Algorithmic Complexity-Based Fractional-Order Derivatives in Computational Biology

Abstract

Fractional calculus approach, providing novel models through the introduction of fractional-order calculus to optimization methods, is employed in machine learning algorithms. This scheme aims to attain optimized solutions by maximizing the accuracy of the model and minimizing the functions like the computational burden. Mathematical-informed frameworks are to be employed to enable reliable, accurate, and robust understanding of various complex biological processes that involve a variety of spatial and temporal scales. This complexity requires a holistic understanding of different biological processes through multi-stage integrative models that are capable of capturing the significant attributes on the related scales. Fractional-order differential and integral equations can provide the generalization of traditional integral and differential equations through the extension of the conceptions with respect to biological processes. In addition, algorithmic complexity (computational complexity), as a way of comparing the efficiency of an algorithm, can enable a better grasping and designing of efficient algorithms in computational biology as well as other related areas of science. It also enables the classification of the computational problems based on their algorithmic complexity, as defined according to the way the resources are required for the solution of the problem, including the execution time and scale with the problem size. Based on a novel mathematical informed framework and multi-staged integrative method concerning algorithmic complexity, this study aims at establishing a robust and accurate model reliant on the combination of fractional-order derivative and Artificial Neural Network (ANN) for the diagnostic and differentiability predictive purposes for the disease, (diabetes, as a metabolic disorder, in our case) which may display various and transient biological properties. Another aim of this study is benefitting from the concept of algorithmic complexity to obtain the fractional-order derivative with the least complexity in order that it would be possible to achieve the optimized solution. To this end, the following steps were applied and integrated. Firstly, the Caputo fractional-order derivative with three-parametric Mittag-Leffler function (α,β,γ) was applied to the diabetes dataset. Thus, new fractional models with varying degrees were established by ensuring data fitting through the fitting algorithm Mittag-Leffler function with three parameters (α,β,γ) based on heavy-tailed distributions. Following this application, the new dataset, named the mfc_diabetes, was obtained. Secondly, classical derivative (calculus) was applied to the diabetes dataset, which yielded the cd_diabetes dataset. Subsequently, the performance of the new dataset as obtained from the first step and of the dataset obtained from the second step as well as of the diabetes dataset was compared through the application of the feed forward back propagation (FFBP) algorithm, which is one of the ANN algorithms. Next, the fractional order derivative model which would be the most optimal for the disease was generated. Finally, algorithmic complexity was employed to attain the Caputo fractional-order derivative with the least complexity, or to achieve the optimized solution. This approach through the application of fractional-order calculus to optimization methods and the experimental results have revealed the advantage of maximizing the model’s accuracy and minimizing the cost functions like the computational costs, which points to the applicability of the method proposed in different domains characterized by complex, dynamic and transient components.