Mittag-Leffler functions with heavy-tailed distributions' algorithm based on different biology datasets to be fit for optimum mathematical models' strategies

Abstract

Complexity of living organisms owing to their inherent functional properties points toward a systems biology approach due to the fact that structural and topological uncertainties exist along with abrupt transitions characterized by unknown inputs, time-varying parameters and unpredictable observation states. The related uncertain, emergent and evolving qualities of organisms along with their varying quantities and states present in the related complex system need to be identified in biological datasets based on mathematical models in a way that enables the structural identification analysis in a reasonable time frame, the detection of nonlinear dependencies among the many parameters involved and practical analysis for the identification of data at stake. Superstatistics, which is concerned with the study of nonlinear systems, has proven to be a significant tool to examine the dynamic aspects of organisms, substances, particles and other biological elements. Superstatistics is characterized by the superposition of varying statistical models to achieve the desired nonlinearity. The challenge of integrating fractional calculus in cases of complexity requires an effective use of empirical, numerical, experimental and analytical methods to tackle complexity. One of the most noteworthy tools in the fractional calculus context is the Mittag-Leffler (ML) functions. Mittag-Leffler distributions have extensive application domains when dealing with irregular and nonhomogeneous environments for dynamic problems' solutions. These distributions can be used in reliability modeling as an alternative for exponential distribution; and thus, the proposed integrated approach in this study addresses the Mittag-Leffler (ML) function with two parameters in order to investigate the dynamics of diseases related to biological elements. Arising in the different solutions of varying complex biological systems, ML function generalizes the exponential function; and to this end, firstly, we applied the ML function with two parameters to biological datasets (cancer cell dataset and diabetes dataset, namely raw datasets) in order to obtain the new datasets (ml_cancer cell dataset and ml_diabetes dataset) with significant attributes for diagnosis, prognosis and classification of diseases. Secondly, heavy-tailed distributions (The Mittag-Leffler distribution, Pareto distribution, Cauchy distribution and Weibull distribution) were applied to the new datasets obtained, and their comparison was made with regard to the performances, by employing the log likelihood value (MLE) and the Akaike Information Criterion (AIC). Fitting algorithm Mittag-Leffler function is based on heavy-tailed distributions. Subsequently, the ML functions that represent the cancer cell and diabetes data were identified so that the two parameters yielding the optimum value based on the distributions fit could be found. By finding the most significant attributes with heavy-tailed distributions (The Mittag-Leffler distribution, Pareto distribution, Cauchy distribution and Weibull distribution) based on Mittag-Leffler function with two parameters the diagnosis, prognosis and classification of the diseases has been enabled in our study. In this way, through this proposed integrative scheme, optimal strategical means have been obtained for accurate and robust mathematical models' strategies concerning the diagnosis and progress of the diseases. The results obtained by the current study for diseases on biological datasets based on mathematical models demonstrate that the integrative approach with Mittag-Leffler with heavy-tailed distributions algorithm is applicable and fits very well to the related data with the robust parameters' values observed and estimated in transient chaotic and unpredictable settings. The analysis results obtained by the data fitting algorithm scheme proposed have demonstrated its criticality for understanding the dynamics of transmission and prevalence operating in the complex biological and epidemiological systems along the Mittag-Leffler function based on distribution scale, with temporal and spatial attributes, to improve applicability and accuracy constituting optimal mathematical models' strategies.