Hurst Parameter Estimation Using Artificial Neural Networks

 

S. Ledesma–Orozco*¹, J. Ruiz–Pinales², G. García–Hernández³, G. Cerda–Villafaña⁴, D. Hernández–Fusilier⁵

 

^1,2,3,4,5 Universidad de Guanajuato, Comunidad de Palo Blanco, C.P.36885 Salamanca, Guanajuato, Mexico. *E–mail: selo@ugto.mx

 

ABSTRACT

The Hurst parameter captures the amount of long–range dependence (LRD) in a time series. There are several methods to estimate the Hurst parameter, being the most popular: the variance–time plot, the R/S plot, the periodogram, and Whittle's estimator. The first three are graphical methods, and the estimation accuracy depends on how the plot is interpreted and calculated. In contrast, Whittle's estimator is based on a maximum likelihood technique and does not depend on a graph reading; however, it is computationally expensive. A new method to estimate the Hurst parameter is proposed. This new method is based on an artificial neural network. Experimental results show that this method outperforms traditional approaches, and can be used on applications where a fast and accurate estimate of the Hurst parameter is required, i.e., computer network traffic control. Additionally, the Hurst parameter was computed on series of different length using several methods. The simulation results show that the proposed method is at least ten times faster than traditional methods.

Keywords: Parameter estimation, time series, network traffic analysis, neural network.

 

RESUMEN

El parámetro de Hurst captura la cantidad de dependencia de rango amplio (LRD) en las series de tiempo. Hay varios métodos para estimar el parámetro de Hurst, siendo los más populares: la gráfica de varianza contra tiempo, la gráfica R/S, el periodograma, y el estimador de Whittle. Los tres primeros son métodos gráficos, y la precisión de la estimación depende de cómo se interprete y calcule la gráfica. Por otro lado, el estimador de Whittle se basa en una técnica de máxima probabilidad y no depende de una lectura gráfica; sin embargo, éste requiere una gran demanda computacional para su cálculo. Se propone un nuevo método para estimar el parámetro de Hurst. Este nuevo método está basado en una red neuronal artificial. Los resultados experimentales muestran que este método supera a los métodos tradicionales, y que puede ser usado en aplicaciones que requieran una estimación precisa y rápida del parámetro de Hurst, por ejemplo en control de tráfico en redes de computadoras. Adicionalmente, el parámetro de Hurst se calculó en series de diferentes tamaños utilizando varios métodos. Los resultados de la simulación muestran que el método propuesto es por lo menos diez veces más rápido que los métodos tradicionales.

 

DESCARGAR ARTÍCULO EN FORMATO PDF

 

References

[1] J. Beran, Statistical methods for data with long–range dependence. Statistical Science, Volume 7, Number 4, pp. 404–427, (1992).         [ Links ]

[2] A. Erramilli, O, Narayan and W. Willinger, Experimental queueing analysis with long–range dependent packet traffic. IEEE/ACM Transactions on Networking, Volume 4, Number 29, pp. 209–223, (1996).         [ Links ]

[3] R. Fox and M. S. Taqqu, Large–sample properties of parameter estimates for strongly dependent stationary Gaussian time series. The Annals of Statistics, Volume 4, Number 2, pp. 517–532, (1985).         [ Links ]

[4] T. Karagiannis, M. Molle, and M. Faloutsos, Long–range dependence: ten years of Internet traffic modeling. IEEE Internet Computing, Volume 8, Number 5, pp. 5764, (2004).         [ Links ]

[5] M. Krunz and I. Matta, Analytical investigation of the bias effect in variance–type estimators for inference of long–range dependence. Computer Networks, Volume 40, Number 3, pp. 445–458, (2002).         [ Links ]

[6] B. B. Mandelbrot and J. R. Wallis, Some long–run properties of geophysical records. Fractals in the earth sciences, New York: Plenum Press, (1969).         [ Links ]

[7] O. Rose, Estimation of the Hurst Parameter of Long–Range Dependent Time Series. Report 137, Institute of Computer Science, University of Wurzburg, (1996).         [ Links ]

[8] S. Stoev, M. S. Taqqu, C. Park, G. Michailidis and J. S. Marron, LASS: a tool for the local analysis of self–similarity. Computational Statistics & Data Analysis, Volume 50, Number 9, pp. 2447–2471, (2006).         [ Links ]

[9] T. Masters, Practical Neural Network Recipes in C++. Preparing Input Data (C–16), Academic Press, Inc., pp. 253–270, (1993).         [ Links ]

[10] S. J. Russel and P. Norvig, Artificial Intelligence: A Modern Approach. Prentice–Hall of India, Second Edition. Statistical Learning Methods (C–20), pp. 712–762, (2006).         [ Links ]

[11] T. Masters, Neural, Novel & Hybrid Algorithms for Time Series Prediction. Neural Network Tools (C–10), John Wiley & Sons Inc., pp. 367–374, (1995).         [ Links ]

[12] T. Masters, Signal and Image Processing With Neural Networks. Data Preparation for Neural Networks (C–3), John Wiley & Sons Inc., pp. 61–80, (1994).         [ Links ]

[13] T. Masters, Advanced Algorithms for Neural Networks. Assessing Generalization Ability (C–9), John Wiley & Sons Inc., pp. 335–380, (1995).         [ Links ]

[14] R. D. Reed and R. J. Marks II, Neural Smithing: Supervised Learning in Feedforward Artificial Neural Networks. Factors Influencing Generalization (C–14), The MIT Press, pp. 239–256, (1999).         [ Links ]

[15] http://en.wikipedia.org/wiki/Neural_Lab        [ Links ]

[16] Y. Chen, R. Sun and A. Zhou, An improved Hurst parameter estimator based on fractional Fourier transform, Telecommunication Systems, Volume 43, Numbers 3–4, pp. 197–206, (2010).         [ Links ]

[17] V. Paxson, Fast, approximate synthesis of fractional Gaussian noise for generating self–similar network traffic, Computer Communications Review, Volume 27, pp. 518, (1997).         [ Links ]

[18] S. Ledesma, D. Liu, and D. Hernandez, Two approximation methods to synthesize the power spectrum of fractional Gaussian noise. Computational Statistics and Data Analysis, Volume 52, Number 2, pp. 1047–1062, (2007).         [ Links ]

[19] S. Ledesma and D. Liu, Synthesis of fractional Gaussian noise using linear approximation for generating self–similar network traffic. ACM Computer Communication Review, Volume 30, Number 2, pp. 4–17, (2000).         [ Links ]

[20] A. V. Oppenheim and R. W. Schafer, Discrete–time signal processing. Sampling of Continuos–Time Signals (C–3), Upper Saddle River, N.J., Prentice Hall, pp. 80148, (1989).         [ Links ]

[21] J. Purczynski and P. Wlodarski, On fast generation of fractional Gaussian noise. Computational Statistics and Data Analysis, Volume 50, Number 10, pp. 25372551, (2005).         [ Links ]

[22] M. Kulikovs, S. Sharkovsky and E. Petersons, Comparative studies of methods for accurate Hurst parameter estimation, Electronics and Electrical Engineering, Volume 7, Number 103, pp. 113–116, (2010).         [ Links ]

[23] J. Park and C. Park, Robust estimation of the Hurst parameter and selection of an onset scaling, Statistica Sinica, Volume 19, pp. 1531–1555, (2009).         [ Links ]

[24] N. Wang, Y. Li and H. Zhang, Hurst exponent estimation based on moving average method, Advances in wireless networks and information systems, Lecture Notes in Electrical Engineering, Volume 72, pp. 137–142, (2010).         [ Links ]

[25] M. T. Jones, AI Application Programming. Introduction to Neural Networks and the Backpropagation Algorithm (C–8), Charles River Media, 2nd edition, pp. 165–204, (2005).         [ Links ]

[26] http://en.wikipedia.org/wiki/Confusion_matrix        [ Links ]

[27] J. C. Ramirez and R. D. Torres, Local and Cumulative Analysis of Self–similar Traffic Communications and Computers, Volume 37, Number 1, pp. 27–33, (2006).         [ Links ]