Emanuela Raffinetti

A crucial issue for the trustworthiness of Artificial Intelligence methods is to ensure that their results are explainable to human beings. The most employed model agnostic measures to assess explainability are Shapley‐values whose application, however, suffers from being computationally intensive. In this paper we overcome this drawback through a new explainable AI tool: the Xi–Lorenz method, which can provide both pre hoc (pre‐model) and post hoc (post‐model) explanations. The advantage of doing so is that we can separate input evaluation (in the pre hoc stage) from output evaluation (in the post hoc stage), enabling to distinguish the influence of data quality from the reasoning capacity of the model. By assessing inputs independently from outputs, one can determine whether observed shortcomings are primarily attributable to noisy, incomplete, or biased data, or whether they reflect limitations in the model's internal mechanisms. We demonstrate the practical advantages of the Xi–Lorenz method resorting to both simulated and real data. Our findings indicate that the proposed method is effective and that it improves the extant explainable AI methods.

Inequality measures are quantitative measures that take values in the unit interval, with a zero value characterizing perfect equality. Although originally proposed to measure economic inequalities, they can be applied to several other situations, in which one is interested in the mutual variability between a set of observations, rather than in their deviations from the mean. While unidimensional measures of inequality, such as the Gini index, are widely known and employed, multidimensional measures, such as Lorenz Zonoids, are difficult to interpret and computationally expensive and, for these reasons, are not much well known. To overcome the problem, in this paper we propose a new scaling invariant multidimensional inequality index, based on the Fourier transform, which exhibits a number of interesting properties, and whose application to the multidimensional case is rather straightforward to calculate and interpret.

A key point to assess statistical forecasts is the evaluation of their predictive accuracy. Recently, a new measure, called Rank Graduation Accuracy (RGA), based on the concordance between the ranks of the predicted values and the ranks of the actual values of a series of observations to be forecast, was proposed for the assessment of the quality of the predictions. In this paper, we demonstrate that, in a classification perspective, when the response to be predicted is binary, the RGA coincides both with the AUROC and the Wilcoxon-Mann–Whitney statistic, and can be employed to evaluate the accuracy of probability forecasts. When the response to be predicted is real valued, the RGA can still be applied, differently from the AUROC, and similarly to measures such as the RMSE. Differently from the RMSE, the RGA measure evaluates point predictions in terms of their ranks, rather than in terms of their values, improving robustness.