Data loading and inspection

We will start by downloading and unzipping the sample data from the GitHub repository:

## Create a folder named data-raw inside the working directory to place downloaded data
if(!dir.exists("./data-raw")) dir.create("./data-raw")

## If you run into download problems try changing: method = "wget"
download.file("https://raw.githubusercontent.com/joaofgoncalves/R_exercises_raster_tutorial/master/data/CLIM_DATA_PT.zip", "./data-raw/CLIM_DATA_PT.zip", method = "auto")

## Uncompress the zip file
unzip("./data-raw/CLIM_DATA_PT.zip", exdir = "./data-raw")

Now, let’s load the raster layers containing the predictive variables used to build the regression model of air temperature:

library(raster)

# GeoTIFF file list
fl <- list.files("./data-raw/climData/rst", pattern = ".tif$", full.names = TRUE)

# Create the raster stack
rst <- stack(fl)

# Change the layer names to coincide with table data
names(rst) <- c("distCoast", "Elev", "Lat", "Lon")
plot(rst)

Next step, let’s read the point data containing annual average temperature values along with location and predictive variables for each weather station:

climDataPT <- read.csv("./data-raw/ClimData/clim_data_pt.csv")

knitr::kable(head(climDataPT, n=10))

Based on the previous data, create a SpatialPointsDataFrame object to store all points and make some preliminary plots:

proj4Str <- "+proj=longlat +ellps=WGS84 +datum=WGS84 +no_defs"

statPoints <- SpatialPointsDataFrame(coords      = climDataPT[,c("Lon","Lat")], 
                                     data        = climDataPT,
                                     proj4string = CRS(proj4Str))
par(mfrow=c(1,2),mar=c(5,6,3,2))

plot(rst[["Elev"]], main="Elevation (meters a.s.l.) for Portugal\n and weather stations",
     xlab = "Longitude", ylab="Latitude")
plot(statPoints, add=TRUE)

hist(climDataPT$AvgTemp, xlab= "Temperature (ºC)", main="Annual avg. temperature")

From the figure we can see that: (i) weather stations tend to cover more the areas close to the coastline and with lower altitude, and, (ii) temperature values are ‘left-skewed’ with a median equal to 15 and a median-absolute deviation (MAD) of 15.

Before proceeding, it is a good idea to inspect the correlation matrix to analyze the strength of association between the response and the predictive variables. For this, we will use the package corrplot with some nit graphical options

library(corrplot)

corMat <- cor(climDataPT[,3:ncol(climDataPT)])

corrplot.mixed(corMat, number.cex=0.8, tl.cex = 0.9, tl.col = "black", 
               outline=FALSE, mar=c(0,0,2,2), upper="square", bg=NA)

The correlation plot evidence that all predictive variables seem to be correlated with the average temperature, especially ‘Elevation’ and ‘Latitude’ which are well-known regional controls of temperature variation. It also shows that (as expected, given the country geometric shape) both ‘Longitude’ and ‘Distance to the coast’ are highly correlated. As such, given that ‘Longitude’ is less associated with temperature and its climatic effect is less “direct” (compared to ‘distCoast’) we will remove it.

Regression-kriging and model comparison

For comparing the different RK algorithms, we will use 10-fold cross-validation and the Root-mean-square error as the evaluation metric.

Kriging parameters nugget, (partial) sill and range will be fit through Ordinary Least Squares (OLS) from a set of previously defined values that were adjusted with the help of some visual inspection and trial-and-error. The Exponential model was selected since it gave generally best results in preliminary analyses.

The functionalities in package gstat were used for all geostatistical analyses.

Now, let’s define some ancillary functions for creating the k-fold train/test data splits and for obtaining the regression residuals out of a random forest object:

# Generate the K-fold train--test splits
# x are the row indices
# Outputs a list with test (or train) indices
kfoldSplit <- function(x, k=10, train=TRUE){
  x <- sample(x, size = length(x), replace = FALSE)
  out <- suppressWarnings(split(x, factor(1:k)))
  if(train) out <- lapply(out, FUN = function(x, len) (1:len)[-x], len=length(unlist(out)))
  return(out)
}

# Regression residuals from RF object
resid.RF <- function(x) return(x$y - x$predicted)

We also need to define some additional parameters, get the test/train splits with the function kfoldSplit and initialize the matrix that will store all RMSE values (one for each training round and modelling technique; evalData object).

set.seed(12345)

k <- 10

kfolds <- kfoldSplit(1:nrow(climDataPT), k = 10, train = TRUE)

evalData <- matrix(NA, nrow=k, ncol=7, 
                   dimnames = list(1:k, c("OK","RF","GLM","GAM","RF_OK","GLM_OK","GAM_OK")))

Now we are ready to start modelling! One code block, inside the ‘for’ loop, will be used for each regression algorithm tested. Notice how (train) residuals are interpolated through kriging and then (test) residuals are added to (test) regression results for evaluation. Use the comments to guide you through the code.

library(randomForest)
library(mgcv)
library(gstat)


for(i in 1:k){
  
  cat("K-fold...",i,"of",k,"....\n")
  
  # TRAIN indices as integer
  idx <- kfolds[[i]]
  
  # TRAIN indices as a boolean vector
  idxBool <- (1:nrow(climDataPT)) %in% idx
  
  # Observed test data for the target variable
  obs.test <- climDataPT[!idxBool, "AvgTemp"]
  
  
  
  ## ----------------------------------------------------------------------------- ##
  ## Ordinary Kriging ----
  ## ----------------------------------------------------------------------------- ##
    
  # Make variogram
  formMod <- AvgTemp ~ 1
  mod <- vgm(model  = "Exp", psill  = 3, range  = 100, nugget = 0.5)
  variog <- variogram(formMod, statPoints[idxBool, ])
  
  # Variogram fitting by Ordinary Least Sqaure
  variogFitOLS<-fit.variogram(variog, model = mod,  fit.method = 6)
  #plot(variog, variogFitOLS, main="OLS Model")
    
  # kriging predictions
  OK <- krige(formula = formMod ,
              locations = statPoints[idxBool, ], 
              model = variogFitOLS,
              newdata = statPoints[!idxBool, ],
              debug.level = 0)
  
  ok.pred.test <- OK@data$var1.pred
  evalData[i,"OK"] <- sqrt(mean((ok.pred.test - obs.test)^2))
  
  
  
  ## ----------------------------------------------------------------------------- ##
  ## RF calibration ----
  ## ----------------------------------------------------------------------------- ##
  
  RF <- randomForest(y = climDataPT[idx, "AvgTemp"], 
                     x = climDataPT[idx, c("Lat","Elev","distCoast")],
                     ntree = 500,
                     mtry = 2)
  
  rf.pred.test <- predict(RF, newdata = climDataPT[-idx,], type="response")
  evalData[i,"RF"] <- sqrt(mean((rf.pred.test - obs.test)^2))
  
  # Ordinary Kriging of Random Forest residuals
  #
  statPointsTMP <- statPoints[idxBool, ]
  statPointsTMP@data <- cbind(statPointsTMP@data, residRF = resid.RF(RF))
  
  formMod <- residRF ~ 1
  mod <- vgm(model  = "Exp", psill  = 0.6, range  = 10, nugget = 0.01)
  variog <- variogram(formMod, statPointsTMP)
  
  # Variogram fitting by Ordinary Least Sqaure
  variogFitOLS<-fit.variogram(variog, model = mod,  fit.method = 6)
  #plot(variog, variogFitOLS, main="OLS Model")
    
  # kriging predictions
  RF.OK <- krige(formula = formMod ,
              locations = statPointsTMP, 
              model = variogFitOLS,
              newdata = statPoints[!idxBool, ],
              debug.level = 0)
  
  rf.ok.pred.test <- rf.pred.test + RF.OK@data$var1.pred
  evalData[i,"RF_OK"] <- sqrt(mean((rf.ok.pred.test - obs.test)^2))
  
  
  
  ## ----------------------------------------------------------------------------- ##
  ## GLM calibration ----
  ## ----------------------------------------------------------------------------- ##

  GLM <- glm(formula = AvgTemp ~ Elev + Lat + distCoast, data = climDataPT[idx, ])
  
  glm.pred.test <- predict(GLM, newdata = climDataPT[-idx,], type="response")
  evalData[i,"GLM"] <- sqrt(mean((glm.pred.test - obs.test)^2))
  
  # Ordinary Kriging of GLM residuals
  #
  statPointsTMP <- statPoints[idxBool, ]
  statPointsTMP@data <- cbind(statPointsTMP@data, residGLM = resid(GLM))
  
  formMod <- residGLM ~ 1
  mod <- vgm(model  = "Exp", psill  = 0.4, range  = 10, nugget = 0.01)
  variog <- variogram(formMod, statPointsTMP)
  
  # Variogram fitting by Ordinary Least Sqaure
  variogFitOLS<-fit.variogram(variog, model = mod,  fit.method = 6)
  #plot(variog, variogFitOLS, main="OLS Model")
    
  # kriging predictions
  GLM.OK <- krige(formula = formMod ,
              locations = statPointsTMP, 
              model = variogFitOLS,
              newdata = statPoints[!idxBool, ],
              debug.level = 0)
  
  glm.ok.pred.test <- glm.pred.test + GLM.OK@data$var1.pred
  evalData[i,"GLM_OK"] <- sqrt(mean((glm.ok.pred.test - obs.test)^2))
  
  
  
  ## ----------------------------------------------------------------------------- ##
  ## GAM calibration ----
  ## ----------------------------------------------------------------------------- ##
  
  GAM <- gam(formula = AvgTemp ~ s(Elev) + s(Lat) + s(distCoast), data = climDataPT[idx, ])
  
  gam.pred.test <- predict(GAM, newdata = climDataPT[-idx,], type="response")
  evalData[i,"GAM"] <- sqrt(mean((gam.pred.test - obs.test)^2))
 
  # Ordinary Kriging of GAM residuals
  #
  statPointsTMP <- statPoints[idxBool, ]
  statPointsTMP@data <- cbind(statPointsTMP@data, residGAM = resid(GAM))
  
  formMod <- residGAM ~ 1
  mod <- vgm(model  = "Exp", psill  = 0.3, range  = 10, nugget = 0.01)
  variog <- variogram(formMod, statPointsTMP)
  
  # Variogram fitting by Ordinary Least Sqaure
  variogFitOLS<-fit.variogram(variog, model = mod,  fit.method = 6)
  #plot(variog, variogFitOLS, main="OLS Model")
    
  # kriging predictions
  GAM.OK <- krige(formula = formMod ,
              locations = statPointsTMP, 
              model = variogFitOLS,
              newdata = statPoints[!idxBool, ],
              debug.level = 0)
  
  gam.ok.pred.test <- gam.pred.test + GAM.OK@data$var1.pred
  evalData[i,"GAM_OK"] <- sqrt(mean((gam.ok.pred.test - obs.test)^2))
  
}
## K-fold... 1 of 10 ....
## K-fold... 2 of 10 ....
## K-fold... 3 of 10 ....
## K-fold... 4 of 10 ....
## K-fold... 5 of 10 ....
## K-fold... 6 of 10 ....
## K-fold... 7 of 10 ....
## K-fold... 8 of 10 ....
## K-fold... 9 of 10 ....
## K-fold... 10 of 10 ....

Let’s check the average and st.-dev. results for the 10-folds CV:

round(apply(evalData,2,FUN = function(x,...) c(mean(x,...),sd(x,...))),3)
##         OK    RF   GLM   GAM RF_OK GLM_OK GAM_OK
## [1,] 1.193 0.678 0.598 0.569 0.613  0.551  0.521
## [2,] 0.382 0.126 0.195 0.186 0.133  0.179  0.163

From the results above we can see that RK performed generally better than the regression techniques alone or than Ordinary Kriging. The GAM-based RK method obtained the best scores with an RMSE of ca. 0.521. These are pretty good results!!

To finalize, we will predict the temperature values for the entire surface of mainland Portugal based on GAM-based Regression Kriging, which was the best performing technique on the test. For this we will not use any test/train partition but the entire dataset:

GAM <- gam(formula = AvgTemp ~ s(Elev) + s(Lat) + s(distCoast), data = climDataPT)
  
rstPredGAM <- predict(rst, GAM, type="response")

Next, we need to obtain a surface with kriging-interpolated residuals. For that, we have to convert the input RasterStack or RasterLayer into a SpatialPixelsDataFrame so that the krige function can use it as a reference:

rstPixDF <- as(rst[[1]], "SpatialPixelsDataFrame")

Like before, we will interpolate the regression residuals with kriging and add them back to the regression results.

# Create a temporary SpatialPointsDF object to store GAM residuals
statPointsTMP <- statPoints
crs(statPointsTMP) <- crs(rstPixDF)
statPointsTMP@data <- cbind(statPointsTMP@data, residGAM = resid(GAM))

# Define the kriging parameters and fit the variogram using OLS
formMod <- residGAM ~ 1
mod <- vgm(model  = "Exp", psill  = 0.15, range  = 10, nugget = 0.01)
variog <- variogram(formMod, statPointsTMP)
variogFitOLS <- fit.variogram(variog, model = mod,  fit.method = 6)

# Plot the results
plot(variog, variogFitOLS, main="Semi-variogram of GAM residuals")

The exponential semi-variogram looks reasonable although some lack-of-convergence problems…

Finally, let’s check the average temperature map obtained from GAM RK:

residKrigMap <- krige(formula = formMod ,
                      locations = statPointsTMP, 
                      model = variogFitOLS,
                      newdata = rstPixDF)

residKrigRstLayer <- as(residKrigMap, "RasterLayer")

gamKrigMap <- rstPredGAM + residKrigRstLayer

plot(gamKrigMap, main="Annual average air temperature\n(GAM regression-kriging)",
     xlab="Longitude", ylab="Latitude", cex.main=0.8, cex.axis=0.7, cex=0.8)

This concludes our exploration of the raster package and regression kriging for this post. Hope you find it useful!