Wavelet-based detection of coherent structures and self-affinity in financial data

Abstract

As a linear superposition of translated and dilated versions of a chosen analyzing wavelet function, the wavelet transform lends itself to the analysis of underlying multi-scale structure in nonstationary time series. In this work, we use the discrete wavelet transform (DWT) to investigate scaling and search for the presence of coherent structures in financial data. Quantitative measurements are given by the DWT of the original time series and wavelet coefficient variance. We find that variations and correlations in the transform coefficients are able to indicate the presence of structure and that measurements based on the DWT allow us to observe scaling directly in the nonstationary time series. PACS. 05.45.Tp Time series analysis -05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion. -02.90.+p Other topics in mathematical methods in physics.

Identification of the Multivariate Fractional Brownian Motion

“…To conclude the identification of the mfBm, parameters σ i , ρ ij , η ij , can be estimated by plugging estimators (20,21,22) into Equation (19). We then obtain…”

“…neuroscience, economy, sociology, physics, etc), multivariate measurements are performed and they involve specific properties such as fractality, long-range dependence, self-similarity, etc. Examples can be found in economic time series (see [11], [14], [15]), genetic sequences [2], multipoint velocity measurements in turbulence, functional Magnetic Resonance Imaging of several regions of the brain [1].…”

Multi-Fractal Spectral Analysis of the 1987 Stock Market Crash

“…Wavelet multiresolution analysis of time series of financial market returns yields a time-frequency analysis with contracted and dilated versions of a chosen prototype wavelet basis (Fleming et al, 2001). The application of wavelet MRA enables us, first, to perform time -scale or time -frequency analysis and, second, to achieve de-noising of the data for better detection of patterns.…”

“…Wavelet multiresolution based on ortgonal wavelet bases is exhaustive and complete and does not lead to statistical "double-counting." Fleming et al (2001) extract the set of wavelet detail coefficients {d j,k } using a wavelet, ψ(t) with N vanishing moments. One can identify the Hurst exponent of the series from the variances of wavelet detail coefficients based on dyadic scaling, as follows…”

“…To measure the Hurst exponents for the time series of index prices, we implemented two methods: the method of Fleming et al (2001) and the method based on the built-in algorithm in Fraclab developed by the French Institute National de Recherche en Informatique et en Automatique (National Institute for Information and Automation Research). Their method is based on discrete wavelet coefficients.…”