A wavelet lifting approach to long-memory estimation

Knight, Marina I.; Nason, Guy P.; Nunes, Matthew

doi:10.1007/s11222-016-9698-2

Cited by 21 publications

(39 citation statements)

References 81 publications

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“…The Hurst exponent H (Hurst 1951;Mandelbrot & van Ness 1968;Katsev & L'Heureux 2003;Tarnopolski 2016;Knight et al 2017) measures the statistical self similarity of a time series x(t). It is said that x(t) is self similar (or self affine) if it satisfies…”

Section: Hurst Exponentmentioning

confidence: 99%

“…For irregularly sampled data, the Hurst exponent can be obtained without any interpolation of the examined time series (Knight et al 2017) with the use of the lifting wavelet transform algorithm called lifting one coefficient at a time (LOCAAT). The algorithm aims at producing a set of wavelet-like coefficients, {d jr } r , whose variance obeys the relation log 2 var(d jr ) = α · j * + const.,…”

Section: Hurst Exponentmentioning

confidence: 99%

“…See also(Veitch & Abry 1999;Tarnopolski 2015 Tarnopolski , 2016Knight et al 2017) and references therein for additional details on the Hurst exponent. For a detailed description of the LOCAAT, we refer the reader toKnight et al (2017) and references therein. 4 https://CRAN.R-project.org/package=liftLRD…”

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confidence: 99%

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Optical Variability Modeling of Newly Identified Blazar Candidates behind Magellanic Clouds

Żywucka

Tarnopolski

Böttcher

et al. 2020

ApJ

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We present an optical variability study of 44 newly identified blazar candidates behind the Magellanic Clouds, including 27 flat spectrum radio quasars (FSRQs) and 17 BL Lacertae objects (BL Lacs). All objects in the sample possess high photometric accuracy and irregularly sampled optical light curves (LCs) in I filter from the long-term monitoring conducted by the Optical Gravitational Lensing Experiment. We investigated the variability properties to look for blazar-like characteristics and to analyze the long-term behaviour. We analyzed the LCs with the Lomb-Scargle periodogram to construct power spectral densities (PSDs), found breaks for several objects, and linked them with accretion disk properties. In this way we constrained the black hole (BH) masses of 18 FSRQs to lie within the range 8.18 ≤ log(M BH /M ) ≤ 10.84, assuming a wide range of possible BH spins. By estimating the bolometric luminosities, we applied the fundamental plane of active galactic nuclei variability as an independent estimate, resulting in 8.4 ≤ log(M BH /M ) ≤ 9.6, with a mean error of 0.3. Many of the objects have very steep PSDs, with high frequency spectral index in the range 3 − 7. An alternative attempt to classify the LCs was made using the Hurst exponent, H, and the A − T plane. Two FSRQs and four BL Lacs yielded H > 0.5, indicating presence of long-term memory in the underlying process governing the variability. Additionally, two FSRQs with exceptional PSDs, stand out also in the A − T plane.

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Section: Hurst Exponentmentioning

confidence: 99%

Section: Hurst Exponentmentioning

confidence: 99%

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Optical Variability Modeling of Newly Identified Blazar Candidates behind Magellanic Clouds

Żywucka

Tarnopolski

Böttcher

et al. 2020

ApJ

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“…Recently, wavelet models for spectral estimation and long-memory (Hurst) parameter estimation have been developed for irregularly-spaced time series (or regularly spaced series subject to missing values) using second-generation wavelets otherwise known as lifting, see [36][37][38]. These methods appear to be particularly promising as their performance seems to match or exceed that of regular wavelets on regularly spaced data for reasons that are not yet fully understood.…”

Section: Time Series and Multiscale Methodsmentioning

confidence: 99%

Case study: shipping trend estimation and prediction via multiscale variance stabilisation

Michis

Nason

2016

Journal of Applied Statistics

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“…constructions, just as for real-valued processes (Hurst 1951;Mandelbrot and Ness 1968), the degree of memory can still be quantified by means of a single parameter, the Hurst exponent parameter (Amblard et al 2012;Sykulski and Percival 2016). Accurate estimation of the Hurst parameter offers valuable insight into a multitude of modelling and analysis tasks, such as model calibration and prediction (Beran et al 2013;Rehman and Siddiqi 2009;Knight et al 2017).…”

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confidence: 99%

Long memory estimation for complex-valued time series

Knight

Nunes

2018

Stat Comput

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Long memory has been observed for time series across a multitude of fields, and the accurate estimation of such dependence, for example via the Hurst exponent, is crucial for the modelling and prediction of many dynamic systems of interest. Many physical processes (such as wind data) are more naturally expressed as a complex-valued time series to represent magnitude and phase information (wind speed and direction). With data collection ubiquitously unreliable, irregular sampling or missingness is also commonplace and can cause bias in a range of analysis tasks, including Hurst estimation. This article proposes a new Hurst exponent estimation technique for complex-valued persistent data sampled with potential irregularity. Our approach is justified through establishing attractive theoretical properties of a new complex-valued wavelet lifting transform, also introduced in this paper. We demonstrate the accuracy of the proposed estimation method through simulations across a range of sampling scenarios and complex-and real-valued persistent processes. For wind data, our method highlights that inclusion of the intrinsic correlations between the real and imaginary data, inherent in our complex-valued approach, can produce different persistence estimates than when using real-valued analysis. Such analysis could then support alternative modelling or policy decisions compared with conclusions based on real-valued estimation. Keywords Complex-valued time series • Hurst exponent • Irregular sampling • Long-range dependence • Wavelets 1 Introduction Complex-valued time series arise in many scientific fields of interest, for example digital communication and signal processing (Curtis 1985; Martin 2004), environmental series (Gonella 1972; Lilly and Gascard 2006; Adali et al. 2011) and physiology (Rowe 2005). Modelling and analysis of such series in the complex domain is not only natural, but also convenient. In addition, complex-valued time series models are often able to represent more realistic behaviour in observed physical processes; see, for example, Mandic and Goh (2009) and Sykulski et al. (2017). A particular modelling Electronic supplementary material The online version of this article (

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A wavelet lifting approach to long-memory estimation

Cited by 21 publications

References 81 publications

Optical Variability Modeling of Newly Identified Blazar Candidates behind Magellanic Clouds

Optical Variability Modeling of Newly Identified Blazar Candidates behind Magellanic Clouds

Case study: shipping trend estimation and prediction via multiscale variance stabilisation

Long memory estimation for complex-valued time series

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