SciKit-GStat

SciKit-GStat provides a variety of semi-variance estimators and variogram model functions. Most estimators are implemented using numba's JIT decorator for performance.

Available Estimators

• matheron
• cressie
• dowd
• genton
• entropy
• quantiles (experimental)
• minmax (experimental)

Available Models

Models are available in both nugget and no-nugget variations:

• spherical
• exponential
• gaussian
• cubic
• stable
• matérn

Geostatistics aims to describe and estimate spatial covariance (correlation) in point data. It assumes that closer observation points show higher covariances than distant points.

Inverse Distance Weighting (IDW)

IDW is a simple interpolation technique where a value at an unobserved location is estimated as a distance-weighted mean of surrounding points. The weight $w_i$ is inversely proportional to the Euclidean distance between the unobserved point and the sample point:

$$w_i = \frac{1}{|\overrightarrow{ux_i}|}$$

Geostatistical Approach

Unlike IDW, which relies exclusively on distance, geostatistics derives weights from the actual (co)variance found in the observations. This allows the interpolation to incorporate the specific spatial structure of the data, though it requires stricter statistical assumptions.

SciKit-GStat provides several binning functions used to spatially aggregate a distance matrix into lag classes (bins). These functions are located in the skgstat.binning module and are typically used to group distances into meaningful intervals for variogram analysis.

Available binning methods include:

• even_width_lags: Creates bins with equal widths.
• uniform_count_lags: Creates bins with an equal number of samples.
• auto_derived_lags: Automatically derives lag classes.
• kmeans: Uses K-means clustering to define bins.
• ward: Uses Ward's linkage method for binning.
• stable_entropy_lags: Uses entropy-based methods to define stable lags.

The precision attribute in OrdinaryKriging is a hyperparameter that controls the accuracy of the semivariance estimation in estimate mode.

• Higher precision: Decreases the lag deviation and results in values closer to the exact calculation, but increases calculation time (potentially by an order of magnitude).
• Lower precision: Increases calculation speed but may lead to visible differences in the reconstructed variogram or prediction weights.

The relationship between range and the uncertainty of the lag distance can be thought of as:

d = range / precision

Increasing the precision value decreases the lag deviation.

Scikit-GStat provides several semi-variance estimator functions in the skgstat.estimators submodule. These functions can be used independently of the Variogram class.

Important Requirement: When using these functions directly (instead of through the Variogram class), you must pass an array of pairwise differences to the function, rather than the raw data values themselves.

A key feature of the Variogram class is the ability to mutate parameters in place. When you update attributes such as n_lags, maxlag, or bin_func, the object automatically invalidates dependent results and recalculates the experimental variogram and fitted model.

# Mutating parameters triggers automatic recalculation
V.n_lags = 25
V.maxlag = 500
V.bin_func = 'kmeans'

# The model is now fitted to the new configuration
V.plot()

When calling functions within the skgstat.data module, the returned object is a dictionary containing two specific keys:

1. 'sample': The actual data values/coordinates used for analysis.
2. 'description': A string describing the dataset.

Always access the data via the 'sample' key when passing it to skgstat estimators or variogram models.

from skgstat import data

dataset = data.pancake()
# Access the data for use in scikit-gstat
sample_data = dataset['sample']
print(dataset['description'])

Geostatistics can be categorized into three main functional areas. While scikit-gstat focuses primarily on the first, it provides interfaces to connect with other libraries for the others:

1. Variography: Focuses on describing, visualizing, and modelling covariance structures in space and time using the variogram as the primary tool. scikit-gstat is primarily a variography tool.
2. Kriging: A family of interpolation methods that use a variogram to estimate weights for spatial estimation.
3. Geostatistical Simulation: Aims to generate random fields that fit a given set of observations or a pre-defined variogram/covariance function.

To perform Kriging or Simulation using a variogram modeled in scikit-gstat, you can use the Variogram.to_gstools() method to export your model to the gstools library.

In geostatistics, spatial interpolation is known as Kriging. It is a method used to estimate a variable at an unobserved location using observations from nearby locations.

Kriging is considered a BLUE (Best Linear Unbiased Estimator) because it utilizes the spatial structure described by a variogram to determine the optimal weights for averaging observations. The estimation at an unobserved location $Z^*(s_0)$ is calculated as a weighted mean of the observations $Z(s_i)$:

$Z^*(s_0) = \sum_{i=0}^N \lambda_i Z(s_i)$

SciKit-GStat is a SciPy-styled module for variogram analysis. Most users will primarily interact with the Variogram class, but the library provides specialized classes for different geostatistical tasks:

• Variogram: The base class for standard variogram analysis.
• DirectionalVariogram: Used for computing directional variograms.
• SpaceTimeVariogram: Used for spatio-temporal variogram analysis.
• OrdinaryKriging: Used for spatial interpolation.
• MetricSpace: Used for working with pre-computed spatial samples.

All variogram classes share a similar interface, allowing them to compute experimental variograms and fit theoretical variogram model functions.