spT.Gibbs: MCMC sampling for the spatio-temporal models.
In spTimer: Spatio-Temporal Bayesian Modelling
spT.Gibbs R Documentation
MCMC sampling for the spatio-temporal models.
Description
This function is used to draw MCMC samples using the Gibbs sampler.
Usage
spT.Gibbs(formula, data = parent.frame(), model = "GP", time.data = NULL,
coords, knots.coords, newcoords = NULL, newdata = NULL, priors = NULL,
initials = NULL, nItr = 5000, nBurn = 1000, report = 1, tol.dist = 0.05,
distance.method = "geodetic:km", cov.fnc = "exponential",
scale.transform = "NONE", spatial.decay = spT.decay(distribution = "FIXED"),
truncation.para = list(at = 0,lambda = 2), annual.aggrn = "NONE",
fitted.values="TRANSFORMED")
Arguments
formula
The symnbolic description of the model equation of the regression part of the space-time model.
data
An optional data frame containing the variables in the model. If omitted, the variables are taken from environment(formula), typically the environment from which spT.Gibbs is called. The data should be ordered first by the time and then by the sites specified by the coords below. One can also supply coordinates through this argument, where coordinate names should be "Latitude" and "Longitude".
model
The spatio-temporal models to be fitted, current choices are: "GP", "truncatedGP", "AR", "GPP", and "truncatedGPP", with the first one as the default.
time.data
Defining the segments of the time-series set up using the function spT.time.
coords
The n by 2 matrix or data frame defining the locations (e.g., longitude/easting, latitude/northing) of the fitting sites, where n is the number of fitting sites. One can also supply coordinates through a formula argument such as ~Longitude+Latitude.
knots.coords
The locations of the knots in similar format to coords above, only required if model="GPP".
newcoords
The locations of the prediction sites in similar format to coords above, only required if fit and predictions are to be performed simultaneously. If omitted, no predictions will be performed.
newdata
The covariate values at the prediction sites specified by newcoords. This should have same space-time structure as the original data frame.
priors
The prior distributions for the parameters. Default distributions are specified if these are not provided. If priors=NULL a flat prior distribution will be used with large variance. See details in spT.priors.
initials
The preferred initial values for the parameters. If omitted, default values are provided automatically. Further details are provided in spT.initials.
nItr
Number of MCMC iterations. Default value is 5000.
nBurn
Number of burn-in samples. This number of samples will be discarded before making any inference. Default value is 1000.
report
Number of reports to display while running the Gibbs sampler. Defaults to number of iterations.
distance.method
The preferred method to calculate the distance between any two locations. The available options are "geodetic:km", "geodetic:mile", "euclidean", "maximum", "manhattan", and "canberra". See details in dist. The default is "geodetic:km".
tol.dist
Minimum separation distance between any two locations out of those specified by coords, knots.coords and pred.coords. The default is 0.005. The programme will exit if the minimum distance is less than the non-zero specified value. This will ensure non-singularity of the covariance matrices.
cov.fnc
Covariance function for the spatial effects. The available options are "exponential", "gaussian", "spherical" and "matern". If "matern" is used then by default the smooth parameter (\nu) is estimated from (0,1) uniform distribution using discrete samples.
scale.transform
The transformation method for the response variable. Currently implemented options are: "NONE", "SQRT", and "LOG" with "NONE" as the deault.
spatial.decay
Provides the prior distribution for the spatial decay parameter \phi. Currently implemented options are "FIXED", "Unif", or "Gamm". Further details for each of these are specified by spT.decay.
truncation.para
Provides truncation parameter \lambda and truncation point "at" using list.
annual.aggrn
This provides the options for calculating annual summary statistics by aggregating different time segments (e.g., annual mean). Currently implemented options are: "NONE", "ave" and "an4th", where "ave" = annual average, "an4th"= annual 4th highest. Only applicable if spT.time inputs more than one segment and when fit and predict are done simultaneously.
fitted.values
This option provides calculating fitted values and corresponding sd in the original scale. Currently implemented options are: "ORIGINAL" and "TRANSFORMED". Only applicable if scale.transform inputs "SQRT" or "LOG". Note that the PMCC (model validation criteria) values will be changed accordingly.
Value
accept
The acceptance rate for the \phi parameter if the "MH" method of sampling is chosen.
phip
MCMC samples for the parameter \phi.
nup
MCMC samples for the parameter \nu. Only available if "matern" covariance function is used.
sig2eps
MCMC samples for the parameter \sigma^2_\epsilon.
sig2etap
MCMC samples for the parameter \sigma^2_\eta.
betap
MCMC samples for the parameter \beta.
rhop
MCMC samples for \rho for the AR or GPP model.
op
MCMC samples for the true observations.
fitted
MCMC summary (mean and sd) for the fitted values.
tol.dist
Minimum tolerance distance limit between the locations.
distance.method
Name of the distance calculation method.
cov.fnc
Name of the covariance function used in model fitting.
scale.transform
Name of the scale.transformation method.
sampling.sp.decay
The method of sampling for the spatial decay parameter \phi.
covariate.names
Name of the covariates used in the model.
Distance.matrix
The distance matrix.
coords
The coordinate values.
n
Total number of sites.
r
Total number of segments in time, e.g., years.
T
Total points of time, e.g., days within each year.
p
Total number of model coefficients, i.e., \beta's including the intercept.
initials
The initial values used in the model.
priors
The prior distributions used in the model.
PMCC
The predictive model choice criteria obtained by minimising the expected value of a loss function, see Gelfand and Ghosh (1998). Results for both goodness of fit and penalty are given.
iterations
The number of samples for the MCMC chain, without burn-in.
nBurn
The number of burn-in period for the MCMC chain.
computation.time
The computation time required for the fitted model.
model
The spatio-temporal model used for analyse the data.
Text Output
This option is only applicable when fit and predictions are done simultaneously.
For GP models:
OutGP_Values_Parameter.txt: (nItr x parameters matrix) has the MCMC samples for the parameters, ordered as: beta's, sig2eps, sig2eta, and phi.
OutGP_Stats_FittedValue.txt: (N x 2) matrix of fitted summary, with 1st column as mean and 2nd column as standard deviations, where N=nrT.
OutGP_Stats_PredValue.txt: ((predsites*r*T) x 2) matrix of prediction summary, with 1st column as mean and 2nd column as standard deviations.
OutGP_Values_Prediction.txt: (nItr x (predsites*r*T)) matrix of MCMC predicted values in the predicted sites.
If annual.aggregation="ave" then we get text output as:
OutGP_Annual_Average_Prediction.txt: (nItr x (predsites*r)) matrix.
If annual.aggregation="an4th" then we get text output as:
OutGP_Annual_4th_Highest_Prediction.txt: (nItr x (predsites*r)) matrix.
For AR models:
OutAR_Values_Parameter.txt: (nItr x parameters matrix) has the MCMC samples for the parameters, ordered as: beta's, rho, sig2eps, sig2eta, mu_l's, sig2l's and phi.
OutAR_Stats_TrueValue.txt: (N x 2) matrix of true summary values, with 1st column as mean and 2nd column as standard deviations.
OutAR_Stats_FittedValue.txt: (N x 2) matrix of fitted summary, with 1st column as mean and 2nd column as standard deviations.
OutAR_Stats_PredValue.txt: ((predsites*r*T) x 2) matrix of prediction summary, with 1st column as mean and 2nd column as standard deviations.
OutAR_Values_Prediction.txt: (nItr x (predsites*r*T)) matrix of MCMC predicted values in the predicted sites.
If annual.aggregation="ave" then we get text output as:
OutAR_Annual_Average_Prediction.txt: (nItr x (predsites*r)) matrix.
If annual.aggregation="an4th" then we get text output as:
OutAR_Annual_4th_Highest_Prediction.txt: (nItr x (predsites*r)) matrix.
For models using GPP approximations:
OutGPP_Values_Parameter.txt: (nItr x parameters matrix) has the MCMC samples for the parameters, ordered as: beta's, rho, sig2eps, sig2eta, and phi.
OutGPP_Stats_FittedValue.txt: (N x 2) matrix of fitted summary, with 1st column as mean and 2nd column as standard deviations.
OutGPP_Stats_PredValue.txt: ((predsites*r*T) x 2) matrix of prediction summary, with 1st column as mean and 2nd column as standard deviations.
OutGPP_Values_Prediction.txt: (nItr x (predsites*r*T)) matrix of MCMC predicted values in the predicted sites.
If annual.aggregation="ave" then we get text output as:
OutGPP_Annual_Average_Prediction.txt: (nItr x (predsites*r)) matrix.
If annual.aggregation="an4th" then we get text output as:
OutGPP_Annual_4th_Highest_Prediction.txt: (nItr x (predsites*r)) matrix.
References
Bakar, K. S. and Sahu, S. K. (2015) spTimer: Spatio-Temporal Bayesian Modelling Using R. Journal of Statistical Software, 63(15). 1–32.
Sahu, S. K. and Bakar, K. S. (2012a) A comparison of Bayesian Models for Daily Ozone Concentration Levels Statistical Methodology, 9, 144-157.
Sahu, S. K. and Bakar, K. S. (2012b) Hierarchical Bayesian auto-regressive models for large space time data with applications to ozone concentration modelling. Applied Stochastic Models in Business and Industry, 28, 395-415.
See Also
spT.priors, spT.initials, spT.geodist, dist, summary.spT, plot.spT, predict.spT.
Examples
##
###########################
## Attach library spTimer
###########################
library(spTimer)
###########################
## The GP models:
###########################
##
## Model fitting
##
# Read data
data(NYdata)
# MCMC via Gibbs using default choices
set.seed(11)
post.gp <- spT.Gibbs(formula=o8hrmax ~cMAXTMP+WDSP+RH,
data=NYdata, model="GP", coords=~Longitude+Latitude,
scale.transform="SQRT")
print(post.gp)
# MCMC via Gibbs not using default choices
# Read data
s<-c(8,11,12,14,18,21,24,28)
DataFit<-spT.subset(data=NYdata, var.name=c("s.index"), s=s, reverse=TRUE)
DataFit<-subset(DataFit, with(DataFit, !(Day %in% c(30, 31) & Month == 8)))
DataValPred<-spT.subset(data=NYdata, var.name=c("s.index"), s=s)
DataValPred<-subset(DataValPred, with(DataValPred, !(Day %in% c(30, 31) & Month == 8)))
# define the time-series
time.data<-spT.time(t.series=60,segment=1)
# hyper-parameters for the prior distributions
priors<-spT.priors(model="GP",inv.var.prior=Gamm(2,1),
beta.prior=Norm(0,10^4))
# initial values for the model parameters
initials<-spT.initials(model="GP", sig2eps=0.01,
sig2eta=0.5, beta=NULL, phi=0.001)
# input for spatial decay, any one approach from below
#spatial.decay<-spT.decay(distribution="FIXED", value=0.01)
spatial.decay<-spT.decay(distribution=Gamm(2,1), tuning=0.08)
#spatial.decay<-spT.decay(distribution=Unif(0.01,0.02),npoints=5)
# Iterations for the MCMC algorithms
nItr<-5000
# MCMC via Gibbs
set.seed(11)
post.gp <- spT.Gibbs(formula=o8hrmax ~ cMAXTMP+WDSP+RH,
data=DataFit, model="GP", time.data=time.data,
coords=~Longitude+Latitude, priors=priors, initials=initials,
nItr=nItr, nBurn=0, report=nItr,
tol.dist=2, distance.method="geodetic:km",
cov.fnc="exponential", scale.transform="SQRT",
spatial.decay=spatial.decay)
print(post.gp)
# Summary and plots
summary(post.gp)
summary(post.gp,pack="coda")
plot(post.gp)
plot(post.gp,residuals=TRUE)
coef(post.gp)
confint(post.gp)
terms(post.gp)
formula(post.gp)
model.frame(post.gp)
model.matrix(post.gp)
# Model selection criteria
post.gp$PMCC
######################################
## The GP model for sp class data
######################################
# Creating sp class data
library(sp)
data(meuse)
summary(meuse)
coordinates(meuse) <- ~x+y
class(meuse)
out<-spT.Gibbs(formula=zinc~sqrt(dist),data=meuse,
model="GP", scale.transform="LOG")
summary(out)
# Create a dataset with spacetime class
library(spTimer)
site<-unique(NYdata[,c("Longitude","Latitude")])
library(spacetime)
row.names(site)<-paste("point",1:nrow(site),sep="")
site <- SpatialPoints(site)
ymd<-as.POSIXct(seq(as.Date("2006-07-01"),as.Date("2006-08-31"),by=1))
# introduce class STFDF
newNYdata<-STFDF(sp=site, time=ymd, data=NYdata) # full lattice
class(newNYdata)
out <- spT.Gibbs(formula=o8hrmax ~cMAXTMP+WDSP+RH,
data=newNYdata, model="GP", scale.transform="SQRT")
summary(out)
###########################
## The AR models:
###########################
##
## Model fitting
##
# Read data
data(NYdata)
# Define the coordinates
coords<-as.matrix(unique(cbind(NYdata[,2:3])))
# MCMC via Gibbs using default choices
set.seed(11)
post.ar <- spT.Gibbs(formula=o8hrmax ~cMAXTMP+WDSP+RH,
data=NYdata, model="AR", coords=coords,
scale.transform="SQRT")
print(post.ar)
# MCMC via Gibbs not using default choices
# define the time-series
time.data<-spT.time(t.series=62,segment=1)
# hyper-parameters for the prior distributions
priors<-spT.priors(model="AR",inv.var.prior=Gamm(2,1),
beta.prior=Norm(0,10^4))
# initial values for the model parameters
initials<-spT.initials(model="AR", sig2eps=0.01,
sig2eta=0.5, beta=NULL, phi=0.001)
# Input for spatial decay
#spatial.decay<-spT.decay(distribution="FIXED", value=0.01)
spatial.decay<-spT.decay(distribution=Gamm(2,1), tuning=0.08)
#spatial.decay<-spT.decay(distribution=Unif(0.01,0.02),npoints=5)
# Iterations for the MCMC algorithms
nItr<-5000
# MCMC via Gibbs
set.seed(11)
post.ar <- spT.Gibbs(formula=o8hrmax~cMAXTMP+WDSP+RH,
data=NYdata, model="AR", time.data=time.data,
coords=coords, priors=priors, initials=initials,
nItr=nItr, nBurn=0, report=nItr,
tol.dist=2, distance.method="geodetic:km",
cov.fnc="exponential", scale.transform="SQRT",
spatial.decay=spatial.decay)
print(post.ar)
# Summary and plots
summary(post.ar)
plot(post.ar)
# Model selection criteria
post.ar$PMCC
#################################
## The GPP approximation models:
#################################
##
## Model fitting
##
# Read data
data(NYdata);
# Define the coordinates
coords<-as.matrix(unique(cbind(NYdata[,2:3])))
# Define knots
knots<-spT.grid.coords(Longitude=c(max(coords[,1]),
min(coords[,1])),Latitude=c(max(coords[,2]),
min(coords[,2])), by=c(4,4))
# MCMC via Gibbs using default choices
set.seed(11)
post.gpp <- spT.Gibbs(formula=o8hrmax ~cMAXTMP+WDSP+RH,
data=NYdata, model="GPP", coords=coords,
knots.coords=knots, scale.transform="SQRT")
print(post.gpp)
# MCMC via Gibbs not using default choices
# define the time-series
time.data<-spT.time(t.series=62,segment=1)
# hyper-parameters for the prior distributions
priors<-spT.priors(model="GPP",inv.var.prior=Gamm(2,1),
beta.prior=Norm(0,10^4))
# initial values for the model parameters
initials<-spT.initials(model="GPP", sig2eps=0.01,
sig2eta=0.5, beta=NULL, phi=0.001)
# input for spatial decay
#spatial.decay<-spT.decay(distribution="FIXED", value=0.001)
spatial.decay<-spT.decay(distribution=Gamm(2,1), tuning=0.05)
#spatial.decay<-spT.decay(distribution=Unif(0.001,0.009),npoints=10)
# Iterations for the MCMC algorithms
nItr<-5000
# MCMC via Gibbs
set.seed(11)
post.gpp <- spT.Gibbs(formula=o8hrmax~cMAXTMP+WDSP+RH,
data=NYdata, model="GPP", time.data=time.data,
coords=coords, knots.coords=knots,
priors=priors, initials=initials,
nItr=nItr, nBurn=0, report=nItr,
tol.dist=2, distance.method="geodetic:km",
cov.fnc="exponential", scale.transform="SQRT",
spatial.decay=spatial.decay)
print(post.gpp)
# Summary and plots
summary(post.gpp)
plot(post.gpp)
# Model selection criteria
post.gpp$PMCC
######################################################
## The Truncated/Censored GP models:
######################################################
##
## Model fitting
##
data(NYdata)
# Truncation at 30 (say)
NYdata$o8hrmax[NYdata$o8hrmax<=30] <- 30
# Read data
s<-c(8,11,12,14,18,21,24,28)
DataFit<-spT.subset(data=NYdata, var.name=c("s.index"), s=s, reverse=TRUE)
DataFit<-subset(DataFit, with(DataFit, !(Day %in% c(30, 31) & Month == 8)))
DataValPred<-spT.subset(data=NYdata, var.name=c("s.index"), s=s)
DataValPred<-subset(DataValPred, with(DataValPred, !(Day %in% c(30, 31) & Month == 8)))
DataValFore<-spT.subset(data=NYdata, var.name=c("s.index"), s=c(8,11,12,14,18,21,24,28))
DataValFore<-subset(DataValFore, with(DataValFore, (Day %in% c(30, 31) & Month == 8)))
DataFitFore<-spT.subset(data=NYdata, var.name=c("s.index"), s=c(8,11,12,14,18,21,24,28),
reverse=TRUE)
DataFitFore<-subset(DataFitFore, with(DataFitFore, (Day %in% c(30, 31) & Month == 8)))
#
nItr <- 5000 # number of MCMC samples for each model
nBurn <- 1000 # number of burn-in from the MCMC samples
# Truncation at 30
# fit truncated GP model
out <- spT.Gibbs(formula=o8hrmax~cMAXTMP+WDSP+RH,data=DataFit,
model="truncatedGP",coords=~Longitude+Latitude,
distance.method="geodetic:km",nItr=nItr,nBurn=nBurn,report=5,
truncation.para = list(at = 30,lambda = 2),
fitted.values="ORIGINAL")
#
summary(out)
head(fitted(out))
plot(out,density=FALSE)
#
head(cbind(DataFit$o8hrmax,fitted(out)[,1]))
plot(DataFit$o8hrmax,fitted(out)[,1])
spT.validation(DataFit$o8hrmax,fitted(out)[,1])
##
## prediction (spatial)
##
pred <- predict(out,newdata=DataValPred, newcoords=~Longitude+Latitude, tol=0.05)
names(pred)
plot(DataValPred$o8hrmax,c(pred$Mean))
spT.validation(DataValPred$o8hrmax,c(pred$Mean))
#pred$prob.below.threshold
##
## forecast (temporal)
##
# unobserved locations
fore <- predict(out,newdata=DataValFore, newcoords=~Longitude+Latitude,
type="temporal", foreStep=2, tol=0.05)
spT.validation(DataValFore$o8hrmax,c(fore$Mean))
plot(DataValFore$o8hrmax,c(fore$Mean))
#fore$prob.below.threshold
# observed locations
fore <- predict(out,newdata=DataFitFore, newcoords=~Longitude+Latitude,
type="temporal", foreStep=2, tol=0.05)
spT.validation(DataFitFore$o8hrmax,c(fore$Mean))
plot(DataFitFore$o8hrmax,c(fore$Mean))
#fore$prob.below.threshold
######################################################
## The Truncated/Censored AR models:
######################################################
##
## Model fitting
##
data(NYdata)
# Truncation at 30 (say)
NYdata$o8hrmax[NYdata$o8hrmax<=30] <- 30
# Read data
s<-c(8,11,12,14,18,21,24,28)
DataFit<-spT.subset(data=NYdata, var.name=c("s.index"), s=s, reverse=TRUE)
DataFit<-subset(DataFit, with(DataFit, !(Day %in% c(30, 31) & Month == 8)))
DataValPred<-spT.subset(data=NYdata, var.name=c("s.index"), s=s)
DataValPred<-subset(DataValPred, with(DataValPred, !(Day %in% c(30, 31) & Month == 8)))
DataValFore<-spT.subset(data=NYdata, var.name=c("s.index"), s=c(8,11,12,14,18,21,24,28))
DataValFore<-subset(DataValFore, with(DataValFore, (Day %in% c(30, 31) & Month == 8)))
DataFitFore<-spT.subset(data=NYdata, var.name=c("s.index"), s=c(8,11,12,14,18,21,24,28),
reverse=TRUE)
DataFitFore<-subset(DataFitFore, with(DataFitFore, (Day %in% c(30, 31) & Month == 8)))
#
nItr <- 5000 # number of MCMC samples for each model
nBurn <- 1000 # number of burn-in from the MCMC samples
# Truncation at 30
# fit truncated AR model
out <- spT.Gibbs(formula=o8hrmax~cMAXTMP+WDSP+RH,data=DataFit,
model="truncatedAR",coords=~Longitude+Latitude,
distance.method="geodetic:km",nItr=nItr,nBurn=nBurn,report=5,
truncation.para = list(at = 30,lambda = 2),
fitted.values="ORIGINAL")
#
summary(out)
head(fitted(out))
plot(out,density=FALSE)
#
head(cbind(DataFit$o8hrmax,fitted(out)[,1]))
plot(DataFit$o8hrmax,fitted(out)[,1])
spT.validation(DataFit$o8hrmax,fitted(out)[,1])
##
## prediction (spatial)
##
pred <- predict(out,newdata=DataValPred, newcoords=~Longitude+Latitude, tol=0.05)
names(pred)
plot(DataValPred$o8hrmax,c(pred$Mean))
spT.validation(DataValPred$o8hrmax,c(pred$Mean))
#pred$prob.below.threshold
##
## forecast (temporal)
##
# unobserved locations
fore <- predict(out,newdata=DataValFore, newcoords=~Longitude+Latitude,
type="temporal", foreStep=2, tol=0.05)
spT.validation(DataValFore$o8hrmax,c(fore$Mean))
plot(DataValFore$o8hrmax,c(fore$Mean))
#fore$prob.below.threshold
# observed locations
fore <- predict(out,newdata=DataFitFore, newcoords=~Longitude+Latitude,
type="temporal", foreStep=2, tol=0.05)
spT.validation(DataFitFore$o8hrmax,c(fore$Mean))
plot(DataFitFore$o8hrmax,c(fore$Mean))
#fore$prob.below.threshold
######################################################
## The Truncated/Censored GPP models:
######################################################
##
## Model fitting
##
data(NYdata)
# Define the coordinates
coords<-as.matrix(unique(cbind(NYdata[,2:3])))
# Define knots
knots<-spT.grid.coords(Longitude=c(max(coords[,1]),
min(coords[,1])),Latitude=c(max(coords[,2]),
min(coords[,2])), by=c(4,4))
# Truncation at 30 (say)
NYdata$o8hrmax[NYdata$o8hrmax<=30] <- 30
# Read data
s<-c(8,11,12,14,18,21,24,28)
DataFit<-spT.subset(data=NYdata, var.name=c("s.index"), s=s, reverse=TRUE)
DataFit<-subset(DataFit, with(DataFit, !(Day %in% c(30, 31) & Month == 8)))
DataValPred<-spT.subset(data=NYdata, var.name=c("s.index"), s=s)
DataValPred<-subset(DataValPred, with(DataValPred, !(Day %in% c(30, 31) & Month == 8)))
DataValFore<-spT.subset(data=NYdata, var.name=c("s.index"), s=c(8,11,12,14,18,21,24,28))
DataValFore<-subset(DataValFore, with(DataValFore, (Day %in% c(30, 31) & Month == 8)))
DataFitFore<-spT.subset(data=NYdata, var.name=c("s.index"), s=c(8,11,12,14,18,21,24,28),
reverse=TRUE)
DataFitFore<-subset(DataFitFore, with(DataFitFore, (Day %in% c(30, 31) & Month == 8)))
#
nItr <- 5000 # number of MCMC samples for each model
nBurn <- 1000 # number of burn-in from the MCMC samples
# Truncation at 30
# fit truncated GPP model
out <- spT.Gibbs(formula=o8hrmax ~cMAXTMP+WDSP+RH,
data=DataFit, model="truncatedGPP",coords=~Longitude+Latitude,
knots.coords=knots, distance.method="geodetic:km",
nItr=nItr,nBurn=nBurn,report=5,fitted="ORIGINAL",
truncation.para = list(at = 30,lambda = 2))
#
summary(out)
head(fitted(out))
plot(out,density=FALSE)
#
head(cbind(DataFit$o8hrmax,fitted(out)[,1]))
plot(DataFit$o8hrmax,fitted(out)[,1])
spT.validation(DataFit$o8hrmax,fitted(out)[,1])
##
## prediction (spatial)
##
pred <- predict(out,newdata=DataValPred, newcoords=~Longitude+Latitude, tol=0.05)
names(pred)
plot(DataValPred$o8hrmax,c(pred$Mean))
spT.validation(DataValPred$o8hrmax,c(pred$Mean))
#pred$prob.below.threshold
##
## forecast (temporal)
##
# unobserved locations
fore <- predict(out,newdata=DataValFore, newcoords=~Longitude+Latitude,
type="temporal", foreStep=2, tol=0.05)
spT.validation(DataValFore$o8hrmax,c(fore$Mean))
plot(DataValFore$o8hrmax,c(fore$Mean))
#fore$prob.below.threshold
# observed locations
fore <- predict(out,newdata=DataFitFore, newcoords=~Longitude+Latitude,
type="temporal", foreStep=2, tol=0.05)
spT.validation(DataFitFore$o8hrmax,c(fore$Mean))
plot(DataFitFore$o8hrmax,c(fore$Mean))
#fore$prob.below.threshold
######################################################
######################################################
##
spTimer documentation built on Sept. 11, 2024, 6:25 p.m.
R Package Documentation
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| spT.Gibbs | R Documentation |
MCMC sampling for the spatio-temporal models.
Description
This function is used to draw MCMC samples using the Gibbs sampler.
Usage
spT.Gibbs(formula, data = parent.frame(), model = "GP", time.data = NULL,
coords, knots.coords, newcoords = NULL, newdata = NULL, priors = NULL,
initials = NULL, nItr = 5000, nBurn = 1000, report = 1, tol.dist = 0.05,
distance.method = "geodetic:km", cov.fnc = "exponential",
scale.transform = "NONE", spatial.decay = spT.decay(distribution = "FIXED"),
truncation.para = list(at = 0,lambda = 2), annual.aggrn = "NONE",
fitted.values="TRANSFORMED")
Arguments
formula |
The symnbolic description of the model equation of the regression part of the space-time model. |
data |
An optional data frame containing the variables in the model. If omitted, the variables are taken from environment(formula), typically the environment from which spT.Gibbs is called. The data should be ordered first by the time and then by the sites specified by the |
model |
The spatio-temporal models to be fitted, current choices are: "GP", "truncatedGP", "AR", "GPP", and "truncatedGPP", with the first one as the default. |
time.data |
Defining the segments of the time-series set up using the function |
coords |
The n by 2 matrix or data frame defining the locations (e.g., longitude/easting, latitude/northing) of the fitting sites, where n is the number of fitting sites. One can also supply coordinates through a formula argument such as ~Longitude+Latitude. |
knots.coords |
The locations of the knots in similar format to coords above, only required if |
newcoords |
The locations of the prediction sites in similar format to coords above, only required if fit and predictions are to be performed simultaneously. If omitted, no predictions will be performed. |
newdata |
The covariate values at the prediction sites specified by |
priors |
The prior distributions for the parameters. Default distributions are specified if these are not provided. If priors=NULL a flat prior distribution will be used with large variance. See details in |
initials |
The preferred initial values for the parameters. If omitted, default values are provided automatically. Further details are provided in |
nItr |
Number of MCMC iterations. Default value is 5000. |
nBurn |
Number of burn-in samples. This number of samples will be discarded before making any inference. Default value is 1000. |
report |
Number of reports to display while running the Gibbs sampler. Defaults to number of iterations. |
distance.method |
The preferred method to calculate the distance between any two locations. The available options are "geodetic:km", "geodetic:mile", "euclidean", "maximum", "manhattan", and "canberra". See details in |
tol.dist |
Minimum separation distance between any two locations out of those specified by coords, knots.coords and pred.coords. The default is 0.005. The programme will exit if the minimum distance is less than the non-zero specified value. This will ensure non-singularity of the covariance matrices. |
cov.fnc |
Covariance function for the spatial effects. The available options are "exponential", "gaussian", "spherical" and "matern". If "matern" is used then by default the smooth parameter ( |
scale.transform |
The transformation method for the response variable. Currently implemented options are: "NONE", "SQRT", and "LOG" with "NONE" as the deault. |
spatial.decay |
Provides the prior distribution for the spatial decay parameter |
truncation.para |
Provides truncation parameter |
annual.aggrn |
This provides the options for calculating annual summary statistics by aggregating different time segments (e.g., annual mean). Currently implemented options are: "NONE", "ave" and "an4th", where "ave" = annual average, "an4th"= annual 4th highest. Only applicable if |
fitted.values |
This option provides calculating fitted values and corresponding sd in the original scale. Currently implemented options are: "ORIGINAL" and "TRANSFORMED". Only applicable if |
Value
accept |
The acceptance rate for the |
phip |
MCMC samples for the parameter |
nup |
MCMC samples for the parameter |
sig2eps |
MCMC samples for the parameter |
sig2etap |
MCMC samples for the parameter |
betap |
MCMC samples for the parameter |
rhop |
MCMC samples for |
op |
MCMC samples for the true observations. |
fitted |
MCMC summary (mean and sd) for the fitted values. |
tol.dist |
Minimum tolerance distance limit between the locations. |
distance.method |
Name of the distance calculation method. |
cov.fnc |
Name of the covariance function used in model fitting. |
scale.transform |
Name of the scale.transformation method. |
sampling.sp.decay |
The method of sampling for the spatial decay parameter |
covariate.names |
Name of the covariates used in the model. |
Distance.matrix |
The distance matrix. |
coords |
The coordinate values. |
n |
Total number of sites. |
r |
Total number of segments in time, e.g., years. |
T |
Total points of time, e.g., days within each year. |
p |
Total number of model coefficients, i.e., |
initials |
The initial values used in the model. |
priors |
The prior distributions used in the model. |
PMCC |
The predictive model choice criteria obtained by minimising the expected value of a loss function, see Gelfand and Ghosh (1998). Results for both goodness of fit and penalty are given. |
iterations |
The number of samples for the MCMC chain, without burn-in. |
nBurn |
The number of burn-in period for the MCMC chain. |
computation.time |
The computation time required for the fitted model. |
model |
The spatio-temporal model used for analyse the data. |
Text Output |
This option is only applicable when fit and predictions are done simultaneously. For GP models: For AR models: For models using GPP approximations: |
References
Bakar, K. S. and Sahu, S. K. (2015) spTimer: Spatio-Temporal Bayesian Modelling Using R. Journal of Statistical Software, 63(15). 1–32.
Sahu, S. K. and Bakar, K. S. (2012a) A comparison of Bayesian Models for Daily Ozone Concentration Levels Statistical Methodology, 9, 144-157.
Sahu, S. K. and Bakar, K. S. (2012b) Hierarchical Bayesian auto-regressive models for large space time data with applications to ozone concentration modelling. Applied Stochastic Models in Business and Industry, 28, 395-415.
See Also
spT.priors, spT.initials, spT.geodist, dist, summary.spT, plot.spT, predict.spT.
Examples
##
###########################
## Attach library spTimer
###########################
library(spTimer)
###########################
## The GP models:
###########################
##
## Model fitting
##
# Read data
data(NYdata)
# MCMC via Gibbs using default choices
set.seed(11)
post.gp <- spT.Gibbs(formula=o8hrmax ~cMAXTMP+WDSP+RH,
data=NYdata, model="GP", coords=~Longitude+Latitude,
scale.transform="SQRT")
print(post.gp)
# MCMC via Gibbs not using default choices
# Read data
s<-c(8,11,12,14,18,21,24,28)
DataFit<-spT.subset(data=NYdata, var.name=c("s.index"), s=s, reverse=TRUE)
DataFit<-subset(DataFit, with(DataFit, !(Day %in% c(30, 31) & Month == 8)))
DataValPred<-spT.subset(data=NYdata, var.name=c("s.index"), s=s)
DataValPred<-subset(DataValPred, with(DataValPred, !(Day %in% c(30, 31) & Month == 8)))
# define the time-series
time.data<-spT.time(t.series=60,segment=1)
# hyper-parameters for the prior distributions
priors<-spT.priors(model="GP",inv.var.prior=Gamm(2,1),
beta.prior=Norm(0,10^4))
# initial values for the model parameters
initials<-spT.initials(model="GP", sig2eps=0.01,
sig2eta=0.5, beta=NULL, phi=0.001)
# input for spatial decay, any one approach from below
#spatial.decay<-spT.decay(distribution="FIXED", value=0.01)
spatial.decay<-spT.decay(distribution=Gamm(2,1), tuning=0.08)
#spatial.decay<-spT.decay(distribution=Unif(0.01,0.02),npoints=5)
# Iterations for the MCMC algorithms
nItr<-5000
# MCMC via Gibbs
set.seed(11)
post.gp <- spT.Gibbs(formula=o8hrmax ~ cMAXTMP+WDSP+RH,
data=DataFit, model="GP", time.data=time.data,
coords=~Longitude+Latitude, priors=priors, initials=initials,
nItr=nItr, nBurn=0, report=nItr,
tol.dist=2, distance.method="geodetic:km",
cov.fnc="exponential", scale.transform="SQRT",
spatial.decay=spatial.decay)
print(post.gp)
# Summary and plots
summary(post.gp)
summary(post.gp,pack="coda")
plot(post.gp)
plot(post.gp,residuals=TRUE)
coef(post.gp)
confint(post.gp)
terms(post.gp)
formula(post.gp)
model.frame(post.gp)
model.matrix(post.gp)
# Model selection criteria
post.gp$PMCC
######################################
## The GP model for sp class data
######################################
# Creating sp class data
library(sp)
data(meuse)
summary(meuse)
coordinates(meuse) <- ~x+y
class(meuse)
out<-spT.Gibbs(formula=zinc~sqrt(dist),data=meuse,
model="GP", scale.transform="LOG")
summary(out)
# Create a dataset with spacetime class
library(spTimer)
site<-unique(NYdata[,c("Longitude","Latitude")])
library(spacetime)
row.names(site)<-paste("point",1:nrow(site),sep="")
site <- SpatialPoints(site)
ymd<-as.POSIXct(seq(as.Date("2006-07-01"),as.Date("2006-08-31"),by=1))
# introduce class STFDF
newNYdata<-STFDF(sp=site, time=ymd, data=NYdata) # full lattice
class(newNYdata)
out <- spT.Gibbs(formula=o8hrmax ~cMAXTMP+WDSP+RH,
data=newNYdata, model="GP", scale.transform="SQRT")
summary(out)
###########################
## The AR models:
###########################
##
## Model fitting
##
# Read data
data(NYdata)
# Define the coordinates
coords<-as.matrix(unique(cbind(NYdata[,2:3])))
# MCMC via Gibbs using default choices
set.seed(11)
post.ar <- spT.Gibbs(formula=o8hrmax ~cMAXTMP+WDSP+RH,
data=NYdata, model="AR", coords=coords,
scale.transform="SQRT")
print(post.ar)
# MCMC via Gibbs not using default choices
# define the time-series
time.data<-spT.time(t.series=62,segment=1)
# hyper-parameters for the prior distributions
priors<-spT.priors(model="AR",inv.var.prior=Gamm(2,1),
beta.prior=Norm(0,10^4))
# initial values for the model parameters
initials<-spT.initials(model="AR", sig2eps=0.01,
sig2eta=0.5, beta=NULL, phi=0.001)
# Input for spatial decay
#spatial.decay<-spT.decay(distribution="FIXED", value=0.01)
spatial.decay<-spT.decay(distribution=Gamm(2,1), tuning=0.08)
#spatial.decay<-spT.decay(distribution=Unif(0.01,0.02),npoints=5)
# Iterations for the MCMC algorithms
nItr<-5000
# MCMC via Gibbs
set.seed(11)
post.ar <- spT.Gibbs(formula=o8hrmax~cMAXTMP+WDSP+RH,
data=NYdata, model="AR", time.data=time.data,
coords=coords, priors=priors, initials=initials,
nItr=nItr, nBurn=0, report=nItr,
tol.dist=2, distance.method="geodetic:km",
cov.fnc="exponential", scale.transform="SQRT",
spatial.decay=spatial.decay)
print(post.ar)
# Summary and plots
summary(post.ar)
plot(post.ar)
# Model selection criteria
post.ar$PMCC
#################################
## The GPP approximation models:
#################################
##
## Model fitting
##
# Read data
data(NYdata);
# Define the coordinates
coords<-as.matrix(unique(cbind(NYdata[,2:3])))
# Define knots
knots<-spT.grid.coords(Longitude=c(max(coords[,1]),
min(coords[,1])),Latitude=c(max(coords[,2]),
min(coords[,2])), by=c(4,4))
# MCMC via Gibbs using default choices
set.seed(11)
post.gpp <- spT.Gibbs(formula=o8hrmax ~cMAXTMP+WDSP+RH,
data=NYdata, model="GPP", coords=coords,
knots.coords=knots, scale.transform="SQRT")
print(post.gpp)
# MCMC via Gibbs not using default choices
# define the time-series
time.data<-spT.time(t.series=62,segment=1)
# hyper-parameters for the prior distributions
priors<-spT.priors(model="GPP",inv.var.prior=Gamm(2,1),
beta.prior=Norm(0,10^4))
# initial values for the model parameters
initials<-spT.initials(model="GPP", sig2eps=0.01,
sig2eta=0.5, beta=NULL, phi=0.001)
# input for spatial decay
#spatial.decay<-spT.decay(distribution="FIXED", value=0.001)
spatial.decay<-spT.decay(distribution=Gamm(2,1), tuning=0.05)
#spatial.decay<-spT.decay(distribution=Unif(0.001,0.009),npoints=10)
# Iterations for the MCMC algorithms
nItr<-5000
# MCMC via Gibbs
set.seed(11)
post.gpp <- spT.Gibbs(formula=o8hrmax~cMAXTMP+WDSP+RH,
data=NYdata, model="GPP", time.data=time.data,
coords=coords, knots.coords=knots,
priors=priors, initials=initials,
nItr=nItr, nBurn=0, report=nItr,
tol.dist=2, distance.method="geodetic:km",
cov.fnc="exponential", scale.transform="SQRT",
spatial.decay=spatial.decay)
print(post.gpp)
# Summary and plots
summary(post.gpp)
plot(post.gpp)
# Model selection criteria
post.gpp$PMCC
######################################################
## The Truncated/Censored GP models:
######################################################
##
## Model fitting
##
data(NYdata)
# Truncation at 30 (say)
NYdata$o8hrmax[NYdata$o8hrmax<=30] <- 30
# Read data
s<-c(8,11,12,14,18,21,24,28)
DataFit<-spT.subset(data=NYdata, var.name=c("s.index"), s=s, reverse=TRUE)
DataFit<-subset(DataFit, with(DataFit, !(Day %in% c(30, 31) & Month == 8)))
DataValPred<-spT.subset(data=NYdata, var.name=c("s.index"), s=s)
DataValPred<-subset(DataValPred, with(DataValPred, !(Day %in% c(30, 31) & Month == 8)))
DataValFore<-spT.subset(data=NYdata, var.name=c("s.index"), s=c(8,11,12,14,18,21,24,28))
DataValFore<-subset(DataValFore, with(DataValFore, (Day %in% c(30, 31) & Month == 8)))
DataFitFore<-spT.subset(data=NYdata, var.name=c("s.index"), s=c(8,11,12,14,18,21,24,28),
reverse=TRUE)
DataFitFore<-subset(DataFitFore, with(DataFitFore, (Day %in% c(30, 31) & Month == 8)))
#
nItr <- 5000 # number of MCMC samples for each model
nBurn <- 1000 # number of burn-in from the MCMC samples
# Truncation at 30
# fit truncated GP model
out <- spT.Gibbs(formula=o8hrmax~cMAXTMP+WDSP+RH,data=DataFit,
model="truncatedGP",coords=~Longitude+Latitude,
distance.method="geodetic:km",nItr=nItr,nBurn=nBurn,report=5,
truncation.para = list(at = 30,lambda = 2),
fitted.values="ORIGINAL")
#
summary(out)
head(fitted(out))
plot(out,density=FALSE)
#
head(cbind(DataFit$o8hrmax,fitted(out)[,1]))
plot(DataFit$o8hrmax,fitted(out)[,1])
spT.validation(DataFit$o8hrmax,fitted(out)[,1])
##
## prediction (spatial)
##
pred <- predict(out,newdata=DataValPred, newcoords=~Longitude+Latitude, tol=0.05)
names(pred)
plot(DataValPred$o8hrmax,c(pred$Mean))
spT.validation(DataValPred$o8hrmax,c(pred$Mean))
#pred$prob.below.threshold
##
## forecast (temporal)
##
# unobserved locations
fore <- predict(out,newdata=DataValFore, newcoords=~Longitude+Latitude,
type="temporal", foreStep=2, tol=0.05)
spT.validation(DataValFore$o8hrmax,c(fore$Mean))
plot(DataValFore$o8hrmax,c(fore$Mean))
#fore$prob.below.threshold
# observed locations
fore <- predict(out,newdata=DataFitFore, newcoords=~Longitude+Latitude,
type="temporal", foreStep=2, tol=0.05)
spT.validation(DataFitFore$o8hrmax,c(fore$Mean))
plot(DataFitFore$o8hrmax,c(fore$Mean))
#fore$prob.below.threshold
######################################################
## The Truncated/Censored AR models:
######################################################
##
## Model fitting
##
data(NYdata)
# Truncation at 30 (say)
NYdata$o8hrmax[NYdata$o8hrmax<=30] <- 30
# Read data
s<-c(8,11,12,14,18,21,24,28)
DataFit<-spT.subset(data=NYdata, var.name=c("s.index"), s=s, reverse=TRUE)
DataFit<-subset(DataFit, with(DataFit, !(Day %in% c(30, 31) & Month == 8)))
DataValPred<-spT.subset(data=NYdata, var.name=c("s.index"), s=s)
DataValPred<-subset(DataValPred, with(DataValPred, !(Day %in% c(30, 31) & Month == 8)))
DataValFore<-spT.subset(data=NYdata, var.name=c("s.index"), s=c(8,11,12,14,18,21,24,28))
DataValFore<-subset(DataValFore, with(DataValFore, (Day %in% c(30, 31) & Month == 8)))
DataFitFore<-spT.subset(data=NYdata, var.name=c("s.index"), s=c(8,11,12,14,18,21,24,28),
reverse=TRUE)
DataFitFore<-subset(DataFitFore, with(DataFitFore, (Day %in% c(30, 31) & Month == 8)))
#
nItr <- 5000 # number of MCMC samples for each model
nBurn <- 1000 # number of burn-in from the MCMC samples
# Truncation at 30
# fit truncated AR model
out <- spT.Gibbs(formula=o8hrmax~cMAXTMP+WDSP+RH,data=DataFit,
model="truncatedAR",coords=~Longitude+Latitude,
distance.method="geodetic:km",nItr=nItr,nBurn=nBurn,report=5,
truncation.para = list(at = 30,lambda = 2),
fitted.values="ORIGINAL")
#
summary(out)
head(fitted(out))
plot(out,density=FALSE)
#
head(cbind(DataFit$o8hrmax,fitted(out)[,1]))
plot(DataFit$o8hrmax,fitted(out)[,1])
spT.validation(DataFit$o8hrmax,fitted(out)[,1])
##
## prediction (spatial)
##
pred <- predict(out,newdata=DataValPred, newcoords=~Longitude+Latitude, tol=0.05)
names(pred)
plot(DataValPred$o8hrmax,c(pred$Mean))
spT.validation(DataValPred$o8hrmax,c(pred$Mean))
#pred$prob.below.threshold
##
## forecast (temporal)
##
# unobserved locations
fore <- predict(out,newdata=DataValFore, newcoords=~Longitude+Latitude,
type="temporal", foreStep=2, tol=0.05)
spT.validation(DataValFore$o8hrmax,c(fore$Mean))
plot(DataValFore$o8hrmax,c(fore$Mean))
#fore$prob.below.threshold
# observed locations
fore <- predict(out,newdata=DataFitFore, newcoords=~Longitude+Latitude,
type="temporal", foreStep=2, tol=0.05)
spT.validation(DataFitFore$o8hrmax,c(fore$Mean))
plot(DataFitFore$o8hrmax,c(fore$Mean))
#fore$prob.below.threshold
######################################################
## The Truncated/Censored GPP models:
######################################################
##
## Model fitting
##
data(NYdata)
# Define the coordinates
coords<-as.matrix(unique(cbind(NYdata[,2:3])))
# Define knots
knots<-spT.grid.coords(Longitude=c(max(coords[,1]),
min(coords[,1])),Latitude=c(max(coords[,2]),
min(coords[,2])), by=c(4,4))
# Truncation at 30 (say)
NYdata$o8hrmax[NYdata$o8hrmax<=30] <- 30
# Read data
s<-c(8,11,12,14,18,21,24,28)
DataFit<-spT.subset(data=NYdata, var.name=c("s.index"), s=s, reverse=TRUE)
DataFit<-subset(DataFit, with(DataFit, !(Day %in% c(30, 31) & Month == 8)))
DataValPred<-spT.subset(data=NYdata, var.name=c("s.index"), s=s)
DataValPred<-subset(DataValPred, with(DataValPred, !(Day %in% c(30, 31) & Month == 8)))
DataValFore<-spT.subset(data=NYdata, var.name=c("s.index"), s=c(8,11,12,14,18,21,24,28))
DataValFore<-subset(DataValFore, with(DataValFore, (Day %in% c(30, 31) & Month == 8)))
DataFitFore<-spT.subset(data=NYdata, var.name=c("s.index"), s=c(8,11,12,14,18,21,24,28),
reverse=TRUE)
DataFitFore<-subset(DataFitFore, with(DataFitFore, (Day %in% c(30, 31) & Month == 8)))
#
nItr <- 5000 # number of MCMC samples for each model
nBurn <- 1000 # number of burn-in from the MCMC samples
# Truncation at 30
# fit truncated GPP model
out <- spT.Gibbs(formula=o8hrmax ~cMAXTMP+WDSP+RH,
data=DataFit, model="truncatedGPP",coords=~Longitude+Latitude,
knots.coords=knots, distance.method="geodetic:km",
nItr=nItr,nBurn=nBurn,report=5,fitted="ORIGINAL",
truncation.para = list(at = 30,lambda = 2))
#
summary(out)
head(fitted(out))
plot(out,density=FALSE)
#
head(cbind(DataFit$o8hrmax,fitted(out)[,1]))
plot(DataFit$o8hrmax,fitted(out)[,1])
spT.validation(DataFit$o8hrmax,fitted(out)[,1])
##
## prediction (spatial)
##
pred <- predict(out,newdata=DataValPred, newcoords=~Longitude+Latitude, tol=0.05)
names(pred)
plot(DataValPred$o8hrmax,c(pred$Mean))
spT.validation(DataValPred$o8hrmax,c(pred$Mean))
#pred$prob.below.threshold
##
## forecast (temporal)
##
# unobserved locations
fore <- predict(out,newdata=DataValFore, newcoords=~Longitude+Latitude,
type="temporal", foreStep=2, tol=0.05)
spT.validation(DataValFore$o8hrmax,c(fore$Mean))
plot(DataValFore$o8hrmax,c(fore$Mean))
#fore$prob.below.threshold
# observed locations
fore <- predict(out,newdata=DataFitFore, newcoords=~Longitude+Latitude,
type="temporal", foreStep=2, tol=0.05)
spT.validation(DataFitFore$o8hrmax,c(fore$Mean))
plot(DataFitFore$o8hrmax,c(fore$Mean))
#fore$prob.below.threshold
######################################################
######################################################
##
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