Tutorial 2D
D. Renard
25 juin 2022
Global variables
verbose = TRUE
graphics = TRUE
OptCst_define(ECst_NTCOL(),6)## NULLReading data
The data are stored in a CSV format in the file called Pollution.dat
At this stage: - it is not possible to pass the list of variable names c(“X”,“Y”). - to pass the FLAGS for DbStringFormat (they are not an ENUM)
dlfile = "https://soft.minesparis.psl.eu/gstlearn/data/Pollution/Pollution.dat"
filepath = "Pollution.dat"
download.file(dlfile, filepath)
mydb = Db_createFromCSV(filepath,CSVformat())
err = mydb$setLocator("X",ELoc_X(),0)
err = mydb$setLocator("Y",ELoc_X(),1)
err = mydb$setLocator("Zn",ELoc_Z())
if (verbose)
{
dbfmt = DbStringFormat()
dbfmt$setFlags(flag_extend = TRUE)
mydb$display(dbfmt)
}##
## Data Base Characteristics
## =========================
##
## Data Base Summary
## -----------------
## File is organized as a set of isolated points
## Space dimension = 2
## Number of Columns = 5
## Maximum Number of UIDs = 5
## Total number of samples = 102
##
## Data Base Extension
## -------------------
## Coor #1 - Min = 109.850 - Max = 143.010 - Ext = 33.16
## Coor #2 - Min = 483.660 - Max = 513.040 - Ext = 29.38
##
## Variables
## ---------
## Column = 0 - Name = rank - Locator = NA
## Column = 1 - Name = X - Locator = x1
## Column = 2 - Name = Y - Locator = x2
## Column = 3 - Name = Zn - Locator = z1
## Column = 4 - Name = Pb - Locator = p1## NULLAccessing to the variable names
cat("List of all variable names =",mydb$getAllNames())## List of all variable names = rank X Y Zn PbExtracting the vector containing the Zn variable in order to perform a selection
tabZn = mydb$getColumn("Zn")
selZn = as.numeric(tabZn < 20)
mydb$addSelection(selZn,"sel")## [1] 5mydb$setLocator('Pb',ELoc_Z())## NULLif (verbose)
mydb$display()##
## Data Base Characteristics
## =========================
##
## Data Base Summary
## -----------------
## File is organized as a set of isolated points
## Space dimension = 2
## Number of Columns = 6
## Maximum Number of UIDs = 6
## Total number of samples = 102
## Number of active samples = 99
##
## Variables
## ---------
## Column = 0 - Name = rank - Locator = NA
## Column = 1 - Name = X - Locator = x1
## Column = 2 - Name = Y - Locator = x2
## Column = 3 - Name = Zn - Locator = NA
## Column = 4 - Name = Pb - Locator = z1
## Column = 5 - Name = sel - Locator = sel## NULLDisplay my Data (with samples represented by color and size)
if (graphics)
ggplot() + plot(mydb,name_color="Pb") + plot.decoration(title="Data Set")
Variograms
We first define the geometry of the variogram calculations
myVarioParamOmni = VarioParam()
mydir = DirParam_create(npas=10,dpas=1.)
myVarioParamOmni$addDir(mydir)## NULLWe use the variogram definition in order to calculate the variogram cloud.
dbcloud = db_variogram_cloud(db=mydb, varioparam=myVarioParamOmni)We recall that the Variogram cloud is calculated by filling an underlying grid where each cell is painted according to the number of pairs at the given distance and given variability. Representing the variogram cloud.
if (graphics)
ggplot() + plot(dbcloud,name_raster="Cloud*") + plot.decoration(title="Variogram Cloud")
Calculating the experimental omni-directional variogram
myVarioOmni = Vario(myVarioParamOmni,mydb)
err = myVarioOmni$compute(ECalcVario_VARIOGRAM())
if (verbose)
myVarioOmni$display()##
## Variogram characteristics
## =========================
## Number of variable(s) = 1
## Number of direction(s) = 1
## Space dimension = 2
## Variance-Covariance Matrix 2.881
##
## Direction #1
## ------------
## Number of lags = 10
## Direction coefficients = 1.000 0.000
## Direction angles (degrees) = 0.000 0.000
## Tolerance on direction = 90.000 (degrees)
## Calculation lag = 1.000
## Tolerance on distance = 50.000 (Percent of the lag value)
##
## For variable 1
## Rank Npairs Distance Value
## 0 3.000 0.389 0.462
## 1 123.000 1.081 1.495
## 2 183.000 2.038 1.620
## 3 205.000 3.006 2.526
## 4 231.000 4.013 2.240
## 5 229.000 5.036 2.524
## 6 198.000 5.962 2.396
## 7 187.000 7.000 2.708
## 8 204.000 7.996 2.772
## 9 184.000 8.990 2.868## NULLThe variogram is represented graphically for a quick check
if (graphics)
ggplot() + plot.varmod(myVarioOmni) +
plot.decoration(title="Omni-directional Variogram for Pb")
Calculate a variogram in several directions
myvarioParam = VarioParam()
mydirs = DirParam_createMultiple(ndir=4, npas=10, dpas=1.)
myvarioParam$addMultiDirs(mydirs)## NULLmyvario = Vario(myvarioParam,mydb)
myvario$compute(ECalcVario_VARIOGRAM())## [1] 0if (verbose)
myvario$display()##
## Variogram characteristics
## =========================
## Number of variable(s) = 1
## Number of direction(s) = 4
## Space dimension = 2
## Variance-Covariance Matrix 2.881
##
## Direction #1
## ------------
## Number of lags = 10
## Direction coefficients = 1.000 0.000
## Direction angles (degrees) = 0.000 0.000
## Tolerance on direction = 22.500 (degrees)
## Calculation lag = 1.000
## Tolerance on distance = 50.000 (Percent of the lag value)
##
## For variable 1
## Rank Npairs Distance Value
## 0 1.000 0.410 0.180
## 1 29.000 1.094 1.634
## 2 47.000 2.079 1.415
## 3 53.000 3.003 2.824
## 4 63.000 3.999 2.348
## 5 66.000 5.035 2.319
## 6 60.000 5.978 3.115
## 7 52.000 7.045 2.746
## 8 52.000 8.020 3.927
## 9 37.000 8.980 2.554
##
## Direction #2
## ------------
## Number of lags = 10
## Direction coefficients = 0.707 0.707
## Direction angles (degrees) = 45.000 0.000
## Tolerance on direction = 22.500 (degrees)
## Calculation lag = 1.000
## Tolerance on distance = 50.000 (Percent of the lag value)
##
## For variable 1
## Rank Npairs Distance Value
## 0 1.000 0.344 0.080
## 1 31.000 1.051 1.113
## 2 50.000 1.960 1.890
## 3 62.000 2.999 2.443
## 4 58.000 4.014 2.701
## 5 51.000 5.016 2.702
## 6 36.000 5.999 1.833
## 7 37.000 7.015 2.130
## 8 50.000 7.997 2.060
## 9 53.000 8.995 2.381
##
## Direction #3
## ------------
## Number of lags = 10
## Direction coefficients = 0.000 1.000
## Direction angles (degrees) = 90.000 0.000
## Tolerance on direction = 22.500 (degrees)
## Calculation lag = 1.000
## Tolerance on distance = 50.000 (Percent of the lag value)
##
## For variable 1
## Rank Npairs Distance Value
## 1 32.000 1.149 1.631
## 2 39.000 2.080 1.670
## 3 39.000 2.979 2.511
## 4 48.000 4.012 2.120
## 5 51.000 5.029 3.055
## 6 47.000 5.939 2.856
## 7 49.000 6.965 2.386
## 8 42.000 7.952 2.708
## 9 41.000 9.018 2.320
##
## Direction #4
## ------------
## Number of lags = 10
## Direction coefficients = -0.707 0.707
## Direction angles (degrees) = 135.000 0.000
## Tolerance on direction = 22.500 (degrees)
## Calculation lag = 1.000
## Tolerance on distance = 50.000 (Percent of the lag value)
##
## For variable 1
## Rank Npairs Distance Value
## 0 1.000 0.411 1.125
## 1 31.000 1.028 1.606
## 2 47.000 2.044 1.496
## 3 51.000 3.040 2.330
## 4 62.000 4.028 1.791
## 5 61.000 5.058 2.155
## 6 55.000 5.939 1.587
## 7 49.000 6.975 3.425
## 8 60.000 8.004 2.408
## 9 53.000 8.972 3.996## NULLif (graphics)
ggplot() + plot.varmod(myvario) +
plot.decoration(title="Multi-Directional Variogram of Pb")
Calculating the Variogram Map
myvmap = db_vmap_compute(db=mydb,calcul_type=ECalcVario_VARIOGRAM(),nxx=c(20,20))
if (verbose)
myvmap$display()##
## Data Base Grid Characteristics
## ==============================
##
## Data Base Summary
## -----------------
## File is organized as a regular grid
## Space dimension = 2
## Number of Columns = 5
## Maximum Number of UIDs = 5
## Total number of samples = 1681
##
## Grid characteristics:
## ---------------------
## Origin : -33.160 -29.380
## Mesh : 1.658 1.469
## Number : 41 41
##
## Variables
## ---------
## Column = 0 - Name = rank - Locator = NA
## Column = 1 - Name = x1 - Locator = x1
## Column = 2 - Name = x2 - Locator = x2
## Column = 3 - Name = VMAP.Pb.Var - Locator = z1
## Column = 4 - Name = VMAP.Pb.Nb - Locator = NA## NULLif (graphics)
ggplot() + plot(myvmap, name_raster="*Var") + plot.decoration(title="Variogram Map")
Model
Fitting a Model. We call the Automatic Fitting procedure providing the list of covariance functions to be tested.
mymodel = Model_createFromDb(mydb)
err = mymodel$fit(vario=myvario,types=ECov_fromKeys(c("EXPONENTIAL","SPHERICAL")))Visualizing the resulting model, overlaid on the experimental variogram
if (graphics)
ggplot() + plot.varmod(myvario,mymodel) + plot.decoration(title="Model for Pb")
Model with equality constraints
We can impose some constraints on the parameters during the fit. For instance here, we impose an equality constraint on the range (range = 1).
myModelConstrained = Model_createFromDb(mydb)
constr = Constraints()
paramid = CovParamId(0,0,EConsElem_RANGE(),0,0)
constr$addItem(ConsItem(paramid,EConsType_EQUAL(),1.))## NULLerr = myModelConstrained$fit(vario=myVarioOmni,
types=ECov_fromKeys(c("EXPONENTIAL","SPHERICAL")),
constraints=constr)
myModelConstrained##
## 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
## ---------------
## Exponential
## - Sill = 1.032
## - Range = 1.000
## - Theo. Range = 0.334
## Spherical
## - Sill = 1.605
## - Range = 5.880
## Total Sill = 2.638We can impose inequality constraints by using EConsType.LOWER or EConsType.UPPER.
Adding a drift
mymodel$addDrift(Drift1(mymodel$getContext()))## NULLif (verbose)
mymodel$display()##
## Model characteristics
## =====================
## Space dimension = 2
## Number of variable(s) = 1
## Number of basic structure(s) = 2
## Number of drift function(s) = 1
## Number of drift equation(s) = 1
##
## Covariance Part
## ---------------
## Exponential
## - Sill = 1.035
## - Ranges = 1.786 0.366
## - Theo. Ranges = 0.596 0.122
## - Angles = 45.023 0.000
## - Rotation Matrix
## [, 0] [, 1]
## [ 0,] 0.707 -0.707
## [ 1,] 0.707 0.707
## Spherical
## - Sill = 1.621
## - Ranges = 7.051 5.132
## - Angles = 136.897 0.000
## - Rotation Matrix
## [, 0] [, 1]
## [ 0,] -0.730 -0.683
## [ 1,] 0.683 -0.730
## Total Sill = 2.656
##
## Drift Part
## ----------
## Universality Condition## NULLDefining the Neighborhood
We initiate a Neigborhood (Moving with a small number of samples for Demonstration)
myneigh = NeighMoving_create(flag_xvalid=FALSE,nmaxi=6,radius=10)
if (verbose)
myneigh$display()##
## Moving Neighborhood
## ===================
## Minimum number of samples = 1
## Maximum number of samples = 6
## Maximum horizontal distance = 10## NULLChecking the Moving Neighborhood
We must first create a Grid which covers the area of interest
mygrid = DbGrid_createCoveringDb(dbin=mydb,nx=c(80,72),dx=c(0.5,0.5),
x0=c(107.,481.),margin=c(2,2))
if (verbose)
mygrid$display()##
## Data Base Grid Characteristics
## ==============================
##
## Data Base Summary
## -----------------
## File is organized as a regular grid
## Space dimension = 2
## Number of Columns = 2
## Maximum Number of UIDs = 2
## Total number of samples = 5913
##
## Grid characteristics:
## ---------------------
## Origin : 105.000 479.000
## Mesh : 0.500 0.500
## Number : 81 73
##
## Variables
## ---------
## Column = 0 - Name = x1 - Locator = x1
## Column = 1 - Name = x2 - Locator = x2## NULLWe can now test the neighborhood characteristics for each node of the previously defined grid.
err = test_neigh(mydb,mygrid,mymodel,myneigh)
if (verbose)
mygrid$display()##
## Data Base Grid Characteristics
## ==============================
##
## Data Base Summary
## -----------------
## File is organized as a regular grid
## Space dimension = 2
## Number of Columns = 7
## Maximum Number of UIDs = 7
## Total number of samples = 5913
##
## Grid characteristics:
## ---------------------
## Origin : 105.000 479.000
## Mesh : 0.500 0.500
## Number : 81 73
##
## Variables
## ---------
## Column = 0 - Name = x1 - Locator = x1
## Column = 1 - Name = x2 - Locator = x2
## Column = 2 - Name = Neigh.Pb.Number - Locator = NA
## Column = 3 - Name = Neigh.Pb.MaxDist - Locator = NA
## Column = 4 - Name = Neigh.Pb.MinDist - Locator = NA
## Column = 5 - Name = Neigh.Pb.NbNESect - Locator = NA
## Column = 6 - Name = Neigh.Pb.NbCESect - Locator = z1## NULLWe can visualize some of the newly created variables, such as:
- the number of points per neighborhood
if (graphics)
{
p = ggplot()
p = p + plot(mygrid,name_raster="Neigh*Number")
p = p + plot.decoration(title="Number of Samples per Neighborhood")
ggPrint(p)
}
- the one giving the maximum distance per neighborhood
if (graphics)
{
p = ggplot()
p = p + plot(mygrid,name_raster="Neigh*MaxDist")
p = p + plot.decoration(title="Maximum Distance per Neighborhood")
ggPrint(p)
}
Cross-validation
We can now process the cross-validation step
err = xvalid(mydb,mymodel,myneigh)
if (verbose)
mydb$display()##
## Data Base Characteristics
## =========================
##
## Data Base Summary
## -----------------
## File is organized as a set of isolated points
## Space dimension = 2
## Number of Columns = 8
## Maximum Number of UIDs = 8
## Total number of samples = 102
## Number of active samples = 99
##
## Variables
## ---------
## Column = 0 - Name = rank - Locator = NA
## Column = 1 - Name = X - Locator = x1
## Column = 2 - Name = Y - Locator = x2
## Column = 3 - Name = Zn - Locator = NA
## Column = 4 - Name = Pb - Locator = NA
## Column = 5 - Name = sel - Locator = sel
## Column = 6 - Name = Xvalid.Pb.esterr - Locator = z1
## Column = 7 - Name = Xvalid.Pb.stderr - Locator = NA## NULLif (graphics)
{
p = ggplot()
p = p + plot.hist(mydb,"Xvalid.Pb.stderr")
p = p + plot.decoration(title="Histogram of Stdandrdized Errors")
ggPrint(p)
}## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
Estimating by Kriging
We now perform the Estimation by Ordinary Kriging. The Neighborhood is changed into a Unique Neighborhood.
mydb$setLocator("Pb",ELoc_Z())## NULLmyneigh = NeighUnique_create()
err = kriging(mydb,mygrid,mymodel,myneigh)
if (verbose)
mygrid$display()##
## Data Base Grid Characteristics
## ==============================
##
## Data Base Summary
## -----------------
## File is organized as a regular grid
## Space dimension = 2
## Number of Columns = 9
## Maximum Number of UIDs = 9
## Total number of samples = 5913
##
## Grid characteristics:
## ---------------------
## Origin : 105.000 479.000
## Mesh : 0.500 0.500
## Number : 81 73
##
## Variables
## ---------
## Column = 0 - Name = x1 - Locator = x1
## Column = 1 - Name = x2 - Locator = x2
## Column = 2 - Name = Neigh.Pb.Number - Locator = NA
## Column = 3 - Name = Neigh.Pb.MaxDist - Locator = NA
## Column = 4 - Name = Neigh.Pb.MinDist - Locator = NA
## Column = 5 - Name = Neigh.Pb.NbNESect - Locator = NA
## Column = 6 - Name = Neigh.Pb.NbCESect - Locator = NA
## Column = 7 - Name = Kriging.Pb.estim - Locator = z1
## Column = 8 - Name = Kriging.Pb.stdev - Locator = NA## NULLVisualizing the results
if (graphics)
{
p = ggplot()
p = p + plot(mygrid,name_raster="Kriging.Pb.estim")
p = p + plot(mydb,name_size="Pb", color="yellow")
p = p + plot.decoration(title="Estimate of Pb")
ggPrint(p)
}
if (graphics)
{
p = ggplot()
p = p + plot(mygrid,name_raster="Kriging.Pb.stdev", show.legend.raster=TRUE)
p = p + plot(mydb,"Pb", color="yellow", flagCst=TRUE)
p = p + plot.decoration(title="St. Deviation of Pb")
ggPrint(p)
}
Simulations
We must first transform the Data into Gaussian
myanamPb = AnamHermite_create(nbpoly=30)
err = myanamPb$fitFromLocator(mydb)
if (verbose)
myanamPb##
## Hermitian Anamorphosis
## ----------------------
## Minimum absolute value for Y = -2.7
## Maximum absolute value for Y = 2.6
## Minimum absolute value for Z = 3.0029
## Maximum absolute value for Z = 12.9777
## Minimum practical value for Y = -2.7
## Maximum practical value for Y = 2.6
## Minimum practical value for Z = 3.0029
## Maximum practical value for Z = 12.9777
## Mean = 5.65758
## Variance = 2.86296
## Number of Hermite polynomials = 30
## Normalized coefficients for Hermite polynomials (punctual variable)
## [, 0] [, 1] [, 2] [, 3] [, 4] [, 5] [, 6]
## [ 0,] 5.658 -1.625 0.440 -0.069 -0.017 0.082 -0.061
## [ 7,] 0.001 0.036 -0.044 0.004 0.047 -0.030 -0.029
## [ 14,] 0.037 0.007 -0.031 0.010 0.018 -0.019 -0.003
## [ 21,] 0.019 -0.010 -0.014 0.019 0.006 -0.023 0.004
## [ 28,] 0.022 -0.013We can produce the Gaussian Anamorphosis graphically within its definition domain.
if (graphics)
ggplot() + plot(myanamPb)
The next step consists in translating the target variable (‘Pb’) into its Gaussian transform
mydb$setLocator("Pb",ELoc_Z())## NULLerr = myanamPb$rawToGaussianByLocator(mydb)
if (verbose)
mydb$display()##
## Data Base Characteristics
## =========================
##
## Data Base Summary
## -----------------
## File is organized as a set of isolated points
## Space dimension = 2
## Number of Columns = 9
## Maximum Number of UIDs = 9
## Total number of samples = 102
## Number of active samples = 99
##
## Variables
## ---------
## Column = 0 - Name = rank - Locator = NA
## Column = 1 - Name = X - Locator = x1
## Column = 2 - Name = Y - Locator = x2
## Column = 3 - Name = Zn - Locator = NA
## Column = 4 - Name = Pb - Locator = NA
## Column = 5 - Name = sel - Locator = sel
## Column = 6 - Name = Xvalid.Pb.esterr - Locator = NA
## Column = 7 - Name = Xvalid.Pb.stderr - Locator = NA
## Column = 8 - Name = Y.Pb - Locator = z1## NULLWe quickly calculate experimental (omni-directional) variograms using the already defined directions.
myvarioParam = VarioParam()
mydir = DirParam_create(npas=10,dpas=1.)
myvarioParam$addDir(mydir)## NULLmyVario = Vario(myvarioParam,mydb)
err = myvario$compute(ECalcVario_VARIOGRAM())We fit the model by automatic fit (with the constraints that the total sill be equal to 1).
mymodelG = Model_createFromDb(mydb)
err = mymodelG$fit(myvario,types=ECov_fromKeys(c("EXPONENTIAL")))
if (graphics)
ggplot() + plot.varmod(myvario,mymodelG) + plot.decoration(title="Model for Gaussian Pb")
We perform a set of 10 conditional simulations using the Turning Bands Method.
err = simtub(mydb,mygrid,mymodel,myneigh,nbsimu=10)
if (verbose)
mygrid$display()##
## Data Base Grid Characteristics
## ==============================
##
## Data Base Summary
## -----------------
## File is organized as a regular grid
## Space dimension = 2
## Number of Columns = 19
## Maximum Number of UIDs = 19
## Total number of samples = 5913
##
## Grid characteristics:
## ---------------------
## Origin : 105.000 479.000
## Mesh : 0.500 0.500
## Number : 81 73
##
## Variables
## ---------
## Column = 0 - Name = x1 - Locator = x1
## Column = 1 - Name = x2 - Locator = x2
## Column = 2 - Name = Neigh.Pb.Number - Locator = NA
## Column = 3 - Name = Neigh.Pb.MaxDist - Locator = NA
## Column = 4 - Name = Neigh.Pb.MinDist - Locator = NA
## Column = 5 - Name = Neigh.Pb.NbNESect - Locator = NA
## Column = 6 - Name = Neigh.Pb.NbCESect - Locator = NA
## Column = 7 - Name = Kriging.Pb.estim - Locator = NA
## Column = 8 - Name = Kriging.Pb.stdev - Locator = NA
## Column = 9 - Name = Simu.Y.Pb.1 - Locator = z1
## Column = 10 - Name = Simu.Y.Pb.2 - Locator = z2
## Column = 11 - Name = Simu.Y.Pb.3 - Locator = z3
## Column = 12 - Name = Simu.Y.Pb.4 - Locator = z4
## Column = 13 - Name = Simu.Y.Pb.5 - Locator = z5
## Column = 14 - Name = Simu.Y.Pb.6 - Locator = z6
## Column = 15 - Name = Simu.Y.Pb.7 - Locator = z7
## Column = 16 - Name = Simu.Y.Pb.8 - Locator = z8
## Column = 17 - Name = Simu.Y.Pb.9 - Locator = z9
## Column = 18 - Name = Simu.Y.Pb.10 - Locator = z10## NULLSome statistics on the Conditional simulations in Gaussian scale
if (verbose)
err = mygrid$statisticsByLocator(locatorType = ELoc_Z(),
opers = EStatOption_fromKeys(c("MEAN","STDV","MINI","MAXI")))## MEAN STDV MINI MAXI
## Simu.Y.Pb.1 -0.052 1.582 -6.234 6.963
## Simu.Y.Pb.2 0.081 1.451 -5.167 6.113
## Simu.Y.Pb.3 0.133 1.529 -5.363 5.639
## Simu.Y.Pb.4 0.453 1.478 -6.322 5.836
## Simu.Y.Pb.5 -0.084 1.577 -6.403 5.578
## Simu.Y.Pb.6 0.316 1.551 -4.816 6.297
## Simu.Y.Pb.7 0.185 1.598 -6.441 6.127
## Simu.Y.Pb.8 0.705 1.599 -4.713 7.175
## Simu.Y.Pb.9 0.159 1.639 -6.660 7.135
## Simu.Y.Pb.10 0.310 1.553 -5.955 4.807We visualize a conditional simulation in Gaussian scale
if (graphics)
{
p = ggplot()
p = p + plot(mygrid,name_raster="Simu.Y.Pb.1")
p = p + plot(mydb,"Pb")
p = p + plot.decoration(title="One Simulation of Pb in Gaussian Scale")
ggPrint(p)
}
We turn the Gaussian conditional simulations into Raw scale (using the Anamorphosis back transform) and get rid of the Gaussian conditional simulations.
err = myanamPb$gaussianToRaw(mygrid,name="Simu.Y.*")
mygrid$deleteColumn("Simu.Y.*")## NULLif (verbose)
mygrid$display()##
## Data Base Grid Characteristics
## ==============================
##
## Data Base Summary
## -----------------
## File is organized as a regular grid
## Space dimension = 2
## Number of Columns = 19
## Maximum Number of UIDs = 29
## Total number of samples = 5913
##
## Grid characteristics:
## ---------------------
## Origin : 105.000 479.000
## Mesh : 0.500 0.500
## Number : 81 73
##
## Variables
## ---------
## Column = 0 - Name = x1 - Locator = x1
## Column = 1 - Name = x2 - Locator = x2
## Column = 2 - Name = Neigh.Pb.Number - Locator = NA
## Column = 3 - Name = Neigh.Pb.MaxDist - Locator = NA
## Column = 4 - Name = Neigh.Pb.MinDist - Locator = NA
## Column = 5 - Name = Neigh.Pb.NbNESect - Locator = NA
## Column = 6 - Name = Neigh.Pb.NbCESect - Locator = NA
## Column = 7 - Name = Kriging.Pb.estim - Locator = NA
## Column = 8 - Name = Kriging.Pb.stdev - Locator = NA
## Column = 9 - Name = Z.Simu.Y.Pb.1 - Locator = z1
## Column = 10 - Name = Z.Simu.Y.Pb.2 - Locator = z2
## Column = 11 - Name = Z.Simu.Y.Pb.3 - Locator = z3
## Column = 12 - Name = Z.Simu.Y.Pb.4 - Locator = z4
## Column = 13 - Name = Z.Simu.Y.Pb.5 - Locator = z5
## Column = 14 - Name = Z.Simu.Y.Pb.6 - Locator = z6
## Column = 15 - Name = Z.Simu.Y.Pb.7 - Locator = z7
## Column = 16 - Name = Z.Simu.Y.Pb.8 - Locator = z8
## Column = 17 - Name = Z.Simu.Y.Pb.9 - Locator = z9
## Column = 18 - Name = Z.Simu.Y.Pb.10 - Locator = z10## NULLWe calculate some statistics on the Conditional Simulations in Raw scale.
if (verbose)
err = mygrid$statisticsByLocator(locatorType = ELoc_Z(),
opers = EStatOption_fromKeys(c("MEAN","STDV","MINI","MAXI")))## MEAN STDV MINI MAXI
## Z.Simu.Y.Pb.1 5.965 2.647 3.003 12.978
## Z.Simu.Y.Pb.2 6.113 2.550 3.003 12.978
## Z.Simu.Y.Pb.3 6.238 2.703 3.003 12.978
## Z.Simu.Y.Pb.4 6.777 2.817 3.003 12.978
## Z.Simu.Y.Pb.5 5.937 2.599 3.003 12.978
## Z.Simu.Y.Pb.6 6.568 2.858 3.003 12.978
## Z.Simu.Y.Pb.7 6.384 2.829 3.003 12.978
## Z.Simu.Y.Pb.8 7.270 3.087 3.003 12.978
## Z.Simu.Y.Pb.9 6.362 2.862 3.003 12.978
## Z.Simu.Y.Pb.10 6.575 2.851 3.003 12.978We visualize a Conditional Simulation in Raw Scale
if (graphics)
{
p = ggplot()
p = p + plot(mygrid,name_raster="Z.Simu.Y.Pb.1")
p = p + plot(mydb,"Pb")
p = p + plot.decoration(title="One simulation of Pb in Raw Scale")
ggPrint(p)
}
Let us now average the conditional simulations in order to have a comparison with the estimation by kriging.
err = mygrid$statisticsByLocator(locatorType = ELoc_Z(),
opers = EStatOption_fromKeys(c("MEAN")),
flagStoreInDb = TRUE,
verbose = FALSE)
if (verbose)
mygrid$display()##
## Data Base Grid Characteristics
## ==============================
##
## Data Base Summary
## -----------------
## File is organized as a regular grid
## Space dimension = 2
## Number of Columns = 20
## Maximum Number of UIDs = 30
## Total number of samples = 5913
##
## Grid characteristics:
## ---------------------
## Origin : 105.000 479.000
## Mesh : 0.500 0.500
## Number : 81 73
##
## Variables
## ---------
## Column = 0 - Name = x1 - Locator = x1
## Column = 1 - Name = x2 - Locator = x2
## Column = 2 - Name = Neigh.Pb.Number - Locator = NA
## Column = 3 - Name = Neigh.Pb.MaxDist - Locator = NA
## Column = 4 - Name = Neigh.Pb.MinDist - Locator = NA
## Column = 5 - Name = Neigh.Pb.NbNESect - Locator = NA
## Column = 6 - Name = Neigh.Pb.NbCESect - Locator = NA
## Column = 7 - Name = Kriging.Pb.estim - Locator = NA
## Column = 8 - Name = Kriging.Pb.stdev - Locator = NA
## Column = 9 - Name = Z.Simu.Y.Pb.1 - Locator = NA
## Column = 10 - Name = Z.Simu.Y.Pb.2 - Locator = NA
## Column = 11 - Name = Z.Simu.Y.Pb.3 - Locator = NA
## Column = 12 - Name = Z.Simu.Y.Pb.4 - Locator = NA
## Column = 13 - Name = Z.Simu.Y.Pb.5 - Locator = NA
## Column = 14 - Name = Z.Simu.Y.Pb.6 - Locator = NA
## Column = 15 - Name = Z.Simu.Y.Pb.7 - Locator = NA
## Column = 16 - Name = Z.Simu.Y.Pb.8 - Locator = NA
## Column = 17 - Name = Z.Simu.Y.Pb.9 - Locator = NA
## Column = 18 - Name = Z.Simu.Y.Pb.10 - Locator = NA
## Column = 19 - Name = Stats.MEAN - Locator = z1## NULLDisplaying the average of the Conditional Simulations
if (graphics)
{
p = ggplot()
p = p + plot(mygrid,name_raster="Stats*MEAN")
p = p + plot(mydb,"Pb")
p = p + plot.decoration(title="Mean of Pb simulations")
ggPrint(p)
}
Multivariate case
The Gaussian transform of the Pb variable has already been calculated. It suffices to perform the Gaussian transform of the Zn variable.
mydb$setLocator("Zn",ELoc_Z())## NULLmyanamZn = AnamHermite(nbpoly=30)
myanamZn$fit(mydb, "Zn")## [1] 0if (verbose)
myanamZn##
## Hermitian Anamorphosis
## ----------------------
## Minimum absolute value for Y = -2.5
## Maximum absolute value for Y = 2.6
## Minimum absolute value for Z = 1.1469
## Maximum absolute value for Z = 12.1276
## Minimum practical value for Y = -2.5
## Maximum practical value for Y = 2.6
## Minimum practical value for Z = 1.1469
## Maximum practical value for Z = 12.1276
## Mean = 2.88061
## Variance = 2.76263
## Number of Hermite polynomials = 30
## Normalized coefficients for Hermite polynomials (punctual variable)
## [, 0] [, 1] [, 2] [, 3] [, 4] [, 5] [, 6]
## [ 0,] 2.881 -1.277 0.877 -0.447 -0.095 0.294 -0.121
## [ 7,] -0.087 0.134 -0.029 -0.087 0.069 0.034 -0.065
## [ 14,] 0.005 0.044 -0.026 -0.020 0.034 0.001 -0.033
## [ 21,] 0.010 0.027 -0.016 -0.019 0.016 0.012 -0.014
## [ 28,] -0.005 0.011if (graphics)
{
p = ggplot()
p = p + plot(myanamZn)
p = p + plot.decoration(title="Gaussian Anamorphosis for Zn")
ggPrint(p)
}
We convert the raw data into its Gaussian equivalent
mydb$setLocator("Zn",ELoc_Z())## NULLerr = myanamZn$rawToGaussianByLocator(mydb)
if (verbose)
mydb$display()##
## Data Base Characteristics
## =========================
##
## Data Base Summary
## -----------------
## File is organized as a set of isolated points
## Space dimension = 2
## Number of Columns = 10
## Maximum Number of UIDs = 20
## Total number of samples = 102
## Number of active samples = 99
##
## Variables
## ---------
## Column = 0 - Name = rank - Locator = NA
## Column = 1 - Name = X - Locator = x1
## Column = 2 - Name = Y - Locator = x2
## Column = 3 - Name = Zn - Locator = NA
## Column = 4 - Name = Pb - Locator = NA
## Column = 5 - Name = sel - Locator = sel
## Column = 6 - Name = Xvalid.Pb.esterr - Locator = NA
## Column = 7 - Name = Xvalid.Pb.stderr - Locator = NA
## Column = 8 - Name = Y.Pb - Locator = NA
## Column = 9 - Name = Y.Zn - Locator = z1## NULLWe now perform the multivariate variogram calculation
mydb$setLocator("Y.Pb",ELoc_Z(),0)## NULLmydb$setLocator("Y.Zn",ELoc_Z(),1)## NULLmyvario = Vario(myvarioParam,mydb)
err = myvario$compute(ECalcVario_VARIOGRAM())
mymodelM = Model_createFromDb(mydb)
err = mymodelM$fit(myvario,ECov_fromKeys(c("EXPONENTIAL")))
if (graphics)
{
p = ggplot()
p = p + plot.varmod(myvario,mymodelM)
p = p + plot.decoration(title="Multivariate Model")
ggPrint(p)
}
We perform 10 bivariate conditional simulations (deleting the previous monovariate simulation outcomes first for better legibility).
mygrid$deleteColumn("Z.Simu*")## NULLerr = simtub(mydb,mygrid,mymodelM,myneigh,nbsimu=10)
if (verbose)
mygrid$display()##
## Data Base Grid Characteristics
## ==============================
##
## Data Base Summary
## -----------------
## File is organized as a regular grid
## Space dimension = 2
## Number of Columns = 30
## Maximum Number of UIDs = 50
## Total number of samples = 5913
##
## Grid characteristics:
## ---------------------
## Origin : 105.000 479.000
## Mesh : 0.500 0.500
## Number : 81 73
##
## Variables
## ---------
## Column = 0 - Name = x1 - Locator = x1
## Column = 1 - Name = x2 - Locator = x2
## Column = 2 - Name = Neigh.Pb.Number - Locator = NA
## Column = 3 - Name = Neigh.Pb.MaxDist - Locator = NA
## Column = 4 - Name = Neigh.Pb.MinDist - Locator = NA
## Column = 5 - Name = Neigh.Pb.NbNESect - Locator = NA
## Column = 6 - Name = Neigh.Pb.NbCESect - Locator = NA
## Column = 7 - Name = Kriging.Pb.estim - Locator = NA
## Column = 8 - Name = Kriging.Pb.stdev - Locator = NA
## Column = 9 - Name = Stats.MEAN - Locator = NA
## Column = 10 - Name = Simu.Y.Pb.1 - Locator = z1
## Column = 11 - Name = Simu.Y.Pb.2 - Locator = z2
## Column = 12 - Name = Simu.Y.Pb.3 - Locator = z3
## Column = 13 - Name = Simu.Y.Pb.4 - Locator = z4
## Column = 14 - Name = Simu.Y.Pb.5 - Locator = z5
## Column = 15 - Name = Simu.Y.Pb.6 - Locator = z6
## Column = 16 - Name = Simu.Y.Pb.7 - Locator = z7
## Column = 17 - Name = Simu.Y.Pb.8 - Locator = z8
## Column = 18 - Name = Simu.Y.Pb.9 - Locator = z9
## Column = 19 - Name = Simu.Y.Pb.10 - Locator = z10
## Column = 20 - Name = Simu.Y.Zn.1 - Locator = z11
## Column = 21 - Name = Simu.Y.Zn.2 - Locator = z12
## Column = 22 - Name = Simu.Y.Zn.3 - Locator = z13
## Column = 23 - Name = Simu.Y.Zn.4 - Locator = z14
## Column = 24 - Name = Simu.Y.Zn.5 - Locator = z15
## Column = 25 - Name = Simu.Y.Zn.6 - Locator = z16
## Column = 26 - Name = Simu.Y.Zn.7 - Locator = z17
## Column = 27 - Name = Simu.Y.Zn.8 - Locator = z18
## Column = 28 - Name = Simu.Y.Zn.9 - Locator = z19
## Column = 29 - Name = Simu.Y.Zn.10 - Locator = z20## NULLWe back-transform each set of simulation outcomes using its own Gaussian Anamorphosis function. Finally we delete the Gaussian variables and ask for the statistics on the simulated variables in the Raw Scale.
err = myanamZn$gaussianToRaw(mygrid,"Simu.Y.Zn*")
err = myanamPb$gaussianToRaw(mygrid,"Simu.Y.Pb*")
mygrid$deleteColumn("Simu.Y*")## NULLif (verbose)
err = mygrid$statisticsByLocator(locatorType = ELoc_Z(),
opers = EStatOption_fromKeys(c("MEAN","STDV","MINI","MAXI")))## MEAN STDV MINI MAXI
## Z.Simu.Y.Pb.1 5.854 1.880 3.003 12.978
## Z.Simu.Y.Pb.2 5.731 1.743 3.003 12.978
## Z.Simu.Y.Pb.3 5.630 1.681 3.003 12.978
## Z.Simu.Y.Pb.4 5.811 1.906 3.003 12.978
## Z.Simu.Y.Pb.5 5.724 1.728 3.003 12.978
## Z.Simu.Y.Pb.6 5.717 1.805 3.003 12.978
## Z.Simu.Y.Pb.7 5.922 1.825 3.003 12.978
## Z.Simu.Y.Pb.8 5.742 1.803 3.003 12.978
## Z.Simu.Y.Pb.9 5.788 1.841 3.003 12.978
## Z.Simu.Y.Pb.10 5.778 1.774 3.003 12.978Categorical Variable¶
We compare the initial variable ‘Pb’ with a set of disjoint intervals. The ‘Pb’ values varying from 3 to 12.7, we consider three classes:
- values below 4
- values between 4 and 6
- values above 6
We first build the indicators for each class.
limits = Limits(c(NA, 4., 6., NA))
if (verbose)
limits$display()## Bound( 1 ) : ] -Inf ; 4 [
## Bound( 2 ) : [ 4 ; 6 [
## Bound( 3 ) : [ 6 ; +Inf [## NULLWe apply the set of limits previously defined in order to transform the input variable into Indicators of the different classes.
err = limits$toIndicator(mydb,name="Pb")
if (verbose)
mydb$display()##
## Data Base Characteristics
## =========================
##
## Data Base Summary
## -----------------
## File is organized as a set of isolated points
## Space dimension = 2
## Number of Columns = 13
## Maximum Number of UIDs = 43
## Total number of samples = 102
## Number of active samples = 99
##
## Variables
## ---------
## Column = 0 - Name = rank - Locator = NA
## Column = 1 - Name = X - Locator = x1
## Column = 2 - Name = Y - Locator = x2
## Column = 3 - Name = Zn - Locator = NA
## Column = 4 - Name = Pb - Locator = NA
## Column = 5 - Name = sel - Locator = sel
## Column = 6 - Name = Xvalid.Pb.esterr - Locator = NA
## Column = 7 - Name = Xvalid.Pb.stderr - Locator = NA
## Column = 8 - Name = Y.Pb - Locator = NA
## Column = 9 - Name = Y.Zn - Locator = NA
## Column = 10 - Name = Indicator.Pb.Class.1 - Locator = z1
## Column = 11 - Name = Indicator.Pb.Class.2 - Locator = z2
## Column = 12 - Name = Indicator.Pb.Class.3 - Locator = z3## NULLWe calculate the variogram of the Indicators for future use
myvarioindParam = VarioParam()
myvarioindParam$addDir(mydir)## NULLmyvarioInd = Vario(myvarioindParam,mydb)
err = myvarioInd$compute(ECalcVario_VARIOGRAM())
if (verbose)
myvarioInd$display()##
## Variogram characteristics
## =========================
## Number of variable(s) = 3
## Number of direction(s) = 1
## Space dimension = 2
## Variance-Covariance Matrix
## [, 0] [, 1] [, 2]
## [ 0,] 0.107 -0.062 -0.044
## [ 1,] -0.062 0.250 -0.187
## [ 2,] -0.044 -0.187 0.231
##
## Direction #1
## ------------
## Number of lags = 10
## Direction coefficients = 1.000 0.000
## Direction angles (degrees) = 0.000 0.000
## Tolerance on direction = 90.000 (degrees)
## Calculation lag = 1.000
## Tolerance on distance = 50.000 (Percent of the lag value)
##
## For variable 1
## Rank Npairs Distance Value
## 0 3.000 0.389 0.000
## 1 123.000 1.081 0.081
## 2 183.000 2.038 0.126
## 3 205.000 3.006 0.156
## 4 231.000 4.013 0.132
## 5 229.000 5.036 0.159
## 6 198.000 5.962 0.152
## 7 187.000 7.000 0.107
## 8 204.000 7.996 0.096
## 9 184.000 8.990 0.068
##
## For variables 2 and 1
## Rank Npairs Distance Value
## 0 3.000 0.389 0.000
## 1 123.000 1.081 -0.065
## 2 183.000 2.038 -0.077
## 3 205.000 3.006 -0.085
## 4 231.000 4.013 -0.093
## 5 229.000 5.036 -0.085
## 6 198.000 5.962 -0.061
## 7 187.000 7.000 -0.045
## 8 204.000 7.996 -0.042
## 9 184.000 8.990 -0.038
##
## For variable 2
## Rank Npairs Distance Value
## 0 3.000 0.389 0.167
## 1 123.000 1.081 0.199
## 2 183.000 2.038 0.221
## 3 205.000 3.006 0.251
## 4 231.000 4.013 0.292
## 5 229.000 5.036 0.258
## 6 198.000 5.962 0.237
## 7 187.000 7.000 0.254
## 8 204.000 7.996 0.228
## 9 184.000 8.990 0.234
##
## For variables 3 and 1
## Rank Npairs Distance Value
## 0 3.000 0.389 0.000
## 1 123.000 1.081 -0.016
## 2 183.000 2.038 -0.049
## 3 205.000 3.006 -0.071
## 4 231.000 4.013 -0.039
## 5 229.000 5.036 -0.074
## 6 198.000 5.962 -0.091
## 7 187.000 7.000 -0.061
## 8 204.000 7.996 -0.054
## 9 184.000 8.990 -0.030
##
## For variables 3 and 2
## Rank Npairs Distance Value
## 0 3.000 0.389 -0.167
## 1 123.000 1.081 -0.134
## 2 183.000 2.038 -0.145
## 3 205.000 3.006 -0.166
## 4 231.000 4.013 -0.199
## 5 229.000 5.036 -0.172
## 6 198.000 5.962 -0.177
## 7 187.000 7.000 -0.209
## 8 204.000 7.996 -0.186
## 9 184.000 8.990 -0.196
##
## For variable 3
## Rank Npairs Distance Value
## 0 3.000 0.389 0.167
## 1 123.000 1.081 0.150
## 2 183.000 2.038 0.194
## 3 205.000 3.006 0.237
## 4 231.000 4.013 0.238
## 5 229.000 5.036 0.247
## 6 198.000 5.962 0.268
## 7 187.000 7.000 0.270
## 8 204.000 7.996 0.240
## 9 184.000 8.990 0.226## NULLif (graphics)
multi.varmod(myvarioInd)
Then we build a categorical variable which gives the index of the class to which each sample belongs
err = limits$toCategory(mydb,"Pb")
if (verbose)
{
dbfmt = DbStringFormat()
dbfmt$setFlags(flag_resume = FALSE,
flag_vars = FALSE,
flag_stats = TRUE)
dbfmt$setNames("Category*")
dbfmt$setMode(mode=2) # Consider the variable categorical
mydb$display(dbfmt)
}##
## Data Base Characteristics
## =========================
##
## Data Base Statistics
## --------------------
## 14 - Name Category.Pb - Locator z1
## Nb of data = 102
## Nb of active values = 99
## Class 1 = 12 ( 12.121%)
## Class 2 = 51 ( 51.515%)
## Class 3 = 36 ( 36.364%)## NULL