Özet:
Being the most complex physical system in the universe, life, at all scales requires the understanding of the massive complexity including its origin, structure, dynamic, adaptation and organization. Both the number of substructures and interacting pathways of each substructure along with other ones and neurons determine the degree of complexity. Neural networks, as descriptive models, in systems biology setting, provide the means to gather, store and use experiential knowledge; and are designed in a way to emulate different operations of the human brain. One of the major ongoing challenges of integrating fractional calculus in cases of complexity requires an effective use of empirical, numerical, experimental and analytical methods to tackle complexity. In that regard, Artificial Neural Networks (ANNs), including a family of nonlinear computational methods, are employed to handle experimental data in differing domains owing to their capability of tackling complex computations so that their progressive application can solve practical problems. One of the other most noteworthy tools which arises 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. They can be used in reliability modeling as an alternative for exponential distribution, particularly this provides upper hand for diagnostic and predictive purposes in precision medicine through novel algorithmic models. To address this, the proposed method in the current study has obtained the generation of optimum model strategies for different biology datasets along with Mittag-Leffler functions with heavy-tailed distributions (see Part I). Within this framework, the proposed integrated approach in this study investigates the dynamics of diseases related to biological elements; and arising in the different solutions of varying complex biological systems, ML function generalizes the exponential function. To this end, firstly, the two-parametric Mittag-Leffler function was applied to biological datasets (cancer cell dataset and diabetes dataset, namely raw datasets), namely cancer cell and diabetes in order to obtain the new datasets (ml_cancer cell dataset and ml_diabetes dataset). Heavy-tailed distributions (The Mittag-Leffler distribution, Pareto distribution, Cauchy distribution and Weibull distribution) were applied to the new datasets obtained with their comparison performed in relation to the performances (by employing the log likelihood value and the Akaike Information Criterion (AIC)). 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. Secondly, one of the ANN algorithms, namely Multi-layer Perceptron (MLP) (along with the accuracy, sensitivity, precision, specificity, F1-score, multi-class classification (MCC), ROC curve), was applied for the diagnosis and prediction of the disease course regarding the optimum ML functions that represent the cancer cell and diabetes datasets obtained and the performances of the ML functions with heavy-tailed distributions were compared with ANN training functions (Levenberg-Marquart, Bayes Regularization and BFGS-Quasi-Newton) accordingly. The integrative modeling scheme proposed herein, which has not been addressed through this sort of approach before, is concerned with the applicability and reliability of the solutions obtained by Mittag-Leffler functions with heavy-tailed distributions. The results obtained by the current study for diseases related to biological datasets based on mathematical models demonstrate that the integrative approach with Mittag-Leffler function and ANN applications is applicable and fits very well to the related data with the robust parameters' values observed and estimated. When the fact that complex biological phenomena involve various intrinsic and extrinsic aspects is considered, it becomes a major difficulty to make identifications and recognition on the basis of a single type of data merely. Thus, the proposed approach of our study corroborates its applicability for diagnostic and predictive purposes in precision medicine through the novel algorithmic model, which plays a significant role in the effective and timely management of unpredictable phenomena in dynamic and nonlinear complex situations.