# Presentation: Evaluating the accuracy of equivalent-source predictions using cross-validation

Presented at the EGU2020 General Assembly.

**Abstract**

We investigate the use of cross-validation (CV) techniques to estimate the
accuracy of equivalent-source (also known as equivalent-layer) models for
interpolation and processing of potential-field data.
Our preliminary results indicate that some common CV algorithms (e.g., random
permutations and k-folds) tend to overestimate the accuracy.
We have found that blocked CV methods, where the data are split along spatial
blocks instead of randomly, provide more conservative and realistic accuracy
estimates.
Beyond evaluating an equivalent-source model's performance, cross-validation
can be used to automatically determine configuration parameters, like source
depth and amount of regularization, that maximize prediction accuracy and avoid
over-fitting.

Widely used in gravity and magnetic data processing,
the equivalent-source technique consists of a linear model (usually point
sources) used to predict the observed field at arbitrary locations.
Upward-continuation, interpolation, gradient calculations, leveling, and
reduction-to-the-pole can be performed simultaneously by using the model
to make predictions (i.e., forward modelling).
Likewise, the use of linear models to make predictions is the backbone of many
machine learning (ML) applications.
The predictive performance of ML models is usually evaluated through
cross-validation, in which the data are split (usually randomly) into a
training set and a validation set.
Models are fit on the training set and their predictions are evaluated using
the validation set using a goodness-of-fit metric, like the mean square error
or the R² coefficient of determination.
Many cross-validation methods exist in the literature, varying in how the data
are split and how this process is repeated.
Prior research from the statistical modelling of ecological data suggests that
prediction accuracy is usually overestimated by traditional CV methods when the
data are spatially auto-correlated.
This issue can be mitigated by splitting the data along spatial blocks
rather than randomly.
We conducted experiments on synthetic gravity data to investigate the use of
traditional and blocked CV methods in equivalent-source interpolation.
We found that the overestimation problem also occurs and that more conservative
accuracy estimates are obtained when applying blocked versions of random
permutations and k-fold.
Further studies need to be conducted to generalize these findings to
upward-continuation, reduction-to-the-pole, and derivative calculation.

Open-source software implementations of the equivalent-source and blocked cross-validation (in progress) methods are available in the Python libraries Harmonica and Verde, which are part of the Fatiando a Terra project (www.fatiando.org).