### Background

In this post, the ninth of the geospatial processing series with raster data, I will focus on interpolating and modeling air surface temperature data recorded at weather stations. For this purpose I will explore **regression-kriging** (RK), a spatial prediction technique commonly used in geostatistics that combines a regression of the dependent variable (air temperature in this case) on auxiliary/predictive variables (e.g., elevation, distance from shoreline) with kriging of the regression residuals. RK is mathematically equivalent to the interpolation method variously called universal kriging and kriging with external drift, where auxiliary predictors are used directly to solve the kriging weights.

**Regression-kriging** is an implementation of the best linear unbiased predictor (BLUP) for spatial data, i.e. the best linear interpolator assuming the *universal model of spatial variation*. Hence, RK is capable of modeling the value of a target variable at some location as a sum of a deterministic component (handled by regression) and a stochastic component (kriging). In RK, both deterministic and stochastic components of spatial variation can be modeled separately. Once the deterministic part of variation has been estimated, the obtained residuals can be interpolated with kriging and added back to the estimated trend.

**Regression-kriging** is used in various fields, including meteorology, climatology, soil mapping, geological mapping, species distribution modeling and similar. The only requirement for using RK is that one or more covariates exist which are significantly correlated with the dependent variable.

Although powerful, RK can perform poorly if the point sample is small and non-representative of the target variable, if the relation between the target variable and predictors is non-linear (although some non-linear regression techniques can help on this aspect), or if the points do not represent feature space or represent only the central part of it.

Seven regression algorithms will be used and compared through cross-validation (10-fold CV):

- Interpolation:
- Ordinary Kriging (OK)

- Regression:
- Generalized Linear Model (GLM)
- Generalized Additive Model (GAM)
- Random Forest (RF)

- Regression-kriging:
- GLM + OK of residuals
- GAM + OK of residuals
- RF + OK of residuals

The sample data used for examples is the annual average air temperature for mainland Portugal which includes and summarizes daily records that range from 1950 to 2000. A total of 95 stations are available, unevenly dispersed throughout the country.

Four auxiliary variables were considered as candidates to model the variation of air temperature:

- Elevation (
*Elev*in meters a.s.l.), - Distance to the coastline (
*distCoast*in degrees); - Latitude (
*Lat*in degrees), and, - Longitude (
*Lon*in degrees).

One raster layer *per* predictive variable, with a spatial resolution of 0.009 deg (ca. 1000m) in WGS 1984 Geographic Coordinate System, is available for calculating a continuous surface of temperature values.

### Model development

#### 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))
```

StationName | StationID | Lat | Lon | Elev | AvgTemp | distCoast |
---|---|---|---|---|---|---|

Sagres | 1 | 36.98 | -8.95 | 40 | 16.3 | 0.0000000 |

Faro | 2 | 37.02 | -7.97 | 8 | 17.0 | 0.0201246 |

Quarteira | 3 | 37.07 | -8.10 | 4 | 16.6 | 0.0090000 |

Vila do Bispo | 4 | 37.08 | -8.88 | 115 | 16.1 | 0.0360000 |

Praia da Rocha | 5 | 37.12 | -8.53 | 19 | 16.7 | 0.0000000 |

Tavira | 6 | 37.12 | -7.65 | 25 | 16.9 | 0.0458912 |

S. Brás de Alportel | 7 | 37.17 | -7.90 | 240 | 15.9 | 0.1853213 |

Vila Real Sto. António | 8 | 37.18 | -7.42 | 7 | 17.1 | 0.0127279 |

Monchique | 9 | 37.32 | -8.55 | 465 | 15.0 | 0.1980000 |

Zambujeira | 10 | 37.50 | -8.75 | 106 | 15.0 | 0.0450000 |

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! 😄 👍 👍

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Harriet says

Hello,

Thank you for this article!!

I’m going through it using my own data it is going sucessfully untill this line of code

RF <- randomForest(y = climDataPT[idx, "AvgTemp"],

x = climDataPT[idx, c("Lat","Elev","distCoast")],

ntree = 500,

mtry = 2)

When implementing my own data I get the error message:

'Error in unique.default(y) : unique() applies only to vectors'

I don't really understand what this error means?

The data I'm inputting is the same format as yours, so I'm a bit stuck.

If you have any ideas, I would greatly appreciate them.

João Gonçalves says

Hi there! Do you still have this problem? Sorry, but with so few information and without a reproducible example with data and code it is difficult to provide a solution. Let me know how I can help further.

— cheers

Harriet says

Hello,

Yes still having the problem, I’ve added it here https://github.com/Hazzular/regression-krig/tree/master

You can find the code under regressio krig code.

I’m still a beginner so I was unable to get the raster files for you to access from the code, but the spatial point data is there, which is what is needed for the bit I’m stuck on.

Thanks

João Gonçalves says

Hi Harriet,

It seems that you are not entering the inputs correctly in the Random Forest function. You have to use the @data to access the data frame inside your ‘soil’ variable (which is a SpatialPointsDataFrame). So actually you were trying to use this kind of spatial object in random forest which will cause it to fail.

Change this:

“`

RF <- randomForest(y = soil@data[idx, "Na"],

x = soil@data[idx, c("slope", "Elevation", "Apspect")],

ntree = 500,

mtry = 2)

“`

Also I noticed that the data is not reading properly (some numeric/double variables are actually loading as factors or integers), so next you will have problems related with this. In short, you need to format your data properly to solve this issue.

— Cheers,

João

Harriet says

Hi João,

Thank you so much for the help! I’m almost there now I’ve ran all the code up to:

round(apply(evalData,2,FUN = function(x,…) c(mean(x,…),sd(x,…))),3)

My issue is that in the output evaldata matrix, some of the cross validations do not have values. https://github.com/Hazzular/regression-krig/blob/master/Screenshot_1.jpg Has something gone wrong?

Finallly with this line of code:

residKrigMap <- krige(formula = formMod ,

+ locations = statPointsTMP,

+ model = variogFitOLS,

+ newdata = rstPixDF)

I've been running it for about an hour now, is this unusual?

Thanks again

Harriet

Harriet says

Hello again,

I solved what was the issue, I’ll post it here just in case anyone else has a similar issue. It turned out I had two data points with the same coordinates. Once I got rid of the copy I was able to produce RMSE values for all the folds.

Harriet

João Gonçalves says

Thanks Harriet! 🙂 Please include a description of the problem and some link to the data.

— cheers