In [1]:
import gstlearn as gl
import gstlearn.test as gt
import gstlearn.document as gdoc
import numpy as np
from IPython.display import Markdown
from scipy.spatial import distance_matrix
In [2]:
Markdown(gdoc.loadDoc("Kriging.md"))
Out[2]:

Kriging¶

Let suppose that

$Z(s_i) = X_i\beta + Y(s_i)$

where $Y$ is a second order stationary random field with mean 0 and covariance function $C$.

If $Z$ is a vector of observations, we denote $Z = X\beta + Y$ with $\Sigma$ the covariance of Y

Simple Kriging¶

If $\beta$ is known, we can obtain the simple kriging

$Z_0^{SK} = X_0\beta + \Sigma_0^t\Sigma^{-1}(Z-X\beta) = X_0\beta + \lambda_{SK}^t(Z-X\beta)$

with:

  • the simple kriging weights

$\lambda_{SK}=\Sigma^{-1}\Sigma_0$

  • the variance of the estimator

$\textrm{Var}(Z_0^{SK})=\lambda_{SK}^t\Sigma\lambda_{SK}=\lambda_{SK}^t\Sigma_0$

  • the estimation variance

$\sigma_{SK}^2 = \textrm{Var}(Z_0-Z_0^{SK}) = \sigma_0^2-\Sigma_0^t\Sigma^{-1}\Sigma_0=\sigma_0^2-\lambda_{SK}^t\Sigma_0$

In matrix notation:¶

Simple Kriging System

$ \begin{bmatrix} \Sigma \end{bmatrix} \times \begin{bmatrix} \lambda_{SK} \end{bmatrix} = \begin{bmatrix} \Sigma_0 \end{bmatrix} $

Estimation

$ Z_0^{SK} = \begin{bmatrix} Z \end{bmatrix}^t \times \begin{bmatrix} \lambda_{SK} \end{bmatrix} + m ({ 1 - \sum{\lambda_{SK}}} ) $

Variance of Estimation error

$ \sigma_{SK}^2 = \sigma_0^2 - \begin{bmatrix} \lambda_{SK} \end{bmatrix}^t \times \begin{bmatrix} \Sigma_0 \end{bmatrix} $

Variance of Estimator

$ \textrm{Var}(Z_0^{SK}) = \begin{bmatrix} \lambda_{SK} \end{bmatrix}^t \times \begin{bmatrix} \Sigma \end{bmatrix} \times \begin{bmatrix} \lambda_{SK} \end{bmatrix} = \begin{bmatrix} \lambda_{SK} \end{bmatrix}^t \times \begin{bmatrix} \Sigma_0 \end{bmatrix} $

Universal kriging¶

If $\beta$ is unknown, we can estimate it by

$\hat\beta = \Sigma_c X^t\Sigma^{-1}Z$

Introducing the notation

$\Sigma_c = (X^t\Sigma^{-1}X)^{-1} $

then

$\hat\beta = \Sigma_c X^t\Sigma^{-1}Z$

$\textrm{Var}(\hat\beta)=\Sigma_c$

The Universal kriging is obtained by first computing $\hat\beta$ and then pluging $\hat\beta$ in the simple kriging procedure.

$Z^{UK}_0 = X_0\hat\beta + \Sigma_0^t\Sigma^{-1}(Z-X\hat\beta)= \Sigma_0^t\Sigma^{-1}Z + (X_0 - \Sigma_0^t\Sigma^{-1}X)\hat\beta$

We can rewrite everything with respect to $Z$

$Z^{UK}_0 = (\Sigma_0^t\Sigma^{-1} + (X_0 - \Sigma_0^t\Sigma^{-1}X)\Sigma_c X^t\Sigma^{-1})Z \\ =(\lambda_{SK}^t+(X_0-\lambda_{SK}^tX) \Sigma_c X^t\Sigma^{-1})Z\\ =\lambda_{UK}^tZ$

with

  • the Universal Kriging Weights

$\lambda_{UK}=\lambda_{SK}+\Sigma^{-1}X \Sigma_c(X_0^t-X^t\lambda_{SK})$

  • the variance of the estimator is

$\textrm{Var}(Z^{UK}_0) = \lambda_{UK}^t\Sigma\lambda_{UK} \\ =\textrm{Var}(Z^{SK}_0) +2\lambda_{SK}^tX \Sigma_c \Sigma_c (X_0^t-X^t\lambda_{SK})+(X_0-\lambda_{SK}^tX)\Sigma_c X^t\Sigma^{-1}X\Sigma_c (X_0^t-X^t\lambda_{SK})\\ =\textrm{Var}(Z^{SK}_0) +2\lambda_{SK}^tX\Sigma_c (X_0^t-X^t\lambda_{SK})+(X_0-\lambda_{SK}^tX)\Sigma_c (X_0^t-X^t\lambda_{SK})\\ =\textrm{Var}(Z^{SK}_0)+(\lambda_{SK}^tX+X_0)\Sigma_c (X_0^t-X^t\lambda_{SK})\\ =\textrm{Var}(Z^{SK}_0)-\lambda_{SK}^tX\Sigma_c X^t\lambda_{SK}+X_0 \Sigma_c X_0^t$

  • the estimation variance

$\sigma_{UK}^2 = \sigma_0^2 - 2\textrm{Cov}(Z_0,Z^{UK}_0)+ \textrm{Var}(Z^{UK}_0)\\ = \sigma_0^2 -2\Sigma_0^t\lambda_{UK}+\textrm{Var}(Z^{UK}_0)\\ = \sigma_0^2 -2\Sigma_0^t(\lambda_{SK}+\Sigma^{-1}X \Sigma_c(X_0^t-X^t\lambda_{SK}))+\textrm{Var}(Z^{SK}_0)-\lambda_{SK}^tX \Sigma_c X^t\lambda_{SK}+X_0 \Sigma_c X_0^t\\ = \sigma_0^2 -\Sigma_0^t\lambda_{SK} -2\Sigma_0^t\Sigma^{-1}X \Sigma_c (X_0^t-X^t\lambda_{SK})-\lambda_{SK}^tX \Sigma_c X^t\lambda_{SK}+X_0 \Sigma_c X_0^t\\ =\sigma_{SK}^2-2\lambda_{SK}^tX \Sigma_c (X_0^t-X^t\lambda_{SK})-\lambda_{SK}^tX \Sigma_c X^t\lambda_{SK}+X_0 \Sigma_c X_0^t\\ =\sigma_{SK}^2+(X_0-\lambda_{SK}^tX) \Sigma_c (X_0^t-X^t\lambda_{SK}) $

In matrix notation:¶

Universal Kriging System

$ \begin{bmatrix} \Sigma & X \\ X^t & 0 \end{bmatrix} \times \begin{bmatrix} \lambda_{UK} \\ -\mu \end{bmatrix} = \begin{bmatrix} \Sigma_0 \\ X_0^t \end{bmatrix} $

Estimation

$ Z_0^{UK} = \begin{bmatrix} Z \\ 0 \end{bmatrix}^t \times \begin{bmatrix} \lambda_{UK} \\ -\mu \end{bmatrix} $

Variance of estimation error

$ \sigma_{UK}^2 = \sigma_0^2 - \begin{bmatrix} \lambda_{UK} \\ -\mu \end{bmatrix}^t \times \begin{bmatrix} \Sigma_0 \\ X_0^t \end{bmatrix} $

Variance of estimator

$ \textrm{Var}(Z^{UK}_0) = \begin{bmatrix} \lambda_{UK} \end{bmatrix}^t \times \begin{bmatrix} \Sigma \end{bmatrix} \times \begin{bmatrix} \lambda_{UK} \end{bmatrix} $

Summary¶

Simple Kriging¶

  • the estimator

$Z_0^{SK} = X_0\beta + \lambda_{SK}^t(Z-X\beta) = \lambda_{SK}^tZ +(X_0 -\lambda_{SK}^tX)\beta$

where

$\lambda_{SK}=\Sigma^{-1}\Sigma_0$

  • the variance of the estimator

$\textrm{Var}(Z_0^{SK})=\lambda_{SK}^t\Sigma\lambda_{SK}=\lambda_{SK}^t\Sigma_0$

  • the variance of the estimation error

$\sigma_{SK}^2 = \textrm{Var}(Z_0-Z_0^{SK}) = \sigma_0^2-\Sigma_0^t\Sigma^{-1}\Sigma_0=\sigma_0^2-\lambda_{SK}^t\Sigma_0$

Universal Kriging¶

$\hat\beta = \Sigma_c X^t\Sigma^{-1}Z$

$\textrm{Var}(\hat\beta)= \Sigma_c $

  • the estimator

$Z^{UK}_0 =\lambda^t_{SK}Z + (X_0 - \lambda_{SK}^tX)\hat\beta= \lambda^t_{UK}Z$

$\lambda_{UK}=\lambda_{SK}+\Sigma^{-1}X \Sigma_c (X_0^t-X^t\lambda_{SK})$

$\mu_{UK}=\Sigma_c (X_0 - \lambda_{SK}^tX)^t$

  • the variance of the estimator

$\textrm{Var}(Z^{UK}_0) = \textrm{Var}(Z^{SK}_0)-\lambda_{SK}^tX\Sigma_c X^t\lambda_{SK}+X_0\Sigma_c X_0^t$

  • the variance of the estimation error

$\sigma_{UK}^2 =\sigma_{SK}^2+(X_0-\lambda_{SK}^tX) \Sigma_c (X_0^t-X^t\lambda_{SK}) $

Collocated Option (Unique Neighborhood)¶

This is an interesting case for:

  • estimating a Target
  • in a multivariate case (say with N variables)
  • based on the information in the input Db (note that all 'N' variable(s) do not have to be known in the 'heterotopic' case)
  • when the Target contains information on some of the 'N' variables (these are the collocated variables)

When working in Unique Neighborhood, the relevant information (from the input data base) do not change for any of the targets. But the Collocated information changes at each target.

Hence the interest of benefiting of the inversion of the covariance matrix (restricted to the information of the Input data base).

In [3]:
Markdown(gdoc.loadDoc("Kriging_Bayes.md"))
Out[3]:

Bayesian framework¶

In the Bayesian framework, we assume that

$\beta\sim\mathcal{N}(\beta_0,S)$

We obtain

$\beta|Z\sim\mathcal{N}(\mu_c,\Sigma_c)$

$\Sigma_c = (X^t\Sigma^{-1}X+S^{-1})^{-1}$

and

$\mu_c=\Sigma_c(S^{-1}\beta_0+X^t\Sigma^{-1}Z)$

We obtain the Bayesian quantities:

  • the estimator

$Z^{Bayes}_0 =\lambda^t_{SK}Z + (X_0 - \lambda_{SK}^tX)\mu_c$

  • the variance of the estimator

$\textrm{Var}(Z^{Bayes}_0) = \textrm{Var}(Z^{SK}_0)-\lambda_{SK}^tX\Sigma_c X^t\lambda_{SK}+X_0\Sigma_c X_0^t$

  • the variance of the estimation error

$\sigma_{Bayes}^2 =\sigma_{SK}^2+(X_0-\lambda_{SK}^tX)\Sigma_c(X_0^t-X^t\lambda_{SK}) $
In [4]:
def cova(x,sills=1):
    return np.kron(sills,np.exp(-x/2))

np.random.seed(1234)
A = np.random.normal(size=(3,3))
sills = gl.VectorDouble((A@A.T).reshape(1,-1)[0])
model = gl.Model.createFromParam(gl.ECov.EXPONENTIAL,range = 2.,flagRange=False,sills=sills)

nx = [10,10]

def matriceReduce(m,ind):
    M = gl.MatrixSquareSymmetric(len(ind))
    for i,j in enumerate(ind):
        for k in range(j,len(ind)):
            M.setValue(i,k,m.getValue(j,ind[k]))
    return M

def modelReduce(model,var):
    ctxt=model.getContext()
    ctxt.setNVar(len(var))
    modeln = gl.Model.create(ctxt)
    covlist = model.getCovAnisoList()
    
    for i in range(covlist.getCovaNumber()):
        cova = covlist.getCova(i)
        sills = matriceReduce(cova.getSill(),var)
        covar = gl.CovAniso.createAnisotropicMulti(ctxt,
                                             cova.getType(),
                                             cova.getRanges(),
                                             sills,
                                             cova.getParam(),
                                             cova.getAnisoAngles())
        modeln.addCov(covar)
    return modeln


def createDbIn(ndat,nvar,percent,ndim=2,selDbin=False,measurement_error=False,ndrift = 0,
               flag_isotopic=False,seed=1234):
    db = gl.Db.create()
    np.random.seed(seed)
    for i in range(ndim):
        db["x" +str(i)] = np.random.uniform(size = ndat)
     
    db.setLocators(["x*"],gl.ELoc.X)
        
    indIn = np.arange(ndat)
    if selDbin:
        np.random.shuffle(indIn)
        indIn = np.sort(indIn)
        indIn = indIn[range(int(ndat/2))]
        sel = np.zeros(shape=ndat)
        sel[indIn] = 1
        db["sel"] = sel
        db.setLocator("sel",gl.ELoc.SEL)
      
    #Creation of an heterotopic data set
    indList = [] 
    for i in range(nvar):
        u = np.array([None for i in range(ndat)])
        ind = np.arange(ndat)
        if not flag_isotopic and nvar>1: 
            np.random.shuffle(ind)
            ind = ind[range(int(ndat*percent[i]))]
        ind = np.sort(list(set(ind) & set(indIn)))
        indList += [ind]
        vect = np.array([None for i in range(ndat)])
        vect[ind] = np.random.normal(size = len(ind))
        db["z"+str(i)]=vect
          
    db.setLocators(["z*"],gl.ELoc.Z)
    
    indF = []
    
    for i in range(nvar):
        indF += list(np.array(indList[i]) + ndat * i)
    
    if measurement_error :
        for i in range(nvar):
            db["err"+str(i)] = 0.1 * np.random.uniform(size = ndat)
            
        db.setLocators(["err*"],gl.ELoc.V)
    
    if ndrift>0:
        for i in range(ndrift):
            db["ff" + str(i)] = np.random.normal(size = ndat)
        db.setLocator("ff*",gl.ELoc.F)
    
    return db,indF


def test_covmat(ndat,nx,nvar,percent,model,cova,
                 irf=None,drift=False,measurement_error=False,compute_vars = True,
                 selDbin = True, selDbout = True,flag_isotopic=True,
                 seed=1234,tol=1e-12,eps=1e-3,test = True,verbose=False):
    
    np.random.seed(seed)
    ndrift = 1 if drift else 0
    modeln = modelReduce(model,range(nvar))
    #### Create the description of the case #####
    casetxt = "case:\n"
    
    inter = ""
    if nvar > 1:
        inter = "co-"
        
    if irf is None and drift:
        return 0
    
    if irf is None and not drift:
        casetxt += "- simple "+ inter+ "kriging\n"
    else :
        if irf is not None :
            casetxt += "- KU with drift of degree " + str(irf) + "\n"
        if drift :
            casetxt +="- with external drift\n"
    if nvar > 1:
        casetxt +="- number of covariables for co-kriging " + str(nvar) + "\n"
        if flag_isotopic:
            casetxt += "- isotopic case\n"
        else:
            casetxt += "- heterotopic case\n"
    if measurement_error:
        casetxt += "- with measurement error\n"
    else:
        casetxt += "- without measurement error\n"
    if compute_vars:
        casetxt += "- no dual form\n"
    else:
        casetxt += "- dual\n"
    casetxt += "- selection on Dbin " + str(selDbin) + "\n"
    casetxt += "- selection on Dbout "+ str(selDbout) + "\n"
    casetxt += "- number of data " + str(ndat) + "\n"
    casetxt += "- nx = ["+str(nx[0]) +"," + str(nx[1]) + "]\n"
    
    if verbose:
        print(casetxt)
        
    ##################################################
    db,indF = createDbIn(ndat,nvar,percent,2,selDbin,measurement_error,ndrift,flag_isotopic,seed)
    
    target = gl.DbGrid.create(nx = nx)
   
    indOut = np.arange(target.getSampleNumber())
    
    if selDbout:
        np.random.shuffle(indOut)
        indOut = indOut[range(int(target.getSampleNumber()/2))]
        indOut = np.sort(indOut)
        sel = np.zeros(shape = target.getSampleNumber())
        sel[indOut] = 1
        target["sel"] = sel
        target.setLocator("sel",gl.ELoc.SEL)
                  
    if irf is not None:
        modeln.setDriftIRF(irf)
    
    if drift :
        target["ff"] = np.random.normal(size = target.getSampleNumber())
        
        target.setLocator("ff",gl.ELoc.F)
        modeln.addDrift(gl.DriftF(0))
      
    v = np.array([db["x0"],db["x1"]]).T
    v0 = np.array([target["x1"][indOut],target["x2"][indOut]]).T
    cov = cova(distance_matrix(v,v),modeln.getSillValues(0).toTL())[indF,:][:,indF]
    c0  = cova(distance_matrix(v,v0),modeln.getSillValues(0).toTL())[indF,:]
    #Creation of a db2 without selection to build the complete covariance matrix
    db2 = db.clone()
    db2.setLocator("sel")
    vect = gl.VectorDouble(nvar**2 * db2.getSampleNumber()**2)
    
    target2 = target.clone()
    target2.setLocator("sel")
    covt = modeln.evalCovMatrixSymmetric(db).toTL()
    c0gl = modeln.evalCovMatrix(db,target).toTL()
    
    if measurement_error:
        err = db["err*"].T.reshape(-1,)
        np.fill_diagonal(cov,np.diag(cov)+err[indF])
    
    vect = gl.VectorDouble(nvar**2 * db2.getSampleNumber() * len(indOut))
    
    neigh = gl.NeighUnique()
    
    errcode = 0
    diff = np.max(np.abs(cov-covt))
    if diff > eps:
        errcode = errcode + 1
        print(f"Difference in Cov  = {round(diff,4)}")
    diff = np.max(np.abs(c0-c0gl))
    if diff > eps:
        errcode = errcode + 1
        print(f"Difference in Cov0 = {round(diff,4)}")
    if irf is not None or drift:
        driftd = np.kron(np.eye(nvar),modeln.getDrifts(db2, False))[:,indF]
        driftt = np.kron(np.eye(nvar),modeln.getDrifts(target, True))
        driftdgl = modeln.evalDriftMatrix(db).toTL().T
        drifttgl = modeln.evalDriftMatrix(target).toTL().T
        diff = np.max(np.abs(driftdgl-driftd))
        if diff > eps:
            errcode = errcode + 1
            print(f"Difference in X  = {round(diff,4)}")
        diff = np.max(np.abs(drifttgl-driftt))
        if diff > eps:
            errcode = errcode + 1
            print(f"Difference in X0 = {round(diff,4)}")

    return errcode
In [5]:
percent = [0.5,0.9,1.]
nerr = 0
ntest = 0
for irf in [None,0,1]:
  for drift in [False,True]:
    for measurement_error in [True, False]:
      for selDbin in [True, False]:
        for selDbout in [True, False]:
          for nx in [[5,5]]:
            for nvar in [1,2,3]:
              isolist = [True]
              if nvar >1 :
                isolist = [True,False]
              for iso in isolist:
                for cv in [False,True]:
                  errcode = test_covmat(40,nx,nvar,percent,
                                        model,cova,compute_vars=cv,
                                        irf=irf,drift=drift,
                                        measurement_error=measurement_error,
                                        selDbin=selDbin,selDbout=selDbout,
                                        flag_isotopic = False,
                                        seed=1234,tol=1e-8,eps=1e-3,verbose=False)
                  nerr = nerr + errcode
                  ntest = ntest + 1
                  
print(ntest,"test(s) have been performed with", nerr, "error(s)")

480 test(s) have been performed with 0 error(s)