Browsing by Author "Karaca, Y."

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Citation - Scopus: 3
Algorithmic Complexity-Based Fractional-Order Derivatives in Computational Biology
(Springer Science and Business Media Deutschland GmbH, 2023) Baleanu, D.; Karaca, Y.; 56389; 02.02. Matematik; 02. Fen-Edebiyat Fakültesi; 01. Çankaya Üniversitesi
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. © 2023, The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.
Citation - Scopus: 6
Artificial Neural Network Modeling of Systems Biology Datasets Fit Based on Mittag-Leffler Functions With Heavy-Tailed Distributions for Diagnostic and Predictive Precision Medicine
(Elsevier, 2022) Baleanu, D.; Karaca, Y.; 56389; 02.02. Matematik; 02. Fen-Edebiyat Fakültesi; 01. Çankaya Üniversitesi
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 Eα,β(z) 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. © 2022 Elsevier Inc. All rights reserved.
Citation - Scopus: 8
Computational Fractional-Order Calculus and Classical Calculus Ai for Comparative Differentiability Prediction Analyses of Complex-Systems Paradigm
(Elsevier, 2022) Baleanu, D.; Karaca, Y.; 56389; 02.02. Matematik; 02. Fen-Edebiyat Fakültesi; 01. Çankaya Üniversitesi
Modern science having embarked on the thorough and accurate interpretation of natural and physical phenomena has proven to provide successful models for the analysis of complex systems and harnessing of control over the various processes therein. Computational complexity, in this regard, comes to the foreground by providing applicable sets of ideas or integrative paradigms to recognize and understand the complex systems' intricate properties. Thus, while making the appropriate, adaptable and evolutive decisions in complex dynamic systems, it is essential to acknowledge different degrees of acceptance of the problems and construct the model it to account for its inherent constraints or limits. In this respect, while hypothesis-driven research has its inherent limitations regarding the investigation of multifactorial and heterogeneous diseases, a data-driven approach enables the examination of the way variables impact one another, which paves the way for the interpretation of dynamic and heterogeneous mechanisms of diseases. Fractional Calculus (FC), in this scope characterized by complexity, provides the applicable means and methods to solve integral, differential and integro-differential equations so FC enables the generalization of integration and differentiation possible in a flexible and consistent manner owing to its capability of reflecting the systems' actual state properties, which exhibit unpredictable variations. The fractional integration and differentiation of fractional-order is capable of providing better characterization of nonstationary and locally self-similar attributes in contrast to constant-order fractional calculus. It becomes possible to model many complex systems by fractional-order derivatives based on fractional calculus so that related syntheses can be realized in a robust and effective way. To this end, our study aims at providing an intermediary facilitating function both for the physicians and individuals by establishing accurate and robust model based on the integration of fractional-order calculus and Artificial Neural Network (ANN) for the diagnostic and differentiability predictive purposes with the diseases which display highly complex properties. The integrative approach we have proposed in this study has a multistage quality the steps of which are stated as follows: first of all, the Caputo fractional-order derivative, one of the fractional-order derivatives, has been used with two-parametric Mittag-Leffler function on the stroke dataset and cancer cell dataset, manifesting biological and neurological attributes. In this way, new fractional models with varying degrees have been established. Mittag-Leffler function, with its distributions of extensive application domains, can address irregular and heterogeneous environments for the solution of dynamic problems; thus, Mittag-Leffler function has been opted for accordingly. Following this application, the new datasets (mlf_stroke dataset and mlf_cancer cell dataset) have been obtained by employing Caputo fractional-order derivative with the two-parametric Mittag-Leffler function (α,β). In addition, classical derivative (calculus) was applied to the raw datasets; and cd_stroke dataset and cd_cancer cell dataset were obtained. Secondly, the performance of the new datasets as obtained from the Caputo fractional derivative with the two-parametric Mittag-Leffler function, the datasets obtained from the classical derivative application and the raw datasets have been compared by using feed forward back propagation (FFBP) algorithm, one of the algorithms of ANN (along with accuracy rate, sensitivity, precision, specificity, F1-score, multiclass classification (MCC), ROC curve). Based on the accuracy rate results obtained from the application with FFBP, the Caputo fractional-order derivative model that is most suitable for the diseases has been generated. The experimental results obtained demonstrate the applicability of the complex-systems-grounded paradigm scheme as proposed through this study, which has no existing counterpart. The integrative multi-stage method based on mathematical-informed framework with comparative differentiability prediction analyses can point toward a new direction in the various areas of applied sciences to address formidable challenges of critical decision making and management of chaotic processes in different complex dynamic systems. © 2022 Elsevier Inc. All rights reserved.
Introduction
(Elsevier, 2022) Karaca, Y.; Baleanu, D.; 02.02. Matematik; 02. Fen-Edebiyat Fakültesi; 01. Çankaya Üniversitesi
Citation - Scopus: 4
Mittag-Leffler Functions With Heavy-Tailed Distributions' Algorithm Based on Different Biology Datasets To Be Fit for Optimum Mathematical Models' Strategies
(Elsevier, 2022) Karaca, Y.; Baleanu, D.; 56389; 02.02. Matematik; 02. Fen-Edebiyat Fakültesi; 01. Çankaya Üniversitesi
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 (α,β) 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 Eα,β(z) 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 (α,β) 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. © 2022 Elsevier Inc. All rights reserved.
Citation - Scopus: 14
Multi-Chaos, Fractal and Multi-Fractional Artificial Intelligence of Different Complex Systems
(Elsevier, 2022) Moonis, M.; Baleanu, D.; Zhang, Y.-D.; Gervasi, O.; Karaca, Y.; 56389; 02.02. Matematik; 02. Fen-Edebiyat Fakültesi; 01. Çankaya Üniversitesi
Multi-Chaos, Fractal and Multi-Fractional Artificial Intelligence of Different Complex Systems addresses different uncertain processes inherent in the complex systems, attempting to provide global and robust optimized solutions distinctively through multifarious methods, technical analyses, modeling, optimization processes, numerical simulations, case studies as well as applications including theoretical aspects of complexity. Foregrounding Multi-chaos, Fractal and Multi-fractional in the era of Artificial Intelligence (AI), the edited book deals with multi- chaos, fractal, multifractional, fractional calculus, fractional operators, quantum, wavelet, entropy-based applications, artificial intelligence, mathematics-informed and data driven processes aside from the means of modelling, and simulations for the solution of multifaceted problems characterized by nonlinearity, non-regularity and self-similarity, frequently encountered in different complex systems. The fundamental interacting components underlying complexity, complexity thinking, processes and theory along with computational processes and technologies, with machine learning as the core component of AI demonstrate the enabling of complex data to augment some critical human skills. Appealing to an interdisciplinary network of scientists and researchers to disseminate the theory and application in medicine, neurology, mathematics, physics, biology, chemistry, information theory, engineering, computer science, social sciences and other far-reaching domains, the overarching aim is to empower out-of-the-box thinking through multifarious methods, directed towards paradoxical situations, uncertain processes, chaotic, transient and nonlinear dynamics of complex systems. © 2022 Elsevier Inc. All rights reserved.
Citation - Scopus: 2
Multicompartmental Mathematical Models of Infectious Dynamic Diseases With Time Fractional-Order Derivatives
(Institute of Electrical and Electronics Engineers Inc., 2023) Baleanu, D.; Rahman, M.U.; Momani, S.; Karaca, Y.; 56389; 02.02. Matematik; 02. Fen-Edebiyat Fakültesi; 01. Çankaya Üniversitesi
Nonlinear dynamic models with multiple compartments are characterized by subtle attributes like high dimensionality and heterogeneity, with fractional-order derivatives and constituting fractional calculus, which can provide a thorough comprehension, control and optimization of the related dynamics and structure. This requirement poses a formidable challenge, and thereby, has gained prominence in different fields where fractional derivatives and nonlinearities interact. Thus, fractional models have become relevant to address phenomena with memory effects, with fractional calculus providing amenities to deal with the time-dependent impacts observed. A novel infectious disease epidemic model with time fractional order and a Caputo fractional derivative type operator is discussed in the current study which is carried out for the considered epidemic model. Accordingly, a method for the semi-analytical solution of the epidemic model of a dynamic infectious disease with fractional order is employed in terms of the Caputo fractional derivative operator in this study. The existence and uniqueness of the solution is constructed with the aid of fixed point theory in particular. Furthermore, the Adams-Bashforth method, an extensively employed technique for the semi-analytical solution of these types of models. The simulation results for various initial data demonstrate that the solution of the considered model is stable and shows convergence toward a single point, and numerical simulations for different fractional orders lying between (0,1) and integer order have been obtained. On both initial approximations, the dynamical behavior of each compartment has shown stability as well as convergence. Consequently, the results obtained from our study based on experimental data can be stated to confirm the accurate total density and capacity for each compartment lying between two different integers considering dynamical processes and systems. © 2023 IEEE.