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Wavelet analysis and scaling properties of time series

Profile image of Prasanta K. PanigrahiPrasanta K. Panigrahi

2005, Physical Review E

Abstract

We propose a wavelet based method for the characterization of the scaling behavior of nonstationary time series. It makes use of the built-in ability of the wavelets for capturing the trends in a data set, in variable window sizes. Discrete wavelets from the Daubechies family are used to illustrate the efficacy of this procedure. After studying binomial multifractal time series with the present and earlier approaches of detrending for comparison, we analyze the time series of averaged spin density in the 2D Ising model at the critical temperature, along with several experimental data sets possessing multi-fractal behavior.

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We study the scaling behavior of the fluctuations, as extracted through wavelet coefficients based on discrete wavelets. The analysis is carried out on a variety of physical data sets, as well as Gaussian white noise and binomial multi-fractal model time series and the results are compared with continuous wavelet based average wavelet coefficient method. It is found that high-pass coefficients of wavelets, belonging to the Daubechies family are quite good in estimating the true power in the fluctuations in a non-stationary time series. Hence, the fluctuation functions based on discrete wavelet coefficients find the Hurst scaling exponents accurately.

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J.-f. Muzy, Jean-françois Muzy

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Multifractal Detrended Fluctuation Analysis of Nonstationary Time Series
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We develop a method for the multifractal characterization of nonstationary time series, which is based on a generalization of the detrended uctuation analysis (DFA). We relate our multifractal DFA method to the standard partition function-based multifractal formalism, and prove that both approaches are equivalent for stationary signals with compact support. By analyzing several examples we show that the new method can reliably determine the multifractal scaling behavior of time series. By comparing the multifractal DFA results for original series with those for shu ed series we can distinguish multifractality due to long-range correlations from multifractality due to a broad probability density function. We also compare our results with the wavelet transform modulus maxima method, and show that the results are equivalent.

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The Wavelet Scalogram in the Study of Time Series
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Advances in Differential Equations and Applications, 2014

Wavelet theory has been proved to be a useful tool in the study of time series [4, 7]. Specifically, the scalogram allows the detection of the most representative scales (or frequencies) of a signal. In this work, we present the scalogram as a tool for studying some aspects of a given signal. Firstly, we introduce a parameter called scale index [2], interpreted as a measure of the degree of the signal's non-periodicity. In this way, it can complement the maximal Lyapunov exponent method for determining chaos transitions of a given dynamical system. Secondly, we introduce a method for comparing different scalograms. This can be applied for determining if two time series follow similar patterns.

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Prasanta K. Panigrahi

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We make use of wavelet transform to study the multi-scale, self similar behavior and deviations thereof, in the stock prices of large companies, belonging to different economic sectors. The stock market returns exhibit multi-fractal characteristics, with some of the companies showing deviations at small and large scales. The fact that, the wavelets belonging to the Daubechies' (Db) basis enables one to isolate local polynomial trends of different degrees, plays the key role in isolating fluctuations at different scales. One of the primary motivations of this work is to study the emergence of the k -3 behavior [1] of the fluctuations starting with high frequency fluctuations. We make use of Db4 and Db6 basis sets to respectively isolate local linear and quadratic trends at different scales in order to study the statistical characteristics of these financial time series. The fluctuations reveal fat tail non-Gaussian behavior, unstable periodic modulations, at finer scales, from which the characteristic k -3 power law behavior emerges at sufficiently large scales. We further identify stable periodic behavior through the continuous Morlet wavelet.

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Understanding the statistical properties of time-occurrence series of seismic sequences is considered one of the most pervasive scientific topics. Investigating into the patterns of seismic sequences reveals evidence of time-scaling features. This is shown in the fractal analysis of the 1986-2001 seismicity of three different seismic zones in Italy. Describing the sequence of earthquakes by means of the series of the interevent times, power-law behaviour has been found applying Hurst analysis and detrended fluctuation analysis (DFA), with consistent values for the scaling exponents. The multifractal analysis has clearly evidenced differences among the earthquake sequences. The multifractal spectrum parameters (maximum a 0 , asymmetry B and width W ), derived from the analysis of the shape of the singularity spectrum, have been used to measure the complexity of seismicity.

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References (20)

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