Cascade processes have been used to model many different self-similar systems, as they are able to accurately describe most of their global statistical properties. The so-called optimal wavelet basis allows to achieve a geometrical representation of the cascade process-named microcanonical cascade- that describes the behavior of local quantities and thus it helps to reveal the underlying dynamics of the system. In this context, we study the benefits of using the optimal wavelet in contrast to other wavelets when used to define cascade variables, and we provide an optimality degree estimator that is appropriate to determine the closest-to-optimal wavelet in real data. Particularizing the analysis to stock market series, we show that they can be represented by microcanonical cascades in both the logarithm of the price and the volatility. Also, as a promising application in forecasting, we derive the distribution of the value of next point of the series conditioned to the knowledge of past points and the cascade structure, i.e., the stochastic kernel of the cascade process.
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Arneodo A, Argoul F, Bacry E, Elezgaray J, Muzy JF (1995) Ondelettes, multifractales et turbulence. Diderot Editeur, Paris
Arneodo A, Muzy J-F, Sornette D (1998) “direct” causal cascade in the stock market. Eur Phys J B 2: 277–282
Buccigrossi RW, Simoncelli EP (1999) Image compression via joint statistical characterization in the wavelet domain. IEEE Trans Image Process 8: 1688–1701
Calvet L, Fisher A (2001) Forecasting multifractal volatility. J Economet 105: 27–58
Daubechies I (1992) Ten lectures on wavelets. CBMS-NSF Series in App. Math. Capital City Press, Montpelier
Falconer K (1990) Fractal geometry: mathematical foundations and applications. Wiley, Chichester
Frisch U (1995) Turbulence. Cambridge University Press, Cambridge
Mallat S (1999) A wavelet tour of signal processing, 2nd ed. Academic Press, New York
Muzy J-F, Sornette D, Delour J, Arneodo A (2001) Multifractal returns and hierarchical portfolio theory. Quant Finance 1: 131–148
Novikov EA (1994) Infinitely divisible distributions in turbulence. Phys Rev E 50: R3303
Perelló J, Masoliver J, Bouchaud J (2004) Multiple time scales in volatility and leverage correlations: a stochastic volatility model. J Math Finance 11: 27–50
Pont O, Turiel A, Pérez-Vicente C (2006) Application of the microcanonical multifractal formalism to monofractal systems. Phys Rev E 74: 061110
Pont O, Turiel A, Perez-Vicente C (2008) On optimal wavelet bases for the realization of microcanonical cascade processes. Under revision in Eur Phys J B, arXiv:0805.4810v1
She ZS, Leveque E (1994) Universal scaling laws in fully developed turbulence. Phys Rev Lett 72: 336–339
Turiel A, Parga N (2000a) The multi-fractal structure of contrast changes in natural images: from sharp edges to textures. Neural Comput 12: 763–793
Turiel A, Parga N (2000b) Multifractal wavelet filter of natural images. Phys Rev Lett 85: 3325–3328
Turiel A, Pérez-Vicente C (2003) Multifractal geometry in stock market time series. Phys A 322: 629–649
Turiel A, Pérez-Vicente C (2005) Role of multifractal sources in the analysis of stock market time series. Phys A 355: 475–496
Turiel A, Nadal J-P, Parga N (2003) Orientational minimal redundancy wavelets: from edge detection to perception. Vis Res 43(9): 1061–1079
Turiel A, Grazzini J, Yahia H (2005) Multiscale techniques for the detection of precipitation using thermal IR satellite images. IEEE Geosci Remote Sens Lett 2(4): 447–450. doi:10.1109/LGRS.2005.852712
Turiel A, Isern-Fontanet J, García-Ladona E, Font J (2005) Multifractal method for the instantaneous evaluation of the stream function in geophysical flows. Phys Rev Lett 95(10): 104502. doi:10.1103/PhysRevLett.95.104502
Turiel A, Pérez-Vicente C, Grazzini J (2006) Numerical methods for the estimation of multifractal singularity spectra on sampled data: a comparative study. J Comput Phys 216(1): 362–390
Turiel A, Solé J, Nieves V, Ballabrera-Poy J, García-Ladona E (2008a) Tracking oceanic currents by singularity analysis of micro-wave sea surface temperature images. Remote Sens Environ 112: 2246–2260
Turiel A, Yahia H, Pérez-Vicente C (2008b) Microcanonical multifractal formalism: a geometrical approach to multifractal systems. Part I: Singularity analysis. J Phys A 41: 015501
Wainwright MJ, Simoncelli EP, Willsky AS (2001) Random cascades on wavelet trees and their use in modeling and analyzing natural images. Appl Comput Harmonic Anal 11: 89–123
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Pont, O., Turiel, A. & Perez-Vicente, C.J. Description, modelling and forecasting of data with optimal wavelets. J Econ Interact Coord 4, 39 (2009). https://doi.org/10.1007/s11403-009-0046-x
- Optimal wavelet
- Cascade processes
- Microcanonical multifractal formalism
- Time series forecasting