Computational Complexity-based Fractional-Order Neural Network Models for the Diagnostic Treatments and Predictive Transdifferentiability of Heterogeneous Cancer Cell Propensity

Year 2023, Volume: 5 Issue: 1, 34 - 51, 31.03.2023

Yeliz Karaca

https://doi.org/10.51537/chaos.1249532

Cited By: 4

Abstract

Neural networks and fractional order calculus are powerful tools for system identification through which there exists the capability of approximating nonlinear functions owing to the use of nonlinear activation functions and of processing diverse inputs and outputs as well as the automatic adaptation of synaptic elements through a specified learning algorithm. Fractional-order calculus, concerning the differentiation and integration of non-integer orders, is reliant on fractional-order thinking which allows better understanding of complex and dynamic systems, enhancing the processing and control of complex, chaotic and heterogeneous elements. One of the most characteristic features of biological systems is their different levels of complexity; thus, chaos theory seems to be one of the most
applicable areas of life sciences along with nonlinear dynamic and complex systems of living and non-living environment. Biocomplexity, with multiple scales ranging from molecules to cells and organisms, addresses complex structures and behaviors which emerge from nonlinear interactions of active biological agents. This sort of emergent complexity is concerned with the organization of molecules
into cellular machinery by that of cells into tissues as well as that of individuals to communities. Healthy systems sustain complexity in their lifetime and are chaotic, so complexity loss or chaos loss results in diseases. Within the mathematics-informed frameworks, fractional-order calculus based Artificial Neural Networks (ANNs) can be employed for accurate understanding of complex biological
processes. This approach aims at achieving optimized solutions through the maximization of the model’s accuracy and minimization of computational burden and exhaustive methods. Relying on a transdifferentiable mathematics-informed framework and multifarious integrative methods concerning computational complexity, this study aims at establishing an accurate and robust model based upon
integration of fractional-order derivative and ANN for the diagnosis and prediction purposes for cancer cell whose propensity exhibits various transient and dynamic biological properties. The other aim is concerned with showing the significance of computational complexity for obtaining the fractional-order derivative with the least complexity in order that optimized solution could be achieved. The multifarious
scheme of the study, by applying fractional-order calculus to optimization methods, the advantageous aspect concerning model accuracy maximization has been demonstrated through the proposed method’s applicability and predictability aspect in various domains manifested by dynamic and nonlinear nature displaying different levels of chaos and complexity.

Keywords

Computational complexity, Complex systems, Artificial Intelligence (AI), Chaos theory, Fractional calculus, Fractional-order derivatives, Mittag-Leffler function, Heavy-tailed distributions, Computational biocomplexity, Nonlinearity and uncertainty, Multilayer perceptron algorithm (MLP), Neural networks, Transdifferentiable mathematics-informed framework, Complex order optimization, Mathematical biology, Data-driven fractional-order biological modeling, Cancer cell propensity

References

• Abdul Hamid, N., N. Mohd Nawi, R. Ghazali, and M. N. Mohd Salleh, 2011 Accelerating learning performance of back propagation algorithm by using adaptive gain together with adaptive momentum and adaptive learning rate on classification problems. In Ubiquitous Computing and Multimedia Applications: Second International Conference, UCMA 2011, Daejeon, Korea, April 13-15, 2011. Proceedings, Part II 2, pp. 559–570, Springer.
• Aguilar, C. Z., J. Gómez-Aguilar, V. Alvarado-Martínez, and H. Romero-Ugalde, 2020 Fractional order neural networks for system identification. Chaos, Solitons & Fractals 130: 109444.
• Al Na’mneh, R. andW. D. Pan, 2007 Five-step fft algorithm with reduced computational complexity. Information processing letters 101: 262–267.
• Almalki, S. J. and S. Nadarajah, 2014 Modifications of the weibull distribution: A review. Reliability Engineering & System Safety 124: 32–55.
• Alsmadi, M., K. B. Omar, and S. A. Noah, 2009 Back propagation algorithm: the best algorithm among the multi-layer perceptron algorithm .
• Arnold, B. C., 2014 Pareto distribution. Wiley StatsRef: Statistics Reference Online pp. 1–10.
• Arnold, B. C. and R. J. Beaver, 2000 The skew-cauchy distribution. Statistics & probability letters 49: 285–290.
• Arora, S. and B. Barak, 2009 Computational complexity: a modern approach. Cambridge University Press.
• Baleanu, D. and Y. Karaca, 2022 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, pp. 117–132, Elsevier.
• Blazewicz, J. and M. Kasprzak, 2012 Complexity issues in computational biology. Fundamenta Informaticae 118: 385–401.
• Boroomand, A. and M. B. Menhaj, 2009 Fractional-based approach in neural networks for identification problem. In 2009 Chinese Control and Decision Conference, pp. 2319–2322, IEEE.
• Camargo, R. F., E. C. de Oliveira, and J. Vaz, 2012 On the generalized mittag-leffler function and its application in a fractional telegraph equation. Mathematical Physics, Analysis and Geometry 15: 1–16.
• Carletti, M. and M. Banerjee, 2019 A backward technique for demographic noise in biological ordinary differential equation models. Mathematics 7: 1204.
• Chakraborty, S. and S. Ong, 2017 Mittag-leffler function distribution-a new generalization of hyper-poisson distribution. Journal of Statistical distributions and applications 4: 1–17.
• Chivers, I., J. Sleightholme, I. Chivers, and J. Sleightholme, 2015 An introduction to algorithms and the big o notation. Introduction to Programming with Fortran: With Coverage of Fortran 90, 95, 2003, 2008 and 77 pp. 359–364.
• D’Agostino, R., 2017 Goodness-of-fit-techniques. Routledge. David, S., J. Linares, and E. Pallone, 2011 Cálculo de ordem fracionária: apologia histórica, conceitos básicos e algumas aplicações. Revista Brasileira de Ensino de Física 33: 4302–4302.
• De Oliveira, E. C. and J. A. Tenreiro Machado, 2014 A review of definitions for fractional derivatives and integral. Mathematical Problems in Engineering 2014.
• Debnath, L., 2003 Recent applications of fractional calculus to science and engineering. International Journal of Mathematics and Mathematical Sciences 2003: 3413–3442.
• Du, D.-Z. and K.-I. Ko, 2011 Theory of computational complexity, volume 58. John Wiley & Sons.
• Du, M., Z. Wang, and H. Hu, 2013 Measuring memory with the order of fractional derivative. Scientific reports 3: 3431. Fan, J. and I. Gijbels, 2018 Local polynomial modelling and its applications. Routledge.
• Fernandez, A. and I. Husain, 2020 Modified mittag-leffler functions with applications in complex formulae for fractional calculus. Fractal and Fractional 4: 45.
• Garrappa, R., 2015 Numerical evaluation of two and three parameter mittag-leffler functions. SIAM Journal on Numerical Analysis 53: 1350–1369.
• Garrappa, R., E. Kaslik, and M. Popolizio, 2019 Evaluation of fractional integrals and derivatives of elementary functions: Overview and tutorial. Mathematics 7: 407.
• Gomolka, Z., 2018 Backpropagation algorithm with fractional derivatives. In ITM web of conferences, volume 21, p. 00004, EDP Sciences. Gorenflo, R., A. A. Kilbas, F. Mainardi, S. V. Rogosin, et al., 2020
• Mittag-Leffler functions, related topics and applications. Springer. Gutierrez, R. E., J. M. Rosário, and J. Tenreiro Machado, 2010
• Fractional order calculus: basic concepts and engineering applications. Mathematical problems in engineering 2010.
• Haykin, S., 2009 Neural networks and learning machines, 3/E. Pearson Education India.
• Herrmann, R., 2011 Fractional calculus: an introduction for physicists. World Scientific.
• Jachowicz, R. E., P. Duch, P. W. Ostalczyk, and D. J. Sankowski, 2022 Fractional order derivatives as an optimization tool for object detection and tracking algorithms. IEEE Access 10: 18619–18630.
• Kadam, P., G. Datkhile, and V. A. Vyawahare, 2019 Artificial neural network approximation of fractional-order derivative operators: analysis and dsp implementation. In Fractional Calculus and Fractional Differential Equations, pp. 93–126, Springer.
• Karaca, Y., 2016 Case study on artificial neural networks and applications. Applied Mathematical Sciences 10: 2225–2237.
• Karaca, Y. and D. Baleanu, 2020 A novel r/s fractal analysis and wavelet entropy characterization approach for robust forecasting based on self-similar time series modeling. Fractals 28: 2040032.
• Karaca, Y. and D. Baleanu, 2022a Algorithmic complexity-based fractional-order derivatives in computational biology. In Advances in Mathematical Modelling, Applied Analysis and Computation: Proceedings of ICMMAAC 2021, pp. 55–89, Springer.
• Karaca, Y. and D. Baleanu, 2022b 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 Multifractional Artificial Intelligence of Different Complex Systems, pp. 133–148, Elsevier.
• Karaca, Y. and D. Baleanu, 2022c 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, pp. 149–168, Elsevier.
• Karaca, Y., D. Baleanu, and R. Karabudak, 2022 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 246: 108694.
• Karaca, Y. and C. Cattani, 2018 Computational methods for data analysis. In Computational Methods for Data Analysis, De Gruyter.
• Karaca, Y., M. Moonis, and D. Baleanu, 2020 Fractal and multifractional-based predictive optimization model for stroke subtypes’ classification. Chaos, Solitons & Fractals 136: 109820.
• KARCI, A. et al., 2014 Fractional order derivative and relationship between derivative and complex functions. Mathematical Sciences and Applications E-Notes 2: 44–54.
• Khan, H., A. Khan, M. Al Qurashi, D. Baleanu, and R. Shah, 2020 An analytical investigation of fractional-order biological model using an innovative technique. Complexity 2020: 1–13.
• Kharazmi, O., 2016 Generalized weighted weibull distribution. Journal of Mathematical Extension 10: 89–118.
• Kochubei, A., Y. Luchko, V. E. Tarasov, and I. Petráš, 2019 Handbook of fractional calculus with applications, volume 1. de Gruyter Berlin, Germany.
• Krishna, B. and K. Reddy, 2008 Active and passive realization of fractance device of order 1/2. Active and passive electronic components 2008.
• Lewis, M. R., P. G. Matthews, and E. M. Hubbard, 2016 Neurocognitive architectures and the nonsymbolic foundations of fractions understanding. In Development of mathematical cognition, pp. 141–164, Elsevier.
• Li, C., D. Qian, Y. Chen, et al., 2011 On riemann-liouville and caputo derivatives. Discrete Dynamics in Nature and Society 2011.
• Lopes, A. M. and J. Tenreiro Machado, 2019 The fractional view of complexity.
• Magin, R. L., 2010 Fractional calculus models of complex dynamics in biological tissues. Computers & Mathematics with Applications 59: 1586–1593.
• Mainardi, F., 2020 Why the mittag-leffler function can be considered the queen function of the fractional calculus? Entropy 22: 1359.
• Mainardi, F. and R. Gorenflo, 2000 On mittag-leffler-type functions in fractional evolution processes. Journal of Computational and Applied mathematics 118: 283–299.
• Mall, S. and S. Chakraverty, 2018 Artificial neural network approach for solving fractional order initial value problems. arXiv preprint arXiv:1810.04992 .
• MATLAB, 2022 version 9.12.0 (R2022a). The MathWorks Inc., Natick, Massachusetts.
• Matusiak, M., 2020 Optimization for software implementation of fractional calculus numerical methods in an embedded system. Entropy 22: 566.
• Mia, M. M. A., S. K. Biswas, M. C. Urmi, and A. Siddique, 2015 An algorithm for training multilayer perceptron (mlp) for image reconstruction using neural network without overfitting. International Journal of Scientific & Technology Research 4: 271–275.
• Michener, W. K., T. J. Baerwald, P. Firth, M. A. Palmer, J. L. Rosenberger, et al., 2001 Defining and unraveling biocomplexity. Bio- Science 51: 1018–1023.
• Mittag-Leffler, G., 1903 Sur la nouvelle fonction ea (x). Comptes rendus de l’Académie des Sciences 137: 554–558.
• Murphy, P. M., 1994 Uci repository of machine learning databases. http://www. ics. uci. edu/˜ mlearn/MLRepository. html .
• Newman, M. E., 2005 Power laws, pareto distributions and zipf’s law. Contemporary physics 46: 323–351.
• Niu, H., Y. Chen, and B. J.West, 2021 Why do big data and machine learning entail the fractional dynamics? Entropy 23: 297.
• Oldham, K. and J. Spanier, 1974 The fractional calculus theory and applications of differentiation and integration to arbitrary order. Elsevier.
• Ouyang, Y. andW.Wang, 2016 Comparison of definition of several fractional derivatives. In 2016 International Conference on Education, Management and Computer Science, pp. 553–557, Atlantis Press.
• Panda, R. and M. Dash, 2006 Fractional generalized splines and signal processing. Signal Processing 86: 2340–2350.
• Pang, D., W. Jiang, and A. U. Niazi, 2018 Fractional derivatives of the generalized mittag-leffler functions. Advances in Difference Equations 2018: 1–9.
• Petrás, I., 2011 Fractional derivatives, fractional integrals, and fractional differential equations in Matlab. IntechOpen.
• Pillai, R. and O. M.-L. Functions, 1990 Related distributions. Ann. Inst. Statist. Math 42: 157–161.
• Raubitzek, S., K. Mallinger, and T. Neubauer, 2022 Combining fractional derivatives and machine learning: A review. Entropy 25: 35.
• Rodríguez-Germá, L., J. J. Trujillo, and M. Velasco, 2008 Fractional calculus framework to avoid singularities of differential equations. Fract. Cal. Appl. Anal 11: 431–441.
• Ross, B., 1977 Fractional calculus. Mathematics Magazine 50: 115– 122.
• Sidelnikov, O., A. Redyuk, and S. Sygletos, 2018 Equalization performance and complexity analysis of dynamic deep neural networks in long haul transmission systems. Optics express 26: 32765–32776.
• Singh, A. P., D. Deb, H. Agrawal, K. Bingi, and S. Ozana, 2021 Modeling and control of robotic manipulators: A fractional calculus point of view. Arabian Journal for Science and Engineering 46: 9541–9552.
• Singhal, G., V. Aggarwal, S. Acharya, J. Aguayo, J. He, et al., 2010 Ensemble fractional sensitivity: a quantitative approach to neuron selection for decoding motor tasks. Computational intelli-gence and neuroscience 2010: 1–10.
• Sommacal, L., P. Melchior, A. Oustaloup, J.-M. Cabelguen, and A. J. Ijspeert, 2008 Fractional multi-models of the frog gastrocnemius muscle. Journal of Vibration and Control 14: 1415–1430.
• Steck, G. P., 1958 A uniqueness property not enjoyed by the normal distribution. Sandia Corporation.
• Stockmeyer, L., 1987 Classifying the computational complexity of problems. The journal of symbolic logic 52: 1–43.
• Tenreiro Machado, J., V. Kiryakova, and F. Mainardi, 2010 A poster about the old history of fractional calculus. Fractional Calculus and Applied Analysis 13: 447–454.
• Tokhmpash, A., 2021 Fractional Order Derivative in Circuits, Systems, and Signal Processing with Specific Application to Seizure Detection. Ph.D. thesis, Northeastern University.
• Toledo-Hernandez, R., V. Rico-Ramirez, G. A. Iglesias-Silva, and U. M. Diwekar, 2014 A fractional calculus approach to the dynamic optimization of biological reactive systems. part i: Fractional models for biological reactions. Chemical Engineering Science 117: 217–228.
• Tzoumas, V., Y. Xue, S. Pequito, P. Bogdan, and G. J. Pappas, 2018 Selecting sensors in biological fractional-order systems. IEEE Transactions on Control of Network Systems 5: 709–721.
• Valentim, C. A., J. A. Rabi, and S. A. David, 2021 Fractional mathematical oncology: On the potential of non-integer order calculus applied to interdisciplinary models. Biosystems 204: 104377.
• Van Rossum, G. and F. L. Drake Jr, 1995 Python tutorial, volume 620. Centrum voorWiskunde en Informatica Amsterdam, The Netherlands.
• Viola, J. and Y. Chen, 2022 A fractional-order on-line self optimizing control framework and a benchmark control system accelerated using fractional-order stochasticity. Fractal and Fractional 6: 549.
• West, B. J., 2016 Fractional calculus view of complexity: tomorrow’s science. CRC Press.
• West, B. J., M. Bologna, and P. Grigolini, 2003 Physics of fractal operators, volume 10. Springer.
• Wiman, A., 1905 Über den fundamentalsatz in der teorie der funktionen e a (x) .
• Wu, A., L. Liu, T. Huang, and Z. Zeng, 2017 Mittag-leffler stability of fractional-order neural networks in the presence of generalized piecewise constant arguments. Neural Networks 85: 118–127.
• Xue, H., Z. Shao, and H. Sun, 2020 Data classification based on fractional order gradient descent with momentum for rbf neural network. Network: Computation in Neural Systems 31: 166–185.
• Zhang, Y.-D. and L.Wu, 2008Weights optimization of neural network via improved bco approach. Progress In Electromagnetics Research 83: 185–198.
• Ziane, D., M. Hamdi Cherif, D. Baleanu, and K. Belghaba, 2020 Non-differentiable solution of nonlinear biological population model on cantor sets. Fractal and Fractional 4: 5.