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spTimer (version 3.3.3)

predict.spT: 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 spT
predict(object, newdata, newcoords, foreStep=NULL, type="spatial", 
        nBurn, tol.dist, predAR=NULL, Summary=TRUE, ...)

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.

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.

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.

See Also

spT.Gibbs, as.forecast.object.

Examples

Run this code
# \dontshow{
# 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)))
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 <- 100
nBurn <- 0
# GP: MCMC via Gibbs using default choices
set.seed(11)
post.gp <- spT.Gibbs(formula=o8hrmax ~cMAXTMP+WDSP+RH,   
        nItr=nItr,  nBurn=nBurn,
        data=DataFit, model="GP", coords=~Longitude+Latitude, 
        scale.transform="SQRT")
pred.gp <- predict(post.gp, newdata=DataValPred, newcoords=~Longitude+Latitude)
fore.gp <- predict(post.gp, newdata=DataValFore, newcoords=~Longitude+Latitude, 
           type="temporal", foreStep=2)
# AR: MCMC via Gibbs using default choices
set.seed(11)
post.ar <- spT.Gibbs(formula=o8hrmax ~cMAXTMP+WDSP+RH,   
        nItr=nItr,  nBurn=nBurn,
        data=DataFit, model="AR", coords=~Longitude+Latitude, 
        scale.transform="SQRT")
pred.ar <- predict(post.ar, newdata=DataValPred, newcoords=~Longitude+Latitude)
fore.ar <- predict(post.ar, newdata=DataValFore, newcoords=~Longitude+Latitude, 
           type="temporal", foreStep=2)
# GPP: MCMC via Gibbs using default choices
# Define knots
coords<-as.matrix(unique(cbind(NYdata[,2:3])))
knots<-spT.grid.coords(Longitude=c(max(coords[,1]),
              min(coords[,1])),Latitude=c(max(coords[,2]),
              min(coords[,2])), by=c(4,4))
set.seed(11)
post.gpp <- spT.Gibbs(formula=o8hrmax ~cMAXTMP+WDSP+RH,
         nItr=nItr,   nBurn=nBurn, 
         data=DataFit, model="GPP", coords=~Longitude+Latitude, 
         knots.coords=knots, scale.transform="SQRT")
pred.gpp <- predict(post.gpp, newdata=DataValPred, newcoords=~Longitude+Latitude)
fore.gpp <- predict(post.gpp, newdata=DataValFore, newcoords=~Longitude+Latitude, 
           type="temporal", foreStep=2)
# GP (truncated): MCMC via Gibbs 
data(NYdata)
# Truncation at 30 (say)
NYdata$o8hrmax[NYdata$o8hrmax<=30] <- 30
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)))
out <- spT.Gibbs(formula=o8hrmax~cMAXTMP+WDSP+RH,data=DataFit,
  model="truncatedGP",coords=~Longitude+Latitude,
  distance.method="geodetic:km",nItr=nItr, nBurn=nBurn,
  truncation.para = list(at = 30,lambda = 4))
pred <- predict(out,newdata=DataValPred, newcoords=~Longitude+Latitude, tol=0.05)
fore <- predict(out,newdata=DataValFore, newcoords=~Longitude+Latitude,
   type="temporal", foreStep=2, tol=0.05)
# GPP (truncated):MCMC via Gibbs 
coords<-as.matrix(unique(cbind(NYdata[,2:3])))
knots<-spT.grid.coords(Longitude=c(max(coords[,1]),
              min(coords[,1])),Latitude=c(max(coords[,2]),
              min(coords[,2])), by=c(4,4))
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,
         truncation.para = list(at = 30,lambda = 4))
pred <- predict(out,newdata=DataValPred, newcoords=~Longitude+Latitude, tol=0.05)
fore <- predict(out,newdata=DataValFore, newcoords=~Longitude+Latitude,
   type="temporal", foreStep=2, tol=0.05)
##	   
# }
# \donttest{
##

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


######################################################
######################################################
##
# }

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