Self-Similarity: Part II—Optimal Estimation of Fractal Processes

2011

Adv Comput Math

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The fractional Laplacian (− ) γ /2 commutes with the primary coordination transformations in the Euclidean space R d : dilation, translation and rotation, and has tight link to splines, fractals and stable Levy processes. For 0 < γ < d, its inverse is the classical Riesz potential I γ which is dilationinvariant and translation-invariant. In this work, we investigate the functional properties (continuity, decay and invertibility) of an extended class of differential operators that share those invariance properties. In particular, we extend the definition of the classical Riesz potential I γ to any noninteger number γ larger than d and show that it is the unique left-inverse of the fractional Laplacian (− ) γ /2 which is dilation-invariant and translationinvariant. We observe that, for any 1 ≤ p ≤ ∞ and γ ≥ d(1 − 1/ p), there exists a Schwartz function f such that I γ f is not p-integrable. We then introduce the new unique left-inverse I γ,p of the fractional Laplacian (− ) γ /2 with the property that I γ,p is dilation-invariant (but not translation-invariant) and that I γ,p f is p-integrable for any Schwartz function f . We finally apply that linear operator I γ,p with p = 1 to solve the stochastic partial differential equation (− ) γ /2 = w with white Poisson noise as its driving term w.

Section: ) Then I γ Is the Unique Continuous Linear Operator From mentioning

confidence: 99%

“…which is the natural L p extension of the fractional integral operator that was introduced in [1,12,13] for p = 2 and γ ∈ Z/2.…”

Section: ) Then I γ Is the Unique Continuous Linear Operator From mentioning

confidence: 99%

Left-inverses of fractional Laplacian and sparse stochastic processes

Sun

2011

Adv Comput Math

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“…We will see that the full denoising process involves a stationary part-digital filtering and interpolation in a fractional spline space-and a non-stationary part which, in particular, ensures that the estimate vanishes at t = 0, just like an fBm [4]. Interestingly, we will establish that the best estimation of an fBm given its noisy samples is a fractional spline of degree 2γ where γ is the Hurst exponent of the fBm.…”

Section: Introductionmentioning

confidence: 82%

“…The proof is somewhat technical and will be given elsewhere [4]. This result shows that the optimal estimation of the non-stationary signal B γ (t) is not a mere filtered version of the y k 's, as it would be the case in the stationary case.…”

Section: Theorem 1 the Optimal Estimate Ofmentioning

confidence: 89%

“…Note however, that if we, quite reasonably, assume that the autocorrelation of the noise is localized around 0 (in the limit white noise case, we would even have r k = 0 for all k = 0) then this second term is guaranteed to vanish when k → ∞ because h k is itself a localized filter [4]. Thus, for large values of t, the best estimate of an fBm is simply a filtered version of its noisy samples, as in the stationary case.…”

Section: Theorem 1 the Optimal Estimate Ofmentioning

confidence: 99%

See 1 more Smart Citation

Optimal Interpolation of Fractional Brownian Motion Given Its Noisy Samples

Blu

2006 IEEE International Conference on Acoustics Speed and Signal Processing Proceedings

We consider the problem of estimating a fractional Brownian motion known only from its noisy samples at the integers. We show that the optimal estimator can be expressed using a digital Wiener-like filter followed by a simple timevariant correction accounting for nonstationarity.Moreover, we prove that this estimate lives in a symmetric fractional spline space and give a practical implementation for optimal upsampling of noisy fBm samples by integer factors.

Scaling Limits of Solutions of Linear Stochastic Differential Equations Driven by Lévy White Noises

Fageot

2018

J Theor Probab

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Correspondence to be sent to: julien.fageot@epfl.ch Consider a random process s solution of the stochastic differential equation Ls = w with L a homogeneous operator and w a multidimensional Lévy white noise. In this paper, we study the asymptotic effect of zooming in or zooming out of the process s. More precisely, we give sufficient conditions on L and w such that a H s(·/a) converges in law to a non-trivial self-similar process for some H, when a → 0 (coarse-scale behavior) and a → ∞ (fine-scale behavior). The parameter H depends on the homogeneity order of the operator L and the Blumenthal-Getoor and Pruitt indices associated to the Lévy white noise w. Finally, we apply our general results to several notorious classes of random processes and random fields and illustrate our results on simulations of Lévy processes. 2 homogeneous of order 1. Our aim is to study the statistical behavior of the rescaling x → s(x/a) of a solution of (2) when a > 0 is varying. Our two main questions are:• What is the asymptotic behavior of s(·/a) when we zoom out the process (i.e., when a → 0)?• What is the local behavior when we zoom in (i.e., when a → ∞)?Our main contribution is to identify sufficient conditions such that the rescaling a H s(·/a) of a solution of (2) has a non-trivial self-similar asymptotic limit as a goes to 0 or ∞. When this limit exists, the parameter H is unique and depends essentially on the degree of homogeneity γ of L and on the Blumenthal-Getoor and Pruitt indices β ∞ and β 0 of w [8,44]. The indices β 0 and β ∞ are used in the literature to characterize the asymptotic and local behaviors of Lévy processes, and more generally Lévy-type processes [9]. We summarize the main results of our paper in Theorem 1.1. Precise definitions and rigorous statements are given later.Theorem 1.1. Let L be a γ-homogeneous operator and w a Lévy process with indices β ∞ and β 0 . Under some technical conditions, the solution s to the equation Ls = w has the following properties.