Air quality data of Bogota, Colombia

Description

Particular material 10 (PM10) of Bogota measure in 10 stations in the city.

Usage

data(AirQualityBogota)

Format

two data.frame of measurements 'PM10' and coordinates 'coord'. Also a map file of class SpatialPolygonDataFrame load from a shape file by rgdal::readOGR

Source

Secretaria de Ambiente de Bogotá

References

Monitor network of air quality of Bogota http://rmcab.ambientebogota.gov.co

Air quality data of Mexico

Description

Particular material 10 (PM10) and NO2 measured in 13 and 18 locations in Mexico. The data correspond to consecutive hours from January 01, 2015 at 1:00 a.m. to May 30, 2015 at 12:00 a.m., at 23 environmental stations. The stations in the air quality network RAMA (Red Automática de Monitoreo Atmosférico), monitor hourly particulate matter up to 10 micrometers in size (PM10) and Nitrogen dioxide (NO2) among others. The particulate matter (PM) is an important component of air pollution. NO2 is a gaseous air pollutant produced by the road traffic and other fossil fuel combustion processes and it contributes to the formation and modification of other air pollutants such as particulate matter.

Usage

data(COKMexico)

Format

four data.frame of measurements 'Mex_PM10','NO2' and their coordinates 'coord_PM10' and 'coord_NO2'. Also a map (map_mex) file of class SpatialPolygonDataFrame

Source

Mexico

Functional cokriging

Description

Linear Spatial functional prediction. Two predictors are possible: scores or lambda.

Usage

COKS_scores_lambdas(SFD, newcoords, model, method = "scores", fill.all=TRUE)

Arguments

Details

Each functional variable is represented in terms of its functional principal components \chi_{\bm s_i}(t)=\bm \xi^T(t)\bm f_{\bm s_i},\;i=1,...,n where \bm f_{\bm s_i}=\left(f_{\bm s_i}^1,...,f_{\bm s_i}^K\right)^T

The goal is the prediction of a spatial functional variable of \chi^r_{\bm s_0}(t)\;1\leq r\leq P at the unsampled site \bm s_0 based on P spatial functional variables. The method performs cokriging directly on the scores chosen for all functional variables involved.

\left(\bm f_{\bm s}^{11},...,f_{\bm s}^{1K_1},...,f_{\bm s}^{P1},...,f_{\bm s}^{PK_P}\right)

Scores predictions are used to build the cokriging functional predictor.

Value

Returns a 'COKS_pred' object with functional cokriging

Author(s)

Valeria Bejarano <vbejaranos@unal.edu.co>

References

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. Spatial prediction and optimal sampling of functional data in Geostatistical Functional Data Analysis: Theory and Methods (2021). John Wiley Sons, Chichester, UK. ISBN: 978-1-119-38784-8. https://www.wiley.com/en-us/Geostatistical+Functional+Data+Analysis-p-9781119387848.

See Also

SpatFD,summary.COKS_pred

Examples

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_scores_lambdas(SFD_PM10_NO2,newcoords,model1)

Leave-One-Out Cross-Validation for Functional cokriging

Description

This function performs leave-one-out cross-validation for functional cokriging. It systematically leaves out one location at a time from the dataset, fits the model to the remaining data, and then makes a prediction for the left-out observation. It is used to assess the predictive performance of the functional cokriging model.

Usage

COK_crossval_loo(object, plot_show, var, show_all)

Arguments

Value

An object containing the results of the leave-one-out cross-validation. Includes:

Author(s)

Venus Puertas vpuertasg@unal.edu.co

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).

See Also

recons_fd, KS_scores_lambdas

Examples

# Load the data and the required packages
library(SpatFD)
library(gstat)
data(COKMexico)

# Create the SpatFD data objects
SFD_PM10_NO2 <- SpatFD(Mex_PM10, coords = coord_PM10, basis = "Fourier", 
                       nbasis = 21, lambda = 0.000001, nharm = 2,
                       name = names(Mex_PM10))
SFD_PM10_NO2 <- SpatFD(NO2, coords = coord_NO2, basis = "Fourier", 
                       nbasis = 27, lambda = 0.000001, nharm = 2,
                       add = SFD_PM10_NO2,name = names(NO2))

# Fit the model
model1 <- gstat::vgm(647677.1,"Gau",23317.05)
model1 <- gstat::vgm(127633,"Wav",9408.63, add.to = model1)

# Perform the cokriging
newcoords <- data.frame(x = 509926,y = 2179149)
coks <- COKS_scores_lambdas(SFD_PM10_NO2,newcoords,model1)

# Perform the cross-validation along NO2
COK_crossval_loo(object = coks, var = 2,show_all=TRUE)

Creates univariate and multivariate CrossSpatFD object to perform crossed validation for functional spatial prediction.

Description

Creates univariate and multivariate CrossSpatFD object considering the base od a SpatFD or FD object to perform crossed validation for functional spatial prediction.

Usage

CrossSpatFD(data,coords,basis,lambda=0,nharm=NULL,name=NULL,add=NULL,...)

Arguments

Details

The CrossSpatFD-objects storage the functional data, its parameters, the functional principal component analysis results, and the spatial coordinates for each variable. Each variable has its own functional data, data-frame or matrix and its spatial coordinates file.

Value

For each variable: The functional data and functional principal components linked with spatial coordinates.

Note

1. This function is for internal use and should not be implemented directly

Author(s)

Diego Sandoval diasandovalsk@unal.edu.co & Angie Villamil acvillamils@unal.edu.co.

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).

See Also

summary.SpatFD

Optimal Spatial Design For Functional Data

Description

Given a variogram model and a set of points in which we want to predict certain variable optimally, this function finds where must be placed the stations in which the information will be collected for functional or scalar data.

Usage

FD_optimal_design(k, s0, model, fixed_stations = NULL,
                   scalar = FALSE, nharm = NULL,
                   method = "lambda", grid = NULL,
                   map = NULL, plt = FALSE)

Arguments

Details

Bohorquez, M., Giraldo, R., Mateu, J. (2016) present several methods for finding the best combination predictor-design according to the kriging prediction error variance for functional data. They show different functional kriging methods and two of them are implemented on this function.

If method is "lambda", optimal spatial sampling using FPCA and simple kriging will be used (see section 3.2 of Bohorquez, M., Giraldo, R., Mateu, J. (2016)). If method is "scores", simple kriging will be applied on each harmonic and the total variance will be minimized. This total variance is computed as follows:

TotVar = \sum_{j=1}^{nharm}V_j

where V_j is the variance of the simple kriging prediction of j-th score.

Value

The function returns an OptimalSpatialDesign object that is a list with the following elements:

Warning

When method is 'lambda', the minimized value is not the variance, but the negative of expression (12) in Bohorquez, M., Giraldo, R., & Mateu, J. (2016), that is

-\sum_{l = 1}^L \varsigma_l'\Omega^{-1}\varsigma_l

Note

'lambda' method tends to be faster than 'scores' method.

Author(s)

Nathaly Vergel Serrano nvergel@unal.edu.co & Samuel Sánchez Gutiérrez ssanchezgu@unal.edu.co.

References

Bohorquez, M., Giraldo, R., & Mateu, J. (2016). Optimal sampling for spatial prediction of functional data. Statistical Methods & Applications, 25(1), 39-54.

See Also

print.OptimalSpatialDesign

Examples

library(gstat)
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)

Functional Kriging

Description

Linear Spatial functional prediction. Two predictors are possible: scores or lambda.

Usage

KS_scores_lambdas(SFD, newcoords, model, method = "lambda", name = NULL, fill.all = NULL)

Arguments

Details

"lambda" option corresponds to the predictor \breve{\chi}_{\bm s_0}(t), given by

\breve{\chi}_{\bm s_0}(t)=\sum\limits_{i=1}^{n}\lambda_i\chi_{\bm s_i}(t)

and weigths are found such that minimize \left\|\bm\chi_{\bm s_0}(t)-\breve{\chi}_{\bm s_0}(t)\right\|^2.

"scores" method performs kriging or cokriging directly on the scores and predictions are used to build the functional prediction

It is used simple cokriging to predict the vector \bm f(\bm s_0)=\left(f_1(\bm s_0),...,f_K(\bm s_0)\right)^T at the unsampled location \bm s_0. The predictor is f^*(\bm s_0), so the prediction of the curve \chi_{\bm s_0}(t) is \chi^*_{\bm s_0}(t)=\bm \xi^T(t)\bm f^*(\bm s_0),\;\;\;\;\;\;i=1,...,n.

Value

Returns a 'KS_pred' object with functional kriging: weights (lambda) using the first method and kriging score predictions using the second method in Bohorquez, M., Giraldo, R., & Mateu, J. (2016).

Author(s)

Diego Sandoval diasandovalsk@unal.edu.co & Angie Villamil acvillamils@unal.edu.co.

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. Spatial prediction and optimal sampling of functional data in Geostatistical Functional Data Analysis: Theory and Methods (2021). John Wiley Sons, Chichester, UK. ISBN: 978-1-119-38784-8. https://www.wiley.com/en-us/Geostatistical+Functional+Data+Analysis-p-9781119387848.

See Also

SpatFD,summary.KS_pred

Examples

library(gstat)
data(AirQualityBogota)

newcoorden=data.frame(X=110000,Y=125000)

# Recibir los datos, suavizarlos y ACP
SFD_PM10 <- SpatFD(PM10, coords = coord[,2:3], basis = "Bsplines",
nbasis = 17,norder=5, lambda = 0.00002, nharm=3)

#Variogram model for each component
modelos <- list(vgm(psill = 2634000, "Exp", range = 2103.25, nugget =  0),
                vgm(psill = 101494.96, "Exp", range = 1484.57, nugget = 0),
                vgm(psill =53673, "Exp", range = 42406, nugget =  0))

#Genera los scores y los lambdas para predecir en nuevas coordenadas

#method = "lambda"
KS_SFD_PM10_l <- KS_scores_lambdas(SFD_PM10, newcoorden ,method = "lambda",
model = modelos)
class(KS_SFD_PM10_l)

#method = "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)

Air quality data of Mexico

Description

Particular material 10 (PM10) measured in 13 locations in Mexico.

Usage

data(COKMexico)

Format

data.frame of measurements 'Mex_PM10'

Source

Mexico

Air quality data of Mexico

Description

NO2 measured in 18 locations in Mexico.

Usage

data(COKMexico)

Format

data.frame of measurement 'NO2'

Source

Mexico

PM10 of Bogota, Colombia

Description

Particular material 10 (PM10) of Bogota measure in 10 stations in the city.

Usage

data(AirQualityBogota)

Format

data.frame of measurements 'PM10'

Source

Secretaria de Ambiente de Bogotá

References

Monitor network of air quality of Bogota http://rmcab.ambientebogota.gov.co

Creates univariate and multivariate SpatFD objects.

Description

Creates an object of the class SpatFD from spatial coordinates, and functions or time-series observed at each spatial location. Time series is a generic term. In fact, observations might be across the frequency or across another spatial dimension such as depth, instead of time.

Usage

SpatFD(data, coords, basis = "Bsplines", nbasis = 4, lambda = 0, nharm = NULL,
name = NULL, add = NULL, ...)

Arguments

Details

The SpatFD-objects storage the functional data, its parameters, the functional principal component analysis results, and the spatial coordinates for each variable. Each variable has its own functional data, data-frame or matrix and its spatial coordinates file.

Value

For each variable: The functional data and functional principal components linked with spatial coordinates.

Note

1. Although there is no limit for the number of variables for functional cokriging, the real limitation is found on the constraints required to find a valid multivariate covariance model. So, it is highly recommended to apply the parsimony principle.

2. Locations must be in the same region of interest to make sense to include all of them in the same prediction model. However, each variable can be observed in different spatial locations and each can have a different number of observations. There is no limit for the number of variables to be included in this object.

Author(s)

Diego Sandoval diasandovalsk@unal.edu.co & Angie Villamil acvillamils@unal.edu.co.

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).

See Also

summary.SpatFD

Examples

# Load data
data(AirQualityBogota)

# Create an univariate object using 2 nharm
SFD_PM10 <- SpatFD(PM10, coords = coord[,2:3], basis = "Bsplines", nbasis = 91,
lambda = 0.00002, nharm = 2)
SFD_PM10

Classification Function for Functional Data

Description

This function classifies new functional data based on PCA results from training data.

Usage

  classification(data.train.pca, new.basis, k, distance, mcov = NULL)

Arguments

Value

The predicted class for the new data.

Examples

  
  data(vowels)
  #### Create parameters and names for the data.
  p = 228 ; nelec = 21 ; nvow = 5
  names_vowels = c("a","e","i","o","u")
  n.basis<-c(14,13,12,13,11)
  s4.gfdata = gfdata(data=vowels,p=p,names=names_vowels,coords=vowels_coords,nbasis=n.basis)
  # Create train and test data
  s4.sep=gfd_clasif_data(s4.gfdata, 0.8,seed = 2910)
  s4.train=s4.sep$train
  s4.test=s4.sep$test
  
  # Classification
  cla<-classification(data.train.pca = s4.train,
                     new.basis=s4.test[[1]]$data_fd[[1]],
                     k=4,
                     distance='euclidean',
                     mcov = mcov
  )
  

Coordinates of measurement stations Bogota, Colombia

Description

Coordinates of 10 stations in the city.

Usage

data(AirQualityBogota)

Format

data.frame of coordinates

Source

Secretaria de Ambiente de Bogotá

References

Monitor network of air quality of Bogota http://rmcab.ambientebogota.gov.co

Coordinates of air quality data of Mexico

Description

18 locations in Mexico where measured NO2.

Usage

data(COKMexico)

Format

data.frame of coordinates 'coord_NO2'

Source

Mexico

Coordinates of air quality data of Mexico

Description

13 locations in Mexico where measured Mex_PM10.

Usage

data(COKMexico)

Format

data.frame of coordinates 'coord_PM10'

Source

Mexico

Create Covariance Matrices given a series of spatial model parameters

Description

This function creates covariance matrices for spatial data based on the provided model parameters.

Usage

  create_mcov(coordenadas, t.models)

Arguments

Value

A list of covariance matrices.

Examples

  
  data(vowels)
  t.models = data.frame(sill = c(1887.47,1447.27,3533.96,1850.27,432.63),
                      phi  = c(106.68,109.1,19.4,150.32,31.52),
                      tausq =c(0,0,0,0,0),
                      model = as.character(c("gaussian","gneiting",
                                             "matern","cubic",
                                             "wave")),
                      kappa = c(0,0,7.3,0,0)
                      
)
create_mcov(vowels_coords,t.models)
  

Leave-One-Out Cross-Validation for Functional Kriging

Description

This function performs leave-one-out cross-validation for functional kriging and cokriging. It systematically leaves out one observation at a time from the dataset, fits the model to the remaining data, and then makes a prediction for the left-out observation. It is used to assess the predictive performance of the functional kriging model.

Usage

crossval_loo(object, plot_show)

Arguments

Value

An object containing the results of the leave-one-out cross-validation. Includes:

Author(s)

Joan Castro jocastroc@unal.edu.co.

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).

See Also

recons_fd, KS_scores_lambdas

Examples

# Example code demonstrating how to use the crossval_loo function
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=TRUE)

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

# 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)
# 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)

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

Creates functional ortogonal basis as fd object.

Description

This function returns the first nth elements of a functional basis as an fd object.

Usage

generate_basis(basis = "Fourier",n_functions = 10,L = NULL,fda_basis = NULL)

Arguments

Details

Fourier basis functions are given by:

f_k(x) = \sqrt{\frac{2}{L}}\text{sin}\left(\frac{k\pi x}{2L}\right)

for k=2,4,6,..., and

f_k(x) = \sqrt{\frac{2}{L}}\text{cos}\left(\frac{(k + 1)\pi x}{2L}\right)

for k=1,3,5,....

Furthermore, Legendre basis functions are given by:

f_k(x) = \frac{1}{2^nn!}\frac{d^n}{dx}(x^2 - 1)^n

for k = 1,2,3,4,....

Value

fda::fd object with n_functions curves.

Note

Generating n Legendre basis functions requires to evaluate \frac{n(n+1)}{2} derivates, so its recomended to use values below 10.

Author(s)

Samuel Sánchez Gutiérrez ssanchezgu@unal.edu.co.

References

Conway, J. B. (2019). A course in functional analysis (Vol. 96). Springer.

See Also

sim_functional_process

Examples

library(fda)

# 10 Fourier functions
res <- generate_basis(L=1)
plot(res)

# 20 Fourier functions
res <- generate_basis(n_functions = 20,L = 3)
plot(res)

# 10 Legendre functions
res <- generate_basis(basis = "Legendre")
plot(res)

# 7 Legendre functions
res <- generate_basis(basis = "Legendre", n_functions = 7)
plot(res)

Divide the data in train and test dataset

Description

This function divides the data in train and test datasets

Usage

  gfd_clasif_data(gfd_data, prop.train, seed = NULL) 

Arguments

Value

gfdata divided object

Author(s)

Diego Sandoval diasandovalsk@unal.edu.co.

References

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).

Examples

  library(SpatFD)
  data(vowels)
  
  #### Create parameters and names for the data.
  p = 228 ; nelec = 21 ; nvow = 5
  names_vowels = c("a","e","i","o","u")
  n.basis<-c(14,13,12,13,11)
  
  s4.gfdata = gfdata(data=vowels,p=p,names=names_vowels,coords=vowels_coords,nbasis=n.basis)
  s4.sep=gfd_clasif_data(s4.gfdata, 0.8,seed = 2910)
  
  s4.train=s4.sep$train
  s4.test=s4.sep$test
  

Generate Variograms for Functional Data from a gfdata object

Description

This function generates variograms for functional data based on PCA results.

Usage

  gfd_variog_geoR(gfd_pca_Data, pairsmin = 2)

Arguments

Value

A list containing geodata objects and variograms.

Examples

  data(vowels)
  
  #### Create parameters and names for the data.
  p = 228 ; nelec = 21 ; nvow = 5
  names_vowels = c("a","e","i","o","u")
  n.basis<-c(14,13,12,13,11)
  s4.gfdata = gfdata(data=vowels,p=p,names=names_vowels,coords=vowels_coords,nbasis=n.basis)
  
  s4.sep=gfd_clasif_data(s4.gfdata, 0.8,seed = 2910)
  
  s4.train=s4.sep$train
  
  s4.var.geoR=gfd_variog_geoR(s4.train)

Creates gfdata objects.

Description

Creates an object of the class gfdata from spatial coordinates, and functions or time-series observed at each spatial location. Time series is a generic term. In fact, observations might be across the frequency or across another spatial dimension such as depth, instead of time.

Usage

  gfdata(data, p, basis = "Bsplines", coords = NULL, nbasis = NULL, 
                      names = NULL, lambda = 0)

Arguments

Details

The gfdata-objects storage the functional data, its parameters, the functional principal component analysis results, and the spatial coordinates for each variable. Each variable has its own functional data, data-frame or matrix and its spatial coordinates file.

Value

For each subject and class: The functional data and functional principal components linked with spatial coordinates.

Author(s)

Venus Puertas vpuertasg@unal.edu.co.

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).

See Also

summary.gfdata

Examples

  
    library(SpatFD)
    data(vowels)
    
    #### Create parameters and names for the data.
    p = 228 ; nelec = 21 ; nvow = 5
    names_vowels = c("a","e","i","o","u")
    n.basis<-c(14,13,12,13,11)
    
    s4.gfdata = gfdata(data=vowels,p=p,names=names_vowels,coords=vowels_coords,nbasis=n.basis)
    
  

Map plot of a 'KS_pred' object

Description

A visualization of the predicted kriging in a colormap for a specific window time.

Usage

ggmap_KS(KS, map_path, window_time = NULL, method = "lambda", map_n = 5000,
zmin = NULL, zmax = NULL, graph = "plotly")

Arguments

Value

Plotly or ggplot image

Author(s)

Diego Sandoval diasandovalsk@unal.edu.co.

References

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).

See Also

KS_scores_lambdas

Examples

library(gstat)
data(AirQualityBogota)
newcoorden=data.frame(X=seq(93000,105000,len=100),Y=seq(97000,112000,len=100))

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

modelos <- list(vgm(psill = 2634000, "Exp", range = 2103.25, nugget =  0),
                vgm(psill = 101494.96, "Exp", range = 1484.57, nugget = 0),
                vgm(psill =53673, "Exp", range = 42406, nugget =  0))

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

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

ggplot of predicted functions

Description

Plot with or without predicted variance in each spatial location of the functional kriging.

Usage

ggplot_KS(KS, show.varpred = FALSE, main = "Functional Data",
main2 = "Functional Data", ylab = "Value", xlab = "Time", ndigits = 2,
palette.plot = c("#440154FF", "#3336FF", "#33FCFF", "#33FF4C", "#FDE725FF"))

Arguments

Value

ggplot. If there are two plots a list of ggplots.

Author(s)

Diego Sandoval diasandovalsk@unal.edu.co.

References

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).

See Also

KS_scores_lambdas

Examples

library(gstat)
data(AirQualityBogota)
newcoorden=data.frame(X=seq(93000,105000,len=10),Y=seq(97000,112000,len=10))
SFD_PM10 <- SpatFD(PM10, coords = coord[, -1], basis = "Bsplines", nbasis = 17,
norder=5, lambda = 0.00002, nharm=3)
modelos <- list(vgm(psill = 2634000, "Exp", range = 2103.25, nugget =  0),
                vgm(psill = 101494.96, "Exp", range = 1484.57, nugget = 0),
                vgm(psill =53673, "Exp", range = 42406, nugget =  0))

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

ggplot_KS(KS_SFD_PM10_both,show.varpred = FALSE,
          main = "Plot 1 - Using Scores",
          main2 = "Plot 2 - Using Lambda",
          ylab = "PM10")
          

map of Bogota, Colombia

Description

Map of Bogota.

Usage

data(AirQualityBogota)

Format

Map file of class SpatialPolygonDataFrame

Source

Unidad Administrativa Especial de Catastro Distrital

References

https://datosabiertos.bogota.gov.co/dataset/sector-catastral

map of Mexico

Description

Map of Mexico.

Usage

data(COKMexico)

Format

Map file of class SpatialPolygonDataFrame

Source

Automatic Monitoring System SEDEMA

References

http://www.aire.df.gob.mx/default.php

Get the mean of means for each class

Description

This function generates multivariate vowel data based on the provided mean functions and basis functions.

Usage

  mclass_data(mean.mean, n.basis, type.basis = "bspline")

Arguments

Value

A data frame containing the generated vowel data.

Examples

  data(vowels)

#### Create parameters and names for the data.
p = 228 ; nelec = 21 ; nvow = 5
names_vowels = c("a","e","i","o","u")
n.basis<-c(14,13,12,13,11)
s4.gfdata = gfdata(data=vowels,p=p,names=names_vowels,coords=vowels_coords,nbasis=n.basis)
s4.sep=gfd_clasif_data(s4.gfdata, 0.8,seed = 2910)
s4.train=s4.sep$train
s4.test=s4.sep$test
mean_mean <- mean_mean(s4.train)
class_mean <- mclass_data(mean_mean,n.basis)

Calculate Mean Functions for Each Class

Description

This function calculates the mean functions for each class based on PCA results from training data.

Usage

  mean_mean(data.train.pca)

Arguments

Value

A list of mean functions for each class.

Examples

  data(vowels)
#### Create parameters and names for the data.
p = 228 ; nelec = 21 ; nvow = 5
names_vowels = c("a","e","i","o","u")
n.basis<-c(14,13,12,13,11)
s4.gfdata = gfdata(data=vowels,p=p,names=names_vowels,coords=vowels_coords,nbasis=n.basis)

s4.sep=gfd_clasif_data(s4.gfdata, 0.8,seed = 2910)

s4.train=s4.sep$train
mean_mean <- mean_mean(s4.train)

Print of OptimalSpatialDesign objects

Description

This functions prints a summary of the main objects of OptimalSpatialDesign objects.

Usage

## S3 method for class 'OptimalSpatialDesign'
print(x, ...)

Arguments

Value

Shows the amount of fixed stations, new stations and the first six new coordinates.

Author(s)

Samuel Sánchez Gutiérrez ssanchezgu@unal.edu.co.

References

Bohorquez, M., Giraldo, R., & Mateu, J. (2016). Optimal sampling for spatial prediction of functional data. Statistical Methods & Applications, 25(1), 39-54.

See Also

FD_optimal_design

Examples

library(gstat)
data(AirQualityBogota)

vgm_model  <- gstat::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,#method = "scores",
               fixed_stations = old_stations) -> res
print(res)

Linear combinations for functional kriging

Description

This is an internal function for functional kriging and cokriging. To perform the linear combinations to obtain functional kriging and cokriging. Once optimization process is finished and scores or lambda are obtained, this function builds the prediction, performing the linear combination between coefficients and basis functions.

Usage

recons_fd(X,name = NULL)

Arguments

Value

Spatial functional predictions. The fd object based on the two functional kriging methods described in Bohorquez, M., Giraldo, R., & Mateu, J. (2016). List of

Author(s)

Diego Sandoval diasandovalsk@unal.edu.co.

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. Spatial prediction and optimal sampling of functional data in Geostatistical Functional Data Analysis: Theory and Methods (2021). John Wiley Sons, Chichester, UK. ISBN: 978-1-119-38784-8. https://www.wiley.com/en-us/Geostatistical+Functional+Data+Analysis-p-9781119387848. ~

See Also

KS_scores_lambdas

Examples

library(gstat)
data(AirQualityBogota)
newcoorden=data.frame(X=110000,Y=125000)
SFD_PM10 <- SpatFD(PM10, coords = coord[,2:3], basis = "Bsplines", nbasis = 17,
norder=5, lambda = 0.00002, nharm=3)
modelos <- list(vgm(psill = 2634000, "Exp", range = 2103.25, nugget =  0),
                vgm(psill = 101494.96, "Exp", range = 1484.57, nugget = 0),
                vgm(psill =53673, "Exp", range = 42406, nugget =  0))

#method = "lambda"
KS_SFD_PM10_l <- KS_scores_lambdas(SFD_PM10, newcoorden ,method = "lambda",
model = modelos)
curves_PM10_l <- recons_fd(KS_SFD_PM10_l)
plot(curves_PM10_l)

#method = "scores"
KS_SFD_PM10_sc <- KS_scores_lambdas(SFD_PM10, newcoorden, method = "scores",
model = modelos)
curves_PM10_sc <- recons_fd(KS_SFD_PM10_sc)
plot(curves_PM10_sc)

Spatial random field of scores

Description

Creates a list of P data-frames, one for each of the P spatial functional variables in the SpatFD-object. Each data-frame contains one column for each harmonic selected with its respective n_p-score values, and two additional columns for horizontal and vertical spatial coordinates.

Usage

scores(X)

Arguments

Details

For a SpatFD object with P spatial functional variables measured on n_p locations each, and nharm=k_p, p=1,..., P. The scores-function builds a data-frame with n_p rows and k_p+2 columns. The first two columns are the horizontal and vertical spatial coordinates x,y. The rest of the columns are the score values for each of the first k_p harmonics selected for the p variable, at each of the n_p locations.

Value

A list with P data-frames, with dimension n_p \times k_p+2, each. p=1,…,P.

Note

Functional principal components (FPCA) are applied to the centered data. The scores are scalar second order stationary Random fields, in virtue of the requirements for FPCA. Hence, the covariance function exists, that is, models are bounded, and always have sill and range parameters. Of course, some models have additional parameters, such as smoothing parameters. From the theoretical perspective, in this case, there is no possibility of non-bounded variogram models. The spatial covariance between two curves is determined by the sum of the covariance between spatial score vectors associated, see Bohorquez, Giraldo and Mateu 2016 and Bohorquez, Giraldo and Mateu 2021. This covariance can be modeled using the usual packages for geostatistical analysis, such as geoR and gstat. Finally, the sill of the variogram model for each dimension is bounded for the respective eigenvalue. Actually, an adequate option is to use this eigenvalue as sill and estimate the rest of the parameters.

Author(s)

Diego Sandoval diasandovalsk@unal.edu.co.

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).

See Also

SpatFD

Examples

data(AirQualityBogota)
newcoorden=data.frame(X=seq(93000,105000,len=100),Y=seq(97000,112000,len=100))
# Building the SpatFD object
SFD_PM10 <- SpatFD(PM10, coords = coord[, -1], basis = "Bsplines", nbasis = 17,
norder=5, lambda = 0.00002, nharm=3)
scores(SFD_PM10)

Simulation of unconditional or conditional functional spatial process.

Description

Given a variogram model, this functions simulates several realizations of a functional spatial process. This simulation can be conditioned to observed data.

Usage

sim_functional_process(nsims,variograms,nbasis,coords,data = NULL,
                      data_coords = NULL,basis = NULL,mu = NULL,L = NULL)

Arguments

Details

When data is passed, conditional simulation is performed. That means that each simulated realization of the process interpolated the observed curves in data. If data is NULL, the realizations of the process are simulated without imterpolation restrictions.

Value

A list of nsims SpatFD objects each one with as much curves as points are in coords.

Author(s)

Samuel Sánchez Gutiérrez ssanchezgu@unal.edu.co.

References

Bohorquez, M., Giraldo, R. & Mateu, J. Multivariate functional random fields: prediction and optimal sampling. Stoch Environ Res Risk Assess 31, 53–70 (2017). https://doi.org/10.1007/s00477-016-1266-y

See Also

generate_basis

Examples

library(gstat)
library(fda)
library(sp)

data("CanadianWeather")
canada.CRS <- CRS("+init=epsg:4608")
coords <- SpatialPoints(CanadianWeather$coordinates,
                        proj4string = CRS("+init=epsg:4326"))
coords <- spTransform(coords,canada.CRS)
obs <- CanadianWeather$dailyAv[,,1] # Temperature

Lfd_obj <- int2Lfd(m = 2)
create.bspline.basis(rangeval = c(1,365),
                     nbasis = 40, norder = 4) -> mi.base
mi.fdPar <- fdPar(mi.base, Lfd_obj, lambda = 7.389)
mi.fd <- smooth.basis(argvals = 1:365,
                      y = obs, fdParobj = mi.fdPar)

nbasis <- 5
canada <- mi.fd$fd
canada.pca <- pca.fd(canada,nharm = 10)
base_ort <- canada.pca$harmonics[1:nbasis]
canada_mean <- canada.pca$meanfd

formula2fd <- function(rango, expresion) {
  # Generate grid
  n <- 500  # length of the grid
  x <- seq(rango[1], rango[2], length.out = n)
  
  # evaluate expression on the grid
  y_vals <- eval(parse(text = expresion))
  
  # convert to fd
  basis <- create.bspline.basis(rangeval = rango, nbasis = 30)
  fd_obj <- Data2fd(x, y_vals,basisobj = basis)
  
  return(fd_obj)
}
media <- formula2fd(c(-1,1),"3*sin(x*4)")

# No conditional
vario <- vgm(.25, "Exp", .5, .05)
nbasis <- 6
sims <- sim_functional_process(10,vario,nbasis,coords,basis = 'Legendre',mu = media)
class(sims)
length(sims)

class(sims[[1]])
# plot(sims[[3]][[1]]$data_fd)

sims <- sim_functional_process(10,vario,nbasis,coords,basis = 'Legendre')
class(sims)
length(sims)
class(sims[[1]])
# plot(sims[[3]][[1]]$data_fd)

# Conditional
vario <- vgm(100, "Exp", 900, 10)
new_coords <- spsample(coords,100,type = "regular")
gridded(new_coords) <- TRUE
length(new_coords)
a <- sim_functional_process(10,vario,nbasis,new_coords,canada,coords)
class(a)
length(a)
class(a[[1]])
#plot(a[[1]][[1]]$data_fd)

vario <- vgm(100, "Wav", 900, 10)
a <- sim_functional_process(10,vario,nbasis,new_coords,canada,coords)
class(a)
length(a)
class(a[[1]])
#plot(a[[1]][[1]]$data_fd)

Summary of COKS_pred objects

Description

This functions shows a summary of the main objects of COKS_pred objects.

Usage

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

Arguments

Value

This functions prints according to method computed: eigenvalues, variance of prediction and each of the models.

Author(s)

Joan Nicolás Castro jocastroc@unal.edu.co.

References

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).

See Also

COKS_scores_lambdas

Examples

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 <- COKS_scores_lambdas(SFD_PM10_NO2,newcoords,model1)
summary(coks)

Summary of KS_pred objects

Description

This functions shows a summary of the main objects of KS_pred objects.

Usage

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

Arguments

Value

This functions prints according to method computed: eigenvalues, variance of prediction and each of the models.

Author(s)

Joan Nicolás Castro jocastroc@unal.edu.co.

References

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).

See Also

KS_scores_lambdas

Examples

library(gstat)
data(AirQualityBogota)

newcoorden=data.frame(X=110000,Y=125000)

# Recibir los datos, suavizarlos y ACP
SFD_PM10 <- SpatFD(PM10, coords = coord[,2:3], basis = "Bsplines", nbasis = 17,
norder=5, lambda = 0.00002, nharm=3)

#Variogram model for each component
modelos <- list(vgm(psill = 2634000, "Exp", range = 2103.25, nugget =  0),
                vgm(psill = 101494.96, "Exp", range = 1484.57, nugget = 0),
                vgm(psill =53673, "Exp", range = 42406, nugget =  0))

#Genera los scores y los lambdas para predecir en nuevas coordenadas

#method = "lambda"
KS_SFD_PM10_l <- KS_scores_lambdas(SFD_PM10, newcoorden ,method = "lambda",
model = modelos)
summary(KS_SFD_PM10_l)

Summary of SpatFD objects

Description

This functions shows a summary of the main objects of SpatFD objects.

Usage

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

Arguments

Value

For each variable included in the SpatFd object, this functions return: Head of data, Coordinates, Eigenvalues, Mean coefficients, Proportion of explained variance by each component

Author(s)

Joan Nicolás Castro jocastroc@unal.edu.co.

References

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).

See Also

SpatFD

Examples

# Load data
data(AirQualityBogota)

# Create an univariate object using 2 nharm
SFD_PM10 <- SpatFD(PM10, coords = coord[,2:3], basis = "Bsplines", nbasis = 91,
lambda = 0.00002, nharm = 2)
summary(SFD_PM10)

Summary of gfdata objects

Description

This functions shows a summary of the main objects of gfdata objects.

Usage

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

Arguments

Value

For each variable included in the gfdata object, this functions return: Head of data, Coordinates, Eigenvalues, Mean coefficients, Proportion of explained variance by each component

Author(s)

Joan Nicolás Castro jocastroc@unal.edu.co, Venus Celeste Puertas vpuertasg@unal.edu.co

References

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).

See Also

gfdata

Examples

data(vowels)

#### Create parameters and names for the data.
p = 228 ; nelec = 21 ; nvow = 5
names_vowels = c("a","e","i","o","u")
n.basis<-c(14,13,12,13,11)
s4.gfdata = gfdata(data=vowels,p=p,names=names_vowels,coords=vowels_coords,nbasis=n.basis)
summary.gfdata(object=s4.gfdata)

Imaginary thinking of the five Spanish vowels

Description

Data consist of EEG signals taken from 21 electrodes from imaginary thinking of the five Spanish vowels, to be applied into a BCI for a hand prosthesis.

Usage

data(vowels)

Format

two data.frame of measurements 'vowels' and coordinates 'vowels_coords'.

Source

https://github.com/carlos-sarmientov/DATABASE-IMAGINED-VOWELS-1

References

Classification techniques for imaginary speech brain signal through spatial functional data https://repositorio.unal.edu.co/handle/unal/85267

Coordinates of electrodes from the vowels data set

Description

Coordinates of the electrodes from the vowels data set.

Usage

data(vowels)

Format

matrix of coordinates

Source

https://github.com/carlos-sarmientov/DATABASE-IMAGINED-VOWELS-1

References

Classification techniques for imaginary speech brain signal through spatial functional data https://repositorio.unal.edu.co/handle/unal/85267