Wavelets for time series analysis - a survey and new results

Physica A-statistical Mechanics and Its Applications, 2006

We study the functional link between the Hurst parameter and the Normalized Total Wavelet Entropy when analyzing fractional Brownian motion (fBm) time seriesthese series are synthetically generated. Both quantifiers are mainly used to identify fractional Brownian motion processes . The aim of this work is understand the differences in the information obtained from them, if any.

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 1999

This article reviews the role of wavelets in statistical time series analysis. We survey work that emphasises scale such as estimation of variance and the scale exponent of a process with a speci c scale behaviour such as 1/f processes. We present some of our own work on locally stationary wavelet (lsw) processes which model both stationary and some kinds of non-stationary processes. Analysis of time series assuming the lsw model permits identi cation of an evolutionary wavelet spectrum (ews) that quanti es the variation in a time series over a particular scale and at a particular time. We address estimation of the ews and show how our methodology reveals phenomena of interest in an infant electrocardiogram series.

IOP Publishing, 2021

In this paper, wavelets were used to study the multivariate fractional Brownian motion through the deviations of the random process to find an efficient estimation of Hurst exponent. The results of simulations experiments were shown that the performance of the proposed estimator was efficient. The estimation process was made by taking advantage of the detail coefficients stationarity from the wavelet transform, as the variance of this coefficient showed the power-low behavior. We use two wavelet filters (Haar and db5) to manage minimizing the mean square error of the model.

Pakistan Journal of Statistics and Operation Research, 2022

In this paper, we propose a method using continuous wavelets to study the multivariate fractional Brownian motion through the deviations of the transformed random process to find an efficient estimate of Hurst exponent using eigenvalue regression of the covariance matrix. The results of simulations experiments shown that the performance of the proposed estimator was efficient in bias but the variance get increase as signal change from short to long memory the MASE increase relatively. The estimation process was made by calculating the eigenvalues for the variance-covariance matrix of Meyer's continuous wavelet details coefficients.

Computational Economics, 2019

In this paper, we investigate the performance of four semi-parametric estimators in the wavelet domain in order to estimate the parameter of stationary long-memory models. The goal is to consider a wavelet estimate for the fractional differencing parameter d where the time series exhibit heavy tails. We show by Monte Carlo experiments that the wavelet Exact Local Whittle-type estimator improves considerably the other suggested wavelet-based estimators in terms of smaller bias, Root Mean Squared Error and variance. Furthermore, the simulation results show that the wavelet periodogram estimators perform better in most cases than wavelet ordinary least square estimate methods when the sample size is increased.

Stochastic Processes and their Applications, 2000

Let 0 ¡ 62 and let T ⊆ R. Let {X (t); t ∈ T } be a linear fractional-stable (0 ¡ 62) motion with scaling index H (0 ¡ H ¡ 1) and with symmetric-stable random measure. Suppose that is a bounded real function with compact support [a; b] and at least one null moment. Let the sequence of the discrete wavelet coe cients of the process X be D j; k = R X (t) j; k (t) dt; j; k ∈ Z : We use a stochastic integral representation of the process X to describe the wavelet coe cients as-stable integrals when H − 1= ¿ − 1. This stochastic representation is used to prove that the stochastic process of wavelet coe cients {D j; k ; k ∈ Z}, with ÿxed scale index j ∈ Z, is strictly stationary. Furthermore, a property of self-similarity of the wavelet coe cients of X is proved. This property has been the motivation of several wavelet-based estimators for the scaling index H .

2011

All praises and thanks be to Almighty, the most beneficent, the most merciful, Who bestowed upon me the courage, patience and strength to embark upon this work and carry it out to its successful completion. After Almighty, I must admire to our last and great Prophet, Huzoor (S.A.W.) whose hfe (as a model) provided me the determination and patience to continue my hard work especially in the period of doubt and despair. Dated: OZ-11' ZOli ^g^Khan) ni Chapter 5 is based on analysis of stock market by using wavelets. In this chapter, we have used wavelet transform to analyze data of stock market. This is one of the application of wavelets in time series analysis. The scatter plots of shares of different companies have been plotted and total variation plots of Sensex versus shares of companies are also shown. At the end, an extensive bibliography of the existing literature related to the subject matter of the dissertation is included.

Operator fractional Brownian motion (OFBM) is the natural vector-valued extension of the univariate fractional Brownian motion. Instead of a scalar parameter, the law of an OFBM scales according to a Hurst matrix that affects every component of the process. In this paper, we develop the wavelet analysis of OFBM, as well as a new estimator for the Hurst matrix of bivariate OFBM. For OFBM, the univariate-inspired approach of analyzing the entry-wise behavior of the wavelet spectrum as a function of the (wavelet) scales is fraught with difficulties stemming from mixtures of power laws. The proposed approach consists of considering the evolution along scales of the eigenstructure of the wavelet spectrum. This is shown to yield consistent and asymptotically normal estimators of the Hurst eigenvalues, and also of the coordinate system itself under assumptions. A simulation study is included to demonstrate the good performance of the estimators under finite sample sizes.

Wavelet based estimators of the H parameter for fractional Brownian motion (fBm) is known to have interesting asymptotical properties. In this communication, we are studying the practical point of view by testing this estimator on finite length fBm signals of N samples. These signals are generated using the circulant embedding method (CEM). CEM is fast and exact such that the synthesis of true fBm signals of long size is easy. Results show that above N=4096 wavelet based estimator is unbiased and very close to the Cramer-Rao lower bound. At the light of this study, and by combining it to recent results, a practical user guide to estimate the H parameter of fBm can be provided: if N is lower than 512 classical maximum likelihood (ML) should be chosen. For N in the interval ]512,4096] Whittle ML is to be preferred. For N above 4096 wavelet based should be selected. Finally, a precise confidence interval of the true H parameter can be given.

The Journal of Fourier Analysis and Applications, 1999

We provide an almost sure convergent expansion of fractional Brownian motion in wavelets which decorrelates the high frequencies. Our approach generalizes L~vy's midpoint displacement technique which is used to generate Brownian motion. The low-frequency terms in the expansion involve an independent fractional Brownian motion evaluated at discrete times or, alternatively, partial sums of a stationary fractional ARIMA time series. The wavelets fill in the gaps and provide the necessary high frequency corrections. We also obtain a way of constructing an arbitrary number of non-Gaussian continuous time processes whose second order properties are the same as those of fractional Brownian motion.