Help for package spTimer

Type:

Package

Title:

Spatio-Temporal Bayesian Modelling

Version:

3.3.3

Date:

2024-09-05

Depends:

R (≥ 4.4.0)

Description:

Fits, spatially predicts and temporally forecasts large amounts of space-time data using [1] Bayesian Gaussian Process (GP) Models, [2] Bayesian Auto-Regressive (AR) Models, and [3] Bayesian Gaussian Predictive Processes (GPP) based AR Models for spatio-temporal big-n problems. Bakar and Sahu (2015) <doi:10.18637/jss.v063.i15>.

License:

GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]

Imports:

coda, sp, spacetime, extraDistr, grDevices, graphics, stats, utils

LazyData:

yes

NeedsCompilation:

yes

Packaged:

2024-09-08 08:33:08 UTC; kbak4671

Author:

K. Shuvo Bakar

[aut, cre], Sujit K. Sahu

[ctb]

Maintainer:

K. Shuvo Bakar <shuvo.bakar@gmail.com>

Repository:

CRAN

Date/Publication:

2024-09-08 09:10:02 UTC

Spatio-Temporal Bayesian Modelling using R

Description

This package uses different hierarchical Bayesian spatio-temporal modelling strategies, namely:
(1) Gaussian processes (GP) models,
(2) Autoregressive (AR) models,
(3) Gaussian predictive processes (GPP) based autoregressive models for big-n problem.

Details

Package:	spTimer
Type:	Package

The back-end code of this package is built under c language.
Main functions used:
> spT.Gibbs
> predict.spT
Some other important functions:
> spT.priors
> spT.initials
> spT.decay
> spT.time
Data descriptions:
> NYdata

Author(s)

K.S. Bakar & S.K. Sahu
Maintainer: K.S. Bakar <shuvo.bakar@gmail.com>

References

1. Bakar, K. S., & Sahu, S. K. (2015). sptimer: Spatio-temporal bayesian modelling using r. Journal of Statistical Software, 63(15), 1-32.
2. Sahu, S.K. & Bakar, K.S. (2012). Hierarchical Bayesian Autoregressive Models for Large Space Time Data with Applications to Ozone Concentration Modelling. Applied Stochastic Models in Business and Industry, 28, 395-415.
3. Sahu, S.K., Gelfand, A.E., & Holland, D.M. (2007). High-Resolution Space-Time Ozone Modelling for Assessing Trends. Journal of the American Statistical Association, 102, 1221-1234.
4. Bakar, K.S. (2012). Bayesian Analysis of Daily Maximum Ozone Levels. PhD Thesis, University of Southampton, Southampton, United Kingdom.

Observations of ozone concentration levels, maximum temperature and wind speed.

Description

This data set contains values of daily 8-hour maximum average ozone concentrations (parts per billion (ppb)), maximum temperature (in degree Celsius), wind speed (knots), and relative humidity, obtained from 28 monitoring sites of New York, USA.

NYgrid: This dataset contains total 6200 rows for 62 days of observations for 10x10 = 100 grid points.

Usage

NYdata

Format

Columns for NYdata: each contains 1798 observations.

1st col = Site index (s.index),
2nd col = Longitude,
3rd col = Latitude,
4th col = Year,
5th col = Month,
6th col = Day,
7th col = Ozone (o8hrmax),
8th col = Maximum temperature (cMAXTMP),
9th col = Wind speed (WDSP).
10th col = Relative humidity (RH).

Source

US EPA

Examples

##
  library("spTimer")
# NY data
  data(NYdata)
  head(NYdata)
# plots in NY map
  NYsite<-unique(cbind(NYdata[,1:3]))
  head(NYsite)
# map
  #library(maps)
  #map(database="state",regions="new york")
  #points(NYsite[,2:3],pch=19)

  # Grid data
  data(NYgrid)
  head(NYgrid)
  grid.coords<-unique(cbind(NYgrid[,8:9]))
  #library(maps)
  plot(grid.coords,pch=19,col=1)
  #map(database="state",regions="new york",add=TRUE)

##

Credible intervals for model parameters.

Description

This function is used to obtain credible intervals for model parameters from the MCMC samples.

Usage

## S3 method for class 'spT'
confint(object, parm, level = 0.95, ...)

##

Arguments

object

Object of class inheriting from "spT".

parm

a specification of which parameters are to be given credible intervals, a vector of names. If missing, all parameters are considered.

level

The required credible interval.

...

other arguments.

Examples

## Not run: 
##

confint(out) # where out is the output from spT class

##

## End(Not run)

Extract model fitted values.

Description

Extract average fitted values and corresponding standard deviations from model.

Usage

## S3 method for class 'spT'
fitted(object, ...)

##

Arguments

object

Object of class inheriting from "spT".

...

Other arguments.

Value

Mean

Fitted mean values obtained from the MCMC samples.

SD

Corresponding standard deviations.

Examples

## Not run: 
##

fitted(out) # where out is the output from spT class

##

## End(Not run)

Plots for spTimer output.

Description

This function is used to obtain MCMC summary, residual and fitted surface plots.

Usage

## S3 method for class 'spT'
plot(x, residuals=FALSE, coefficient=NULL, ...)

##

Arguments

x

Object of class inheriting from "spT".

residuals

If TRUE then plot residual vs. fitted and normal qqplot of the residuals. If FALSE then plot MCMC samples of the parameters using coda package. Defaults value is FALSE.

coefficient

Only applicable for package "spTDyn" (see details: https://cran.r-project.org/web/packages/spTDyn/index.html).

...

Other arguments.

Examples

## Not run: 
##

plot(out) # where out is the output from spT class
plot(out, residuals=TRUE) # where out is the output from spT class

##

## End(Not run)

Spatial and temporal predictions for the spatio-temporal models.

Description

This function is used to obtain spatial predictions in the unknown locations and also to get the temporal forecasts using MCMC samples.

Usage

## S3 method for class 'spT'
predict(object, newdata, newcoords, foreStep=NULL, type="spatial", 
        nBurn, tol.dist, predAR=NULL, Summary=TRUE, ...)

Arguments

object

Object of class inheriting from "spT".

newdata

The data set providing the covariate values for spatial prediction or temporal forecasts. This data should have the same space-time structure as the original data frame.

newcoords

The coordinates for the prediction or forecast sites. The locations are in similar format to coords, see spT.Gibbs.

foreStep

Number of K-step (time points) ahead forecast, K=1,2, ...; Only applicable if type="temporal".

type

If the value is "spatial" then only spatial prediction will be performed at the newcoords which must be different from the fitted sites provided by coords. When the "temporal" option is specified then forecasting will be performed and in this case the newcoords may also contain elements of the fitted sites in which case only temporal forecasting beyond the last fitted time point will be performed.

nBurn

Number of burn-in. Initial MCMC samples to discard before making inference.

tol.dist

Minimum tolerance distance limit between fitted and predicted locations.

predAR

The prediction output, if forecasts are in the prediction locations. Only applicable if type="forecast" and data fitted with the "AR" model.

Summary

To obtain summary statistics for the posterior predicted MCMC samples. Default is TRUE.

...

Other arguments.

Value

pred.samples or fore.samples

Prediction or forecast MCMC samples.

pred.coords or fore.coords

prediction or forecast coordinates.

Mean

Average of the MCMC predictions

Median

Median of the MCMC predictions

SD

Standard deviation of the MCMC predictions

Low

Lower limit for the 95 percent CI of the MCMC predictions

Up

Upper limit for the 95 percent CI of the MCMC predictions

computation.time

The computation time.

model

The model method used for prediction.

type

"spatial" or "temporal".

...

Other values "obsData", "fittedData" and "residuals" are provided only for temporal prediction. This is to analyse the spTimer forecast output using package forecast through function as.forecast.object.

References

Bakar, K. S. and Sahu, S. K. (2014) spTimer: Spatio-Temporal Bayesian Modelling Using R. Technical Report, University of Southampton, UK. To appear in the Journal of Statistical Software.
Sahu, S. K. and Bakar, K. S. (2012) A comparison of Bayesian Models for Daily Ozone Concentration Levels Statistical Methodology , 9, 144-157.
Sahu, S. K. and Bakar, K. S. (2012) 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.

Examples



##

###########################
## The GP models:
###########################

##
## Spatial prediction/interpolation
##

# Read data
data(NYdata)
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=c(8,11,12,14,18,21,24,28)) 
DataValPred<-subset(DataValPred, with(DataValPred, !(Day %in% c(30, 31) & Month == 8)))

# MCMC via Gibbs using default choices
set.seed(11)
post.gp <- spT.Gibbs(formula=o8hrmax ~cMAXTMP+WDSP+RH,   
        data=DataFit, model="GP", coords=~Longitude+Latitude, 
        scale.transform="SQRT")
print(post.gp)

# Define prediction coordinates
pred.coords<-as.matrix(unique(cbind(DataValPred[,2:3])))

# Spatial prediction using spT.Gibbs output
set.seed(11)
pred.gp <- predict(post.gp, newdata=DataValPred, newcoords=pred.coords)
print(pred.gp)
names(pred.gp)

# validation criteria
spT.validation(DataValPred$o8hrmax,c(pred.gp$Mean))  

##
## Temporal  prediction/forecast 
## 1. In the unobserved locations
##

# Read data
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)))

# define forecast coordinates
fore.coords<-as.matrix(unique(cbind(DataValFore[,2:3])))

# Two-step ahead forecast, i.e., in day 61 and 62 
# in the unobserved locations using output from spT.Gibbs
set.seed(11)
fore.gp <- predict(post.gp, newdata=DataValFore, newcoords=fore.coords, 
           type="temporal", foreStep=2)
print(fore.gp)
names(fore.gp)

# Forecast validations 
spT.validation(DataValFore$o8hrmax,c(fore.gp$Mean)) 

# Use of "forecast" class
#library(forecast)
#tmp<-as.forecast.object(fore.gp, site=1) # default for site 1
#plot(tmp)
#summary(tmp)

##
## Temporal  prediction/forecast 
## 2. In the observed/fitted locations
##

# Read data
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)))

# Define forecast coordinates
fore.coords<-as.matrix(unique(cbind(DataFitFore[,2:3])))

# Two-step ahead forecast, i.e., in day 61 and 62, 
# in the fitted locations using output from spT.Gibbs
set.seed(11)
fore.gp <- predict(post.gp, newdata=DataFitFore, newcoords=fore.coords, 
           type="temporal", foreStep=2)
print(fore.gp)
names(fore.gp)

# Forecast validations 
spT.validation(DataFitFore$o8hrmax,c(fore.gp$Mean)) # 

# Use of "forecast" class
#library(forecast)
#tmp<-as.forecast.object(fore.gp, site=5) # for site 5
#plot(tmp)

###########################
## The AR models:
###########################

##
## Spatial prediction/interpolation
##

# Read data
data(NYdata)
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=c(8,11,12,14,18,21,24,28)) 
DataValPred<-subset(DataValPred, with(DataValPred, !(Day %in% c(30, 31) & Month == 8)))

# MCMC via Gibbs using default choices
set.seed(11)
post.ar <- spT.Gibbs(formula=o8hrmax ~cMAXTMP+WDSP+RH,   
         data=DataFit, model="AR", coords=~Longitude+Latitude, 
         scale.transform="SQRT")
print(post.ar)

# Define prediction coordinates
pred.coords<-as.matrix(unique(cbind(DataValPred[,2:3])))

# Spatial prediction using spT.Gibbs output
set.seed(11)
pred.ar <- predict(post.ar, newdata=DataValPred, newcoords=pred.coords)
print(pred.ar)
names(pred.ar)

# validation criteria
spT.validation(DataValPred$o8hrmax,c(pred.ar$Mean))  

##
## Temporal  prediction/forecast 
## 1. In the unobserved locations
##

# Read data
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)))

# define forecast coordinates
fore.coords<-as.matrix(unique(cbind(DataValFore[,2:3])))

# Two-step ahead forecast, i.e., in day 61 and 62 
# in the unobserved locations using output from spT.Gibbs
set.seed(11)
fore.ar <- predict(post.ar, newdata=DataValFore, newcoords=fore.coords, 
           type="temporal", foreStep=2, predAR=pred.ar)
print(fore.ar)
names(fore.ar)

# Forecast validations 
spT.validation(DataValFore$o8hrmax,c(fore.ar$Mean)) 

# Use of "forecast" class
#tmp<-as.forecast.object(fore.ar, site=1) # default for site 1
#plot(tmp)


##
## Temporal  prediction/forecast 
## 2. In the observed/fitted locations
##

# Read data
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)))

# Define forecast coordinates
fore.coords<-as.matrix(unique(cbind(DataFitFore[,2:3])))

# Two-step ahead forecast, i.e., in day 61 and 62, 
# in the fitted locations using output from spT.Gibbs
set.seed(11)
fore.ar <- predict(post.ar, newdata=DataFitFore, newcoords=fore.coords, 
           type="temporal", foreStep=2)
print(fore.ar)
names(fore.ar)

# Forecast validations 
spT.validation(DataFitFore$o8hrmax,c(fore.ar$Mean)) # 

# Use of "forecast" class
#tmp<-as.forecast.object(fore.ar, site=1) # default for site 1
#plot(tmp)

#################################
## The GPP approximation models:
#################################

##
## Spatial prediction/interpolation
##

# Read data
data(NYdata)
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=c(8,11,12,14,18,21,24,28)) 
DataValPred<-subset(DataValPred, with(DataValPred, !(Day %in% c(30, 31) & Month == 8)))
DataValPred<-spT.subset(data=NYdata, var.name=c("s.index"), s=c(8,11,12,14,18,21,24,28)) 
DataValPred<-subset(DataValPred, with(DataValPred, !(Day %in% c(30, 31) & Month == 8)))

# 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=DataFit, model="GPP", coords=~Longitude+Latitude, 
         knots.coords=knots, scale.transform="SQRT")
print(post.gpp)

# Define prediction coordinates
pred.coords<-as.matrix(unique(cbind(DataValPred[,2:3])))

# Spatial prediction using spT.Gibbs output
set.seed(11)
pred.gpp <- predict(post.gpp, newdata=DataValPred, newcoords=pred.coords)
print(pred.gpp)
names(pred.gpp)

# validation criteria
spT.validation(DataValPred$o8hrmax,c(pred.gpp$Mean))  

##
## Temporal  prediction/forecast 
## 1. In the unobserved locations
##

# Read data
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)))

# define forecast coordinates
fore.coords<-as.matrix(unique(cbind(DataValFore[,2:3])))

# Two-step ahead forecast, i.e., in day 61 and 62 
# in the unobserved locations using output from spT.Gibbs
set.seed(11)
fore.gpp <- predict(post.gpp, newdata=DataValFore, newcoords=fore.coords, 
           type="temporal", foreStep=2)
print(fore.gpp)
names(fore.gpp)

# Forecast validations 
spT.validation(DataValFore$o8hrmax,c(fore.gpp$Mean)) 

# Use of "forecast" class
#tmp<-as.forecast.object(fore.gpp, site=1) # default for site 1
#plot(tmp)

##
## Temporal  prediction/forecast 
## 2. In the observed/fitted locations
##

# Read data
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)))

# Define forecast coordinates
fore.coords<-as.matrix(unique(cbind(DataFitFore[,2:3])))

# Two-step ahead forecast, i.e., in day 61 and 62, 
# in the fitted locations using output from spT.Gibbs
set.seed(11)
fore.gpp <- predict(post.gpp, newdata=DataFitFore, newcoords=fore.coords, 
           type="temporal", foreStep=2)
print(fore.gpp)
names(fore.gpp)

# Forecast validations 
spT.validation(DataFitFore$o8hrmax,c(fore.gpp$Mean)) # 

# Use of "forecast" class
#tmp<-as.forecast.object(fore.gpp, site=1) # default for site 1
#plot(tmp)

##

######################################################
## 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 = 4),
  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 = 4))
#  
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


######################################################
######################################################
##

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.

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


######################################################
######################################################
##

Choice for sampling spatial decay parameter `\phi`.

Description

This function initialises the sampling method for the spatial decay parameter \phi.

Usage

spT.decay(distribution=Gamm(a=2,b=1), tuning=NULL, npoints=NULL, value=NULL)

Arguments

distribution

Prior distribution for \phi. Currently available methods are, Gamm(a,b) and Unif(low,up). One can also used "FIXED" value for \phi parameter.

tuning

If the Gamma prior distribution is used then we need to define the tuning parameter for sampling \phi. The tuning is the standard deviation for the normal proposal distribution of the random-walk Metropolis algorithm used to sample \phi on the log-scale.

npoints

If Unif distribution is used then need to define the number of segments for the range of limits by npoints. Default value is 5.

value

If distribution="FIXED" type is used then need to define the value for \phi. The default value is 3/dmax where dmax is the maximum distance between the fitting sites provided by coords.

Examples

## 

# input for random-walk Metropolis within Gibbs 
# sampling for phi parameter
spatial.decay<-spT.decay(distribution=Gamm(2,1), tuning=0.08)

# input for discrete sampling of phi parameter 
# with uniform prior distribution
spatial.decay<-spT.decay(distribution=Unif(0.01,0.02),npoints=5)

# input for spatial decay if FIXED is used
spatial.decay<-spT.decay(distribution="FIXED", value=0.01)

##

Geodetic/geodesic Distance

Description

This geodetic distance provides the distance between the locations in Kilometers (k.m.) and Miles, using spherical law of Cosines.

Usage

spT.geodist(Lon, Lat, KM = TRUE)

spT.geo.dist(point1, point2)
spT.geo_dist(points)

Arguments

Lon

The longitude position.

Lat

The latitude position.

KM

A logical value, if 'TRUE' then output is in ‘kilometers’, otherwise in ‘miles’.

point1

In the form of (longitude, latitude) position.

point2

In the form of (longitude, latitude) position.

points

In the form of points 1:(longitude, latitude) 2:(longitude, latitude) positions.

Details

spT.geodist is used to get geodetic distance in both miles and kilometers. spT.geo.dist is only used to get geodetic distance in kilometers with a different format. spT.geo_dist is only used to get geodetic distance in kilometers with a different format.

Examples

##

# Load 28 ozone monitoring locations of New York.
data(NYdata)	
head(NYdata)
NYsite<-unique(NYdata[,1:3])	

# Find the geodetic distance in km
spT.geodist(Lon=NYsite$Longitude, Lat=NYsite$Latitude, KM=TRUE)
   
# Find the geodetic distance in miles
spT.geodist(Lon=NYsite$Longitude, Lat=NYsite$Latitude, KM=FALSE)

##

# using spT.geo.dist
point1<-c(-73.757,42.681)
point2<-c(-73.881,40.866)
spT.geo.dist(point1,point2)

# using spT.geo_dist
points<-c(point1,point2)
spT.geo_dist(points)

##

Grid Coordinates

Description

This function is used to obtain Longitude/x and Latitude/y coordinates in a grid set.

Usage

spT.grid.coords(Longitude = c(max, min),
   Latitude = c(max, min), by = c(NA,NA))

Arguments

Longitude

The maximum and minimum longitude position.

Latitude

The maximum and minimum latitude position.

by

The number of x and y points in each axis.

Examples

##

# Load 29 ozone monitoring locations in New York.

data(NYdata)	
coords <- as.matrix(NYdata[,c(2,3)])

# Find the knots coordinates

knots.coords <- spT.grid.coords(Longitude=c(max(coords[,1]),
          min(coords[,1])), Latitude=c(max(coords[,2]),
          min(coords[,2])),by=c(4,4))      
knots.coords

##

Initial values for the spatio-temporal models.

Description

This command is useful to assign the initial values of the hyper-parameters of the prior distributions.

Usage

spT.initials(model, sig2eps=0.01, sig2eta=NULL, rho=NULL, beta=NULL, phi=NULL)

Arguments

model

The spatio-temporal models, current options are: "GP", "AR", and "GPP".

sig2eps

Initial value for the parameter \sigma^2_\epsilon.

sig2eta

Initial value for the parameter \sigma^2_\eta.

rho

Initial value for the parameter \rho.

beta

Initial value for the parameter \beta.

phi

Initial value for the parameter \phi.

Note

Initial values are automatically given if the user does not provide these.

Examples

## 

initials<-spT.initials(model="GPP", sig2eps=0.01, 
        sig2eta=0.5, beta=NULL, phi=0.001)
initials

##

Nominal Coverage

Description

This function is used to obtain nominal coverage.

Usage

spT.pCOVER(z=NULL,zup=NULL,zlow=NULL,zsample=NULL,level=95)

Arguments

z

The original values (matrix or vector).

zup

The predicted values for upper interval (matrix or vector).

zlow

The predicted values for lower interval (matrix or vector).

zsample

Predicted MCMC samples.

level

Level of coverages.

Examples

##

# Create `x': the true values.
# Create `yup': the upper interval.
# Create `ylow': the lower interval.

x <- rnorm(1000,5,0.1)
yup <- rnorm(1000,7,2)
ylow <- rnorm(1000,3,2)
	
# The pCOVER is:

spT.pCOVER(z=x, zup=yup, zlow=ylow)

# create predicted MCMC samples

y <- matrix(rnorm(1000*5000,5,1),1000,5000)

# The pCOVER is:

spT.pCOVER(z=x, zsample=y)
spT.pCOVER(z=x, zsample=y, level=50)

##

Priors for the spatio-temporal models.

Description

This command is useful to assign the hyper-parameters of the prior distributions.

Usage

spT.priors(model="GP", inv.var.prior=Gamm(a=2,b=1),
  beta.prior=Norm(0,10^10), rho.prior=Norm(0,10^10))

Arguments

model

The spatio-temporal models, current input: "GP", "AR", and "GPP".

inv.var.prior

The hyper-parameter for the Gamma prior distribution (with mean = a/b) of the precision (inverse variance) model parameters (e.g., 1/\sigma2_\epsilon, 1/\sigma2_\eta).

beta.prior

The hyper-parameter for the Normal prior distribution of the \beta model parameters.

rho.prior

The hyper-parameter for the Normal prior distribution of the \rho model parameter.

Note

If no prior information are given (assigned as NULL), then it use flat prior values of the corresponding distributions.
Gamm and Norm refers to Gamma and Normal distributions respectively.

Examples

## 

priors<-spT.priors(model="GPP",inv.var.prior=Gamm(2,1),
      beta.prior=Norm(0,10^4))
priors

##

Utility plot for prediction/forecast

Description

This function is used to obtain scatter plots with 95 percent CI for predictions/forecasts.

Usage

spT.segment.plot(obs, est, up, low, limit = NULL)

Arguments

obs

Observed values.

est

Estimated values.

up

Upper limit of the estimated values.

low

Lower limit of the estimated values.

limit

x-axis and y-axis limits.

Examples

##

obs<-rnorm(10,15,1)
est<-rnorm(10,15,1.5)
up<-rnorm(10,25,0.5)
low<-rnorm(10,5,0.5)
spT.segment.plot(obs,est,up,low,limit=c(0,30)) 

##

Select a subset of Spatial data.

Description

This command selects a subset of the dataset using the site numbers.

Usage

spT.subset(data, var.name, s = NULL, reverse = FALSE)

Arguments

data

The dataset.

var.name

The name of the variable for which data will be sub-setted, e.g., "s.index".

s

The site numbers to be selected/deselected based on the argument reverse, e.g., c(2,8,12).

reverse

Logical value: if TRUE then num.rs will be discarded from the data.

Examples

##

# Load ozone concentration data for New York.
data(NYdata)	
NYdata	
# Choose sites 2, 8, and 12.
subdata<-spT.subset(data=NYdata, var.name=c("s.index"), s=c(2,8,12))
# Do not choose purposively defined sites numbered as 2, 8, and 12.
subdata<-spT.subset(data=NYdata, var.name=c("s.index"), s=c(2,8,12), reverse=TRUE)
   
##

Timer series information.

Description

This function defines the time series in the spatio-temporal data.

Usage

spT.time(t.series, segments=1)

Arguments

t.series

Number of times within each segment in each series. It could be either a scalar or a vector. It should be a scalar if the segments are of equal length and should be a vector of length segments whose entries give the length of the segments.

segments

Number of segments in each time series. This should be a scalar.

Examples

## 

# Equal length time-series in each of 3 years
time.data<-spT.time(t.series=365,segments=3)

# Un-equal length time-series in 5 years
time.data<-spT.time(t.series=c(366, 365, 365, 365, 366),segments=5)

##

Validation Commands

Description

The following function is used to validate the predicted observations with the actual values.

Usage

spT.validation(z, zhat, names=FALSE)

Arguments

z

The original values (matrix or vector).

zhat

The predicted values (matrix or vector).

names

Logical, if TRUE then print the names of the validation statistics.

Value

MSE

Mean Squared Error.

RMSE

Root Mean Squared Error.

MAE

Mean Absolute Error.

MAPE

Mean Absolute Percentage Error.

BIAS

Bias.

rBIAS

Relative Bias.

rMSEP

Relative Mean Separation.

Examples

##

# Create `x', which is the true values.
# Create `y', which is the predicted values.

x <- rnorm(10,5,0.1)
y <- rnorm(10,5,1)
spT.validation(x, y)

##

Validation Commands

Description

The following function is used to validate the predicted observations with the actual values based on some threshold.

Usage

spT.validation2(z,zhat,cutoff,names=FALSE)

Arguments

z

The original values (matrix or vector).

zhat

The predicted values (matrix or vector).

cutoff

The threshold value or cut-off point.

names

Logical, if TRUE then print the names of the validation statistics.

Value

TPR

True Positive Rate, Sensitivity, Hit rate, Recall

FPR

False Positive Rate, False alarm

FNR

False Negative Rate, Miss rate

TNR

True Negative Rate, Specificity

Prevalence

Prevalence

Accuracy

Accuracy

Precision

Precision, Positive Predictive Value

FOR

False Ommission Rate

LRp

Positive Likelihood Ratio

LRn

Negative Likelihood Ratio

FDR

False Discovery Rate

NPV

Negative Predictive Value

DOR

Diagnostic Odds Ratio

F1score

F1 score

Heidke.Skill

Heidke Skill

Examples

##

# Create `x', which is the true values.
# Create `y', which is the predicted values.

x <- rnorm(100,0,0.1)
y <- rnorm(100,0,1)
spT.validation2(x, y, cutoff=0,names=TRUE)

##

Service functions and some undocumented functions for the spTimer library

Description

Internal spTimer functions

Summary statistics of the parameters.

Description

This function is used to obtain MCMC summary statistics.

Usage

## S3 method for class 'spT'
summary(object, digits=4, package="spTimer", coefficient=NULL, ...)

##

Arguments

object

Object of class inheriting from "spT".

digits

Rounds the specified number of decimal places (default 4).

package

If "coda" then summary statistics are given using coda package. Defaults value is "spTimer". One can also use "spTDyn" for obtaining spatially varying and temporal dynamic models (see details: https://cran.r-project.org/web/packages/spTDyn/index.html)

coefficient

Only applicable for package "spTDyn".

...

Other arguments.

Value

sig2eps

Summary statistics for \sigma_\epsilon^2.

sig2eta

Summary statistics for \sigma_\eta^2.

phi

Summary statistics for spatial decay parameter \phi, if estimated using spT.decay.

...

Summary statistics for other parameters used in the models.

Examples

## Not run: 
##

summary(out) # where out is the output from spT class
summary(out, digit=2) # where out is the output from spT class
summary(out, pack="coda") # where out is the output from spT class

##

## End(Not run)

Spatio-Temporal Bayesian Modelling using R

Description

Details

Author(s)

References

See Also

Observations of ozone concentration levels, maximum temperature and wind speed.

Description

Usage

Format

Source

See Also

Examples

Credible intervals for model parameters.

Description

Usage

Arguments

See Also

Examples

Extract model fitted values.

Description

Usage

Arguments

Value

See Also

Examples

Plots for spTimer output.

Description

Usage

Arguments

See Also

Examples

Spatial and temporal predictions for the spatio-temporal models.

Description

Usage

Arguments

Value

References

See Also

Examples

MCMC sampling for the spatio-temporal models.

Description

Usage

Arguments

Value

References

See Also

Examples

Choice for sampling spatial decay parameter \phi.

Description

Usage

Arguments

See Also

Examples

Geodetic/geodesic Distance

Description

Usage

Arguments

Details

See Also

Examples

Grid Coordinates

Description

Usage

Arguments

See Also

Examples

Initial values for the spatio-temporal models.

Description

Usage

Arguments

Note

See Also

Examples

Nominal Coverage

Description

Usage

Arguments

See Also

Examples

Choice for sampling spatial decay parameter `\phi`.