Abstract
Fractal Analysis is the well developed theory in the data analysis of non-linear time series. Especially Fractal Dimension is a powerful mathematical tool for modeling many physical and biological time signals with high complexity and irregularity. Fractal dimension is a suitable tool for analyzing the nonlinear behaviour and state of the many chaotic systems. Particularly in analysis of chaotic time series such as electroencephalograms (EEG), this feature has been used to identify and distinguish specific states of physiological function.Epilepsy is the main fatal neurological disorder in our brain, which is analyzed by the biomedical signal called Electroencephalogram (EEG). The detection of Epileptic seizures in the EEG Signals is an important tool in the diagnosis of epilepsy. So we made an attempt to analyze the EEG in depth for knowing the mystery of human consciousness. EEG has more fluctuations recorded from the human brain due to the spontaneous electrical activity. Hence EEG Signals are represented as Fractal Time Series.The algorithms of fractal dimension methods have weak ability to the estimation of complexity in the irregular graphs. Divider method is widely used to obtain the fractal dimension of curves embedded into a 2-dimensional space. The major problem is choosing initial and final step length of dividers. We propose a new algorithm based on the size measure relationship (SMR) method, quantifying the dimensional behaviour of irregular rectifiable graphs with minimum time complexity. The evidence for the suitability (equality with the nature of dimension) of the algorithm is illustrated graphically.We would like to demonstrate the criterion for the selection of dividers (minimum and maximum value) in the calculation of fractal dimension of the irregular curves with minimum time complexity. For that we design a new method of computing fractal dimension (FD) of biomedical waveforms. Compared to Higuchi’s algorithm, advantages of this method include greater speed and the criterion to choose the maximum and minimum values for time intervals. Comparisons with the other waveform fractal dimension algorithms are also demonstrated. In order to discriminate the Healthy and the Epileptic EEGs, an improved method of Multifractal Measure such as Generalized Fractal Dimensions (GFD) is also proposed. Finally we conclude that there are significant differences between the Healthy and Epileptic Signals in the designed method than the GFD through graphical and statistical tools. The improved multifractal measure is very efficient technique to analyze the EEG Signals and to compute the state of illness of the Epileptic patients.
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References
Accardo, A., Affinito, M., Carrozzi, M., Bouquet, F.: Use of the fractal dimension for the analysis of electroencephalographic time series. Biol. Cybern. 77, 339–350 (1997)
Acharya, U., Faust, O., Kannathal, N., et al.: Non-linear analysis of EEG signals at various sleep stages. Comp. Meth. Prog. Biomed. 80, 37–45 (2005)
Adeli, H., Zhou, Z., Dadmehr, N.: Analysis of EEG records in an epileptic patient using wavelet transform. J. Neurosci. Meth. 123(1), 69–87 (2003)
Andrzejak, R., Lehnertz, K., Mormann, F., et al.: Indications of nonlinear deterministic and finite-dimensional structures in time series of brain electrical activity: Dependence on recording region and brain state. Phys. Rev. E 64, 061907-1–061907-8 (2001)
Arle, J., Simon, R.: An application of fractal dimension to the detection of transients in the electroencephalogram. Electroencephalogr. Clin. Neurophysiol. 75, 296–305 (1990)
Ayala, M., Cabrerizo, M., Tito, M., Barreto, A., Adjouadi, M.: A spreadsheet application for processing long-term EEG recordings. Comp. Biol. Med. 39, 844–851 (2009)
Barnsley, M.: Fractals Everywhere, 2 edn. Academic, USA (1993)
Bullmore, E., Brammer, M., Bourlon, P., Alarcon, G., Polkey, C., Elwes, R., Binnie, C.: Fractal analysis of electroencephalographic signals intracerebrally recorded during 35 epileptic seizures: Evaluation of a new method for synoptic visualisation of ictal events. Electroencephalogr. Clin. Neurophysiol. 91, 337–345 (1994)
Burlaga, L., Klein, L.: Fractal structure of the interplanetary magnetic field. J. Geophys. Res. 91, 347 (1986)
Carlin, M.: Measuring the complexity of non-fractal shapes by a fractal method. Pattern Recognit. Lett. 21(11), 1013–1017 (2000)
Clinic: Department of Epileptology University of Bonn Medical Centre, B.G.: http://www.epileptologie-bonn.de/. Accessed January 2010
Crevecoeur, G., Hallez, H., Hese, P.V., Asseler, Y., Dupre, L., de Walle, R.V.: EEG source analysis using space mapping techniques. J. Comput. Appl. Math. 215, 339–347 (2008)
Cuffin, B.: A method for localizing EEG sources in realistic head models. IEEE Trans. Biomed. Eng. 42, 68–71 (1995)
Diambra, L., Figueiredo, J., Malta, C.: Epileptic activity recognition in EEG recording. J. Phys. A: Math. Gen. 15, L611–L615 (1999)
Easwaramoorthy, D., Uthayakumar, R.: Analysis of biomedical EEG signals using wavelet transforms and multifractal analysis. In: Proceedings of the 1st IEEE International Conference on Communication Control and Computing Technologies, pp. 544–549. IEEE Xplore Digital Library, IEEE, USA (2010)
Easwaramoorthy, D., Uthayakumar, R.: Analysis of EEG signals using advanced generalized fractal dimensions. In: Proceedings of the Second International Conference on Computing, Communication and Networking Technologies, pp. 1–6. IEEE Xplore Digital Library, IEEE, USA (2010)
Easwaramoorthy, D., Uthayakumar, R.: Estimating the complexity of biomedical signals by multifractal analysis. In: Proceedings of the IEEE Students’ Technology Symposium, pp. 6–11. IEEE Xplore Digital Library, IEEE, USA (2010)
Easwaramoorthy, D., Uthayakumar, R.: Improved generalized fractal dimensions in the discrimination between healthy and epileptic EEG signals. J. Comput. Sci. 2(1), 31–38 (2011)
Elger, C., Lehnertz, K.: Seizure prediction by non-linear time series analysis of brain electrical activity. Eur. J. Neurosci. 10, 786–789 (1998)
Esteller, R., Vachtsevanos, G., Echauz, J., Litt, B.: A comparison of waveform fractal dimension algorithms. IEEE Trans. Circ. Syst. 48(2), 177–183 (2001)
Falconer, K.: Fractal Geometry: Mathematical Foundations and Applications, 2 edn. Wiley, England (2003)
Gevins, A., Remonds, A.: Handbook of Electroencephalography and Clinical Neurophysiology, vol. 1. Elsevier, University of Chicago (1987)
Goldberger, A., West, B.: Fractals in physiology and medicine. Yale J. Biol. Med. 60, 421–435 (1987)
Grassberger, P.: Generalized dimensions of strange attractors. Phys. Lett. A 97, 227–320 (1983)
Grassberger, P., Proccacia, I.: Characterization of strange attractors. Phys. Rev. Lett. 50(5), 346–349 (1983)
Grassberger, P., Proccacia, I.: Measuring the strangeness of strange attractors. Phys. D 9D, 189–208 (1983)
Hazarika, N., Chen, J., Tsoi, A., Sergejew, A.: Classification of EEG signals using the wavelet transform. Signal Process. 59(1), 61–72 (1997)
Hentschel, H., Procaccia, I.: The infinite number of generalized dimensions of fractals and strange attractors. Physica 8D, 435–444 (1983)
Higuchi, T.: Approach to an irregular time series on the basis of the fractal theory. Phys. D 31, 277–283 (1988)
Iasemidis, L.: Epileptic seizure prediction and control. IEEE Trans. Biomed. Eng. 50, 549–558 (2003)
Iasemidis, L., Sackellares, J.: The evolution with time of the spatial distribution of the largest lyapunov exponent on the human epileptic cortex. In: Measuring Chaos in the Human Brain. pp. 49–82. World Scientific Publishing, Singapore (1991)
Iasemidis, L., Zaveri, H., Sackellares, J., Williams, W.: Phase space topography of the electrocarticogram and the lyapunov exponent in partial seizures. Brain Topogr. 2, 187–201 (1990)
Iasemidis, L., Shiau, D., Sackellares, J., Pardalos, P.: Transition to epileptic seizures: Optimization. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pp. 55–74. American Mathematical Society, USA (1999)
Iasemidis, L., Pardalos, P., Sackellares, J., Shiau, D.: Quadratic binary programming and dynamical system approach to determine the predictability of epileptic seizures. J. Combin. Optim. 5, 9–26 (2001)
Iasemidis, L., Shiau, D., Pardalos, J.S.P., Prasad, A.: Dynamical resetting of the human brain at epileptic seizures: Application of nonlinear dynamics and global optimization techniques. IEEE Trans. Biomed. Eng. 51(3), 493–506 (2004)
Jain, A., Dubes, R.: Algorithm for Clustering Data. Prentice-Hall, Englewood Cliffs (1988)
Kalauzi, A., Spasic, S.: Consecutive differences as a method of signal fractal analysis. Fractals 13(4), 283–292 (2005)
Kannathal, N., Acharya, U., Lim, C., et al.: Characterization of EEG - A comparative study. Comp. Meth. Progr. Biomed. 80, 17–23 (2005)
Kannathal, N., Lim, C., Acharya, U., et al.: Entropies for detection of epilepsy in EEG. Comp. Meth. Progr. Biomed. 80, 187–194 (2005)
Katz, M.: Fractals and the analysis of waveforms. Comp. Biol. Med. 18(3), 145–156 (1988)
Kulish, V., Sourin, A., Sourina, O.: Human electroencephalograms seen as fractal time series: Mathematical analysis and visualization. Comp. Biol. Med. 36, 291–302 (2006)
Lakshmanan, M., Rajasekar, S.: Nonlinear Dynamics: Integrability, Chaos and Patterns. Springer, Berlin (2003)
Lehnertz, K., Elger, C.: Can epileptic seizures be predicted? evidence from nonlinear time series analysis of brain electrical activity. Phys. Rev. Lett. 80, 5019–5022 (1998)
Litt, B., Esteller, R., Echauz, J., Maryann, D., Shor, R., Henry, T., Pennell, P., Epstein, C., Bakay, R., Dichter, M., Vachtservanos, G.: Epileptic seizures may begin hours in advance of clinical onset: A report of five patients. Neuron 30, 51–64 (2001)
Maiwald, T., Winterhalder, M., Aschenbrenner-Scheibe, R., Voss, H., Schulze-Bonhage, A., Timmer, J.: Comparison of three nonlinear seizure prediction methods by means of the seizure prediction characteristics. Phys. D 194, 357–368 (2004)
Mandelbrot, B.: The Fractal Geometry of Nature. W.H. Freeman and Company, New York (1983)
Mandelbrot, B.: Negative fractal dimensions and multifractals. Phys. A 163, 306–315 (1990)
Meyer-Lindenberg, A.: The evolution of complexity in human brain development: An EEG study. Electroencephalogr. Clin. Neurophysiol. 99, 405–411 (1996)
Munck, J., Goncalves, S., Mammoliti, R., Heethaar, R., Silva, F.: Interactions between different EEG frequency bands and their effect on alpha-fMRI correlations. Neuroimage 47, 69–76 (2009)
Nan, X., Jinghua, X.: The fractal dimension of EEG as a physical measure of conscious human brain activities. Bull. Math. Biol. 50(5), 559–565 (1988)
Normant, F., Tricot, C.: Method for evaluating the fractal dimension of curves using convex hulls. Phys. Rev. A 43, 6518–6525 (1991)
Ocak, H.: Automatic detection of epileptic seizures in EEG using discrete wavelet transform and approximate entropy. Expert Syst. Appl. 36, 2027–2036 (2009)
Ouyang, G., Li, X., Guan, X.: Application of wavelet based similarity analysis to epileptic seizure prediction. Comp. Biol. Med. 37, 430–437 (2007)
Pachori, R.: Discrimination between ictal and seizure-free EEG signals using empirical mode decomposition. Res. Lett. Signal Process. Article ID 293056 (2008)
Paivinen, N., Lammi, S., Pitkanen, A., et al.: Epileptic seizure detection: A nonlinear view point. Comp. Meth. Progr. Biomed. 79, 151–159 (2005)
Paramanathan, P., Uthayakumar, R.: An algorithm for computing the fractal dimension of waveforms. Appl. Math. Comput. 195(2), 598–603 (2008)
Paramanathan, P., Uthayakumar, R.: Application of fractal theory in analysis of human electroencephalogram signals. Comp. Biol. Med. 38(3), 372–378 (2008)
Paramanathan, P., Uthayakumar, R.: Size measure relationship method for fractal analysis of signals. Fractals 16, 235–241 (2008)
Pardalos, P., Sackellares, J., Carney, P., Iasemidis, L.: Quantitative Neuroscience: Models, Algorithms, Diagnostics and Therapeutic Applications. Springer, Berlin (2004)
Petrosian, A.: Kolmogorov complexity of finite sequences and recognition of different preictal EEG patterns. In: Proceedings of IEEE Symposium on Computer Based Medical Systems, pp. 212–217. IEEE Xplore Digital Library, IEEE, USA (1995)
Qiong, X., Xiong, W.: Fractal dimension of voice signal waveforms. Wuhan Univ. J. Nat. Sci. 7(4), 399–402 (2002)
Quyen, M., Martinerie, J., Baulac, M., Varela, F.: Anticipating epileptic seizures in real time by non-linear analysis of similarity between EEG recordings. Neuro Rep. 10, 2149–2155 (1999)
Renyi, A.: On a new axiomatic theory of probability. Acta Math. Hung. 6, 285–335 (1955)
Sackellares, J., Iasemidis, L., Shiau, D.: Detection of the preictal transition in scalp EEG. Epilepsia 40, 176 (1999)
Sackellares, J., Iasemidis, L., Gilmore, R., Roper, S.: Epilepsy - When Chaos Fails, Chaos in the Brain? World Scientific, Singapore (2002)
Sebastian, M., Navascues, M., Valdizan, J.: Surface laplacian and fractal brain mapping. J. Comput. Appl. Math. 189(1), 132–141 (2004)
Shannon, C.: The Mathematical Theory of Communication. University of Illinois Press, Champaign (1998)
Shelberg, M.: The development of a curve and surface algorithm to measure fractal dimensions. Master’s thesis, Ohio State University (1982)
Spasic, S., Kalauzi, A., Culic, M., Grbic, G., Martac, L.: Fractal analysis of rat brain activity injury. Med. Biol. Eng. Comput. 43(4), 345–348 (2005)
Tricot, C.: Curves and Fractal Dimension. Springer, New York (1995)
Ubeyli, E.: Analysis of eeg signals using lyapunov exponents. Neural Netw. World 16(3), 257–273 (2006)
Ubeyli, E.: Statistics over features: EEG signals analysis. Comp. Biol. Med. 39, 733–741 (2009)
Uthayakumar, R., Paramanathan, P.: An algorithm for computing fractal dimension of rectifiable irregular graphs. Appl. Math. Comput. 190(1), 305–308 (2007)
Uthayakumar, R., Paramanathan, P.: Fractal dimension of irregular digitalized curves by divider method. Appl. Math. Comput. 189(1), 68–71 (2007)
Wu, L., Gotman, J.: Segmentation and classification of EEG during epileptic seizures. Electroencephalogr. Clin. Neurophysiol. 106, 344–356 (1998)
Zhang, J., Yang, X., Luo, L., Shao, J., Zhang, C., Ma, J., Wang, G., Liu, Y., Peng, C., Fang, J.: Assessing severity of obstructive sleep apnea by fractal dimension sequence analysis of sleep EEG. Phys. A 388, 4407–4414 (2009)
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The research work has been supported by University Grants Commission (UGC - MRP & SAP), Government of India, New Delhi, India.
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Uthayakumar, R. (2013). Fractal Dimension in Epileptic EEG Signal Analysis. In: Banerjee, S., Rondoni, L. (eds) Applications of Chaos and Nonlinear Dynamics in Science and Engineering - Vol. 3. Understanding Complex Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34017-8_4
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