Super Kriging¶

This file is meant to demonstrate the use of gstlearn for Super Kriging. It is run on several 2-D cases.

In [1]:
import numpy as np
import pandas as pd
import sys
import os
import gstlearn as gl
import gstlearn.plot as gp
import gstlearn.document as gdoc
import matplotlib.pyplot as plt

gdoc.setNoScroll()

Setting some global variables

In [2]:
# Set the Global Options
verbose = True
flagGraphic = True

# Define the Space Dimension
ndim = 2
gl.defineDefaultSpace(gl.ESpaceType.RN, ndim)

# Set the Seed for the Random Number generator
gl.law_set_random_seed(5584)

# Next option performs a verbose kriging of one target node (lots of outputs)
flagDebug = True

We define the grid on which all calculations will be performed. The grid is constituted of square meshes and has a rectangular extension of 200 by 150.

In [3]:
grid = gl.DbGrid.create(nx=[200,150],dx=[1.,1.])
grid
Out[3]:
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      = 30000

Grid characteristics:
---------------------
Origin :      0.000     0.000
Mesh   :      1.000     1.000
Number :        200       150

Variables
---------
Column = 0 - Name = rank - Locator = NA
Column = 1 - Name = x1 - Locator = x1
Column = 2 - Name = x2 - Locator = x2

Poisson Data Set¶

In this paragraph, we generate a Poisson Data Set (data uniformy distributed along each space dimension) which covers the Grid expansion area. The Data Set contains 100 samples.

In [4]:
coormin = grid.getCoorMinimum()
coormax = grid.getCoorMaximum()
nech = 100
data = gl.Db.createFromBox(nech, coormin, coormax)

A variable is generated on this data set, as the result of a non-conditional simulation with a build-in model.

In [5]:
model = gl.Model()
model.addCovFromParam(gl.ECov.SPHERICAL, range=40, sill=0.7)
model.addCovFromParam(gl.ECov.NUGGET, sill = 0.3)
err = gl.simtub(None, dbout=data, model=model)
data
Out[5]:
Data Base Characteristics
=========================

Data Base Summary
-----------------
File is organized as a set of isolated points
Space dimension              = 2
Number of Columns            = 4
Total number of samples      = 100

Variables
---------
Column = 0 - Name = rank - Locator = NA
Column = 1 - Name = x-1 - Locator = x1
Column = 2 - Name = x-2 - Locator = x2
Column = 3 - Name = Simu - Locator = z1
In [6]:
gp.setDefaultGeographic(dims=[7,7], aspect=1)
ax = data.plot(nameColor="Simu")

Defining standard neighborhood¶

We first define a standard Moving Neighborhood.

In [7]:
nmini = 1
nmaxi = 10
radius = 30.
nsect = 8
nsmax = 3
neigh = gl.NeighMoving.create(flag_xvalid=False, nmaxi=nmaxi, radius=radius, nmini=nmini, 
                              nsect=nsect, nsmax=nsmax)
neigh
Out[7]:
Moving Neighborhood
===================
Minimum number of samples           = 1
Maximum number of samples           = 10
Number of angular sectors           = 8
Maximum number of points per sector = 3
Maximum horizontal distance         = 30

Checking the neighborhood around a central grid node

In [8]:
node = 15300
target = grid.getSampleCoordinates(node)
neigh.attach(data, grid)
ranks = neigh.select(node)

dataSel = data.clone()
dum = dataSel.addSelectionByRanks(ranks)
In [9]:
ax = data.plot()
ax = dataSel.plot(color='blue')
ax = gp.sample(target, color='black')
ax = gp.curve(grid.getCellEdges(node))
ax = gp.curve(neigh.getEllipsoid(target))
ax = gp.multisegments(target,neigh.getSectors(target))

Point Kriging¶

Performing a Point Kriging with the current Neighborhood feature

In [10]:
err = gl.kriging(data, grid, model, neigh, 
                 namconv=gl.NamingConvention("Point_Kriging"))
In [11]:
if flagDebug:
    err = gl.krigtest(data, grid, model, neigh, node)

 Target location
 ---------------
 Sample #15301 (from 30000)
 Coordinate #1 = 100.000000
 Coordinate #2 = 76.000000
 
 Data selected in neighborhood
 -----------------------------
        Rank      Sample          x1          x2      Sector 
           1          20     114.631      72.031           4 
           2          52     116.444      65.590           4 
           3          56     114.720      79.035           5 
           4          65     116.816      94.871           6 
           5          69      89.960      69.482           1 
 
 LHS of Kriging matrix (compressed)
 ==================================
 Number of active samples    = 5
 Total number of equations   = 5
 Reduced number of equations = 5
 
        Rank                       1           2           3           4           5 
                    Flag           1           2           3           4           5 
           1           1       1.000       0.526       0.518       0.164       0.132 
           2           2       0.526       1.000       0.358       0.069       0.102 
           3           3       0.518       0.358       1.000       0.303       0.106 
           4           4       0.164       0.069       0.303       1.000       0.006 
           5           5       0.132       0.102       0.106       0.006       1.000 
 
 RHS of Kriging matrix (compressed)
 ==================================
 Number of active samples    = 5
 Total number of equations   = 5
 Reduced number of equations = 5
 Number of right-hand sides  = 1
 Punctual Estimation
 
        Rank        Flag           1 
           1           1       0.321 
           2           2       0.229 
           3           3       0.324 
           4           4       0.125 
           5           5       0.395 
 
 (Co-) Kriging weights
 =====================
        Rank          x1          x2        Data         Z1* 
           1     114.631      72.031       0.057       0.150 
           2     116.444      65.590       1.275       0.048 
           3     114.720      79.035      -0.061       0.180 
           4     116.816      94.871       1.144       0.040 
           5      89.960      69.482       0.308       0.351 
 Sum of weights                                     0.769 
 
 (Co-) Kriging results
 =====================
 Target Sample = 15301
 Variable Z1 
  - Estimate  =        0.213 
  - Std. Dev. =        0.860 
  - Variance  =        0.739 
  - Cov(h=0)  =        1.000

The Point Kriging results are displayed (overlaying the control data points)

In [12]:
ax = grid.plot("Point_Kriging*estim")
ax = data.plot(color="white")
ax.decoration(title="Point Kriging with Standard Neighborhood")

We also display the standard deviation map of the Estimation error

In [13]:
ax = grid.plot("Point_Kriging*stdev")
ax = data.plot(color="white")
ax.decoration(title="Error for Point Kriging with Standard Neighborhood")

Block Kriging¶

Performing a Block Kriging with the current Neighborhood feature. Note that the discretization parameters have been set to small numbers in order to let the calculations be performed in a reasonable time frame (for a demonstration file).

In [14]:
err = gl.kriging(data, grid, model, neigh, calcul=gl.EKrigOpt.BLOCK, ndisc=[5,5],
                namconv=gl.NamingConvention("Block_Kriging"))
In [15]:
if flagDebug:
    err = gl.krigtest(data, grid, model, neigh, node, calcul=gl.EKrigOpt.BLOCK, ndisc=[5,5])

 Target location
 ---------------
 Sample #15301 (from 30000)
 Coordinate #1 = 100.000000
 Coordinate #2 = 76.000000
 
 Data selected in neighborhood
 -----------------------------
        Rank      Sample          x1          x2      Sector 
           1          20     114.631      72.031           4 
           2          52     116.444      65.590           4 
           3          56     114.720      79.035           5 
           4          65     116.816      94.871           6 
           5          69      89.960      69.482           1 
 
 LHS of Kriging matrix (compressed)
 ==================================
 Number of active samples    = 5
 Total number of equations   = 5
 Reduced number of equations = 5
 
        Rank                       1           2           3           4           5 
                    Flag           1           2           3           4           5 
           1           1       1.000       0.526       0.518       0.164       0.132 
           2           2       0.526       1.000       0.358       0.069       0.102 
           3           3       0.518       0.358       1.000       0.303       0.106 
           4           4       0.164       0.069       0.303       1.000       0.006 
           5           5       0.132       0.102       0.106       0.006       1.000 
 
 RHS of Kriging matrix (compressed)
 ==================================
 Number of active samples    = 5
 Total number of equations   = 5
 Reduced number of equations = 5
 Number of right-hand sides  = 1
 Block Estimation : Discretization =  5  x  5 
 
        Rank        Flag           1 
           1           1       0.321 
           2           2       0.229 
           3           3       0.324 
           4           4       0.125 
           5           5       0.395 
 
 (Co-) Kriging weights
 =====================
        Rank          x1          x2       Size1       Size2        Data         Z1* 
           1     114.631      72.031       1.000       1.000       0.057       0.150 
           2     116.444      65.590       1.000       1.000       1.275       0.048 
           3     114.720      79.035       1.000       1.000      -0.061       0.180 
           4     116.816      94.871       1.000       1.000       1.144       0.040 
           5      89.960      69.482       1.000       1.000       0.308       0.351 
 Sum of weights                                                           0.769 
 
 (Co-) Kriging results
 =====================
 Target Sample = 15301
 Variable Z1 
  - Estimate  =        0.213 
  - Std. Dev. =        0.652 
  - Variance  =        0.425 
  - Cov(h=0)  =        0.686

The Block Kriging results are displayed (overlaying the control data points)

In [16]:
ax = grid.plot("Block_Kriging*estim")
ax = data.plot(color="white")
ax.decoration(title="Block Kriging with Standard Neighborhood")

We also display the standard deviation map of the Estimation error

In [17]:
ax = grid.plot("Block_Kriging*stdev")
ax = data.plot(color="white")
ax.decoration(title="Error for Block Kriging with Standard Neighborhood")

Comparing the Estimation maps

In [18]:
ax = gp.correlation(grid,namex="Point_Kriging*estim",namey="Block_Kriging*estim", bins=100)

Comparing the Error Estimation maps

In [19]:
ax = gp.correlation(grid,namex="Point_Kriging*stdev",namey="Block_Kriging*stdev", bins=100)

The difference is not very impressive due to the small size of block extensions

Defining variable Block Extensions¶

In this section, we will generate variables in the Grid File, which contain the cell extension. The square Block size is fixed to 50.

In [20]:
size = 50.
iuid = grid.addColumnsByConstant(1, size, "X-ext", gl.ELoc.BLEX, 0)
iuid = grid.addColumnsByConstant(1, size, "Y-ext", gl.ELoc.BLEX, 1)

The now check the neighborhood feature which consists in forcing any sample located within the cell extension centered on the target grid node.

In [21]:
nmini = 1
neighC = gl.NeighCell.create(flag_xvalid=False, nmini=nmini)
neighC.display()

Cell Neighborhood
=================
Reject samples which do not belong to target Block

We check the new neighborhood on the same target grid node as before

In [22]:
neighC.attach(data, grid)
ranks = neighC.select(node)
dataSel = data.clone()
dum = dataSel.addSelectionByRanks(ranks)

The next figure displays the samples selected in the neighborhood of the target node (same as before). As expected all samples lying within the super_block centered on the target node are considered (i.e. 34) rather than the samples which would have been considered in the standard neighborhood case (i.e. 15).

In [23]:
ax = data.plot()
ax = dataSel.plot(color='blue')
ax = gp.sample(target, color='black')
ax = gp.curve(grid.getCellEdges(node), color='black')

We now perform the Super Kriging which is nothing but a standard Kriging with the new neighborhood feature (demonstrated above).

In [24]:
err = gl.kriging(data, grid, model, neighC, calcul=gl.EKrigOpt.BLOCK, ndisc=[5,5],
                 namconv=gl.NamingConvention("Super_Kriging"))
In [25]:
if flagDebug:
    err = gl.krigtest(data, grid, model, neighC, node, calcul=gl.EKrigOpt.BLOCK, ndisc=[5,5])

 Target location
 ---------------
 Sample #15301 (from 30000)
 Coordinate #1 = 100.000000
 Coordinate #2 = 76.000000
 
 Data selected in neighborhood
 -----------------------------
        Rank      Sample          x1          x2 
           1          20     114.631      72.031 
           2          52     116.444      65.590 
           3          56     114.720      79.035 
           4          65     116.816      94.871 
           5          69      89.960      69.482 
 
 LHS of Kriging matrix (compressed)
 ==================================
 Number of active samples    = 5
 Total number of equations   = 5
 Reduced number of equations = 5
 
        Rank                       1           2           3           4           5 
                    Flag           1           2           3           4           5 
           1           1       1.000       0.526       0.518       0.164       0.132 
           2           2       0.526       1.000       0.358       0.069       0.102 
           3           3       0.518       0.358       1.000       0.303       0.106 
           4           4       0.164       0.069       0.303       1.000       0.006 
           5           5       0.132       0.102       0.106       0.006       1.000 
 
 RHS of Kriging matrix (compressed)
 ==================================
 Number of active samples    = 5
 Total number of equations   = 5
 Reduced number of equations = 5
 Number of right-hand sides  = 1
 Block Estimation : Discretization =  5  x  5 
 
        Rank        Flag           1 
           1           1       0.205 
           2           2       0.179 
           3           3       0.206 
           4           4       0.140 
           5           5       0.224 
 
 (Co-) Kriging weights
 =====================
        Rank          x1          x2       Size1       Size2        Data         Z1* 
           1     114.631      72.031      50.000      50.000       0.057       0.075 
           2     116.444      65.590      50.000      50.000       1.275       0.082 
           3     114.720      79.035      50.000      50.000      -0.061       0.089 
           4     116.816      94.871      50.000      50.000       1.144       0.094 
           5      89.960      69.482      50.000      50.000       0.308       0.196 
 Sum of weights                                                           0.535 
 
 (Co-) Kriging results
 =====================
 Target Sample = 15301
 Variable Z1 
  - Estimate  =        0.270 
  - Std. Dev. =        0.266 
  - Variance  =        0.071 
  - Cov(h=0)  =        0.176

The results of the Super Kriging are visualized in the next figure, together with the ones of the standard neighborhood.

In [26]:
fig, axs = plt.subplots(1,2, figsize=(16,8))

axs[0].gstgrid(grid,"Super_Kriging*estim") 
axs[0].gstpoint(data,color="white")
axs[0].decoration(title="Super Block Kriging")

axs[1].gstgrid(grid,"Block_Kriging*estim")
axs[1].gstpoint(data,color="white")
axs[1].decoration(title="Block Kriging with Standard Neighborhood")

We also display the standard deviation map of the Super Kriging Estimation error

In [27]:
fig, axs = plt.subplots(1,2,figsize=(16,8))

axs[0].gstgrid(grid,"Super_Kriging*stdev")
axs[0].gstpoint(data,color="white")
axs[0].decoration(title="Error for Super Block Kriging")

axs[1].gstgrid(grid,"Block_Kriging*stdev")
axs[1].gstpoint(data,color="white")
axs[1].decoration(title="Error for Block Kriging with Standard Neighborhood")

Comparing with Block Kriging with standard block extension (equal to the grid mesh)

In [28]:
ax = gp.correlation(grid,namex="Block_Kriging*estim",namey="Super_Kriging*estim", bins=100)

Comparing the error maps

In [29]:
ax = gp.correlation(grid,namex="Block_Kriging*stdev",namey="Super_Kriging*stdev", bins=100)