In geostatistical analysis the mean function of the spatial process is specified by a parametric model, $\mu(s; \beta)$. The most commonly used parametric mean model is a linear function, given by

$$\mu(\mathbf{s}; \mathbf{\beta}) = \mathbf X(s)^T\mathbf{\beta}\, ,$$

where $X(s)$ is a vector of covariates (explanatory variables) observed at $s$, and $\mathbf{\beta}$ is a parameter vector. The covariates may also include the geographic coordinates (e.g., latitude and longitude) of $s$, mathematical functions (such as polynomials) of those coordinates, and attribute variables.

The standard method for fitting a provisional linear mean function to geostatistical data is ordinary least squares (OLS). This method yields the OLS estimator $\hat{\beta}_{OLS}$ of $\beta$, given by

$$\hat{\beta}_{OLS} = \text{argmin} \sum_{i=1}^n[Z(\mathbf s_i) -\mathbf X(\mathbf s_i)^T\mathbf{\beta}]^2 \, .$$

Fitted values and fitted residuals at data locations are given by $\hat z = \mathbf{X^T}\mathbf{\hat{\beta}}_{OLS}$ and $\hat e= z-\hat z$, respectively. The latter are passed to the second stage of the geostatistical analysis, to be described in the next section.

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The E-Learning project SOGA-Py was developed at the Department of Earth Sciences by Annette Rudolph, Joachim Krois and Kai Hartmann. You can reach us via mail by soga[at]zedat.fu-berlin.de.

You may use this project freely under the Creative Commons Attribution-ShareAlike 4.0 International License.

Please cite as follow: Rudolph, A., Krois, J., Hartmann, K. (2023): Statistics and Geodata Analysis using Python (SOGA-Py). Department of Earth Sciences, Freie Universitaet Berlin.