Preamble

In this preamble, we load the gstlearn library, and clean the workspace.

rm(list=ls())
library(gstlearn)
library(ggplot2)
library(ggpubr)
library(ggnewscale)

Then we download the data base dat.

fileNF = loadData("Scotland", "Scotland_Temperatures.NF")
dat = Db_createFromNF(fileNF)

We also create a Db object containing a grid covering the data points in the data base dat. To do so, we start by displaying the minimal and maximal coordinates of the points in dat using the getExtremas() method from the Db class.

dat$getExtremas()

## [[1]]
## [1]  78.2 460.7
## 
## [[2]]
## [1]  530.4 1208.9

The first (resp. second) element of the list contains the min and max coordinates in the first (resp. second) space dimension. Based on this information, we create a grid covering all the points using the DbGrid_create function. We specify the coordinates of the origin (i.e. lower left corner) of the grid (argument x0), the step sizes in each dimension (argument dx) and the number of points in each dimension (argument nx).

grid = DbGrid_create(x0=c(65,530),dx=c(4.94, 4.96),nx=c(82,138))

We then print a summary of the content of the grid using the display method of Db class, which we supply with a DbStringFormat object specifying that we would like information about the extent of the grid (argument flag_extend in the DbStringFormat_createFromFlags function).

dbfmt = DbStringFormat_createFromFlags(flag_extend=TRUE) 
grid$display(dbfmt)

## 
## Data Base Grid Characteristics
## ==============================
## 
## Data Base Summary
## -----------------
## File is organized as a regular grid
## Space dimension              = 2
## Number of Columns            = 3
## Total number of samples      = 11316
## 
## Grid characteristics:
## ---------------------
## Origin :     65.000   530.000
## Mesh   :      4.940     4.960
## Number :         82       138
## 
## Data Base Extension
## -------------------
## Coor #1 - Min =     65.000 - Max =    465.140 - Ext = 400.14
## Coor #2 - Min =    530.000 - Max =   1209.520 - Ext = 679.52
## 
## Variables
## ---------
## Column = 0 - Name = rank - Locator = NA
## Column = 1 - Name = x1 - Locator = x1
## Column = 2 - Name = x2 - Locator = x2

## NULL

Experimental Variogram and fitted Model

We compute the experimental variogram vario2dir (in 2 directions) (cf. Variography for more details).

varioParamMulti = VarioParam_createMultiple(ndir=2, npas=15, dpas=15.)
vario2dir = Vario(varioParamMulti)
err = vario2dir$compute(dat)

We then the fit a model fitmod

fitmod = Model()
types = ECov_fromKeys(c("NUGGET","EXPONENTIAL","GAUSSIAN"))
err = fitmod$fit(vario2dir,types=types)

ggplot() + plot.varmod(vario2dir, fitmod)

Simple kriging prediction

Kriging

Let suppose that

\(Z(s_i) = X_i\beta + Y(s_i)\)

where \(Y\) is a second order stationary random field with mean 0 and covariance function \(C\).

If \(Z\) is a vector of observations, we denote \(Z = X\beta + Y\) with \(\Sigma\) the covariance of Y

Simple Kriging

If \(\beta\) is known, we can obtain the simple kriging

\(Z_0^{SK} = X_0\beta + \Sigma_0^t\Sigma^{-1}(Z-X\beta) = X_0\beta + \lambda_{SK}^t(Z-X\beta)\)

with:

the simple kriging weights

\(\lambda_{SK}=\Sigma^{-1}\Sigma_0\)

the variance of the estimator

\(\textrm{Var}(Z_0^{SK})=\lambda_{SK}^t\Sigma\lambda_{SK}=\lambda_{SK}^t\Sigma_0\)

the estimation variance

\(\sigma_{SK}^2 = \textrm{Var}(Z_0-Z_0^{SK}) = \sigma_0^2-\Sigma_0^t\Sigma^{-1}\Sigma_0=\sigma_0^2-\lambda_{SK}^t\Sigma_0\)

In matrix notation:

Simple Kriging System

\[ \begin{bmatrix} \Sigma \end{bmatrix} \times \begin{bmatrix} \lambda_{SK} \end{bmatrix} = \begin{bmatrix} \Sigma_0 \end{bmatrix} \]

Estimation

\[ Z_0^{SK} = \begin{bmatrix} Z \end{bmatrix}^t \times \begin{bmatrix} \lambda_{SK} \end{bmatrix} + m ({ 1 - \sum{\lambda_{SK}}} ) \]

Variance of Estimation error

\[ \sigma_{SK}^2 = \sigma_0^2 - \begin{bmatrix} \lambda_{SK} \end{bmatrix}^t \times \begin{bmatrix} \Sigma_0 \end{bmatrix} \]

Variance of Estimator

\[ \textrm{Var}(Z_0^{SK}) = \begin{bmatrix} \lambda_{SK} \end{bmatrix}^t \times \begin{bmatrix} \Sigma \end{bmatrix} \times \begin{bmatrix} \lambda_{SK} \end{bmatrix} = \begin{bmatrix} \lambda_{SK} \end{bmatrix}^t \times \begin{bmatrix} \Sigma_0 \end{bmatrix} \]

To perform simple kriging, we use the function called kriging. We specify:

the Db object containing the data points (argument dbin) : the variable used for kriging is the (unique) variable of the data base with a z locator (i.e. it should have locator z1 and the other variables should not have a locator starting with z)
the Db object containing the target points, i.e. the points where the kriging predictor will be computed (argument dbout)
the Model object containing the model used to define the kriging predictor (argument model): in particular, the mean used to define the predictor is the one set in the Model object
the type of neighborhood used in the prediction (argument neigh), eg. unique neighborhood (to use all the data points for each predictor) or moving neighborhood (to use only the data points in the vicinity of the target point in the prediction). This argument is defined using a “neighborhood” object (see example below).

Additionally, it is possible to specify whether we wish to compute, at each target point, the kriging predictor (argument flag_est, default=TRUE), the kriging standard-deviation (argument flag_std, default=TRUE) and the kriging variance (argument flag_varz, default=FALSE).

The kriging function then adds new variables to the Db entered in the dbout argument corresponding to these variables. The names of these newly created variables will start by Kriging, but this prefix can be changed using the namconv argument of the kriging function

In the next example, we perform a simple kriging prediction (with unique neighborhood) of the variable January_temp in the dat data base, on the grid defined in the grid data base. To do so, we start by selecting the variable January_temp in the dat data base (i.e. we make ensure that it is the only variable with a z locator).

dat$setLocator("January_temp",ELoc_Z())

## NULL

dat

## 
## Data Base Characteristics
## =========================
## 
## Data Base Summary
## -----------------
## File is organized as a set of isolated points
## Space dimension              = 2
## Number of Columns            = 5
## Total number of samples      = 236
## 
## Variables
## ---------
## Column = 0 - Name = rank - Locator = NA
## Column = 1 - Name = Longitude - Locator = x1
## Column = 2 - Name = Latitude - Locator = x2
## Column = 3 - Name = Elevation - Locator = NA
## Column = 4 - Name = January_temp - Locator = z1

We then create a “neighborhood” object corresponding to the specification of a unique neighborhood: this is done using the NeighUnique function.

uniqueNeigh = NeighUnique()

We now call the kriging function to perform the kriging prediction. We use the model fitmod that we previously fitted on our data, require to compute the kriging predictor and its standard-deviation (but not its variance), and change the prefix of the newly created variables to “SK”.

err = kriging(dbin=dat, dbout=grid, model=fitmod, 
              neigh=uniqueNeigh,
              flag_est=TRUE, flag_std=TRUE, flag_varz=FALSE,
              namconv=NamingConvention("SK")
              )

We see that the kriging predictor and its standard deviation have been added to the grid data base.

grid

## 
## Data Base Grid Characteristics
## ==============================
## 
## Data Base Summary
## -----------------
## File is organized as a regular grid
## Space dimension              = 2
## Number of Columns            = 5
## Total number of samples      = 11316
## 
## Grid characteristics:
## ---------------------
## Origin :     65.000   530.000
## Mesh   :      4.940     4.960
## Number :         82       138
## 
## Variables
## ---------
## Column = 0 - Name = rank - Locator = NA
## Column = 1 - Name = x1 - Locator = x1
## Column = 2 - Name = x2 - Locator = x2
## Column = 3 - Name = SK.January_temp.estim - Locator = z1
## Column = 4 - Name = SK.January_temp.stdev - Locator = NA

Finally, we plot the kriging prediction over the grid using the plot.grid function and the data points.

p = ggDefaultGeographic()
p = p + plot.grid(grid,
                  flagLegendRaster = TRUE, palette="Spectral",legendNameRaster="°C") 
p = p + plot.point(dat,flagCst = T,pch=18,cex=1.5)
p = p + plot.decoration(title="Simple Kriging over whole Grid")
ggPrint(p)

By default, the plotting function plots the variable with locator z1, which in our case corresponds to the kriging predictor (as the kriging function automatically assigns the locator z1 to it). To plot another variable, we can simply specify their name.

For instance, we can plot the kriging standard deviation using the following code.

p = ggDefaultGeographic()
p = p + plot.grid(grid,nameRaster="SK.January_temp.stdev",
                  flagLegendRaster = TRUE, palette="Spectral",legendNameRaster="°C")
p = p + plot.point(dat,flagCst = T,pch=18,cex=1.5)
p = p + plot.decoration(title="Simple Kriging std-dev over whole Grid")
ggPrint(p)

As mentioned above, the mean used in the simple kriging predictor is the one set in the Model object supplied in the kriging function. By default, this mean is zero. It can be changed using the setMean method of the Model object.

For instance, considering the model fitmod previously fitted on the data, we can clone it (using the clone method), and assign it a new mean (equal to 4) as follows.

fitmodSK = fitmod$clone()
err = fitmodSK$setMean(mean=4)

Then, simple kriging is performed using the same command as before, but with the newly created model.

err = kriging(dbin=dat, dbout=grid, model=fitmodSK, 
              neigh=uniqueNeigh,
              flag_est=TRUE, flag_std=TRUE, flag_varz=FALSE,
              namconv=NamingConvention("Mean4_SK")
              )

Finally, we plot the new kriging prediction over the grid and the data points.

p = ggDefaultGeographic()
p = p + plot.grid(grid,
                  flagLegendRaster = TRUE, palette="Spectral",legendNameRaster="°C") 
p = p + plot.point(dat,flagCst = T,pch=18,cex=1.5)
p = p + plot.decoration(title="Simple Kriging over whole Grid: Mean=4")
ggPrint(p)

Ordinary kriging

In gstlearn, ordinary kriging is seen as the formulation of a kriging predictor on a model with a constant drift. Hence, to perform ordinary kriging, we use the kriging function with the same syntax as for simple kriging, but call it with a model that includes a constant drift. Adding a constant drift to a model is done with the addDrift() method.

Note: Removing a drift from a model can be done using the delDrift method (while specifying the index of the drift we wish to remove) or using the delAllDrifts() (to remove all the drifts at once).

Let us go back to our running example. Considering the model fitmod previously fitted on the data, we can clone it (using the clone method), and add a constant drift as follows.

fitmodOK = fitmod$clone()
err = fitmodOK$addDrift(DriftM())

Then, ordinary kriging is performed using the same command as before, but with the newly created model.

err = kriging(dbin=dat, dbout=grid, model=fitmodOK, 
              neigh=uniqueNeigh,
              flag_est=TRUE, flag_std=TRUE, flag_varz=FALSE,
              namconv=NamingConvention("OK")
              )

Finally, we plot the new kriging prediction over the grid and the data points.

p = ggDefaultGeographic()
p = p + plot.grid(grid,
                  flagLegendRaster = TRUE, palette="Spectral",legendNameRaster="°C") 
p = p + plot.point(dat,flagCst = T,pch=18,cex=1.5)
p = p + plot.decoration(title="Ordinary Kriging over whole Grid")
ggPrint(p)

We also plot the kriging standard deviations.

p = ggDefaultGeographic()
p = p + plot.grid(grid,nameRaster="OK.January_temp.stdev",
                  flagLegendRaster = TRUE, palette="Spectral",legendNameRaster="°C")
p = p + plot.point(dat,flagCst = T,pch=18,cex=1.5)
p = p + plot.decoration(title="Ordinary Kriging std-dev over whole Grid")
ggPrint(p)

Let us compare the results from the simple and ordinary kriging predictors. To do so, we create a correlation plot between the two predictors.

p = ggplot()
p = p + plot.correlation(grid,namex="OK.January_temp.estim",namey="SK.January_temp.estim", 
                     flagBiss=TRUE, flagSameAxes=TRUE, bins=100)
p = p + plot.decoration(title="Estimation Simple vs. Ordinary", 
                    xlab="Ordinary Kriging", ylab="Simple Kriging")
ggPrint(p)

We also compare the kriging standard-deviations obtained in both cases.

p = ggplot()
p = p + plot.correlation(grid,namex="OK.January_temp.stdev",namey="SK.January_temp.stdev", 
                     flagBiss=TRUE, flagSameAxes=TRUE, bins=100)
p = p + plot.decoration(title="St. dev. Simple vs. Ordinary", 
                    xlab="Ordinary Kriging", ylab="Simple Kriging")
ggPrint(p)

Working with selections

We now load new grid.

fileNF = loadData("Scotland", "Scotland_Elevations.NF")
grid = DbGrid_createFromNF(fileNF)
grid$display(dbfmt)

## 
## Data Base Grid Characteristics
## ==============================
## 
## Data Base Summary
## -----------------
## File is organized as a regular grid
## Space dimension              = 2
## Number of Columns            = 4
## Total number of samples      = 11097
## Number of active samples     = 3092
## 
## Grid characteristics:
## ---------------------
## Origin :     65.000   535.000
## Mesh   :      4.938     4.963
## Number :         81       137
## 
## Data Base Extension
## -------------------
## Coor #1 - Min =     65.000 - Max =    455.123 - Ext = 390.123
## Coor #2 - Min =    535.000 - Max =   1200.109 - Ext = 665.109
## 
## Variables
## ---------
## Column = 0 - Name = Longitude - Locator = x1
## Column = 1 - Name = Latitude - Locator = x2
## Column = 2 - Name = Elevation - Locator = f1
## Column = 3 - Name = inshore - Locator = sel

## NULL

This new grid data base contains a variable called inshore, with a locator sel. This indicates that this variable is a selection, i.e. a binary variable allowing to select some points, called active cells, in a data base (by setting them to 1, while the other points will be 0). Selections allow to restrict computations and operations made on a data base to only the active cells. For, when plotting a data base with a selection, only the active cells are represented.

p = ggDefaultGeographic()
p = p + plot.grid(grid,nameRaster="Longitude",
                  flagLegendRaster = TRUE, palette="Spectral",legendNameRaster="°C")
p = p + plot.point(dat,flagCst = T,pch=18,cex=1.5)
p = p + plot.decoration(title="Longitude values (restricted to the selection)")
ggPrint(p)

When calling the kriging function on a data base that contains a selection, the predictors are only computed on the active cells (while the other cells are left undefined).

err = kriging(dbin=dat, dbout=grid, model=fitmodOK, neigh=uniqueNeigh,
              flag_est=TRUE, flag_std=TRUE, flag_varz=FALSE,
              namconv=NamingConvention("OK"))

p = ggDefaultGeographic()
p = p + plot.grid(grid,nameRaster="OK*estim",
                  flagLegendRaster = TRUE, palette="Spectral",legendNameRaster="°C")
p = p + plot.point(dat,flagCst = T,pch=18,cex=1.5)
p = p + plot.decoration(title="Estimation by Ordinary Kriging (restricted to the selection)")
ggPrint(p)

Cross-validation

The function xvalid can be used to perform leave-one-out cross-validation with kriging: each point of the data base is separately predicted by kriging, using all the other points in the data base. This function is called in a similar way as the kriging function. We specify:

the Db object containing the data points on which we want to perform the cross-validation (argument dbin) : once again, the variable used for the study is the (unique) variable of the data base with a z locator
the Model object containing the model used to define the kriging predictor (argument model)
the type of neighborhood used in the prediction (argument neigh)
whether we wish to return, for each point, only the predictor (argument flag_xvalid_est=-1, returns a variable with name ending with “estim”), the cross-validation error defined as the difference between the predictor and the true value (argument flag_xvalid_est=1, returns a variable with name ending with “esterr”) or neither (argument flag_xvalid_est=0)
whether we wish to return, for each point, only the kriging standard-deviation at each point (argument flag_xvalid_std=-1, returns a variable with name ending with “stdev”), a standardized error defined as the ratio (Cross-validation error)/(Kriging standard-deviation) (argument flag_xvalid_std=1, returns a variable with name ending with “stderr”) or neither (argument flag_xvalid_std=0)

Going back to our running example, we perform a cross-validation on our data base dat. In particular we ask, at each point, for the cross-validation error and for the standardized error described above. We also specify, through the nameconv that we do not wish to modify the current locators in the data base (otherwise, the locator z1 is “moved” to the variable containing the cross-validation error).

err = xvalid(db=dat, model=fitmodOK, neigh=uniqueNeigh, 
             flag_xvalid_est=1, flag_xvalid_std=1,  
             namconv=NamingConvention_create("Xvalid", flag_locator = FALSE)
            )

We plot the histogram of cross-validation errors.

p = ggplot()
p = p + plot.hist(dat,name="*esterr*",bins=30,fill="blue")
p = p + plot.decoration(xlab="Estimation Errors", title="Cross-Validation")
ggPrint(p)

We plot the histogram of standardized errors.

p = ggplot()
p = p + plot.hist(dat,name="*stderr*",bins=30,fill="blue")
p = p + plot.decoration(xlab="Standardized Errors", title="Cross-Validation")
ggPrint(p)

Finally, we compute a few statistics about these errors.

print(c("Mean cross-validation error:",round(mean(dat$getColumn("*esterr*"),na.rm=TRUE),5)))

## [1] "Mean cross-validation error:" "-0.00417"

print(c("Mean squared cross-validation error:",round(mean(dat$getColumn("*esterr*")^2,na.rm=TRUE),5)))

## [1] "Mean squared cross-validation error:"
## [2] "0.23937"

print(c("Mean standardized error:",round(mean(dat$getColumn("*stderr*")^2,na.rm=TRUE),5)))

## [1] "Mean standardized error:" "0.91178"

We now plot the absolute value of the cross-validation errors at each point on top of the grid map. We use the argument flagAbsSize = TRUE in the plot.point function to specify that we want the size of the points to be proportional to the absolute value of the variable specified in the nameSize argument (here, the cross-validation errors).

p = ggDefaultGeographic()
p = p + plot.grid(grid,"inshore")
p = p + plot.point(dat,nameSize="*esterr",sizmax=3,flagAbsSize = TRUE)
p = p + plot.decoration(title="Cross-Validation scores")
ggPrint(p)

Kriging with moving neighborhood

Up until now, we only considered kriging with a unique neighborhood. To work with a moving neighborhood, we first need to define it by creating “neighborhood” object describing its characteristics. This is done using the NeighMoving_create function. We can specify:

a neighborhood radius (argument radius, default=1.234e30): note that the default radius value is taken very large so that the neighborhood radius can basically be seen as infinite if the argument is not set by the user
the minimum and maximum number of data points (within the specified radius) that should be included in the neighborhood (respectively through the arguments nmini and nmaxi): for a given target point, if the number of data points within the neighborhood radius is smaller that the specified minimum, then no prediction is performed at this target (it is set to undefined)

For instance, to design a small Moving Neighborhood with only 1 sample per neighborhood (irregardless of its distance to the target point), we use the following command:

smallNeigh = NeighMoving_create(nmini=1, nmaxi=1)

Then, (ordinary) kriging with moving neighborhood is performed using the same commands as before, but replacing the unique neighborhood in the neigh argument by our custom moving neighborhood object.

err = kriging(dbin=dat, dbout=grid, model=fitmodOK, neigh=smallNeigh,
              flag_est=TRUE, flag_std=TRUE, 
              namconv=NamingConvention("Small"))

p = ggDefaultGeographic()
p = p + plot.grid(grid,nameRaster="Small*estim",
                  flagLegendRaster = TRUE, palette="Spectral",legendNameRaster="°C")
p = p + plot.point(dat,flagCst = T,pch=18,cex=1.5)
p = p + plot.decoration(title="Estimation by Ordinary Kriging (Small Neigh.)")
ggPrint(p)

To create a moving neighborhood with radius 20 and containing between 1 and 10 points, we use the following command:

movingNeigh = NeighMoving_create(nmini=1, nmaxi=10, radius=20)

Running the Ordinary Kriging then gives:

err = kriging(dat,grid,fitmodOK,movingNeigh,
              flag_est=TRUE, flag_std=TRUE, 
              namconv=NamingConvention("Reduced"))

p = ggDefaultGeographic()
p = p + plot.grid(grid,nameRaster="Reduced*estim",
                  flagLegendRaster = TRUE, palette="Spectral",legendNameRaster="°C")
p = p + plot.point(dat,flagCst = T,pch=18,cex=1.5)
p = p + plot.decoration(title="Estimation by Ordinary Kriging (Reduced Moving Neigh.)")
ggPrint(p)

Note that some of the target sites are not predicted as no sample is found within their neighborhood.

Let us then consider a moving neighborhood with a bigger radius (150) and containing between 1 and 10 points.

movingNeigh = NeighMoving_create(nmini=1, nmaxi=10, radius=150)

Running the Ordinary Kriging then gives:

err = kriging(dat,grid,fitmodOK,movingNeigh,
              flag_est=TRUE, flag_std=TRUE, 
              namconv=NamingConvention("Moving"))

p = ggDefaultGeographic()
p = p + plot.grid(grid,nameRaster="Moving*estim",
                  flagLegendRaster = TRUE, palette="Spectral",legendNameRaster="°C")
p = p + plot.point(dat,flagCst = T,pch=18,cex=1.5)
p = p + plot.decoration(title="Estimation by Ordinary Kriging (Moving Neigh.)")
ggPrint(p)

And plotting the associated kriging standard-deviations gives:

p = ggDefaultGeographic()
p = p + plot.grid(grid,nameRaster="Moving*stdev",
                  flagLegendRaster = TRUE, palette="Spectral",legendNameRaster="°C")
p = p + plot.point(dat,flagCst = T,pch=18,cex=1.5)
p = p + plot.decoration(title="St. dev. by Ordinary Kriging (Moving Neigh.)")
ggPrint(p)

Finally, let us compare the results obtained with a unique and a moving neighborhood (through correlation plot). First, we compare the kriging predictions in both cases.

p = ggplot()
p = p + plot.correlation(grid,namex = "OK*estim",namey="Moving*estim", 
                         bins=100, flagDiag=TRUE)
p = p + plot.decoration(title="Ordinary Kriging Estimation", 
                    xlab="Unique", ylab="Moving")
ggPrint(p)

Then, we compare the kriging standard-deviations in both cases.

p = ggplot()
p = p + plot.correlation(grid,namex = "OK*stdev",namey="Moving*stdev", 
                         bins=100, flagDiag=TRUE)
p = p + plot.decoration(title="Ordinary Kriging St. Dev.", 
                    xlab="Unique", ylab="Moving")
ggPrint(p)

Tutorial on Estimation

gstlearn Team