Correlation properties of (discrete) fractional Gaussian noise and fractional Brownian motion. (English) Zbl 1393.60036
Summary: The fractional Gaussian noise/fractional Brownian motion framework (fGn/fBm) has been widely used for modeling and interpreting physiological and behavioral data. The concept of \(1/f\) noise, reflecting a kind of optimal complexity in the underlying systems, is of central interest in this approach. It is generally considered that fGn and fBm represent a continuum, punctuated by the boundary of “ideal” \(1/f\) noise. In the present paper, we focus on the correlation properties of discrete-time versions of these processes (dfGn and dfBm). We especially derive a new analytical expression of the autocorrelation function (ACF) of dfBm. We analyze the limit behavior of dfGn and dfBm when they approach their upper and lower limits, respectively. We show that, as \(H\) approaches 1, the ACF of dfGn tends towards 1 at all lags, suggesting that dfGn series tend towards straight line. Conversely, as \(H\) approaches 0, the ACF of dfBm tends towards 0 at all lags, suggesting that dfBm series tend towards white noise. These results reveal a severe breakdown of correlation properties around the \(1/f\) boundary and challenge the idea of a smooth transition between dfGn and dfBm processes. We discuss the implications of these findings for the application of the dfGn/dfBm model to experimental series, in terms of theoretical interpretation and modeling.
MSC:
| 60G22 | Fractional processes, including fractional Brownian motion |
Software:
longmemoReferences:
| [1] | Goldberger, A. L.; Amaral, L. A. N.; Hausdorff, J. M.; Ivanov, P. C.; Peng, C.-K.; Stanley, H. E., Fractal dynamics in physiology: alterations with disease and aging, Proceedings of the National Academy of Sciences of the United States of America, 99, 1, 2466-2472, (2002) · doi:10.1073/pnas.012579499 |
| [2] | Wallot, S.; Fusaroli, R.; Tylén, K.; Jegindø, E.-M., Using complexity metrics with R-R intervals and BPM heart rate measures, Frontiers in Physiology, 4, article 211, (2013) · doi:10.3389/fphys.2013.00211 |
| [3] | Herman, P.; Sanganahalli, B. G.; Hyder, F.; Eke, A., Fractal analysis of spontaneous fluctuations of the BOLD signal in rat brain, NeuroImage, 58, 4, 1060-1069, (2011) · doi:10.1016/j.neuroimage.2011.06.082 |
| [4] | Montez, T.; Poil, S.-S.; Jones, B. F.; Manshanden, I.; Verbunt, J. P. A.; van Dijk, B. W.; Brussaard, A. B.; van Ooyen, A.; Stam, C. J.; Scheltens, P.; Linkenkaer-Hansen, K., Altered temporal correlations in parietal alpha and prefrontal theta oscillations in early-stage Alzheimer disease, Proceedings of the National Academy of Sciences of the United States of America, 106, 5, 1614-1619, (2009) · doi:10.1073/pnas.0811699106 |
| [5] | Fadel, P. J.; Barman, S. M.; Phillips, S. W.; Gebber, G. L., Fractal fluctuations in human respiration, Journal of Applied Physiology, 97, 6, 2056-2064, (12004) · doi:10.1152/japplphysiol.00657.2004 |
| [6] | Latka, M.; Glaubic-Latka, M.; Latka, D.; West, B. J., Fractal rigidity in migraine, Chaos, Solitons and Fractals, 20, 1, 165-170, (2004) · Zbl 1071.92016 · doi:10.1016/S0960-0779(03)00440-5 |
| [7] | Sosnoff, J. J.; Valantine, A. D.; Newell, K. M., The adaptive range of 1/f isometric force production, Journal of Experimental Psychology: Human Perception and Performance, 35, 2, 439-446, (2009) · doi:10.1037/a0012731 |
| [8] | Stephen, D. G.; Anastas, J., Fractal fluctuations in gaze speed visual search, Attention, Perception, and Psychophysics, 73, 3, 666-677, (2011) · doi:10.3758/s13414-010-0069-3 |
| [9] | Gilden, D. L.; Thornton, T.; Mallon, M. W., 1/F Noise in human cognition, Science, 267, 5205, 1837-1839, (1995) · doi:10.1126/science.7892611 |
| [10] | Lemoine, L.; Torre, K.; Delignières, D., Testing for the presence of 1/f noise in continuation tapping data, Canadian Journal of Experimental Psychology, 60, 4, 247-257, (2006) · doi:10.1037/cjep2006023 |
| [11] | Torre, K.; Delignières, D.; Lemoine, L., 1/f β fluctuations in bimanual coordination: an additional challenge for modeling, Experimental Brain Research, 183, 2, 225-234, (2007) · doi:10.1007/s00221-007-1035-8 |
| [12] | Mafahim, J. U.; Lambert, D.; Zare, M.; Grigolini, P., Complexity matching in neural networks, New Journal of Physics, 17, (2015) · Zbl 1452.92003 · doi:10.1088/1367-2630/17/1/015003 |
| [13] | Mandelbrot, B. B.; Van Ness, J. W., Fractional Brownian motions, fractional noises and applications, SIAM Review, 10, 4, 422-437, (1968) · Zbl 0179.47801 · doi:10.1137/1010093 |
| [14] | Hurst, B. E., Long-term storage capacity of reservoirs, Transactions of the American Society of Civil Engineers, 116, 770-799, (1951) |
| [15] | Matsuzaki, M., Fractals in earthquakes, Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences, 348, 449-457, (1994) · Zbl 0860.73051 · doi:10.1098/rsta.1994.0104 |
| [16] | Leland, W. E.; Taqqu, M. S.; Willinger, W.; Wilson, D. V., On the self-similar nature of Ethernet traffic (extended version), IEEE/ACM Transactions on Networking, 2, 1, 1-15, (1994) · doi:10.1109/90.282603 |
| [17] | Lipsitz, L. A.; Goldberger, A. L., Loss of ‘complexity’ and aging: potential applications of fractals and chaos theory to senescence, The Journal of the American Medical Association, 267, 13, 1806-1809, (1992) · doi:10.1001/jama.267.13.1806 |
| [18] | Beran, J., Statistics for Long-Memory Processes, (1994), CRC Press · Zbl 0869.60045 |
| [19] | Qian, H.; Rangarajan, G.; Ding, M., Fractional Brownian motion and fractional Gaussian noise, Processes with Long-Range Correlations: Theory and Applications. Processes with Long-Range Correlations: Theory and Applications, Lecture Notes in Physics, 621, 22-33, (2003), Berlin, Germany: Springer, Berlin, Germany · doi:10.1007/3-540-44832-2_2 |
| [20] | Magre, O.; Guglielmi, M., Approximation de l’autocorrélation des incréments du (fbm) de Mandelbrot par modélisation de Barnes et Allan, Colloques sur le Traitement du Signal et des Images, 15, 5-8, (1995) |
| [21] | Wing, A. M.; Kristofferson, A. B., Response delays and the timing of discrete motor responses, Perception & Psychophysics, 14, 1, 5-12, (1973) · doi:10.3758/bf03198607 |
| [22] | Delignières, D.; Marmelat, V., Theoretical and methodological issues in serial correlation analysis, Progress in Motor Control: Neural, Computational and Dynamic Approaches. Progress in Motor Control: Neural, Computational and Dynamic Approaches, Advances in Experimental Medicine and Biology, 782, 127-148, (2013), Berlin, Germany: Springer, Berlin, Germany · Zbl 1331.34084 · doi:10.1007/978-1-4614-5465-6_7 |
| [23] | Eke, A.; Hermán, P.; Bassingthwaighte, J. B.; Raymond, G. M.; Percival, D. B.; Cannon, M.; Balla, I.; Ikrényi, C., Physiological time series: distinguishing fractal noises from motions, Pflügers Archiv—European Journal of Physiology, 439, 4, 403-415, (2000) · doi:10.1007/s004240050957 |
| [24] | Wagenmakers, E. J.; Farrell, S.; Ratcliff, R., Estimation and interpretation of 1/f(alpha) noise in human cognition, Psychonomic Bulletin & Review, 11, 579-615, (2004) · doi:10.3758/BF03196615 |
| [25] | West, B. J.; Geneston, E. L.; Grigolini, P., Maximizing information exchange between complex networks, Physics Reports, 468, 1–3, 1-99, (2008) · Zbl 1211.94005 · doi:10.1016/j.physrep.2008.06.003 |
| [26] | Peng, C.-K.; Mietus, J.; Hausdorff, J. M.; Havlin, S.; Stanley, H. E.; Goldberger, A. L., Long-range anticorrelations and non-Gaussian behavior of the heartbeat, Physical Review Letters, 70, 9, 1343-1346, (1993) · doi:10.1103/PhysRevLett.70.1343 |
| [27] | Marmelat, V.; Torre, K.; Delignières, D., Relative roughness: an index for testing the suitability of the monofractal model, Frontiers in Physiology, 3, article 208, (2012) · doi:10.3389/fphys.2012.00208 |
| [28] | Das, S., Functional Fractional Calculus, (2011), Berlin, Germany: Springer, Berlin, Germany · Zbl 1225.26007 · doi:10.1007/978-3-642-20545-3 |
| [29] | Delignieres, D.; Ramdani, S.; Lemoine, L.; Torre, K.; Fortes, M.; Ninot, G., Fractal analyses for ‘short’ time series: a re-assessment of classical methods, Journal of Mathematical Psychology, 50, 6, 525-544, (2006) · Zbl 1105.62085 · doi:10.1016/j.jmp.2006.07.004 |
| [30] | Kantelhardt, J. W.; Meyers, R. A., Fractal and multifractal time series, Mathematics of Complexity and Dynamical Systems, 463-487, (2011), Berlin, Germany: Springer, Berlin, Germany |
| [31] | Davies, R. B.; Harte, D. S., Tests for hurst effect, Biometrika, 74, 1, 95-101, (1987) · Zbl 0612.62123 · doi:10.1093/biomet/74.1.95 |
| [32] | Saupe, D.; Peitgen, H. O.; Saupe, D., Algorithms for random fractals, The Science of Fractal Images, 71-136, (1988), Berlin, Germany: Springer, Berlin, Germany · doi:10.1007/978-1-4612-3784-6_2 |
| [33] | Farag, A. F.; El-Metwally, S. M.; Morsy, A. A. A.; Balas, V. E.; Fodor, J.; Várkonyi-Kóczy, A. R.; Dombi, J.; Jain, L. C., Automated sleep staging using detrended fluctuation analysis of sleep EEG, Soft Computing Applications. Soft Computing Applications, Advances in Intelligent Systems and Computing, 501-510, (2013), Berlin, Germany: Springer, Berlin, Germany · doi:10.1007/978-3-642-33941-7_44 |
| [34] | Yeh, R.-G.; Shieh, J.-S.; Han, Y.-Y.; Wang, Y.-J.; Tseng, S.-C., Detrended fluctuation analyses of short-term heart rate variability in surgical intensive care units, Biomedical Engineering—Applications, Basis and Communications, 18, 2, 67-72, (2006) · doi:10.4015/s1016237206000130 |
| [35] | Hartmann, A.; Mukli, P.; Nagy, Z.; Kocsis, L.; Hermán, P.; Eke, A., Real-time fractal signal processing in the time domain, Physica A: Statistical Mechanics and its Applications, 392, 1, 89-102, (2013) · doi:10.1016/j.physa.2012.08.002 |
| [36] | Kamath, C., A new approach to detect congestive heart failure using sequential spectrum of electrocardiogram signals, Medical Engineering and Physics, 34, 10, 1503-1509, (2012) · doi:10.1016/j.medengphy.2012.03.001 |
| [37] | Kiefer, A. K.; Rhéa, C. K.; Warren, W. H.; Steinicke, F.; Visell, Y.; Campos, J.; Lecuyer, A., VR-based assessment and rehabilitation of functional mobility, Human Walking in Virtual Environments, 333-350, (2013), Berlin, Germany: Springer, Berlin, Germany |
| [38] | Katsavelis, D.; Mukherjee, M.; Decker, L.; Stergiou, N., The effect of virtual reality on gait variability, Nonlinear Dynamics, Psychology, and Life Sciences, 14, 3, 239-256, (2010) |
| [39] | Rhea, C. K.; Kiefer, A. W.; Wittstein, M. W.; Leonard, K. B.; MacPherson, R. P.; Wright, W. G.; Haran, F. J., Fractal gait patterns are retained after entrainment to a fractal stimulus, PLoS ONE, 9, 9, (2014) · doi:10.1371/journal.pone.0106755 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
