On the inferential consequences in the selection of fractional derivatives

Abstract

The challenge of model selection arises when multiple models appear suitable for modeling the same phenomena. Deciding on the best fractional derivative for models based on fractional differential equations relies on the fit with experimental data. However, data fit does not consider the inferential performance of the models. This paper proposes an enhancement of the standard approach by incorporating statistics and statistical tests to evaluate additional inferential aspects. From this statistical perspective, we evaluate and compare fractional models using the Caputo, conformable, and Caputo-Fabrizio derivatives. We present three model scenarios based on linear, exponential, and logistic growth. For each model scenario, we compare the coverage frequency and length of the confidence intervals; the bias and the mean square error of the estimators; the trade-off between estimation precision and efficiency; and the performance of Type I error and the power when considering hypothesis tests. The statistical methods we use to compare the three competing fractional models include a likelihood-based approach, an identifiability analysis, and simulation studies. Our results show identifiability problems for the Caputo-Fabrizio derivative. For the Caputo and conformable derivatives, we demonstrate a better or comparable performance of the statistical estimators, confidence intervals, and hypothesis tests, depending on the model scenario under study although the efficiency shows evidence for selecting any fractional derivative. We conclude by presenting a real-world problem in an ecological context involving body mass growth, where the ordinary model is rejected based on the data, although efficiency does not reflect this decision.