent and vague priors.

Author(s)

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

References

Berger, J.O, De Oliveira, V. and Sanso, B. (2001). Objective Bayesian Analysis of Spatially Correlated Data. Journal of the American Statistical Association., 96, 1361 – 1374.

See Also

dnsrposoba,dtsrprioroba,dnsrprioroba

Examples

data(dataca20)

set.seed(25)
data(dataelev)## data using by Berger et. al (2001)

######### Using reference prior ###########
m1=intmnorm(prior="reference",formula=elevation~1,
kappa=0.5,cov.model="matern",data=dataelev,maxEval=1000)

log(m1)


######### Using reference prior kappa=1 ###########
m2=intmnorm(prior="reference",formula=elevation~1,
kappa=1,cov.model="matern",data=dataelev,maxEval=1000)
log(m2)

######### Using reference prior kappa=1.5 ###########
m3=intmnorm(prior="reference",formula=elevation~1
,kappa=1.5,cov.model="matern",data=dataelev,maxEval=1000)
log(m3)

tot=m1+m2+m3

########posterior probabilities: higher probability:
#########prior="reference", kappa=1
p1=m1/tot
p2=m2/tot
p3=m3/tot

Bayesian estimation for the NSR model.

Description

This function performs Bayesian estimation of \theta=(\bold{\beta},\sigma^2,\phi) for the NSR model using the based reference, Jeffreys' rule ,Jeffreys' independent and vague priors.

Usage

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

Arguments

Details

For the "unif" proposal, it was considered the structure where a priori, \phi 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.

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

Value

Author(s)

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

References

Berger, J.O, De Oliveira, V. and Sanso, B. (2001). Objective Bayesian Analysis of Spatially Correlated Data. Journal of the American Statistical Association., 96, 1361 – 1374.

See Also

dnsrposoba,dtsrprioroba,dnsrprioroba,tsroba

Examples

set.seed(25)
data(dataelev)


######covariance matern: kappa=0.5
res=nsroba(elevation~1, kappa = 0.5, cov.model = "matern", data=dataelev,
ini.pars=c(10,390))

summary(res)

######covariance matern: kappa=1
res1=nsroba(elevation~1, kappa = 1, cov.model = "matern", data=dataelev,
ini.pars=c(10,390))

summary(res1)

######covariance matern: kappa=1.5
res2=nsroba(elevation~1, kappa = 1.5, cov.model = "matern", data=dataelev,
ini.pars=c(10,390))

summary(res2)

Prediction under Normal Objective Bayesian Analysis (OBA).

Description

This function uses the sampling distribution of parameters obtained from the function tsroba to predict values at unknown locations.

Usage

nsrobapred1(xpred, coordspred, obj)

Arguments

Details

This function predicts using the sampling distribution of parameters obtained from the function nsroba and the conditional normal distribution of the predicted values given the data.

Value

This function returns a vector with the predicted values at the specified locations.

Author(s)

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

References

Berger, J.O, De Oliveira, V. and Sanso, B. (2001). Objective Bayesian Analysis of Spatially Correlated Data. Journal of the American Statistical Association., 96, 1361 – 1374.

Diggle, P. and P. Ribeiro (2007).Model-Based Geostatistics. Springer Series in Statistics.

See Also

nsroba,tsrobapred

Examples

set.seed(25)
data(dataelev)
d1=dataelev[1:42,]

reselev=nsroba(elevation~1, kappa = 0.5, cov.model = "matern", data=d1,
ini.pars=c(10,3),intphi=c(0.8,10))

datapred1=dataelev[43:52,]
coordspred1=datapred1[,1:2]
nsrobapred1(obj=reselev,coordspred=coordspred1,xpred=rep(1,10))

Summary of a nsroba object

Description

summary method for class "nsroba".

Usage

## S3 method for class 'nsroba'
summary(object,...)

Arguments

Value

Author(s)

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

References

Berger, J.O, De Oliveira, V. and Sanso, B. (2001). Objective Bayesian Analysis of Spatially Correlated Data. Journal of the American Statistical Association., 96, 1361 – 1374.

See Also

dnsrposoba,dtsrprioroba,dnsrprioroba,tsroba

Examples

set.seed(25)
data(dataelev)


######covariance matern: kappa=0.5
res=nsroba(elevation~1, kappa = 0.5, cov.model = "matern", data=dataelev,
ini.pars=c(10,3))

summary(res)

Summary of a nsroba object

Description

summary method for class "tsroba".

Usage

## S3 method for class 'tsroba'
summary(object, ...)

Arguments

Value

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(10,3,10))

summary(res)

Bayesian estimation for the TSR model.

Description

This function performs Bayesian estimation of \theta=(\bold{\beta},\sigma^2,\phi) 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

Details

For the prior proposal, it was considered the structure \pi(\phi,\nu,\lambda)=\phi(\phi)\pi(\nu|\lambda)\pi(\lambda). For the vague prior, \phi 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, \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, the sampling procedure generates improper posterior distribution when intercept is considered for the mean function.

Value

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)

Prediction under Student-t Objective Bayesian Analysis (OBA).

Description

This function uses the sampling distribution of parameters obtained from the function tsroba to predict values at unknown locations.

Usage

tsrobapred(obj,xpred,coordspred)

Arguments

Details

This function predicts using the sampling distribution of parameters obtained from the function tsroba and the conditional Student-t distribution of the predicted values given the data.

Value

This function returns a vector with the predicted values at the specified locations.

Author(s)

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

References

Diggle, P. and P. Ribeiro (2007).Model-Based Geostatistics. Springer Series in Statistics.

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

tsroba,nsrobapred1

Examples

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

######covariance matern: kappa=0.3 prior:reference
res=tsroba(calcont~altitude+area, kappa = 0.3, data=d1,
ini.pars=c(10,3,10),iter=50,thin=1,burn=5)

datapred=dataca20[159:178,]
formula=calcont~altitude+area
xpred=model.matrix(formula,data=datapred)

tsrobapred(res,xpred=xpred,coordspred=dataca20[159:178,1:2])