Symbolic regression (SR) methods attempt to learn mathematical expressions that approximate the behavior of an observed system. However, when dealing with multivariate systems, they often fail to identify the functional form that explains the relationship between each variable and the system's response. To begin to address this, we propose an explainable neural SR method that generates univariate symbolic skeletons that aim to explain how each variable influences the system's response. By analyzing multiple sets of data generated artificially, where one input variable varies while others are fixed, relationships are modeled separately for each input variable. The response of such artificial data sets is estimated using a regression neural network (NN). Finally, the multiple sets of input-response pairs are processed by a pre-trained Multi-Set Transformer that solves a problem we termed Multi-Set Skeleton Prediction and outputs a univariate symbolic skeleton. Thus, such skeletons represent explanations of the function approximated by the regression NN. Experimental results demonstrate that this method learns skeleton expressions matching the underlying functions and outperforms two GP-based and two neural SR methods.

In Precision Agriculture, the utilization of management zones (MZs) that take into account within-field variability facilitates effective fertilizer management. This approach enables the optimization of nitrogen (N) rates to maximize crop yield production and enhance agronomic use efficiency. However, existing works often neglect the consideration of responsivity to fertilizer as a factor influencing MZ determination. In response to this gap, we present a MZ clustering method based on fertilizer responsivity. We build upon the statement that the responsivity of a given site to the fertilizer rate is described by the shape of its corresponding N fertilizer-yield response (N-response) curve. Thus, we generate N-response curves for all sites within the field using a convolutional neural network (CNN). The shape of the approximated N-response curves is then characterized using functional principal component analysis. Subsequently, a counterfactual explanation (CFE) method is applied to discern the impact of various variables on MZ membership. The genetic algorithm-based CFE solves a multi-objective optimization problem and aims to identify the minimum combination of features needed to alter a site's cluster assignment. Results from two yield prediction datasets indicate that the features with the greatest influence on MZ membership are associated with terrain characteristics that either facilitate or impede fertilizer runoff, such as terrain slope or topographic aspect.

Response curves exhibit the magnitude of the response of a sensitive system to a varying stimulus. However, response of such systems may be sensitive to multiple stimuli (i.e., input features) that are not necessarily independent. As a consequence, the shape of response curves generated for a selected input feature (referred to as "active feature") might depend on the values of the other input features (referred to as "passive features"). In this work, we consider the case of systems whose response is approximated using regression neural networks. We propose to use counterfactual explanations (CFEs) for the identification of the features with the highest relevance on the shape of response curves generated by neural network black boxes. CFEs are generated by a genetic algorithm-based approach that solves a multi-objective optimization problem. In particular, given a response curve generated for an active feature, a CFE finds the minimum combination of passive features that need to be modified to alter the shape of the response curve. We tested our method on a synthetic dataset with 1-D inputs and two crop yield prediction datasets with 2-D inputs. The relevance ranking of features and feature combinations obtained on the synthetic dataset coincided with the analysis of the equation that was used to generate the problem. Results obtained on the yield prediction datasets revealed that the impact on fertilizer responsivity of passive features depends on the terrain characteristics of each field.