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Ordinary Kriging

%matplotlib widget
import matplotlib.pyplot as plt
plt.ioff()
# turn of warnings
import warnings
warnings.filterwarnings('ignore')

Ordinary kriging will estimate an appropriate mean of the field, based on the given observations/conditions and the covariance model used.

The resulting system of equations for WW is given by:

(Wμ)=(C(x1,x1)C(x1,xn)1C(xn,x1)C(xn,xn)1110)1(C(x1,x0)C(xn,x0)1)\begin{align}\begin{pmatrix}W\\\mu\end{pmatrix} = \begin{pmatrix} C(x_1,x_1) & \cdots & C(x_1,x_n) &1 \\ \vdots & \ddots & \vdots & \vdots \\ C(x_n,x_1) & \cdots & C(x_n,x_n) & 1 \\ 1 &\cdots& 1 & 0 \end{pmatrix}^{-1} \begin{pmatrix}C(x_1,x_0) \\ \vdots \\ C(x_n,x_0) \\ 1\end{pmatrix}\end{align}
(1)#

Here, C(xi,xj)C(x_i,x_j) is the directional covariance of the given observations and μ\mu is a Lagrange multiplier to minimize the kriging error and estimate the mean.

#Example

Here we use ordinary kriging in 1D (for plotting reasons) with 5 given observations/conditions. The estimated mean can be accessed by krig.mean.

import numpy as np
import gstools as gs

# condtions
cond_pos = [0.3, 1.9, 1.1, 3.3, 4.7]
cond_val = [0.47, 0.56, 0.74, 1.47, 1.74]
# resulting grid
gridx = np.linspace(0.0, 15.0, 151)
# spatial random field class
model = gs.Gaussian(dim=1, var=0.5, len_scale=2)
krig = gs.Krige(model, cond_pos=cond_pos, cond_val=cond_val, unbiased=True)
field, var = krig(gridx)
ax = krig.plot()
ax.scatter(cond_pos, cond_val, color="k", zorder=10, label="Conditions")
ax.legend()
krig.get_mean()
array(0.39077137)