Functional Geostatistics: Univariate and Multivariate Functional Spatial Prediction.

SpatFD

Functional Geostatistics: Univariate and Multivariate Functional Spatial Prediction

The R package 'SpatFD' carries out functional kriging, cokriging, optimal sampling and simulation for spatial prediction of functional data. The framework of spatial prediction, optimal sampling and simulation are extended from scalar to functional data. 'SpatFD' is based on the Karhunen-Loève expansion that allows to represent the observed functions in terms of its empirical functional principal components. Based on this approach, the functional auto-covariances and cross-covariances required for spatial functional predictions and optimal sampling, are completely determined by the sum of the spatial auto-covariances and cross-covariances of the respective score components.

The package provides new classes of data and functions for modeling spatial dependence structure among curves. The spatial prediction of curves at unsampled locations can be carried out using two types of predictors, and both of them report, the respective variances of the prediction error. In addition, there is a function for the determination of spatial locations sampling configuration that ensures minimum variance of spatial functional prediction. There are also two functions for plotting predicted curves at each location and mapping the surface at each time point, respectively.

Installation

You can install the development version from Github.

install.packages("devtools")
devtools::install_github("mpbohorquezc/SpatFD-Functional-Geostatistics", ref = "main")

Overview

The objects class and usage in the different functions are listed.

• SpatFD return an object class 'SpatFD' which is used in functional kriging and to obtain the spatial random field of scores.
• KS_scores_lambdas,COKS_scores_lambdas return an object class 'KS_pred','COKS_pred' to use in the linear combinations to obtain functional kriging and cokriging. Plots are made with ggplot_KS and ggmap_KS.
• FD_optimal_design return an object class 'OptimalSpatialDesign' that can be used with print.

Example of use

library(SpatFD)
library(gstat)

# Load data and coordinates
data(AirQualityBogota)

#s_0 nonsampled location. It could be data.frame or matrix and one or more locations of interest
newcoorden=data.frame(X=seq(93000,105000,len=100),Y=seq(97000,112000,len=100))
#newcoorden=data.frame(X=110000,Y=126000)
#newcoorden=matrix(c(110000.23,109000,109500,130000.81,129000,131000),nrow=3,ncol=2,byrow=T)

# Building the SpatFD object
SFD_PM10 <- SpatFD(PM10, coords = coord[, -1], basis = "Bsplines", nbasis = 17,norder=5, lambda = 0.00002, nharm=3)
summary(SFD_PM10)

# Semivariogram models for each spatial random field of scores
modelos <- list(vgm(psill = 2199288.58, "Wav", range = 1484.57, nugget =  0),
                vgm(psill = 62640.74, "Mat", range = 1979.43, nugget = 0,kappa=0.68),
                vgm(psill =37098.25, "Exp", range = 6433.16, nugget =  0))

# Functional kriging. Functional spatial prediction at each location of interest
#method = "lambda"
#Computation of lambda_i
KS_SFD_PM10_l <- KS_scores_lambdas(SFD_PM10, newcoorden ,method = "lambda", model = modelos)
class(KS_SFD_PM10_l)

# method = "scores"
#Simple kriging of scores
KS_SFD_PM10_sc <- KS_scores_lambdas(SFD_PM10, newcoorden, method = "scores", model = modelos)

# method = "both"
KS_SFD_PM10_both <- KS_scores_lambdas(SFD_PM10, newcoorden, method = "both", model = modelos)

summary(KS_SFD_PM10_l)
summary(KS_SFD_PM10_sc)
summary(KS_SFD_PM10_both)

# Linear combinations among weigths predictions and eigenfunctions
recons_fd(KS_SFD_PM10_l)
recons_fd(KS_SFD_PM10_sc)
recons_fd(KS_SFD_PM10_both)

# Curve and variance prediction plots
ggplot_KS(KS_SFD_PM10_l)
ggplot_KS(KS_SFD_PM10_l, show.varpred = T) 
ggplot_KS(KS_SFD_PM10_sc)
ggplot_KS(KS_SFD_PM10_sc, show.varpred = T) 
#Curve and variance prediction for both methods
PlotKS=ggplot_KS(KS_SFD_PM10_both,
          main = "Plot 1 - Using Scores",
          main2 = "Plot 2 - Using Lambda",
          ylab = "PM10")
PlotKS[[1]]
PlotKS[[2]]

# Smoothed prediction maps for the given specific times 
ggmap_KS(KS_SFD_PM10_l,
         map_path = map,
         window_time = c(3500),
         zmin = 25,
         zmax = 100)

ggmap_KS(KS_SFD_PM10_both,
         map_path = map,
         window_time = c(2108),
         method = "lambda",
         zmin = 50,
         zmax = 120)

ggmap_KS(KS_SFD_PM10_both,
         map_path = map,
         window_time = c(5108,5109,5110),
         method = "scores",
         zmin = 50,
         zmax = 120)

# Cross Validation 
crossval_loo(KS_SFD_PM10_l)
crossval_loo(KS_SFD_PM10_sc)
crossval_loo(KS_SFD_PM10_both)

# Optimal spatial design
s0 <- cbind(2*runif(100),runif(100)) # random coordinates on (0,2)x(0,1)
fixed_stations <- cbind(2*runif(4),runif(4))
x_grid <- seq(0,2,length = 30)
y_grid <- seq(0,1,length = 30)
grid <- cbind(rep(x_grid,each = 30),rep(y_grid,30))
model  <- vgm(psill = 5.665312,
                  model = "Exc",
                  range = 8000,
                  kappa = 1.62,
                  add.to = vgm(psill = 0.893,
                               model = "Nug",
                               range = 0,
                               kappa = 0))
FD_optimal_design(k = 10, s0 = s0, model = model,
                  grid = grid, nharm = 2, plt = TRUE,
                  fixed_stations = fixed_stations) -> OSD
OSD$new_stations
OSD$fixed_stations
OSD$plot
class(OSD)

#### Real Data Example ####
vgm_model  <- vgm(psill = 5.665312,
                  model = "Exc",
                  range = 8000,
                  kappa = 1.62,
                  add.to = vgm(psill = 0.893,
                               model = "Nug",
                               range = 0,
                               kappa = 0))

my.CRS <- sp::CRS("EPSG:21899") # https://epsg.io/21899
map <- as(map, "Spatial")

bogota_shp <- sp::spTransform(map,my.CRS)
target <- sp::spsample(bogota_shp,n = 100, type = "random")
# The set of points in which we want to predict optimally.
old_stations <- sp::spsample(bogota_shp,n = 3, type = "random")
# The set of stations that are already fixed.

FD_optimal_design(k = 10, s0 = target,model = vgm_model,
               map = map,plt = TRUE,
               fixed_stations = old_stations) -> res
res
class(res)

# Functional cokriging
data(COKMexico)
SFD_PM10_NO2 <- SpatFD(Mex_PM10, coords = coord_PM10, basis = "Fourier", nbasis = 21, lambda = 0.000001, nharm = 2)
SFD_PM10_NO2 <- SpatFD(NO2, coords = coord_NO2, basis = "Fourier", nbasis = 27, lambda = 0.000001, nharm = 2,
                      add = SFD_PM10_NO2)
model1 <- gstat::vgm(647677.1,"Gau",23317.05)
model1 <- gstat::vgm(127633,"Wav",9408.63, add.to = model1)
newcoords <- data.frame(x = 509926,y = 2179149)
COKS_Mex <- COKS_scores_lambdas(SFD_PM10_NO2,newcoords,model1)
summary(COKS_Mex)
ggplot_KS(COKS_Mex)
ggmap_KS(COKS_Mex,map_path = map_mex,method = 'scores')

For more information

You can read:

• https://onlinelibrary.wiley.com/doi/book/10.1002/9781119387916
• https://link.springer.com/article/10.1007/s10260-015-0340-9
• https://link.springer.com/article/10.1007/s00477-016-1266-y

License

This package is free and open source software, licensed under GPL-3.

References

• Bohorquez, M., Giraldo, R., & Mateu, J. (2016). Optimal sampling for spatial prediction of functional data. Statistical Methods & Applications, 25(1), 39-54.
• Bohorquez, M., Giraldo, R., & Mateu, J. (2016). Multivariate functional random fields: prediction and optimal sampling. Stochastic Environmental Research and Risk Assessment, 31, pages53–70 (2017).
• Bohorquez M., Giraldo R. and Mateu J. (2021). Spatial prediction and optimal sampling of functional data in Geostatistical Functional Data Analysis: Theory and Methods. John Wiley Sons, Chichester, UK. ISBN: 978-1-119-38784-8. https://www.wiley.com/en-us/Geostatistical+Functional+Data+Analysis-p-9781119387848.