SPDE¶

This document aims at demonstrating the use of SPDE for performing Estimation. It is based on Scotland Data

In [1]:
%%javascript
IPython.OutputArea.prototype._should_scroll = function(lines) {
return false;
}
In [2]:
import gstlearn as gl
import matplotlib.pyplot as plt
import gstlearn.plot as gp
import os 
import numpy as np
import urllib.request

flagInternetAvailable = True

Getting the Data Bases from the official website.

In [3]:
fileNF='Scotland_Temperatures.NF'
if flagInternetAvailable:
    temp_nf, head = urllib.request.urlretrieve('https://soft.minesparis.psl.eu/gstlearn/data/Scotland/'+fileNF,'./'+fileNF)
else:
    temp_nf='./'+fileNF
dat = gl.Db.createFromNF(temp_nf)

The input Data Base (called temperatures) contains the target variable (January_temp). Note that this data base also contains the elevation variable (called Elevation) which is assigned the locator f (for external drift). We finally add a selection (called sel) which only consider the samples where the temperature is actually calculated.

In [4]:
temperatures = gl.Db.createFromNF(temp_nf)
temperatures.setLocator("January_temp", gl.ELoc.Z)
temperatures.setLocator("Elevation",gl.ELoc.F)
iuid = temperatures.addSelection(np.invert(np.isnan(temperatures["J*"])),"sel")
temperatures.display()
ax = temperatures.plot("January_temp")
ax.decoration(title="Temperature data")

Data Base Characteristics
=========================

Data Base Summary
-----------------
File is organized as a set of isolated points
Space dimension              = 2
Number of Columns            = 6
Total number of samples      = 236
Number of active samples     = 151

Variables
---------
Column = 0 - Name = rank - Locator = NA
Column = 1 - Name = Longitude - Locator = x1
Column = 2 - Name = Latitude - Locator = x2
Column = 3 - Name = Elevation - Locator = f1
Column = 4 - Name = January_temp - Locator = z1
Column = 5 - Name = sel - Locator = sel

The output file is a grid (called grid). It contains an active selection (inshore) as well as the elevation over the field (called Elevation). Note that this variable is assigned the locator f (external drift).

In [5]:
fileNF='Scotland_Elevations.NF'
if flagInternetAvailable:
    elev_nf, head = urllib.request.urlretrieve('https://soft.minesparis.psl.eu/gstlearn/data/Scotland/'+fileNF,'./'+fileNF)
else:
    elev_nf='./'+fileNF
    

grid = gl.DbGrid.createFromNF(elev_nf)
grid.display()
ax = grid.plot("Elevation")
ax.decoration(title="Elevation")

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

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

Calculate the omni-directional variogram of the temperature for 18 lags of 25.

In [6]:
vparam = gl.VarioParam.createOmniDirection(npas=18, dpas=25)
vario = gl.Vario(vparam,temperatures)
vario.compute()
vario.display()
ax = gp.variogram(vario)
ax.decoration(title="Variogram of Temperature (Raw)")

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              = 18
Direction coefficients      =      1.000     0.000
Direction angles (degrees)  =      0.000     0.000
Tolerance on direction      =     90.000 (degrees)
Calculation lag             =     25.000
Tolerance on distance       =     50.000 (Percent of the lag value)

For variable 1
      Rank    Npairs  Distance     Value
         0    78.000     7.681     0.137
         1   512.000    26.905     0.376
         2   915.000    50.556     0.614
         3  1135.000    74.820     0.855
         4  1285.000   100.070     0.935
         5  1190.000   124.927     1.050
         6  1117.000   149.550     1.099
         7  1004.000   175.060     1.218
         8   924.000   199.603     1.175
         9   769.000   224.493     1.502
        10   594.000   249.105     1.377
        11   438.000   274.605     1.316
        12   363.000   298.685     1.071
        13   236.000   323.920     1.190
        14   153.000   349.725     1.321
        15   105.000   373.088     1.219
        16    76.000   399.464     0.802
        17    58.000   426.095     0.677

We calculate the variogram (using the same calculation parameters) based on the residuals after a trend has been removed. This trend is considered as a linear combinaison of the external drift information.

In [7]:
vparam = gl.VarioParam.createOmniDirection(npas=18, dpas=25)
vario = gl.Vario(vparam,temperatures)
md = gl.Model()
md.setDriftIRF(order=0, nfex=1)
vario.compute(model=md)
vario.display()
ax = gp.variogram(vario)

Variogram characteristics
=========================
Number of variable(s)       = 1
Number of direction(s)      = 1
Space dimension             = 2
Variance-Covariance Matrix     0.363

Direction #1
------------
Number of lags              = 18
Direction coefficients      =      1.000     0.000
Direction angles (degrees)  =      0.000     0.000
Tolerance on direction      =     90.000 (degrees)
Calculation lag             =     25.000
Tolerance on distance       =     50.000 (Percent of the lag value)

For variable 1
      Rank    Npairs  Distance     Value
         0    78.000     7.681     0.102
         1   512.000    26.905     0.163
         2   915.000    50.556     0.211
         3  1135.000    74.820     0.264
         4  1285.000   100.070     0.265
         5  1190.000   124.927     0.321
         6  1117.000   149.550     0.363
         7  1004.000   175.060     0.444
         8   924.000   199.603     0.466
         9   769.000   224.493     0.547
        10   594.000   249.105     0.561
        11   438.000   274.605     0.582
        12   363.000   298.685     0.472
        13   236.000   323.920     0.565
        14   153.000   349.725     0.424
        15   105.000   373.088     0.345
        16    76.000   399.464     0.410
        17    58.000   426.095     0.411

Fit the variogram of residuals in a model having drifts. Some constraints have been added during the fitting step.

In [8]:
model = md

structs = [gl.ECov.NUGGET,gl.ECov.BESSEL_K]

consNug = gl.ConsItem.define(gl.EConsElem.SILL,0, type = gl.EConsType.UPPER,value = 0.1)

cons1P  = gl.ConsItem.define(gl.EConsElem.PARAM,1, type = gl.EConsType.EQUAL,value = 1)
cons1Rm = gl.ConsItem.define(gl.EConsElem.RANGE,1, type = gl.EConsType.LOWER,value = 100)
cons1RM = gl.ConsItem.define(gl.EConsElem.RANGE,1, type = gl.EConsType.UPPER,value = 350)

a = gl.Constraints()
a.addItem(consNug)
a.addItem(cons1P)
a.addItem(cons1Rm)
a.addItem(cons1RM)

err = model.fit(vario,structs,constraints=a)
model.display()
ax = gp.varmod(vario,model)
ax.decoration(title="Vario of the residuals")

Model characteristics
=====================
Space dimension              = 2
Number of variable(s)        = 1
Number of basic structure(s) = 2
Number of drift function(s)  = 2
Number of drift equation(s)  = 2

Covariance Part
---------------
Nugget Effect
- Sill         =      0.100
K-Bessel (Third Parameter = 1)
- Sill         =      0.420
- Range        =    282.586
- Theo. Range  =     81.575
Total Sill     =      0.520

Drift Part
----------
Universality_Condition
External_Drift:0

Derive the parameter of a Global Trend (composed of the Elevation as a drift function) using the SPDE formalism.

In [9]:
spde = gl.SPDE(model,grid,temperatures,gl.ESPDECalcMode.KRIGING)
coeffs = spde.getCoeffs()
print("Trend coefficients:", coeffs)

Trend coefficients: [ 3.97005867 -0.00659771]

Represent the scatter plot of the Temperature given the Elevation and add the Global Trend (calculated beforehand)

In [10]:
ax = gp.correlation(temperatures, namex="Elevation", namey="*temp", asPoint=True)
if len(coeffs)>1:
    plt.plot([0,400], [coeffs[0],coeffs[0]+coeffs[1]*400])

We perform the Estimation in the SPDE framework (considering the Elevation as the Global Trend)

In [11]:
err = gl.krigingSPDE(temperatures, grid, model)
ax = grid.plot(useSel=True)
ax.decoration(title="Temperature (using Elevation as global Trend)")

We also perform the Estimation by Kriging (using Elevation as External Drift). This estimation is performed in Unique Neighborhood.

In [12]:
neighU = gl.NeighUnique.create();
gl.kriging(temperatures, grid, model, neighU);
ax = grid.plot(useSel=True)
ax.decoration(title="Temperature (with Elevation as External Drift)")

Comparing both estimates

In [13]:
grid
Out[13]:
Data Base Grid Characteristics
==============================

Data Base Summary
-----------------
File is organized as a regular grid
Space dimension              = 2
Number of Columns            = 7
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

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
Column = 4 - Name = KrigingSPDE.January_temp.estim - Locator = NA
Column = 5 - Name = Kriging.January_temp.estim - Locator = z1
Column = 6 - Name = Kriging.January_temp.stdev - Locator = NA
In [14]:
ax = gp.correlation(grid,namex="Kriging.January_temp.estim",namey="KrigingSPDE.January_temp.estim",
                    asPoint=True, diagLine=True)
ax.decoration(xlabel="Kriging with External Drift", ylabel="SPDE Kriging")