Tempered fractional Brownian motion: wavelet estimation, modeling and testing. (English) Zbl 1461.62152
Summary: The Davenport spectrum is a modification of the classical Kolmogorov spectrum for the inertial range of turbulence that accounts for non-scaling low frequency behavior. Like the classical fractional Brownian motion vis-à-vis the Kolmogorov spectrum, tempered fractional Brownian motion (tfBm) is a new model that displays the Davenport spectrum. The autocorrelation of the increments of tfBm displays semi-long range dependence (hyperbolic and quasi-exponential decays over moderate and large scales, respectively), a phenomenon that has been observed in a wide range of applications from wind speeds to geophysics to finance. In this paper, we use wavelets to construct the first estimation method for tfBm and a simple and computationally efficient test for fBm vs tfBm alternatives. The properties of the wavelet estimator and test are mathematically and computationally established. An application of the methodology shows that tfBm is a better model than fBm for a geophysical flow data set.
MSC:
| 62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |
| 62M15 | Inference from stochastic processes and spectral analysis |
| 60G18 | Self-similar stochastic processes |
| 60G22 | Fractional processes, including fractional Brownian motion |
| 42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |
Keywords:
fractional Brownian motion; semi-long range dependence; tempered fractional Brownian motion; turbulence; waveletsReferences:
| [1] | Abry, P.; Didier, G., Wavelet estimation for operator fractional Brownian motion, Bernoulli, 24, 2, 895-928 (2018) · Zbl 1432.60045 |
| [2] | Abry, P.; Flandrin, P., On the initialization of the discrete wavelet transform algorithm, IEEE Signal Process. Lett., 1, 2, 32-34 (1994) |
| [3] | Abry, P.; Didier, G.; Li, H., Two-step wavelet-based estimation for Gaussian mixed fractional processes, Stat. Inference Stoch. Process., 22, 157-185 (2019) · Zbl 1420.62366 |
| [4] | Abry, P.; Flandrin, P.; Taqqu, M.; Veitch, D., Wavelets for the analysis, estimation and synthesis of scaling data, (Self-Similar Network Traffic and Performance Evaluation (2000), Wiley), 3988 |
| [5] | Abry, P.; Flandrin, P.; Taqqu, M. S.; Veitch, D., Self-similarity and long-range dependence through the wavelet lens, (Theory and Applications of Long-Range Dependence (2003), Birkhäuser), 527-556 · Zbl 1029.60028 |
| [6] | Anh, V.; Angulo, J.; Ruiz-Medina, M., Possible long-range dependence in fractional random fields, J. Statist. Plann. Inference, 80, 1, 95-110 (1999) · Zbl 1039.62090 |
| [7] | Anh, V.; Heyde, C.; Tieng, Q., Stochastic models for fractal processes, J. Statist. Plann. Inference, 80, 1, 123-135 (1999) · Zbl 0954.62099 |
| [8] | Baeumer, B.; Meerschaert, M. M., Tempered stable Lévy motion and transient super-diffusion, J. Comput. Appl. Math., 233, 10, 2438-2448 (2010) · Zbl 1423.60079 |
| [9] | Bardet, J.-M., Statistical study of the wavelet analysis of fractional Brownian motion, IEEE Trans. Inform. Theory, 48, 4, 991-999 (2002) · Zbl 1061.60036 |
| [10] | Bardet, J.-M.; Lang, G.; Moulines, E.; Soulier, P., Wavelet estimator of long-range dependent processes, Stat. Inference Stoch. Process., 3, 1-2, 85-99 (2000) · Zbl 1054.62579 |
| [11] | Bardet, J.-M.; Lang, G.; Oppenheim, G.; Philippe, A.; Stoev, S.; Taqqu, M., Semi-parametric estimation of the long-range dependence parameter: a survey, (Doukhan, P.; Oppenheim, G.; Taqqu, M. S., Theory and Applications of Long-Range Dependence (2003), Birkhäuser: Birkhäuser Boston), 557-577 · Zbl 1032.62077 |
| [12] | Beaupuits, J.; Otárola, A.; Rantakyrö, F. T.; Rivera, R. C.; Radford, S. J.E.; Nyman, L., Analysis of wind data gathered at Chajnantor, (ALMA Memo 497 (2004), National Radio Astronomy Observatory) |
| [13] | Bianchi, M. L.; Rachev, S. T.; Kim, Y. S.; Fabozzi, F. J., Tempered stable distributions and processes in finance: numerical analysis, (Mathematical and Statistical Methods for Actuarial Sciences and Finance (2010)), 33-42 |
| [14] | Boniece, B. C.; Sabzikar, F.; Didier, G., Tempered fractional Brownian motion: wavelet estimation and modeling of geophysical flows, (IEEE Statistical Signal Processing Workshop. IEEE Statistical Signal Processing Workshop, Freiburg, Germany (2018), IEEE), 1-5 |
| [15] | Brockwell, P.; Davis, R., Time Series: Theory and Methods (1991), Springer · Zbl 0709.62080 |
| [16] | Chen, Y.; Wang, X.; Deng, W., Localization and ballistic diffusion for the tempered fractional Brownian-Langevin motion, J. Stat. Phys., 169, 18-37 (2017) · Zbl 1397.82042 |
| [17] | Chen, Y.; Wang, X.; Deng, W., Resonant behavior of the generalized Langevin system with tempered Mittag-Leffler memory kernel, J. Phys. A: Math. Theor., 51, 18, Article 185201 pp. (2018) · Zbl 1392.82044 |
| [18] | Chevillard, L., Regularized fractional Ornstein-Uhlenbeck processes and their relevance to the modeling of fluid turbulence, Phys. Rev. E, 96, Article 033111 pp. (2017) |
| [19] | Cohen, A., Numerical Analysis of Wavelet Methods, vol. 32 (2003), North-Holland: North-Holland Amsterdam · Zbl 1038.65151 |
| [20] | Cohen, S.; Rosiński, J., Gaussian approximation of multivariate Lévy processes with applications to simulation of tempered stable processes, Bernoulli, 195-210 (2007) · Zbl 1121.60049 |
| [21] | Cont, R.; Potters, M.; Bouchaud, J.-P., Scaling in stock market data: stable laws and beyond, (Scale Invariance and Beyond, vol. 7 (1997)), 75-85 · Zbl 0979.91037 |
| [22] | Craigmile, P.; Guttorp, P.; Percival, D., Wavelet-based parameter estimation for polynomial contaminated fractionally differenced processes, IEEE Trans. Signal Process., 53, 8, 3151-3161 (2005) · Zbl 1373.62434 |
| [23] | Cramér, H.; Leadbetter, M. R., Stationary and Related Stochastic ProcessesSample Function Properties and Their Applications (1967), Courier Dover Publications · Zbl 0162.21102 |
| [24] | Dacorogna, M. M.; Müller, U. A.; Nagler, R. J.; Olsen, R. B.; Pictet, O. V., A geographical model for the daily and weekly seasonal volatility in the foreign exchange market, J. Int. Money Financ., 12, 4, 413-438 (1993) |
| [25] | Daubechies, I., Ten Lectures on Wavelets, vol. 61 (1992), Society for Industrial and Applied Mathematics: Society for Industrial and Applied Mathematics Philadelphia, PA · Zbl 0776.42018 |
| [26] | Davenport, A., The spectrum of horizontal gustiness near the ground in high winds, Q. J. R. Meteorol. Soc., 87, 372, 194-211 (1961) |
| [27] | Davies, R.; Harte, D., Tests for Hurst effect, Biometrika, 74, 1, 95-101 (1987) · Zbl 0612.62123 |
| [28] | Didier, G.; Pipiras, V., Adaptive wavelet decompositions of stationary time series, J. Time Series Anal., 31, 3, 182-209 (2010) · Zbl 1224.60071 |
| [29] | Embrechts, P.; Maejima, M., Selfsimilar Processes, Princeton Series in Applied Mathematics (2002), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 1008.60003 |
| [30] | Flandrin, P., Wavelet analysis and synthesis of fractional Brownian motion, IEEE Trans. Inform. Theory, 38, 910-917 (1992) · Zbl 0743.60078 |
| [31] | Frecon, J.; Didier, G.; Pustelnik, N.; Abry, P., Non-linear wavelet regression and branch & bound optimization for the full identification of bivariate operator fractional Brownian motion, IEEE Trans. Signal Process., 64, 15, 4040-4049 (2016) · Zbl 1414.94201 |
| [32] | Friedlander, S. K.; Topper, L., Turbulence: Classic Papers on Statistical Theory (1961), Interscience Publishers · Zbl 0098.40902 |
| [33] | Gajda, J.; Magdziarz, M., Fractional Fokker-Planck equation with tempered a-stable waiting times: Langevin picture and computer simulation, Phys. Rev. E, 82, Article 011117 pp. (2010) |
| [34] | Gao, J.; Anh, V.; Heyde, C.; Tieng, Q., Parameter estimation of stochastic processes with long-range dependence and intermittency, J. Time Series Anal., 22, 5, 517-535 (2001) · Zbl 0979.62071 |
| [35] | Giraitis, L.; Kokoszka, P.; Leipus, R., Stationary ARCH models: dependence structure and central limit theorem, Econometric Theory, 16, 1, 3-22 (2000) · Zbl 0986.60030 |
| [36] | Giraitis, L.; Kokoszka, P.; Leipus, R.; Teyssière, G., On the power of r/s-type tests under contiguous and semi-long memory alternatives, Acta Appl. Math., 78, 1-3, 285-299 (2003) · Zbl 1029.62075 |
| [37] | Gradshteyn, I. S.; Ryzhik, I. M., Table of Integrals, Series, and Products (2007), Academic Press: Academic Press New York, NY · Zbl 1208.65001 |
| [38] | Granger, C. W.J.; Ding, Z., Varieties of long memory models, J. Econometrics, 73, 1, 61-77 (1996) · Zbl 0854.62100 |
| [39] | Graves, T.; Gramacy, R.; Watkins, N.; Franzke, C., A brief history of long memory: Hurst, Mandelbrot and the road to ARFIMA, 1951-1980, Entropy, 19, 9, 437 (2017) |
| [40] | Kallenberg, O., Foundations of Modern Probability, Probability and Its Applications (New York) (2002), Springer-Verlag: Springer-Verlag New York · Zbl 0996.60001 |
| [41] | Kawai, R.; Masuda, H., Infinite variation tempered stable Ornstein-Uhlenbeck processes with discrete observations, Comm. Statist. Simulation Comput., 41, 1, 125-139 (2012) · Zbl 1489.62261 |
| [42] | Kienitz, J., Tempered Stable Process (2010), Wiley Online Library |
| [43] | Kolmogorov, A. N., The Wiener spiral and some other interesting curves in Hilbert space, Dokl. Akad. Nauk SSSR, 26, 115-118 (1940) · Zbl 0022.36001 |
| [44] | Kolmogorov, A. N., The local structure of turbulence in an incompressible fluid at very high Reynolds numbers, Dokl. Akad. Nauk SSSR, 30, 299-303 (1941) |
| [45] | Küchler, U.; Tappe, S., Tempered stable distributions and processes, Stochastic Process. Appl., 123, 12, 4256-4293 (2013) · Zbl 1352.60021 |
| [46] | Li, Y.; Kareem, A., ARMA systems in wind engineering, Probab. Eng. Mech., 5, 2, 49-59 (1990) |
| [47] | Liemert, A.; Sandev, T.; Kantz, H., Generalized Langevin equation with tempered memory kernel, Phys. A, 466, 356-369 (2017) · Zbl 1400.82195 |
| [48] | Ling, S.; Li, W. K., Asymptotic inference for nonstationary fractionally integrated autoregressive moving-average models, Econometric Theory, 17, 4, 738-764 (2001) · Zbl 1018.62075 |
| [49] | Mallat, S., A Wavelet Tour of Signal Processing (1999), Academic Press · Zbl 0998.94510 |
| [50] | Mandelbrot, B.; Van Ness, J., Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10, 4, 422-437 (1968) · Zbl 0179.47801 |
| [51] | Masry, E., The wavelet transform of stochastic processes with stationary increments and its application to fractional Brownian motion, IEEE Trans. Inform. Theory, 39, 1, 260-264 (1993) · Zbl 0768.60036 |
| [52] | Meerschaert, M.; Sabzikar, F., Tempered fractional Brownian motion, Statist. Probab. Lett., 83, 10, 2269-2275 (2013) · Zbl 1287.60050 |
| [53] | Meerschaert, M.; Sabzikar, F., Stochastic integration for tempered fractional Brownian motion, Stochastic Process. Appl., 124, 7, 2363-2387 (2014) · Zbl 1329.60166 |
| [54] | Meerschaert, M. M.; Zhang, Y.; Baeumer, B., Tempered anomalous diffusion in heterogeneous systems, Geophys. Res. Lett. (2008) |
| [55] | Meerschaert, M.; Sabzikar, F.; Phanikumar, M.; Zeleke, A., Tempered fractional time series model for turbulence in geophysical flows, J. Stat. Mech. Theory Exp., 2014, 9, Article P09023 pp. (2014) |
| [56] | Moulines, E.; Roueff, F.; Taqqu, M., Central limit theorem for the log-regression wavelet estimation of the memory parameter in the Gaussian semi-parametric context, Fractals, 15, 4, 301-313 (2007) · Zbl 1141.62073 |
| [57] | Moulines, E.; Roueff, F.; Taqqu, M., On the spectral density of the wavelet coefficients of long-memory time series with application to the log-regression estimation of the memory parameter, J. Time Series Anal., 28, 2, 155-187 (2007) · Zbl 1150.62058 |
| [58] | Moulines, E.; Roueff, F.; Taqqu, M., A wavelet Whittle estimator of the memory parameter of a nonstationary Gaussian time series, Ann. Statist., 1925-1956 (2008) · Zbl 1142.62062 |
| [59] | Norton, D.; Wolff, C., Mobile offshore platform wind loads, (Offshore Technology Conference (1981)) |
| [60] | Olver, F.; Lozier, D.; Boisvert, R.; Clark, C. W., NIST Handbook of Mathematical Functions (2010), Cambridge University Press · Zbl 1198.00002 |
| [61] | Percival, D. B.; Walden, A., Wavelet Methods for Time Series Analysis, vol. 4 (2006), Cambridge University Press · Zbl 1129.62080 |
| [62] | Phillips, P. C.B.; Moon, H. R.; Xiao, Z., How to estimate autoregressive roots near unity, Econometric Theory, 17, 1, 29-69 (2001) · Zbl 0997.62071 |
| [63] | Piryatinska, A.; Sanchev, A.; Woyczynski, W. A., Models of anomalous diffusion: the subdiffusive case, Phys. A, 349, 375-420 (2005) |
| [64] | Rosiński, J., Tempering stable processes, Stochastic Process. Appl., 117, 6, 677-707 (2007) · Zbl 1118.60037 |
| [65] | Rosiński, J.; Sinclair, J., Generalized tempered stable processes, (Stability in Probability, vol. 90 (2010)), 153-170 · Zbl 1210.60048 |
| [66] | Sabzikar, F., Tempered Hermite process, Modern Stoch. Theory Appl., 2, 327-341 (2015) · Zbl 1352.60035 |
| [67] | Sabzikar, F.; Surgailis, D., Tempered fractional Brownian and stable motions of second kind, Statist. Probab. Lett., 132, 17-27 (2018) · Zbl 1380.60047 |
| [68] | Sabzikar, F.; Meerschaert, M.; Chen, J., Tempered fractional calculus, J. Comput. Phys., 293, 14-28 (2015) · Zbl 1349.26017 |
| [69] | Sabzikar, F.; Meerschaert, M.; McLeod, A. I., Parameter estimation for ARTFIMA time series, J. Statist. Plann. Inference, 200, 129-145 (2019) · Zbl 1421.62130 |
| [70] | Sabzikar, F.; Wang, Q.; Phillips, P. C.B., Asymptotic theory for near integrated processes driven by tempered linear processes, (Cowles Foundation Discussion Paper No. 2131 (2018)), 1-27 |
| [71] | Sandev, T.; Chechkin, A.; Kantz, H.; Metzler, R., Diffusion and Fokker-Planck-Smoluchowski equations with generalized memory kernel, Fract. Calc. Appl. Anal., 18, 4, 1006-1038 (2015) · Zbl 1338.60199 |
| [72] | Shiryaev, A. N., Kolmogorov and the Turbulence (1999), University of Aarhus: Centre for Mathematical Physics and Stochastics |
| [73] | Stanislavsky, A.; Weron, K.; Weron, A., Diffusion and relaxation controlled by tempered α-stable processes, Phys. Rev. E, 78, 5, Article 051106 pp. (2008) |
| [74] | Stoev, S.; Pipiras, V.; Taqqu, M., Estimation of the self-similarity parameter in linear fractional stable motion, Signal Process., 82, 1873-1901 (2002) · Zbl 1098.94589 |
| [75] | Taqqu, M. S., Fractional Brownian motion and long range dependence, (Doukhan, P.; Oppenheim, G.; Taqqu, M. S., Theory and Applications of Long-Range Dependence (2003), Birkhäuser: Birkhäuser Boston), 5-38 · Zbl 1039.60041 |
| [76] | Van der Vaart, A. W., Asymptotic Statistics, vol. 3 (1998), Cambridge University Press · Zbl 0910.62001 |
| [77] | Veitch, D.; Abry, P., A wavelet-based joint estimator of the parameters of long-range dependence, IEEE Trans. Inform. Theory, 45, 3, 878-897 (1999) · Zbl 0945.94006 |
| [78] | Vignat, C., A generalized Isserlis theorem for location mixtures of Gaussian random vectors, Statist. Probab. Lett., 82, 1, 67-71 (2012) · Zbl 1241.60011 |
| [79] | Wendt, H.; Didier, G.; Combrexelle, S.; Abry, P., Multivariate Hadamard self-similarity: testing fractal connectivity, Phys. D, Nonlinear Phenom., 356-357, 1-36 (2017) · Zbl 1381.62121 |
| [80] | Wood, A.; Chan, G., Simulation of stationary Gaussian processes in \([ 0 , 1 ]^d\), J. Comput. Graph. Statist., 3, 4, 409-432 (1994) |
| [81] | Wornell, G.; Oppenheim, A., Estimation of fractal signals from noisy measurements using wavelets, IEEE Trans. Signal Process., 40, 3, 611-623 (1992) |
| [82] | Wu, X.; Deng, W.; Barkai, E., Tempered fractional Feynman-Kac equation: theory and examples, Phys. Rev. E, 93, 3, Article 032151 pp. (2016) |
| [83] | Zeng, C.; Yang, Q.; Chen, Y. Q., Bifurcation dynamics of the tempered fractional Langevin equation, Chaos, 26, 8, Article 084310 pp. (2016) · Zbl 1378.60066 |
| [84] | Zhang, X.; Xiao, W., Arbitrage with fractional Gaussian processes, Phys. A, 471, 620-628 (2017) · Zbl 1400.91692 |
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.
