&lt;title&gt;Scattered data smoothing by empirical Bayesian shrinkage of second-generation wavelet coefficients&lt;/title&gt;

Jansen, Maarten; Nason, Guy P.; Silverman, Bernard W.

doi:10.1117/12.449738

Cited by 25 publications

(49 citation statements)

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“…where the weights (b n i ) i∈I n are obtained from the requirement that the algorithm preserves the signal mean value (Jansen et al 2001(Jansen et al , 2009). The interval lengths associated with the neighbouring points are also updated to account for the decreasing number of unlifted coefficients that remain.…”

Section: Wavelet Lifting Transforms For Irregular Datamentioning

confidence: 99%

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A wavelet lifting approach to long-memory estimation

2016

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Reliable estimation of long-range dependence parameters is vital in time series. For example, in environmental and climate science such estimation is often key to understanding climate dynamics, variability and often prediction. The challenge of data collection in such disciplines means that, in practice, the sampling pattern is either irregular or blighted by missing observations. Unfortunately, virtually all existing Hurst parameter estimation methods assume regularly sampled time series and require modification to cope with irregularity or missing data. However, such interventions come at the price of inducing higher estimator bias and variation, often worryingly ignored. This article proposes a new Hurst exponent estimation method which naturally copes with data sampling irregularity. The new method is based on a multiscale lifting transform exploiting its ability to produce wavelet-like coefficients on irregular data and, simultaneously, to effect a necessary powerful decorrelation of those coefficients. Simulations show that our method is accurate and effective, performing well against competitors

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Section: Wavelet Lifting Transforms For Irregular Datamentioning

confidence: 99%

“…Our Hurst exponent estimation method makes use of a recently developed lifting transform called the lifting one coefficient at a time (LOCAAT) transform proposed by Jansen et al (2001Jansen et al ( , 2009) which works as follows.…”

Section: Wavelet Lifting Transforms For Irregular Datamentioning

confidence: 99%

A wavelet lifting approach to long-memory estimation

2016

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“…We presented above the lifting scheme using a data split into odd and even indices (Sweldens, 1996(Sweldens, , 1998, while Jansen et al (2001Jansen et al ( , 2004 propose generating just one wavelet coefficient at each step. Claypoole et al (1998Claypoole et al ( , 2003 and Piella and Heijmans (2002) use a split-update-predict algorithm through which they adaptively build wavelets.…”

Section: Lifting Schemementioning

confidence: 99%

“…The lifting algorithm still aims to provide a wavelet-like representation of a signal using few (wavelet) coefficients, and the wavelet functions obtained through the lifting construction are referred to in the current literature as 'second generation'. In our work we shall use the version of the lifting scheme that 'removes one coefficient at a time', as proposed by Jansen et al (2001Jansen et al ( , 2004). This will also allow us to use adaptivity in our nondecimated lifting transform, see Nunes et al (2006) for details on an adaptive lifting scheme.…”

Section: Introductionmentioning

confidence: 99%

A ‘nondecimated’ lifting transform

Knight

Nason

2008

Stat Comput

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Classical nondecimated wavelet transforms are attractive for many applications. When the data comes from complex or irregular designs, the use of second generation wavelets in nonparametric regression has proved superior to that of classical wavelets. However, the construction of a nondecimated second generation wavelet transform is not obvious. In this paper we propose a new 'nondecimated' lifting transform, based on the lifting algorithm which removes one coefficient at a time, and explore its behaviour. Our approach also allows for embedding adaptivity in the transform, i.e. wavelet functions can be constructed such that their smoothness adjusts to the local properties of the signal. We address the problem of nonparametric regression and propose an (averaged) estimator obtained by using our nondecimated lifting technique teamed with empirical Bayes shrinkage. Simulations show that our proposed method has higher performance than competing techniques able to work on irregular data. Our construction also opens avenues for generating a 'best' representation, which we shall explore.

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“…Other approaches that particularly apply in two or higher dimensions are based on the construction of second generation wavelets [17] by the lifting scheme, see e.g. [2,7,11,19]. These wavelet constructions adaptively depend on the scattered points and the corresponding function values and therefore lose much of the simplicity and efficiency of the traditional wavelet transform.…”

Section: Introductionmentioning

confidence: 99%

Wavelet Shrinkage on Paths for Denoising of Scattered Data

Heinen

Plonka

2012

Results. Math.

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Abstract. We propose a new algorithm for denoising of multivariate function values given at scattered points in R d . The method is based on the one-dimensional wavelet transform that is applied along suitably chosen path vectors at each transform level. The idea can be seen as a generalization of the relaxed easy path wavelet transform by Plonka (Multiscale Model Simul 7:1474-1496, 2009) to the case of multivariate scattered data. The choice of the path vectors is crucial for the success of the algorithm. We propose two adaptive path constructions that take the distribution of the scattered points as well as the corresponding function values into account. Further, we present some theoretical results on the wavelet transform along path vectors in order to indicate that the wavelet shrinkage along path vectors can really remove noise. The numerical results show the efficiency of the proposed denoising method.

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<title>Scattered data smoothing by empirical Bayesian shrinkage of second-generation wavelet coefficients</title>

Cited by 25 publications

References 0 publications

A wavelet lifting approach to long-memory estimation

A wavelet lifting approach to long-memory estimation

A ‘nondecimated’ lifting transform

Wavelet Shrinkage on Paths for Denoising of Scattered Data

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