Bootstrap-based Bias Reduction for the Estimation of the Self-similarity Exponents of Multivariate Time Series

Abstract

Self-similarity has become a well-established modeling framework in several fields of application and its multivariate formulation is of ever-increasing importance in the Big Data era. Multivariate Hurst exponent estimation has thus received a great deal of attention recently, in particular by means of the wavelet eigenvalue regression method. The present work tackles the issue of the presence of significant finite-sample bias in wavelet eigenvalue regression stemming from the eigenvalue repulsion effect, whose origin and impact are analyzed and quantified. Furthermore, an original wavelet domain bias reduction technique is developed assuming a single multivariate time series is available. The protocol consists of a bootstrap resampling scheme that preserves the joint covariance structure of multivariate wavelet coefficients. Extensive numerical simulations show that this proposed method is effective in counteracting the bias at the price of a small increase in variance. This leads to wavelet eigenanalysisbased estimation of multivariate Hurst exponents with significantly improved finite-sample performance than earlier state-of-the-art formulations.

Wavelet eigenvalue regression in high dimensions

“…(i) In applications, the results in this paper naturally pave the way for the investigation of scaling behavior in high-dimensional data from fields such as physics, neuroscience and signal processing; (ii) Modeling requires a deeper study, in the wavelet domain, of the so-named eigenvalue repulsion effect (e.g., Tao (2012)), which may severely skew the observed scaling laws when the assumptions of Theorem 3.2 are violated. This is particularly important in the context of instances where all scaling eigenvalues are close to equal, with the same holding for the asymptotic rescaled eigenvalues ξ q (2 j ) (see Wendt et al (2019) on preliminary computational studies). In those cases, it is of great interest to develop efficient testing procedures for the statistical identification of distinct scaling eigenvalues in real data; (iii) An interesting direction of extension is the construction of statistical methodology for instances where the r largest eigenvalues of wavelet random matrices exhibit non-Gaussian fluctuations (cf.…”

Counting the Number of Different Scaling Exponents in Multivariate Scale-Free Dynamics: Clustering by Bootstrap in the Wavelet Domain

“…In practice, the estimation of the selfsimilarity exponent is thus central and can be efficiently and robustly performed by means of wavelet transforms [8,9]. The multivariate time series encountered in many modern applications call both for multivariate selfsimilarity models, such as the recently proposed operator fractional Brownian motion (ofBm) [10][11][12][13], and for multivariate wavelet representation based estimation procedures, such as those developed in [2,3,14,15]. These estimation procedures output as many selfsimilarity parameter estimates as there are time series in the data.…”