Preamble

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

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

flagInternetAvailable=TRUE

Then the necessary data set is downloaded and named dat: the target variable is January_temp

fileNF = "Scotland_Temperatures.NF"
if(flagInternetAvailable){
  download.file(paste0("https://soft.minesparis.psl.eu/gstlearn/data/Scotland/",fileNF), fileNF)
}
dat = Db_createFromNF(fileNF)

Variogram Cloud

The data is modeled as samples of a regionalized variable \(z\), i.e. as evaluations at locations \(x_1,..,x_n\) of a variable \(z\) defined across a spatial domain: \[\lbrace z_i = z(x_i) : i = 1, ..., n\rbrace.\]

The variogram cloud is the set of pair of points defined as \[ \big\lbrace \big( \Vert x_i - x_j\Vert, \big\vert z(x_i)-z(x_j)\big\vert^2 \big) \quad\text{where}\quad 1\le i\le j\le n \big\rbrace \] In gstlearn, variogram clouds are computed as grids. We compute the variogram cloud of the dataset, i.e. 

varioParamOmni = VarioParam_createOmniDirection(npas = 100)
grid.cloud = db_variogram_cloud(dat, varioParamOmni)
grid.cloud$display()
## 
## Data Base Grid Characteristics
## ==============================
## 
## Data Base Summary
## -----------------
## File is organized as a regular grid
## Space dimension              = 2
## Number of Columns            = 4
## Maximum Number of UIDs       = 4
## Total number of samples      = 10000
## 
## Grid characteristics:
## ---------------------
## Origin :      0.000     0.000
## Mesh   :      7.789     0.068
## Number :        100       100
## 
## Variables
## ---------
## Column = 0 - Name = rank - Locator = NA
## Column = 1 - Name = x1 - Locator = x1
## Column = 2 - Name = x2 - Locator = x2
## Column = 3 - Name = Cloud.January_temp - Locator = NA
## NULL
p = ggplot()
p = p + plot.grid(grid.cloud, "Cloud.January*")
p = p + plot.geometry(asp=0)
ggPrint(p)

Experimental (isotropic) variograms

The experimental (isotropic) variogram \(\gamma\) is a function defined as

\[\gamma(h)=\frac{1}{2\vert N(h)\vert}\sum_{(i,j) \in N(h)}\big\vert z(x_i)-z(x_j)\big\vert^2, \quad h\ge 0,\] where \(N(h)\) is set of all pairs of data points separated by a distance \(h\) (called lag): \[ N(h) = \bigg\lbrace (i,j) : \Vert x_j-x_i\Vert = h\bigg\rbrace_{1\le i\le j\le n},\] and \(\vert N(h)\vert\) is the cardinal of \(N(h)\). In practice, when computing \(\gamma(h)\), we look for pairs of data points separated by a distance \(h \pm \tau h\) where \(\tau > 0\) is a tolerance on the separation distance \(h\). In other words, \(N(h)\) is replaced by \[ \widehat N(h) = \bigg\lbrace (i,j) : (1-\tau)h \le \Vert x_j-x_i\Vert \le (1+\tau) h\bigg\rbrace_{1\le i\le j\le n}\] To compute an experimental variogram, we start by creating a VarioParam object containing the parameters of the variogram. This is done using the function VarioParam_createOmniDirection. We can specify the number of lags \(h\) for which the experimental variogram is computed (argument npas), and the distance between these lags (argument dpas), as well as the tolerance \(\tau\) on the lags (argument toldis).

Then, the experimental variogram is computed in two steps. First, a Vario object is initialized from the VarioParam object and the Db containing the data points. Then, the values of the experimental variogram at the lags specified by the VarioParam object are computed using the method compute of the Vario object (which returns an error code, 0 meaning that no error was detected).

Note : The variable \(z\) for which we wish to define the experimental variogram should be the only variable in the Db with a z locator (i.e. it should have locator z1 and the other variables should not have a locator starting with z). This can be done bu using the method setLocator of the Db object containing the data. If several variables with z locators are present in the Db, then cross-variograms between are also computed (this subject will be covered in the course on multivariate analysis).

In the next example, we compute an experimental variogram with \(40\) lags separated by a distance \(10\) (meaning that we take \(h =10i\) for \(i=0, ..., 39\)), and consider a tolerance \(\tau = 10\%\) for the variogram computations. We use the Db dat, and select the variable January_temp as our variable of interest (by setting its locator to “z”).

varioParamOmni = VarioParam_createOmniDirection(npas=40, dpas=10,toldis = 0.1) 

dat$setLocator("January_temp",ELoc_Z())
varioexp = Vario(varioParamOmni, dat)
err = varioexp$compute()

We now print the contents of the newly created experimental variogram. The \(40\) experimental variogram values are displayed (Column Value), together with the number \(\vert \widehat N(h)\vert\) of pairs used to compute the value (Column Npairs) and the average distance between the points forming these pairs (Column Distance).

varioexp
## 
## Variogram characteristics
## =========================
## Number of variable(s)       = 1
## Number of direction(s)      = 1
## Space dimension             = 2
## Variance-Covariance Matrix     1.020
## 
## Direction #1
## ------------
## Number of lags              = 40
## Direction coefficients      =      1.000     0.000
## Direction angles (degrees)  =      0.000     0.000
## Tolerance on direction      =     90.000 (degrees)
## Calculation lag             =     10.000
## Tolerance on distance       =     10.000 (Percent of the lag value)
## 
## For variable 1
##       Rank    Npairs  Distance     Value
##          0     2.000     0.141     0.002
##          1    12.000     9.973     0.129
##          2    31.000    20.131     0.270
##          3    48.000    30.003     0.470
##          4    61.000    40.019     0.599
##          5    72.000    50.011     0.582
##          6    82.000    59.995     0.586
##          7    77.000    69.926     0.907
##          8    92.000    80.027     0.899
##          9    96.000    89.985     0.980
##         10    96.000   100.013     0.856
##         11    96.000   109.991     0.905
##         12    95.000   120.064     1.013
##         13    69.000   129.945     1.247
##         14   101.000   139.943     1.001
##         15    98.000   150.020     0.942
##         16    80.000   159.974     1.022
##         17    81.000   170.051     1.330
##         18    75.000   179.943     1.058
##         19    85.000   189.976     1.185
##         20    78.000   200.100     0.957
##         21    73.000   210.005     1.117
##         22    68.000   220.027     1.685
##         23    78.000   230.014     1.405
##         24    49.000   239.927     1.406
##         25    48.000   250.034     1.248
##         26    34.000   260.056     1.364
##         27    39.000   269.806     1.728
##         28    28.000   279.716     1.308
##         29    40.000   289.998     1.223
##         30    29.000   299.826     0.981
##         31    19.000   310.007     1.461
##         32    18.000   319.904     1.154
##         33    13.000   330.008     1.329
##         34    12.000   340.078     1.396
##         35    19.000   349.855     1.428
##         36    11.000   359.903     1.128
##         37    13.000   370.069     0.959
##         38     9.000   379.623     0.629
##         39     7.000   389.619     0.731

We now plot the experimental variogram. In the resulting figure, the experimental variogram is plotted in blue, and the dashed blacked line corresponds to the value of the variance of the data.

ggplot() + plot.varmod(varioexp)

We can also adapt the size of experimental variogram points in the plot so that it is proportional to the number of pairs of points used to compute the value.

ggplot() + plot.varmod(varioexp,draw_psize=TRUE)

Automatic Model Fitting

Fitting a variogram model on an experimental variogram is done in two steps. First, we create Model object. These objects aim at containing all the necessary information about the covariance structure of a random field. In particular, it is assumed that this covariance structure is a superposition of basic elementary covariance structures: the Model objects then contains the covariance types and parameters of each one of these basic covariance structures.

In our case, we wish to build our Model object from an experimental variogram, meaning that we want to find a composition of basic covariance structures which would result in a variogram “close” to the experimental variogram that we computed from the data. This is done by calling the method fit of the Model object, while providing it with the experimental variogram.

In the next example, we create a Model object, that we fit on the experimental variogram the we computed earlier. We then plot both the experimental variogram and the variogram model resulting from the fitting using the plot.varmod function. In the figure we obtain, In the figure above, the dashed blue line corresponds to the experimental variogram, and the solid blue line corresponds to the fitted variogram model.

fitmod = Model()
err = fitmod$fit(varioexp)
ggplot() + plot.varmod(varioexp, fitmod)

We now print the content of our newly created model. As we can see, only one basic covariance structure is used to define the model (namely, a Spherical covariance function whose range and sill are printed).

fitmod
## 
## Model characteristics
## =====================
## Space dimension              = 2
## Number of variable(s)        = 1
## Number of basic structure(s) = 1
## Number of drift function(s)  = 0
## Number of drift equation(s)  = 0
## 
## Covariance Part
## ---------------
## Spherical
## - Sill         =      1.123
## - Range        =    129.766
## Total Sill     =      1.123

Model Fitting with pre-defined basic structures

It is also possible to guide the model fitting by proposing a list of basic covariance structures from which the model is to be built. The list of available basic covariance structures is obtained by running the following command:

ECov_printAll()
##   -2 -     UNKNOWN : Unknown covariance
##   -1 -    FUNCTION : External covariance function
##    0 -      NUGGET : Nugget effect
##    1 - EXPONENTIAL : Exponential
##    2 -   SPHERICAL : Spherical
##    3 -    GAUSSIAN : Gaussian
##    4 -       CUBIC : Cubic
##    5 -     SINCARD : Sine Cardinal
##    6 -    BESSEL_J : Bessel J
##    7 -    BESSEL_K : Bessel K
##    8 -       GAMMA : Gamma
##    9 -      CAUCHY : Cauchy
##   10 -      STABLE : Stable
##   11 -      LINEAR : Linear
##   12 -       POWER : Power
##   13 -   ORDER1_GC : First Order Generalized covariance
##   14 -   SPLINE_GC : Spline Generalized covariance
##   15 -   ORDER3_GC : Third Order Generalized covariance
##   16 -   ORDER5_GC : Fifth Order Generalized covariance
##   17 -     COSINUS : Cosine
##   18 -    TRIANGLE : Triangle
##   19 -      COSEXP : Cosine Exponential
##   20 -       REG1D : 1-D Regular
##   21 -       PENTA : Pentamodel
##   22 -  SPLINE2_GC : Order-2 Spline
##   23 -     STORKEY : Storkey covariance in 1-D
##   24 -   WENDLAND0 : Wendland covariance (2,0)
##   25 -   WENDLAND1 : Wendland covariance (3,1)
##   26 -   WENDLAND2 : Wendland covariance (4,2)
##   27 -      MARKOV : Markovian covariances
## NULL

In practice, we start by creating a list of basic structures using the ECov_fromKeys function which we supply with a vector containing the names of the basic structures we would like to see in the model. To fit the model, we then once again call the fit method and supply it with both the experimental variogram and the newly created list of basic structures (argument types). Then the fitting procedures tries find the composition of models from the supplied list that best fits the experimental variogram.

Note that by default, the fitting algorithm tries to be parsimonious and can therefore “drop” some of the structures that we supply if it deems that a model with less structures provides a better fit. To force the fitting algorithm to keep all the structures from the list, we simply need to add the argument optvar=Option_VarioFit(TRUE) to the fit method.

In the next example, we once again define a model by fitting it on our experimental variogram. But this time, we specify that we want the resulting model to be a composition of basic structures restricted to these choices: a Nugget effect, a Cubic covariance and a Spherical covariance.

types = ECov_fromKeys(c("NUGGET","CUBIC","SPHERICAL"))
err = fitmod$fit(varioexp, types=types)
ggplot() + plot.varmod(varioexp, fitmod)

When printing the contents of the model, we now notice that it consists of a superposition of a Cubic covariance and a Spherical covariance, as intended. Note that the Nugget effect does not appear (it has been dropped by the fitting algorithm).

fitmod
## 
## Model characteristics
## =====================
## Space dimension              = 2
## Number of variable(s)        = 1
## Number of basic structure(s) = 2
## Number of drift function(s)  = 0
## Number of drift equation(s)  = 0
## 
## Covariance Part
## ---------------
## Cubic
## - Sill         =      0.371
## - Range        =     58.088
## Spherical
## - Sill         =      0.904
## - Range        =    237.071
## Total Sill     =      1.275

Model Fitting with constraints

It is possible to impose (in)equality constraints on the covariance parameters of the basic structures used in the model fitting procedure. This is done by creating a Constraints object that is used to specify the constraints we wish to impose on the parameters of the different basic structures composing the model. To add a constraint to the object, we can use the method addItemFromParamId, which takes as arguments the type of parameter for which the constraint applies (given as an EConsElem object: run EConsElem_printAll() for the list of available options), the index of the basic structure for which the constraint applies (argument icov), the type of constraint we wish to apply (argument type, given as an EConsType object: run EConsType_printAll() for the list of available options) and finally the numerical value (argument value) defining the constraint.

In the next example, we start from a list of three basic structures (a Nugget effect, a Cubic covariance and a Spherical covariance), and create a Constraints object containing two constraints. The first one applies to the basic structure of index \(1\) (the cubic structure), and sets an upper-bound of \(20\) for its range. The second one also applies to the basic structure of index \(1\) (the cubic structure), and sets an lower-bound of \(0.03\) for its sill. Finally, the fit method is called to fit the model on the experimental variogram. Note that we also added the option optvar=Option_VarioFit(TRUE) to force the fitting algorithm to keep the three basic structures that we supplied.

types = ECov_fromKeys(c("NUGGET","CUBIC","SPHERICAL"))
constraints = Constraints()
err = constraints$addItemFromParamId(EConsElem_RANGE(),icov=1,type=EConsType_UPPER(),value=20.)
err = constraints$addItemFromParamId(EConsElem_SILL(),icov=1,type=EConsType_LOWER(),value=0.03)
err = fitmod$fit(varioexp, types=types, constraints=constraints, optvar=Option_VarioFit(TRUE))
ggplot() + plot.varmod(varioexp, fitmod)

When printing the content of the fitted model, we see that the constraints are indeed satisfied (and that the three basic structures are present).

fitmod
## 
## Model characteristics
## =====================
## Space dimension              = 2
## Number of variable(s)        = 1
## Number of basic structure(s) = 3
## Number of drift function(s)  = 0
## Number of drift equation(s)  = 0
## 
## Covariance Part
## ---------------
## Nugget Effect
## - Sill         =      0.001
## Cubic
## - Sill         =      0.115
## - Range        =     20.000
## Spherical
## - Sill         =      0.989
## - Range        =    144.544
## Total Sill     =      1.104

In the following example, we now apply equality constraints to the parameters. The first one applies to the basic structure of index \(1\) (the cubic structure), and sets its range to the value \(1000\). The second one also applies to the basic structure of index \(1\) (the cubic structure), and sets its sill to the value \(0.4\).

constraints = Constraints()
err = constraints$addItemFromParamId(EConsElem_RANGE(),icov=1,type=EConsType_EQUAL(),value=1000.)
err = constraints$addItemFromParamId(EConsElem_SILL(),icov=1,type=EConsType_EQUAL(),value=0.4)
err = fitmod$fit(varioexp, types=types, constraints=constraints, optvar=Option_VarioFit(TRUE))
ggplot() + plot.varmod(varioexp, fitmod)

When printing the content of the fitted model, we see that the constraints are once again satisfied (and that the three basic structures are present).

fitmod
## 
## Model characteristics
## =====================
## Space dimension              = 2
## Number of variable(s)        = 1
## Number of basic structure(s) = 3
## Number of drift function(s)  = 0
## Number of drift equation(s)  = 0
## 
## Covariance Part
## ---------------
## Nugget Effect
## - Sill         =      0.001
## Cubic
## - Sill         =      0.400
## - Range        =   1000.000
## Spherical
## - Sill         =      0.994
## - Range        =    112.870
## Total Sill     =      1.395

Directional Variograms

The experimental directional variogram \(\gamma\) is a function defined as \[\gamma(\theta,h)=\frac{1}{2\vert N(\theta, h)\vert}\sum_{(i,j) \in N(\theta, h)}\big\vert z(x_i)-z(x_j)\big\vert^2, \quad 0^{\circ}\le \theta <360^{\circ}, \quad h\ge 0\] where \(N(\theta, h)\) is set of all pairs of data points separated by a vector of size \(h\) and along the direction \(\theta\) (in degrees): \[ N(\theta, h) = \bigg\lbrace (i,j) : \Vert x_j-x_i\Vert = h \quad\text{and the vector } \vec{u}=(x_j-x_i) \text{ is along the direction } \theta\bigg\rbrace_{1\le i\le j\le n},\] In practice, when computing \(\gamma(\theta, h)\), we once gain consider a tolerance \(\tau\) on the separation distance \(h\), and also consider a tolerance \(\eta>0\) is also considered for the direction angle. In other words, \(N(h)\) is replaced by \[\widehat N(\theta, h) = \bigg\lbrace (i,j) : (1-\tau)h \le \Vert x_j-x_i\Vert \le (1+\tau) h \quad\text{and the vector } \vec{u}=(x_j-x_i) \text{ is along the direction } \theta \pm \eta \bigg\rbrace_{1\le i\le j\le n},\]

Much like their isotropic counterparts, experimental directional variograms are computed as Vario objects, which can be created from he VarioParam object (containing the parameters of the variogram) and a Db containing the data points.

This time, the VarioParam object is created using the function VarioParam_createMultiple. There, we specify the number \(K\) of directions \(\theta\) for which we wish to compute the an experimental variogram (argument ndir), as well as the reference angle \(\theta_0\) of the first direction (argument angref, default = \(0\)) so that the directions \(\theta\) = \(\theta_0 + i(180/K)\) for \(i=0,..., K-1\) are considered. We can also specify the number of lags \(h\) for which the experimental variogram is computed (argument npas), and the distance between these lags (argument npas), as well as the tolerance \(\tau\) on the lags (argument toldis). Then, the experimental variogram is computed just as in the isotropic case.

Note: When initializing the VarioParam object as described above, the angle tolerance \(\eta\) is automatically set to \(\eta = (90/K)\), meaning that we span the set of possible directions.

In the following example, we create an experimental variogram in the \(4\) directions \(\theta = 0^{\circ}, 45^{\circ}, 90^{\circ}, 135^{\circ}\).

varioParamMulti = VarioParam_createMultiple(ndir=4, npas=15, dpas=15.)
vario.4dir = Vario(varioParamMulti, dat)
err = vario.4dir$compute()
ggplot() + plot.varmod(vario.4dir)

Then, fitting a model onto the resulting experimental variogram is done using the same commands as in the isotropic case.

model.4dir = Model()
err = model.4dir$fit(vario.4dir,types=types)
ggplot() + plot.varmod(vario.4dir, model.4dir)

Variogram Maps

Experimental variogram map

The experimental variogram map is a map centered at the origin, which represents the value of experimental directional variogram across all directions \(0^{\circ} \le \theta< 360^{\circ}\).

To compute an experimental variogram map, we use the function db_vmap_compute which we supply with the Db containing the data. The output is a Db containing a grid representing the variogram map values.

grid.vmap = db_vmap_compute(dat)
p1 = ggplot() + plot.grid(grid.vmap, show.legend.raster=TRUE, legend.name.raster="T°C Var")
p2 = ggplot() + plot.grid(grid.vmap, name_raster="VMAP.January_temp.Nb", show.legend.raster=TRUE, legend.name.raster="# pairs")
ggarrange(p1,p2,nrow=1,ncol=2)

It is then possible to fit a model directly on the experimental variogram map. This if done with the method fitFromVMap from the Model class. This method is called in the same way as the fit method considered up until now (the experimental variograms being now replaced by the experimental variogram map).

modelVM = Model()
err = modelVM$fitFromVMap(grid.vmap, types=types)
modelVM
## 
## Model characteristics
## =====================
## Space dimension              = 2
## Number of variable(s)        = 1
## Number of basic structure(s) = 2
## Number of drift function(s)  = 0
## Number of drift equation(s)  = 0
## 
## Covariance Part
## ---------------
## Nugget Effect
## - Sill         =      0.257
## Cubic
## - Sill         =      0.943
## - Ranges       =    158.007   214.374
## - Angles       =    334.704     0.000
## - Rotation Matrix
##                [,  0]    [,  1]
##      [  0,]     0.904     0.427
##      [  1,]    -0.427     0.904
## Total Sill     =      1.200

Fitting of variogram map

It is then possible to plot the variogram map resulting from the fitted model. To do so, we start by evaluating the fitted variogram model on the the experimental variogram map grid. This is done using the function dbgrid_model which we supply with both the experimental variogram map and the fitted model. This function adds a additional variable to the Db containing the experimental variogram map corresponding to the evaluations of the variogram model.

err = dbgrid_model(grid.vmap, modelVM)
grid.vmap
## 
## Data Base Grid Characteristics
## ==============================
## 
## Data Base Summary
## -----------------
## File is organized as a regular grid
## Space dimension              = 2
## Number of Columns            = 6
## Maximum Number of UIDs       = 6
## Total number of samples      = 1681
## 
## Grid characteristics:
## ---------------------
## Origin :   -382.500  -678.500
## Mesh   :     19.125    33.925
## Number :         41        41
## 
## Variables
## ---------
## Column = 0 - Name = rank - Locator = NA
## Column = 1 - Name = x1 - Locator = x1
## Column = 2 - Name = x2 - Locator = x2
## Column = 3 - Name = VMAP.January_temp.Var - Locator = NA
## Column = 4 - Name = VMAP.January_temp.Nb - Locator = NA
## Column = 5 - Name = VMAP.Model - Locator = z1
p1 = ggplot() + plot.grid(grid.vmap, name_raster="VMAP.January_temp.Var")
p2 = ggplot() + plot.grid(grid.vmap, name_raster="VMAP.Model")
ggarrange(p1,p2,nrow=1,ncol=2)

Finally, we plot together the experimental directional variograms and the model obtained from fitting the variogram map.

ggplot() + plot.varmod(vario.4dir, modelVM)