The subpackage `gstools.krige`

provides routines for Gaussian process regression,
also known as kriging.
Kriging is a method of data interpolation based on predefined covariance models.

The aim of kriging is to derive the value of a field at some point $x_0$, when there are fixed observed values $z(x_1)\ldots z(x_n)$ at given points $x_i$.

The resluting value $z_0$ at $x_0$ is calculated as a weighted mean:

$z_0 = \sum_{i=1}^n w_i \cdot z_i$

The weights $W = (w_1,\ldots,w_n)$ depent on the given covariance model and the location of the target point.

The different kriging approaches provide different ways of calculating $W$.

The `Krige`

class provides everything in one place and you can switch on/off
the features you want:

`unbiased`

: the weights have to sum up to`1`

. If true, this results in`Ordinary`

kriging, where the mean is estimated, otherwise it will result in`Simple`

kriging, where the mean has to be given.`drift_functions`

: you can give a polynomial order or a list of self defined functions representing the internal drift of the given values. This drift will be fitted internally during the kriging interpolation. This results in`Universal`

kriging.`ext_drift`

: You can also give an external drift per point to the routine. In contrast to the internal drift, that is evaluated at the desired points with the given functions, the external drift has to given for each point form an "external" source. This results in`ExtDrift`

kriging.`trend`

,`mean`

,`normalizer`

: These are used to pre- and post-process data. If you already have fitted a trend model that is provided as a callable function, you can give it to the kriging routine. Normalizer are power-transformations to gain normality.`mean`

behaves similar to`trend`

but is applied at another position:

- conditioning data is de-trended (substracting trend)
- detrended conditioning data is then normalized (in order to follow a normal distribution)
- normalized conditioning data is set to zero mean (subtracting mean)

Cosequently, when there is no normalizer given, trend and mean are the same thing
and only one should be used.
`Detrended`

kriging is a shortcut to provide only a trend and simple kriging
with normal data.

`exact`

and`cond_err`

: To incorporate the nugget effect and/or measurement errors, one can set`exact`

to`False`

and provide either individual measurement errors for each point or set the nugget as a constant measurement error everywhere.`pseudo_inv`

: Sometimes the inversion of the kriging matrix can be numerically unstable. This occurs for examples in cases of redundant input values. In this case we provide a switch to use the pseudo-inverse of the matrix. Then redundant conditional values will automatically be averaged.

## Note¶

All mentioned features can be combined within the `Krige`

class.
All other kriging classes are just shortcuts to this class with a limited list of input parameters.

The routines for kriging are almost identical to the routines for spatial random fields,
with regard to their handling.
First you define a covariance model, as described in :ref:`tutorial_02_cov`

,
then you initialize the kriging class with this model:

```
import gstools as gs
# condtions
cond_pos = [...]
cond_val = [...]
model = gs.Gaussian(dim=1, var=0.5, len_scale=2)
krig = gs.krige.Simple(model, cond_pos=cond_pos, cond_val=cond_val, mean=1)
```

The resulting field instance `krig`

has the same methods as the
`SRF`

class.
You can call it to evaluate the kriged field at different points,
you can plot the latest field or you can export the field and so on.

## Provided Kriging Methods¶

The following kriging methods are provided within the
submodule `gstools.krige`

.

`Krige`

: swiss army knife for kriging`Simple`

:`Krige`

shortcut for simple kriging`Ordinary`

:`Krige`

shortcut for ordinary kriging`Universal`

:`Krige`

shortcut for universal kriging`ExtDrift`

:`Krige`

shortcut for external drift kriging`Detrended`

:`Krige`

shortcut for detrended kriging