ϕ-Extropy complexity measure: Extensions and applications

In this paper, we first introduce the ϕ-extropy measure and the ϕ-extropy divergence (or relative ϕ-extropy) measure. We then explore their theoretical properties, highlighting that the ϕ-extropy encompasses the Gini–Simpson index as a special case. Furthermore, we establish the Bregman divergence in terms of two discrete complementary distributions and the Jensen-ϕ-extopy divergence. It is shown that the Jensen-ϕ-extropy divergence can be expressed as a mixture of the proposed Bregman divergences defined for two discrete complementary distributions. We also introduce two cumulative versions of the ϕ-extropy divergence: one defined in terms of the cumulative distribution function and the other defined in terms of the survival function. A mixture of these two cumulative versions of the ϕ-extropy divergence is proposed to demonstrate performance in real-world applications. Finally, we apply the proposed ϕ-extropy and mixture ϕ-extropy divergence to four cases. The first three cases are related to embedded time series representation used for the computation of permutation entropy, where proposed measures are considered for logistic and Chebyshev chaotic maps, and the impact of number of deaths produced by Covid-19 in India, the USA, and China during the period 2020–2022. The fourth case is related to a binary classification task using heart disease data and compares its performance against ten machine learning models, evaluating accuracy, precision, recall, and F1-score based on a 75%–25% train–test split of the heart disease dataset.