Cross-validation

apbase.cross_validate is the model-selection step behind apbase.Map. It compares IDW and ordinary kriging on the same source data using leave-one-out cross-validation (LOO-CV). The pipeline then uses the method with the lowest validation error for that dataset.

Leave-one-out procedure

For each evaluated source point $x_i$ :

Temporarily remove $x_i$ (and its value $z_i$ ) from the source set.
Re-estimate a value $\hat{z}_i$ at $x_i$ 's location using only the remaining $n - 1$ points, with the method under evaluation (IDW or kriging).
Compare $\hat{z}_i$ to the true, held-out $z_i$ .

Repeating this for every evaluated point gives prediction errors $e_i = z_i - \hat{z}_i$ . These errors show how well the method may generalize to a new observation at each location because each point is predicted without using its own value.

Error metrics

Given $m$ evaluated points:

\mathrm{MAE} = \frac{1}{m}\sum_{i=1}^{m} \left| e_i \right| \qquad\qquad \mathrm{RMSE} = \sqrt{\frac{1}{m}\sum_{i=1}^{m} e_i^{2}}

R^2 = 1 - \frac{\displaystyle\sum_{i=1}^{m} e_i^{2}} {\displaystyle\sum_{i=1}^{m} \left(z_i - \bar{z}\right)^{2}}

RMSE penalizes large individual errors more than MAE, which helps reveal a method that is usually accurate but occasionally very wrong. $R^2$ expresses how much of the source data's variance the method explains compared with always predicting the mean.

Shared variogram, shared neighborhood

Both methods use the same fitted variogram, search radius, and spatial index. Differences in MAE, RMSE, and $R^2$ therefore reflect the interpolation method rather than a difference in configuration.

Bounded runtime

Evaluating every source point requires $O(n)$ leave-one-out fits, which can be expensive for large datasets. max_points caps the number of points tested. If there are more finite points than the limit, APbase draws a random subsample of size max_points, using seed when provided. This keeps runtime bounded as the dataset grows.

apbase.Map runs this comparison automatically and interpolates with whichever method scored the lower RMSE. apbase.create_map is the functional shortcut for the same pipeline.

Cokriging: three methods in one pass

cross_validate_cokriging applies the same leave-one-out procedure to collocated, ICM, and LMC cokriging together. Only the primary value is held out one point at a time. Each method's coregionalization model is fitted once from the full dataset and kept fixed throughout, matching how the model is used in practice. All three methods share one primary spatial index and one radius, so their results remain directly comparable.

Selecting the best of more than two methods

select_best_model compares scores on the samples where every candidate method produced a finite prediction. With more than two methods, it first identifies the winner and runner-up from the point estimates, then uses a paired bootstrap for that pair. With a plain cross_validate(...) result, which compares IDW and kriging, it keeps the two-method behavior.

A paired bootstrap over the common-valid sample estimates whether the margin between the winner and runner-up is likely genuine rather than an artifact of which points were held out. n_bootstrap resamples the common-valid indices, recomputes the score difference for each resample, and sets significant=True when the resulting confidence interval excludes zero.