Factorial Kriging Analysis¶
In [1]:
%%javascript
IPython.OutputArea.prototype._should_scroll = function(lines) {
return false;
}
In [2]:
import gstlearn as gl
import gstlearn.plot as gp
import matplotlib.pyplot as plt
import numpy as np
import os
The Grid containing the information is downloaded from the distribution.
The loaded file (called **grid **) contains 3 variables:
- P (phosphorus) which is the variable of interest
- Cr (chromium) is an auxiliary variable
- Ni (nickel) another auxiliary variable
In [3]:
!wget -q https://soft.minesparis.psl.eu/gstlearn/data/FKA/Image.ascii
In [4]:
fileNF = "Image.ascii"
grid = gl.DbGrid.createFromNF(fileNF)
ndim = 2
gl.defineDefaultSpace(gl.ESpaceType.RN, ndim)
In [5]:
dbfmt = gl.DbStringFormat()
dbfmt.setFlags(flag_resume=False,flag_vars=False,flag_stats=True, names="P")
grid.display(dbfmt)
Data Base Grid Characteristics ============================== Data Base Statistics -------------------- 6 - Name P - Locator NA Nb of data = 262144 Nb of active values = 242306 Minimum value = 0.000 Maximum value = 314.000 Mean value = 31.767 Standard Deviation = 21.759 Variance = 473.457
Note that some pixels are not informed for variable P.
Statistics on auxiliary variables
In [6]:
dbfmt.setFlags(flag_vars=False, flag_resume=True, flag_stats=True, names=["Cr", "Ni"])
grid.display(dbfmt)
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 = 262144 Grid characteristics: --------------------- Origin : 0.000 0.000 Mesh : 1.000 1.000 Number : 512 512 Data Base Statistics -------------------- 4 - Name Cr - Locator NA Nb of data = 262144 Nb of active values = 262144 Minimum value = 2591.000 Maximum value = 24982.000 Mean value = 16800.231 Standard Deviation = 936.213 Variance = 876495.558 5 - Name Ni - Locator NA Nb of data = 262144 Nb of active values = 262144 Minimum value = 1840.000 Maximum value = 12593.000 Mean value = 10111.444 Standard Deviation = 884.996 Variance = 783217.898
Correlation between variables
In [7]:
ax = gp.correlation(grid, namex="Cr", namey="P", bins=100)
In [8]:
ax = gp.correlation(grid, namex="Ni", namey="P", bins=100)
In [9]:
ax = gp.correlation(grid, namex="Ni", namey="Cr", bins=100)
Using inverse square distance for completing the variable P
In [10]:
grid.setLocator("P", gl.ELoc.Z)
err = gl.db_grid_fill(grid)
We concentrate on the variable of interest P completed (Fill.P) in the next operations
In [11]:
gp.setDefaultGeographic(dims=[8,8])
ax = grid.plot("Fill.P")
Variogram Calculation along Grid main axes
In [12]:
varioparam = gl.VarioParam.createMultipleFromGrid(grid, npas=100)
varioP = gl.Vario(varioparam, grid)
err = varioP.compute()
modelP = gl.Model()
err = modelP.fit(varioP, types=[gl.ECov.NUGGET, gl.ECov.SPHERICAL, gl.ECov.POWER],
optvar=gl.Option_VarioFit(True, False))
In [13]:
ax = gp.varmod(varioP, modelP)
We must define the Neighborhood
In [14]:
neigh = gl.NeighImage([10,10])
The image neighborhood is based on $(2*10+1)^2=441$ pixels (centered on the target pixel).
During the estimation, only the contribution of second and third basic structures are kept (Nugget Effect is filtered out): ** Factorial Kriging Analysis**.
In [15]:
modelP.setCovaFiltered(0, True)
means = gl.dbStatisticsMono(grid,["Fill.P"],[gl.EStatOption.MEAN])
modelP.setMeans(means)
modelP
Out[15]:
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 = 273.080 (This component is Filtered) Spherical - Sill = 56.089 - Range = 3.600 Power (Third Parameter = 0.0879341) - Slope = 1.187
In [16]:
err = gl.krimage(grid,modelP,neigh,namconv=gl.NamingConvention("Mono"))
In [17]:
ax = grid.plot("Mono*.P")
ax.decoration(title="P denoised (monovariate)")
Correlation for P variable between Initial image (completed) and its Filtered version (monovariate FKA)
In [18]:
grid
Out[18]:
Data Base Grid Characteristics ============================== Data Base Summary ----------------- File is organized as a regular grid Space dimension = 2 Number of Columns = 8 Maximum Number of UIDs = 8 Total number of samples = 262144 Grid characteristics: --------------------- Origin : 0.000 0.000 Mesh : 1.000 1.000 Number : 512 512 Variables --------- Column = 0 - Name = rank - Locator = NA Column = 1 - Name = x1 - Locator = x1 Column = 2 - Name = x2 - Locator = x2 Column = 3 - Name = Cr - Locator = NA Column = 4 - Name = Ni - Locator = NA Column = 5 - Name = P - Locator = NA Column = 6 - Name = Fill.P - Locator = NA Column = 7 - Name = Mono.Fill.P - Locator = z1
In [19]:
ax = gp.correlation(grid, namex="Fill.P", namey="Mono.Fill.P", bins=100)
ax.decoration(xlabel="P Filled",ylabel="P Filtered (Mono)")
Multivariate approach¶
In [20]:
grid.setLocators(["Fill.P", "Cr", "Ni"], gl.ELoc.Z)
varioM = gl.Vario(varioparam, grid)
err = varioM.compute()
modelM = gl.Model()
err = modelM.fit(varioM, types=[gl.ECov.NUGGET, gl.ECov.SPHERICAL, gl.ECov.POWER],
optvar=gl.Option_VarioFit(True, False))
In [21]:
ax = gp.varmod(varioM, modelM)
Printing the contents of the fitted Multivariate Mpdel
In [22]:
modelM.setCovaFiltered(0, True)
means = gl.dbStatisticsMono(grid,["Fill.P", "Cr", "Ni"],[gl.EStatOption.MEAN])
modelM.setMeans(means)
modelM
Out[22]:
Model characteristics
=====================
Space dimension = 2
Number of variable(s) = 3
Number of basic structure(s) = 3
Number of drift function(s) = 0
Number of drift equation(s) = 0
Covariance Part
---------------
Nugget Effect
- Sill matrix:
[, 0] [, 1] [, 2]
[ 0,] 376.850 452.534 -476.811
[ 1,] 452.534194188.109-11524.845
[ 2,] -476.811-11524.845145939.572
(This component is Filtered)
Spherical
- Sill matrix:
[, 0] [, 1] [, 2]
[ 0,] 57.513 5031.290 -4489.149
[ 1,] 5031.290636076.559-583291.704
[ 2,] -4489.149-583291.704616673.997
- Range = 12.375
Power (Third Parameter = 1.99)
- Slope matrix:
[, 0] [, 1] [, 2]
[ 0,] 0.263 -0.414 6.133
[ 1,] -0.414 145.976 44.478
[ 2,] 6.133 44.478 163.446
Multivariable Factorial Kriging Analysis
In [23]:
err = gl.krimage(grid,modelM,neigh,namconv=gl.NamingConvention("Multi"))
Note that, using the same neigh as in monovariate, the dimension of the Kriging System is now $3 * 441 = 1323$
In [24]:
ax = grid.plot("Multi*.P")
ax.decoration(title="P denoised (multivariate)")
Correlation for P variable between Initial image and its Filtered version (multivariate FKA)
In [25]:
ax = gp.correlation(grid, namex="Fill.P", namey="Multi.Fill.P", bins=100)
ax.decoration(xlabel="P Filled",ylabel="P Filtered (Multi)")
Correlation for P filtered variable between he Monovariate and the Multivariate case
In [26]:
ax = gp.correlation(grid, namex="Mono.Fill.P", namey="Multi.Fill.P", bins=100)
ax.decoration(xlabel="P Filtered (Mono)",ylabel="P Filtered (Multi)")
