Browsing by Author "Karaca, Yeliz"

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Citation Count: Al-Qurashi, Maysaa...et al. (2021). "ACHIEVING MORE PRECISE BOUNDS BASED ON DOUBLE AND TRIPLE INTEGRAL AS PROPOSED BY GENERALIZED PROPORTIONAL FRACTIONAL OPERATORS IN THE HILFER SENSE", Fractals-Complex Geometry Patterns and Scaling in Nature and Society, Vol. 29, No. 05.
ACHIEVING MORE PRECISE BOUNDS BASED ON DOUBLE AND TRIPLE INTEGRAL AS PROPOSED BY GENERALIZED PROPORTIONAL FRACTIONAL OPERATORS IN THE HILFER SENSE
(2021) Al-Qurashi, Maysaa; Rashid, Saima; Karaca, Yeliz; Hammouch, Zakia; Baleanu, Dumitru; Chu, Yu-Ming; 56389
A user-friendly approach depending on nonlocal kernel has been constituted in this study to model nonlocal behaviors of fractional differential and difference equations, which is known as a generalized proportional fractional operator in the Hilfer sense. It is deemed, for differentiable functions, by a fractional integral operator applied to the derivative of a function having an exponential function in the kernel. This operator generalizes a novel version of Cebysev-type inequality in two and three variables sense and furthers the result of existing literature as a particular case of the Cebysev inequality is discussed. Some novel special cases are also apprehended and compared with existing results. The outcome obtained by this study is very broad in nature and fits in terms of yielding an enormous number of relating results simply by practicing the proportionality indices included therein. Furthermore, the outcome of our study demonstrates that the proposed plans are of significant importance and computationally appealing to deal with comparable sorts of differential equations. Taken together, the results can serve as efficient and robust means for the purpose of investigating specific classes of integrodifferential equations.
Citation Count: Baleanu, Dumitru...et al. (2023). "Advanced fractional calculus, differential equations and neural networks: analysis, modeling and numerical computations", Physica Scripta, Vol. 98, No. 11.
Advanced fractional calculus, differential equations and neural networks: analysis, modeling and numerical computations
(2023) Baleanu, Dumitru; Karaca, Yeliz; Vázquez, Luis; Macías-Díaz, Jorge E.; 56389
Most physical systems in nature display inherently nonlinear and dynamical properties; hence, it would be difficult for nonlinear equations to be solved merely by analytical methods, which has given rise to the emerging of engrossing phenomena such as bifurcation and chaos. Conjointly, due to nonlinear systems’ exhibiting more exotic behavior than harmonic distortion, it becomes compelling to test, classify and interpret the results in an accurate way. For this reason, avoiding preconceived ideas of the way the system is likely to respond is of pivotal importance since this facet would have effect on the type of testing run and processing techniques used in nonlinear systems. Paradigms of nonlinear science may suggest that it is ‘the study of every single phenomenon’ due to its interdisciplinary nature, which is another challenge encountered and needs to be addressed by generating and designing a systematic mathematical framework where the complexity of natural phenomena hints the requirement of identifying their commonalties and classifying their various manifestations in different nonlinear systems. Studying such common properties, concepts or paradigms can enable one to gain insight into nonlinear problems, their essence and consequences in a broad range of disciplines all forthwith. Fractional differential equations associated with non-local phenomena in physics have arisen as a powerful mathematical tool within a multidisciplinary research framework. Fractional differential equations, as one extension of the fractional calculus theory, can yield the evolution of various systems properly, which reinforces its position in mathematics and science while setting stage for the description of dynamic, complicated and nonlinear events. Through the reflection of the systems’ actual properties, fractional calculus manifests unforeseeable and hidden variations, and thus, enables integration and differentiation, with the solutions to be approximated by numerical methods along with modeling and predicting the dynamics of multiphysics, multiscale and physical systems. Neural Networks (NNs), consisting of hidden layers with nonlinear functions that have vector inputs and outputs, are also considerably employed owing to their versatile and efficient characteristics in classification problems as well as their sophisticated neural network architectures, which make them capable of tackling complicated governing partial differential equation problems. Furthermore, partial differential equations are used to provide comprehensive and accurate models for many scientific phenomena owing to the advancements of data gathering and machine learning techniques which have raised opportunities for data-driven identification of governing equations derived from experimentally observed data. Given these considerations, while many problems are solvable and have been solved, efforts are still needed to be able to respond to the remaining open questions in the fields that have a broad range of spectrum ranging from mathematics, physics, biology, virology, epidemiology, chemistry, engineering, social sciences to applied sciences. With a view of different aspects of such questions, our special issue provides a collection of recent research focusing on the advances in the foundational theory, methodology and topical applications of fractals, fractional calculus, fractional differential equations, differential equations (PDEs, ODEs, to name some), delay differential equations (DDEs), chaos, bifurcation, stability, sensitivity, machine learning, quantum machine learning, and so forth in order to expound on advanced fractional calculus, differential equations and neural networks with detailed analyses, models, simulations, data-driven approaches as well as numerical computations.
Citation Count: Karaca, Y.; Baleanu, D. "Algorithmic Complexity-Based Fractional-Order Derivatives in Computational Biology", Advances in Mathematical Modelling, Applied Analysis and Computation,ICMMAAC 2021, Proceedings, pp.55-89, 2023.
Algorithmic Complexity-Based Fractional-Order Derivatives in Computational Biology
(2023) Karaca, Yeliz; Baleanu, Dumitru; 56389
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.
Citation Count: Karaca, Yeliz; Baleanu, Dumitru. Artificial neural network modeling of systems biology datasets fit based on Mittag-Leffler functions with heavy-tailed distributions for diagnostic and predictive precision medicine, in Multi-Chaos, Fractal and Multi-Fractional Artificial Intelligence of Different Complex Systems, Academic Press, pp. 133-148, 2022.
Artificial neural network modeling of systems biology datasets fit based on Mittag-Leffler functions with heavy-tailed distributions for diagnostic and predictive precision medicine
(2022) Karaca, Yeliz; Baleanu, Dumitru; 56389
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.
Citation Count: Karaca, Yeliz; Baleanu, Dumitru. Computational fractional-order calculus and classical calculus AI for comparative differentiability prediction analyses of complex-systems-grounded paradigm, in Multi-Chaos, Fractal and Multi-Fractional Artificial Intelligence of Different Complex Systems, Academic Press, pp. 149-168, 2022.
Computational fractional-order calculus and classical calculus AI for comparative differentiability prediction analyses of complex-systems-grounded paradigm
(2022) Karaca, Yeliz; Baleanu, Dumitru; 56389
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 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.
Citation Count: Karaca, Yeliz;...et.al. (2022). "Editorial: Special issue section on fractal ai-based analyses and applications to complex systems: Part iii", Fractals, Vol.30, No.5.
Editorial: Special issue section on fractal ai-based analyses and applications to complex systems: Part iii
(2022) Karaca, Yeliz; Baleanu, Dumitru; Moonis, Majaz; Zhang, Yu-Dong; Gervasi, Osvaldo; 56389
Citation Count: Karaca, Yeliz; Baleanu, D. (2022). "Evolutionary Mathematical Science, Fractional Modeling and Artificial Intelligence of Nonlinear Dynamics in Complex Systems", Chaos Theory and Applications, Vol.4, No.3, pp.111-118.
Evolutionary Mathematical Science, Fractional Modeling and Artificial Intelligence of Nonlinear Dynamics in Complex Systems
(2022) Karaca, Yeliz; Baleanu, Dumitru; 56389
Complex problems in nonlinear dynamics foreground the critical support of artificial phenomena so that each domain of complex systems can generate applicable answers and solutions to the pressing challenges. This sort of view is capable of serving the needs of different aspects of complexity by minimizing the problems of complexity whose solutions are based on advanced mathematical foundations and analogous algorithmic models consisting of numerous applied aspects of complexity. Evolutionary processes, nonlinearity and all the other dimensions of complexity lie at the pedestal of time, reveal time and occur within time. In the ever-evolving landscape and variations, with causality breaking down, the idea of complexity can be stated to be a part of unifying and revolutionary scientific framework to expound complex systems whose behavior is perplexing to predict and control with the ultimate goal of attaining a global understanding related to many branches of possible states as well as high-dimensional manifolds, while at the same time keeping abreast with actuality along the evolutionary and historical path, which itself, has also been through different critical points on the manifold. In view of these, we put forth the features of complexity of varying phenomena, properties of evolution and adaptation, memory effects, nonlinear dynamic system qualities, the importance of chaos theory and applications of related aspects in this study. In addition, processes of fractional dynamics, differentiation and systems in complex systems as well as the dynamical processes and dynamical systems of fractional order with respect to natural and artificial phenomena are discussed in terms of their mathematical modeling. Fractional calculus and fractional-order calculus approach to provide novel models with fractional-order calculus as employed in machine learning algorithms to be able to attain optimized solutions are also set forth besides the justification of the need to develop analytical and numerical methods. Subsequently, algorithmic complexity and its goal towards ensuring a more effective handling of efficient algorithms in computational sciences is stated with regard to the classification of computational problems. We further point out the neural networks, as descriptive models, for providing the means to gather, store and use experiential knowledge as well as Artificial Neural Networks (ANNs) in relation to their employment for handling experimental data in different complex domains. Furthermore, the importance of generating applicable solutions to problems for various engineering areas, medicine, biology, mathematical science, applied disciplines and data science, among many others, is discussed in detail along with an emphasis on power of predictability, relying on mathematical sciences, with Artificial Intelligence (AI) and machine learning being at the pedestal and intersection with different fields which are characterized by complex, chaotic, nonlinear, dynamic and transient components to validate the significance of optimized approaches both in real systems and in related realms.
Citation Count: Karaca, Yeliz; Moonis, Majaz; Baleanu, Dumitru (2020). "Fractal and multifractional-based predictive optimization model for stroke subtypes’ classification", Chaos, Solitons and Fractals, Vol. 136.
Fractal and multifractional-based predictive optimization model for stroke subtypes’ classification
(2020) Karaca, Yeliz; Moonis, Majaz; Baleanu, Dumitru; 56389
Numerous natural phenomena display repeating self-similar patterns. Fractal is used when a pattern seems to repeat itself. Fractal and multifractal methods have extensive applications in neurosciences in which the prevalence of fractal properties like self-similarity in the brain, equipped with a complex structure, in medical data analysis at various levels of observation is admitted. The methods come to the fore since subtle details are not always detected by physicians, but these are critical particularly in neurological diseases like stroke which may be life-threatening. The aim of this paper is to identify the self-similar, significant and efficient attributes to achieve high classification accuracy rates for stroke subtypes. Accordingly, two approaches were implemented. The first approach is concerned with application of the fractal and multifractal methods on the stroke dataset in order to identify the regular, self-similar, efficient and significant attributes from the dataset, with these steps: a) application of Box-counting dimension generated BC_stroke dataset b) application of Wavelet transform modulus maxima generated WTMM_stroke dataset. The second approach involves the application of Feed Forward Back Propagation (FFBP) for stroke subtype classification with these steps: (i) FFBP algorithm was applied on the stroke dataset, BC_stroke dataset and WTMM_stroke dataset. (ii) Comparative analyses were performed based on accuracy, sensitivity and specificity for the three datasets. The main contribution is that the study has obtained the identification of self-similar, regular and significant attributes from the stroke subtypes datasets by following multifarious and integrated methodology. The study methodology is based on the singularity spectrum which provides a value concerning how fractal a set of points are in the datasets (BC_stroke dataset and WTMM_stroke dataset). The experimental results reveal the applicability, reliability and accuracy of our proposed integrated method. No earlier work exists in the literature with the relevant stroke datasets and the methods employed. Therefore, the study aims at pointing a new direction in the relevant fields concerning the complex dynamic systems and structures which display multifractional nature. © 2020 Elsevier Ltd
Citation Count: Karaca, Yeliz; Moonis, Majaz; Baleanu, Dumitru (2020). "Fractal and multifractional-based predictive optimization model for stroke subtypes? classification", Chaos Solitons & Fractals, Vol. 136.
Fractal and multifractional-based predictive optimization model for stroke subtypes? classification
(2020) Karaca, Yeliz; Moonis, Majaz; Baleanu, Dumitru; 56389
Numerous natural phenomena display repeating self-similar patterns. Fractal is used when a pattern seems to repeat itself. Fractal and multifractal methods have extensive applications in neurosciences in which the prevalence of fractal properties like self-similarity in the brain, equipped with a complex structure, in medical data analysis at various levels of observation is admitted. The methods come to the fore since subtle details are not always detected by physicians, but these are critical particularly in neurological diseases like stroke which may be life-threatening. The aim of this paper is to identify the self-similar, significant and efficient attributes to achieve high classification accuracy rates for stroke subtypes. Accordingly, two approaches were implemented. The first approach is concerned with application of the fractal and multifractal methods on the stroke dataset in order to identify the regular, self-similar, efficient and significant attributes from the dataset, with these steps: a) application of Box-counting dimension generated BC_stroke dataset b) application of Wavelet transform modulus maxima generated WTMM_stroke dataset. The second approach involves the application of Feed Forward Back Propagation (FFBP) for stroke subtype classification with these steps: (i) FFBP algorithm was applied on the stroke dataset, BC_stroke dataset and WTMM_stroke dataset. (ii) Comparative analyses were performed based on accuracy, sensitivity and specificity for the three datasets. The main contribution is that the study has obtained the identification of self-similar, regular and significant attributes from the stroke subtypes datasets by following multifarious and integrated methodology. The study methodology is based on the singularity spectrum which provides a value concerning how fractal a set of points are in the datasets (BC_stroke dataset and WTMM_stroke dataset). The experimental results reveal the applicability, reliability and accuracy of our proposed integrated method. No earlier work exists in the literature with the relevant stroke datasets and the methods employed. Therefore, the study aims at pointing a new direction in the relevant fields concerning the complex dynamic systems and structures which display multifractional nature.
Citation Count: Karaca, Yeliz; Rahman, Mati ur; Baleanu, Dumitru. Fractional Order Computing and Modeling with Portending Complex Fit Real-World Data, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 23rd International Conference on Computational Science and Its Applications, ICCSA 2023, 3 July 2023through 6 July 2023, Vol. 14104 LNCS, pp. 144 - 159,
Fractional Order Computing and Modeling with Portending Complex Fit Real-World Data
(2023) Karaca, Yeliz; Rahman, Mati ur; Baleanu, Dumitru; 56389
Fractional computing models identify the states of different systems with a focus on formulating fractional order compartment models through the consideration of differential equations based on the underlying stochastic processes. Thus, a systematic approach to address and ensure predictive accuracy allows that the model remains physically reasonable at all times, providing a convenient interpretation and feasible design regarding all the parameters of the model. Towards these manifolding processes, this study aims to introduce new concepts of fractional calculus that manifest crossover effects in dynamical models. Piecewise global fractional derivatives in sense of Caputo and Atangana-Baleanu-Caputo (ABC) have been utilized, and they are applied to formulate the Zika Virus (ZV) disease model. To have a predictive analysis of the behavior of the model, the domain is subsequently split into two subintervals and the piecewise behavior is investigated. Afterwards, the fixed point theory of Schauder and Banach is benefited from to prove the existence and uniqueness of at least one solution in both senses for the considered problem. As for the numerical simulations as per the data, Newton interpolation formula has been modified and extended for the considered nonlinear system. Finally, graphical presentations and illustrative examples based on the data for various compartments of the systems have been presented with respect to the applicable real-world data for different fractional orders. Based on the impact of fractional order reducing the abrupt changes, the results obtained from the study demonstrate and also validate that increasing the fractional order brings about a greater crossover effect, which is obvious from the observed data, which is critical for the effective management and control of abrupt changes like infectious diseases, viruses, among many more unexpected phenomena in chaotic, uncertain and transient circumstances.
Citation Count: Karaca, Y.; Rahman, M., Baleanu, D. "Fractional Order Computing and Modeling with Portending Complex Fit Real-World Data", 23rd International Conference on Computational Science and Its Applications, Proceedings, pp.144-159, 2023.
Fractional Order Computing and Modeling with Portending Complex Fit Real-World Data
(2023) Karaca, Yeliz; Rahman, Mati ur; Baleanu, Dumitru; 56389
Fractional computing models identify the states of different systems with a focus on formulating fractional order compartment models through the consideration of differential equations based on the underlying stochastic processes. Thus, a systematic approach to address and ensure predictive accuracy allows that the model remains physically reasonable at all times, providing a convenient interpretation and feasible design regarding all the parameters of the model. Towards these manifolding processes, this study aims to introduce new concepts of fractional calculus that manifest crossover effects in dynamical models. Piecewise global fractional derivatives in sense of Caputo and Atangana-Baleanu-Caputo (ABC) have been utilized, and they are applied to formulate the Zika Virus (ZV) disease model. To have a predictive analysis of the behavior of the model, the domain is subsequently split into two subintervals and the piecewise behavior is investigated. Afterwards, the fixed point theory of Schauder and Banach is benefited from to prove the existence and uniqueness of at least one solution in both senses for the considered problem. As for the numerical simulations as per the data, Newton interpolation formula has been modified and extended for the considered nonlinear system. Finally, graphical presentations and illustrative examples based on the data for various compartments of the systems have been presented with respect to the applicable real-world data for different fractional orders. Based on the impact of fractional order reducing the abrupt changes, the results obtained from the study demonstrate and also validate that increasing the fractional order brings about a greater crossover effect, which is obvious from the observed data, which is critical for the effective management and control of abrupt changes like infectious diseases, viruses, among many more unexpected phenomena in chaotic, uncertain and transient circumstances.
Citation Count: Karaca, Yeliz; Baleanu, Dumitru; Karabudak, Rana. (2021). "Hidden Markov Model and multifractal method-based predictive quantization complexity models vis-á-vis the differential prognosis and differentiation of Multiple Sclerosis’ subgroups", Knowledge-Based Systems, Vol.246.
Hidden Markov Model and multifractal method-based predictive quantization complexity models vis-á-vis the differential prognosis and differentiation of Multiple Sclerosis’ subgroups
(2021) Karaca, Yeliz; Baleanu, Dumitru; Karabudak, Rana; 56389
Hidden Markov Model (HMM) is a stochastic process where implicit or latent stochastic processes can be inferred indirectly through a sequence of observed states. HMM as a mathematical model for uncertain phenomena is applicable for the description and computation of complex dynamical behaviours enabling the mathematical formulation of neural dynamics across spatial and temporal scales. The human brain with its fractal structure demonstrates complex dynamics and fractals in the brain are characterized by irregularity, singularity and self-similarity in terms of form at different observation levels, making detection difficult as observations in real-time occurrences can be time variant, discrete, continuous or noisy. Multiple Sclerosis (MS) is an autoimmune degenerative disease with time and space related dissemination, leading to neuronal apoptosis, coupled with some subtle features that could be overlooked by physicians. This study, through the proposed integrated approach with multi-source complex spatial data, aims to attain accurate prediction, diagnosis and prognosis of MS subgroups by HMM with Viterbi algorithm and Forward–Backward algorithm as the dynamic and efficient products of knowledge-based and Artificial Intelligence (AI)-based systems within the framework of precision medicine. Multifractal Bayesian method (MFM) accordingly applied to identify and eliminate “insignificant” irregularities while maintaining “significant” singularities. An efficient modelling of HMM is proposed to diagnose and predict the course of MS while using MFM method. Unlike the methods employed in previous studies, our proposed integrated novel method encompasses the subsequent approaches based on reliable MS dataset (X) collected: (i) MFM method was applied (X) to MS dataset to characterize the irregular, self-similar and significant attributes, thus, attributes with “insignificant” irregularities were eliminated and “significant” singularities were maintained. MFM-MS dataset (Xˆ) was generated. (ii) The continuous values in the MS dataset (X) and MFM-MS dataset (Xˆ) were converted into discrete values through vector quantization method of the HMM (iii) Through transitional matrices, different observation matrices were computed from the both datasets. (v) Computational complexity has been computed for both datasets. (vi) The results of the HMM models based on observation matrices obtained from both datasets were compared. In terms of the integrated HMM model proposed and the MS dataset handled, no earlier work exists in the literature. The experimental results demonstrate the applicability and accuracy of our novel proposed integrated method, HMM and Multifractal (HMM-MFM) method, for the application to the MS dataset (X). Compared with conventional methods, our novel method has achieved more superiority regarding extracting subtle and hidden details, which are significant for distinguishing different dynamic and complex systems including engineering and other related applied sciences. Thus, we have aimed at pointing a new frontier by providing a novel alternative mathematical model to facilitate the critical decision-making, management and prediction processes among the related areas in chaotic, dynamic complex systems with intricate and transient states. ©
Citation Count: Baleanu, Dumitru; Karaca, Yeliz. "Mittag-Leffler functions with heavy-tailed distributions' algorithm based on different biology datasets to be fit for optimum mathematical models' strategies", in Multi-Chaos, Fractal and Multi-Fractional Artificial Intelligence of Different Complex Systems, Academic Press, pp. 117-132, 2022.
Mittag-Leffler functions with heavy-tailed distributions' algorithm based on different biology datasets to be fit for optimum mathematical models' strategies
(Academic Press, 2022) Baleanu, Dumitru; Karaca, Yeliz; 56389
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.
Citation Count: Karaca, Yeliz...et al. Multi-Chaos, Fractal and Multi-Fractional Artificial Intelligence of Different Complex Systems, Elsevier, p. 332, 2022.
Multi-Chaos, Fractal and Multi-Fractional Artificial Intelligence of Different Complex Systems
(Elsevier, 2022) Karaca, Yeliz; Moonis, Majaz; Baleanu, Dumitru; Zhang, Yu-Dong; Gervasi, Osvaldo; 56389
Citation Count: Karaca, Y.;...et.al. "Multicompartmental Mathematical Models of Infectious Dynamic Diseases with Time Fractional-order Derivatives", 2023 International Conference on Fractional Differentiation and Its Applications, ICFDA 2023, Proceedings, 2023.
Multicompartmental Mathematical Models of Infectious Dynamic Diseases with Time Fractional-order Derivatives
(2023) Karaca, Yeliz; Baleanu, Dumitru; Rahman, Mati ur; Momani, Shaher; 56389
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.
Citation Count: Karaca, Yeliz; Baleanu, Dumitru (2020). "Multifractional Gaussian Process Based on Self-similarity Modelling for MS Subgroups' Clustering with Fuzzy C-Means", COMPUTATIONAL SCIENCE AND ITS APPLICATIONS - ICCSA 2020, PT II, Vol. 12250, pp. 426-441.
Multifractional Gaussian Process Based on Self-similarity Modelling for MS Subgroups' Clustering with Fuzzy C-Means
(2020) Karaca, Yeliz; Baleanu, Dumitru; 56389
Multifractal analysis is a beneficial way to systematically characterize the heterogeneous nature of both theoretical and experimental patterns of fractal. Multifractal analysis tackles the singularity structure of functions or signals locally and globally. While Holder exponent at each point provides the local information, the global information is attained by characterization of the statistical or geometrical distribution of Holder exponents occurring, referred to as multifractal spectrum. This analysis is time-saving while dealing with irregular signals; hence, such analysis is used extensively. Multiple Sclerosis (MS), is an auto-immune disease that is chronic and characterized by the damage to the Central Nervous System (CNS), is a neurological disorder exhibiting dissimilar and irregular attributes varying among patients. In our study, the MS dataset consists of the Expanded Disability Status Scale (EDSS) scores and Magnetic Resonance Imaging (MRI) (taken in different years) of patients diagnosed with MS subgroups (relapsing remitting MS (RRMS), secondary progressive MS (SPMS) and primary progressive MS (PPMS)) while healthy individuals constitute the control group. This study aims to identify similar attributes in homogeneous MS clusters and dissimilar attributes in different MS subgroup clusters. Thus, it has been aimed to demonstrate the applicability and accuracy of the proposed method based on such cluster formation. Within this framework, the approach we propose follows these steps for the classification of the MS dataset. Firstly, Multifractal denoising with Gaussian process is employed for identifying the critical and significant self-similar attributes through the removal of MS dataset noise, by which, mFd MS dataset is generated. As another step, Fuzzy C-means algorithm is applied to the MS dataset for the classification purposes of both datasets. Based on the experimental results derived within the scheme of the applicable and efficient proposed method, it is shown that mFd MS dataset yielded a higher accuracy rate since the critical and significant self-similar attributes were identified in the process. This study can provide future direction in different fields such as medicine, natural sciences and engineering as a result of the model proposed and the application of alternative mathematical models. As obtained based on the model, the experimental results of the study confirm the efficiency, reliability and applicability of the proposed method. Thus, it is hoped that the derived results based on the thorough analyses and algorithmic applications will be assisting in terms of guidance for the related studies in the future.
Citation Count: Wang, Miao-Kun...et al. (2021). "NEW MULTI-FUNCTIONAL APPROACH for κ TH-ORDER DIFFERENTIABILITY GOVERNED by FRACTIONAL CALCULUS VIA APPROXIMATELY GENERALIZED (ψ, ?) -CONVEX FUNCTIONS in HILBERT SPACE", Fractals, Vol. 29, No. 5.
NEW MULTI-FUNCTIONAL APPROACH for κ TH-ORDER DIFFERENTIABILITY GOVERNED by FRACTIONAL CALCULUS VIA APPROXIMATELY GENERALIZED (ψ, ?) -CONVEX FUNCTIONS in HILBERT SPACE
(2021) Wang, Miao-Kun; Rashid, Saima; Karaca, Yeliz; Baleanu, Dumitru; Chu, Yu-Ming; 56389
This work addresses several novel classes of convex function involving arbitrary non-negative function, which is known as approximately generalized (ψ, ?)-convex and approximately ψ-quasiconvex function, with respect to Raina's function, which are elaborated in Hilbert space. To ensure the feasibility of the proposed concept and with the discussion of special cases, it is presented that these classes generate other classes of generalized (ψ, ?)-convex functions such as higher-order strongly (HOS) generalized (ψ, ?)-convex functions and HOS generalized ψ-quasiconvex function. The core of the proposed method is a newly developed Simpson's type of identity in the settings of Riemann-Liouville fractional integral operator. Based on the HOS generalized (ψ, ?)-convex function representation, we established several theorems and related novel consequences. The presented results demonstrate better performance for HOS generalized ψ-quasiconvex functions where we can generate several other novel classes for convex functions that exist in the relative literature. Accordingly, the assortment in this study aims at presenting a direction in the related fields. © 2021 The Author(s).
Citation Count: Wang, Miao-Kun;...et.al. (2021). "New Multi-Functional Approach For Κ Th-Order Differentiability Governed By Fractional Calculus Via Approximately Generalized (Ψ, ?) -Convex Functions İn Hilbert Space", Fractals, Vol.29, No.5.
New Multi-Functional Approach For Κ Th-Order Differentiability Governed By Fractional Calculus Via Approximately Generalized (Ψ, ?) -Convex Functions İn Hilbert Space
(2021) Wang, Miao-Kun; Rashid, Saima; Karaca, Yeliz; Chu, Yu-Ming; 56389
This work addresses several novel classes of convex function involving arbitrary non-negative function, which is known as approximately generalized (ψ, ?)-convex and approximately ψ-quasiconvex function, with respect to Raina's function, which are elaborated in Hilbert space. To ensure the feasibility of the proposed concept and with the discussion of special cases, it is presented that these classes generate other classes of generalized (ψ, ?)-convex functions such as higher-order strongly (HOS) generalized (ψ, ?)-convex functions and HOS generalized ψ-quasiconvex function. The core of the proposed method is a newly developed Simpson's type of identity in the settings of Riemann-Liouville fractional integral operator. Based on the HOS generalized (ψ, ?)-convex function representation, we established several theorems and related novel consequences. The presented results demonstrate better performance for HOS generalized ψ-quasiconvex functions where we can generate several other novel classes for convex functions that exist in the relative literature. Accordingly, the assortment in this study aims at presenting a direction in the related fields.
Citation Count: Karaca, Yeliz...et al. (2021). "SPECIAL ISSUE SECTION ON FRACTAL AI-BASED ANALYSES AND APPLICATIONS TO COMPLEX SYSTEMS: PART I", FRACTALS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY, Vol. 29, No. 5.
SPECIAL ISSUE SECTION ON FRACTAL AI-BASED ANALYSES AND APPLICATIONS TO COMPLEX SYSTEMS: PART I
(2021) Karaca, Yeliz; Baleanu, Dumitru; Moonis, Majaz; Muhammad, Khan; Zhang, Yu-Dong; Gervasi, Osvaldo; 56389
Citation Count: Karaca, Yeliz...et al. (2020). "Theory, Analyses and Predictions of Multifractal Formalism and Multifractal Modelling for Stroke Subtypes’ Classification", Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 20th International Conference on Computational Science and Its Applications, ICCSA 2020, Cagliari, 1 July 2020through 4 July 2020, Vol. 12250, pp. 410-425.
Theory, Analyses and Predictions of Multifractal Formalism and Multifractal Modelling for Stroke Subtypes’ Classification
(2020) Karaca, Yeliz; Baleanu, Dumitru; Moonis, Majaz; Zhang, Yu-Dong; 56389
Fractal and multifractal analysis interplay within complementary methodology is of pivotal importance in ubiquitously natural and man-made systems. Since the brain as a complex system operates on multitude of scales, the characterization of its dynamics through detection of self-similarity and regularity presents certain challenges. One framework to dig into complex dynamics and structure is to use intricate properties of multifractals. Morphological and functional points of view guide the analysis of the central nervous system (CNS). The former focuses on the fractal and self-similar geometry at various levels of analysis ranging from one single cell to complicated networks of cells. The latter point of view is defined by a hierarchical organization where self-similar elements are embedded within one another. Stroke is a CNS disorder that occurs via a complex network of vessels and arteries. Considering this profound complexity, the principal aim of this study is to develop a complementary methodology to enable the detection of subtle details concerning stroke which may easily be overlooked during the regular treatment procedures. In the proposed method of our study, multifractal regularization method has been employed for singularity analysis to extract the hidden patterns in stroke dataset with two different approaches. As the first approach, decision tree, Naïve bayes, kNN and MLP algorithms were applied to the stroke dataset. The second approach is made up of two stages: i) multifractal regularization (kulback normalization) method was applied to the stroke dataset and mFr_stroke dataset was generated. ii) the four algorithms stated above were applied to the mFr_stroke dataset. When we compared the experimental results obtained from the stroke dataset and mFr_stroke dataset based on accuracy (specificity, sensitivity, precision, F1-score and Matthews Correlation Coefficient), it was revealed that mFr_stroke dataset achieved higher accuracy rates. Our novel proposed approach can serve for the understanding and taking under control the transient features of stroke. Notably, the study has revealed the reliability, applicability and high accuracy via the methods proposed. Thus, the integrated method has revealed the significance of fractal patterns and accurate prediction of diseases in diagnostic and other critical-decision making processes in related fields.