Spatial structure of uncertainty
At what resolution does uncertainty operate? Per-pixel uncertainty is the default assumption, but whether it is meaningful depends on the model’s effective receptive field and the spatial scale of the prediction errors. A model with a \(256 \times 256\) pixel input patch does not make independent predictions at each pixel as neighboring pixels share the same input context and will have correlated uncertainties. Reporting per-pixel uncertainty in this setting is not wrong exactly but it may overstate the spatial resolution at which the uncertainty is informative. In reality, map errors are spatially correlated. If the model overpredicts biomass at one pixel, it likely overpredicts at neighbouring pixels as well, because the same model biases, input errors, and training data gaps affect nearby locations.
This has a consequence for any user who aggregates pixel-level predictions into regional estimates such as total biomass in a country, total deforested area in a watershed, average canopy height in a management unit. If pixel errors were independent, the uncertainty of the aggregate would shrink as \(1/\sqrt{n}\) with the number of pixels. But when errors are positively correlated, they do not cancel upon averaging and the uncertainty of the aggregate is much larger than the naive \(1/\sqrt{n}\) calculation suggests.
[1] formalized this problem and demonstrated that a number of recent high-profile studies have reported uncertainty for spatial aggregates without accounting for spatial autocorrelation of map errors, leading to substantially underestimated confidence intervals. They propose block kriging as the geostatistically rigorous solution, which requires fitting a variogram to the spatial structure of the model residuals and using it to compute the variance of the spatial average.
The practical importance of this effect has been studied empirically. [2] produced model based uncertainty estimates for biomass aggregated to ownership parcels in New York State, incorporating all four recognized uncertainty components (reference data, sample variability, residual variability, and auxiliary data) as well as spatial autocorrelation of model residuals. They found that residual variance, largely resulting from spatial correlation of residuals, dominated all other sources of uncertainty for most parcels. This means that even if one perfectly characterized label noise, model variance, and input errors, ignoring spatial correlation would still produce dramatically underestimated aggregate uncertainty. [3] reached a similar conclusion in the forest mapping context. When errors are spatially correlated, they introduce domain-level bias in aggregated estimates that must be explicitly modeled, for instance via random area effects or distance-dependent correlation structures.
This remains one of the largest gaps between current practice in ML-based mapping and what is needed for rigorous area-based reporting. The methodology exists as block kriging on model residuals provides a rigorous estimate of aggregate uncertainty [1], [2]. However, fitting a variogram to residuals requires spatially distributed validation data dense enough to characterize the correlation structure, and that structure is unlikely to be stationary. Most mapping efforts lack the validation density to estimate this, let alone estimate it regionally. Until validation protocols are designed with spatial error correlation in mind by sampling densely enough in representative regions to fit at least regional variograms, aggregate uncertainty will remain underestimated by default. As a practical minimum, we recommend that mapping projects characterize residual spatial correlation in a small number of ecologically distinct validation regions, even if global estimation is infeasible. This provides at least an approximate bound on the aggregation effect and prevents the false precision that arises from treating pixel errors as independent.