Ordinary kriging

Kriging implements local ordinary kriging. For each target, it combines nearby source values using the spatial correlation described by the fitted variogram. The weights give the best linear unbiased estimator (BLUE) under that model (Matheron, 1971; Cressie, 1993; Isaaks and Srivastava, 1989).

Setup

Given $n$ source points with values $z_1, \dots, z_n$ , the estimate at a target $x_0$ is a weighted sum

\hat{z}(x_0) = \sum_{i=1}^{n} \lambda_i \, z_i

The weights $\lambda_i$ are chosen so that the estimator is:

Unbiased: $\sum_{i=1}^{n} \lambda_i = 1$ . In ordinary kriging, the process mean is unknown but treated as constant within the local neighborhood. This constraint accounts for that unknown mean.
Minimum variance: among all weights that satisfy the unbiasedness constraint, the chosen weights minimize $\operatorname{Var}[\hat{z}(x_0) - z(x_0)]$ . The semivariance $\gamma(h)$ describes how similar values tend to be at separation $h$ .

The kriging system

Minimizing variance subject to a linear equality constraint is a Lagrange-multiplier problem. Introducing multiplier $\mu$ for the unbiasedness constraint and solving the resulting stationarity conditions gives a linear system in the $n$ weights plus $\mu$ :

\begin{bmatrix} \gamma(d_{11}) & \cdots & \gamma(d_{1n}) & 1 \\ \vdots & \ddots & \vdots & \vdots \\ \gamma(d_{n1}) & \cdots & \gamma(d_{nn}) & 1 \\ 1 & \cdots & 1 & 0 \end{bmatrix} \begin{bmatrix} \lambda_1 \\ \vdots \\ \lambda_n \\ \mu \end{bmatrix} = \begin{bmatrix} \gamma(d_{10}) \\ \vdots \\ \gamma(d_{n0}) \\ 1 \end{bmatrix}

where $d_{ij}$ is the distance between source points $i$ and $j$ , and $d_{i0}$ is the distance from source point $i$ to the target $x_0$ . The $(n+1) \times (n+1)$ matrix on the left is symmetric except for the constraint row/column; the bottom row and right column (all ones, with a zero corner) are exactly what enforces $\sum \lambda_i = 1$ .

Solving this system gives both the kriging weights $\lambda_i$ and the kriging variance -- an estimate of the uncertainty at $x_0$ :

\sigma_K^2(x_0) = \sum_{i=1}^{n} \lambda_i \, \gamma(d_{i0}) + \mu

Unlike IDW's fixed distance-decay weights, kriging accounts for redundancy between clustered source points. Two nearby points that are also close to each other contribute less independent information than two points spread farther apart at the same average distance from the target.

Local kriging

Solving the full system over every source point would become expensive as the dataset grows. Kriging instead builds and solves the system for each target using only the nearest max_neighbors source points within radius, just as IDW does. min_neighbors sets a floor below which a target returns NaN rather than a poorly constrained estimate.

Supplying a variogram directly

model_values lets you pass a precomputed variogram model instead of fitting one from x/y/z. This is useful when several interpolation runs should share exactly one model, as in apbase.cross_validate, which fits one variogram and reuses it for both IDW and kriging.