The Experts below are selected from a list of 633 Experts worldwide ranked by ideXlab platform
Neumann, Michael H. - One of the best experts on this subject based on the ideXlab platform.
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Locally adaptive fitting of semiparametric models to nonstationary time Series
Elsevier Science B.V., 2001Co-Authors: Dahlhaus Rainer, Neumann, Michael H.Abstract:AbstractWe fit a class of semiparametric models to a nonstationary process. This class is parametrized by a mean function μ(·) and a p-dimensional function θ(·)=(θ(1)(·),…,θ(p)(·))′ that parametrizes the time-varying spectral density fθ(·)(λ). Whereas the mean function is estimated by a usual kernel estimator, each component of θ(·) is estimated by a nonlinear Wavelet method. According to a truncated Wavelet Series Expansion of θ(i)(·), we define empirical versions of the corresponding Wavelet coefficients by minimizing an empirical version of the Kullback–Leibler distance. In the main smoothing step, we perform nonlinear thresholding on these coefficients, which finally provides a locally adaptive estimator of θ(i)(·). This method is fully automatic and adapts to different smoothness classes. It is shown that usual rates of convergence in Besov smoothness classes are attained up to a logarithmic factor
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Locally Adaptive Fitting of Semiparametric Models to Nonstationary Time Series
'American Association for Cancer Research (AACR)', 2001Co-Authors: Dahlhaus Rainer, Neumann, Michael H.Abstract:We fit a class of semiparametric models to a nonstationary process. This class is parametrized by a mean function µ( · ) and a p-dimensional function theta ( · ) = (theta(1)( · ) , ..., theta(p) ( · ))´ that parametrizes the time-varying spectral density ftheta( · ) (lambda). Whereas the mean function is estimated by a usual kernel estimator, each component of theta ( · ) is estimated by a nonlinear Wavelet method. According to a truncated Wavelet Series Expansion of theta(i) ( · ), we define empirical versions of the corresponding Wavelet coefficients by minimizing an empirical version of the Kullback-Leibler distance. In the main smoothing step, we perform nonlinear thresholding on these coefficients, which finally provides a locally adaptive estimator of theta(i) ( · ). This method is fully automatic and adapts to different smoothness classes. It is shown that usual rates of convergence in Besov smoothness classes are attained up to a logarithmic factor
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A Wavelet-Based Test for Stationarity
1997Co-Authors: Sachs, Rainer Von, Neumann, Michael H.Abstract:We develop a test for stationarity of a time Series against the alternative of a time-changing covariance structure. Using localized versions of the periodogram, we obtain empirical versions of a reasonable notion of a time-varying spectral density. Coefficients w.r.t. a Haar Wavelet Series Expansion of such a time-varying periodogram are a possible indicator whether there is some deviation from covariance stationarity. We propose a test based on the limit distribution of these empirical coefficients
Croci M - One of the best experts on this subject based on the ideXlab platform.
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Multilevel quasi Monte Carlo methods for elliptic PDEs with random field coefficients via fast white noise sampling
'The Japan Society for Industrial and Applied Mathematics', 2021Co-Authors: Croci M, Giles MAbstract:When solving partial differential equations (PDEs) with random fields as coefficients, the efficient sampling of random field realizations can be challenging. In this paper we focus on the fast sampling of Gaussian fields using quasi-random points in a finite element and multilevel quasi Monte Carlo (MLQMC) setting. Our method uses the stochastic PDE (SPDE) approach of Lindgren et al. combined with a new fast algorithm for white noise sampling which is tailored to (ML)QMC. We express white noise as a Wavelet Series Expansion that we divide into two parts. The first part is sampled using quasi-random points and contains a finite number of terms in order of decaying importance to ensure good quasi Monte Carlo (QMC) convergence. The second part is a correction term which is sampled using standard pseudo-random numbers. We show how the sampling of both terms can be performed in linear time and memory complexity in the number of mesh cells via a supermesh construction, yielding an overall linear cost. Furthermore, our technique can be used to enforce the MLQMC coupling even in the case of nonnested mesh hierarchies. We demonstrate the efficacy of our method with numerical experiments
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Multilevel quasi Monte Carlo methods for elliptic PDEs with random field coefficients via fast white noise sampling
2021Co-Authors: Croci M, Giles M. B., Farrell P. E.Abstract:When solving partial differential equations with random fields as coefficients the efficient sampling of random field realisations can be challenging. In this paper we focus on the fast sampling of Gaussian fields using quasi-random points in a finite element and multilevel quasi Monte Carlo (MLQMC) setting. Our method uses the SPDE approach of Lindgren et al.~combined with a new fast algorithm for white noise sampling which is taylored to (ML)QMC. We express white noise as a Wavelet Series Expansion that we divide in two parts. The first part is sampled using quasi-random points and contains a finite number of terms in order of decaying importance to ensure good QMC convergence. The second part is a correction term which is sampled using standard pseudo-random numbers. We show how the sampling of both terms can be performed in linear time and memory complexity in the number of mesh cells via a supermesh construction, yielding an overall linear cost. Furthermore, our technique can be used to enforce the MLQMC coupling even in the case of non-nested mesh hierarchies. We demonstrate the efficacy of our method with numerical experiments
Yimin Xiao - One of the best experts on this subject based on the ideXlab platform.
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Wavelet Series representation and geometric properties of harmonizable fractional stable sheets
Stochastics, 2020Co-Authors: Antoine Ayache, Narnrueih Shieh, Yimin XiaoAbstract:ABSTRACTLet ZH={ZH(t),t∈RN} be a real-valued N-parameter harmonizable fractional stable sheet with index H=(H1,…,HN)∈(0,1)N. We establish a random Wavelet Series Expansion for ZH which is almost su...
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Wavelet Series representation and geometric properties of harmonizable fractional stable sheets
arXiv: Probability, 2019Co-Authors: Antoine Ayache, Narnrueih Shieh, Yimin XiaoAbstract:Let $Z^H= \{Z^H(t), t \in \R^N\}$ be a real-valued $N$-parameter harmonizable fractional stable sheet with index $H = (H_1, \ldots, H_N) \in (0, 1)^N$. We establish a random Wavelet Series Expansion for $Z^H$ which is almost surely convergent in all the Holder spaces $C^\gamma ([-M,M]^N)$, where $M>0$ and $\gamma\in (0, \min\{H_1,\ldots, H_N\})$ are arbitrary. One of the main ingredients for proving the latter result is the LePage representation for a rotationally invariant stable random measure. Also, let $X=\{X(t), t \in \R^N\}$ be an $\R^d$-valued harmonizable fractional stable sheet whose components are independent copies of $Z^H$. By making essential use of the regularity of its local times, we prove that, on an event of positive probability, the formula for the Hausdorff dimension of the inverse image $X^{-1}(F)$ holds for all Borel sets $F \subseteq \R^d$. This is referred to as a uniform Hausdorff dimension result for the inverse images.
Giles M - One of the best experts on this subject based on the ideXlab platform.
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Multilevel quasi Monte Carlo methods for elliptic PDEs with random field coefficients via fast white noise sampling
'The Japan Society for Industrial and Applied Mathematics', 2021Co-Authors: Croci M, Giles MAbstract:When solving partial differential equations (PDEs) with random fields as coefficients, the efficient sampling of random field realizations can be challenging. In this paper we focus on the fast sampling of Gaussian fields using quasi-random points in a finite element and multilevel quasi Monte Carlo (MLQMC) setting. Our method uses the stochastic PDE (SPDE) approach of Lindgren et al. combined with a new fast algorithm for white noise sampling which is tailored to (ML)QMC. We express white noise as a Wavelet Series Expansion that we divide into two parts. The first part is sampled using quasi-random points and contains a finite number of terms in order of decaying importance to ensure good quasi Monte Carlo (QMC) convergence. The second part is a correction term which is sampled using standard pseudo-random numbers. We show how the sampling of both terms can be performed in linear time and memory complexity in the number of mesh cells via a supermesh construction, yielding an overall linear cost. Furthermore, our technique can be used to enforce the MLQMC coupling even in the case of nonnested mesh hierarchies. We demonstrate the efficacy of our method with numerical experiments
Helga Schack - One of the best experts on this subject based on the ideXlab platform.
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an optimal Wavelet Series Expansion of the riemann liouville process
Journal of Theoretical Probability, 2009Co-Authors: Helga SchackAbstract:Consider the Riemann–Liouville process R α ={R α (t)} t∈[0,1] with parameter α>1/2. Depending on α, Wavelet Series representations for R α (t) of the form ∑ k=1 ∞ u k (t)e k are given, where the u k are deterministic functions, and {e k } k≥1 is a sequence of i.i.d. standard normal random variables. The Expansion is based on a modified Daubechies Wavelet family, which was originally introduced in Meyer (Rev. Mat. Iberoam. 7:115–133, 1991). It is shown that these Wavelet Series representations are optimal in the sense of Kuhn–Linde (Bernoulli 8:669–696, 2002) for all values of α>1/2.
