r"""
Ordinary Kriging
----------------

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 :math:`W` is given by:

.. math::

   \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}

Thereby :math:`c(x_i,x_j)` is the covariance of the given observations
and :math:`\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

from gstools import Gaussian, krige

# 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 = Gaussian(dim=1, var=0.5, len_scale=2)

###############################################################################
krig = krige.Ordinary(model, cond_pos=cond_pos, cond_val=cond_val)
krig(gridx)

###############################################################################
ax = krig.plot()
ax.scatter(cond_pos, cond_val, color="k", zorder=10, label="Conditions")
ax.legend()