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B. Wang - One of the best experts on this subject based on the ideXlab platform.
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Lagrangian advection scheme with Shape Matrix (LASM) v0.2: interparcel mixing, physics–dynamics coupling and 3-D extension
Geoscientific Model Development, 2015Co-Authors: L. Dong, B. Wang, L. Liu, Yongjie HuangAbstract:Abstract. The interparcel mixing algorithm in the Lagrangian advection scheme with Shape Matrix (LASM) is updated to make the scheme more robust. The linear degeneration criterion is replaced by the maximum deviation of the skeleton points so that the new algorithm is more effective in controlling the Shape of parcels, which is vital for long time simulation. LASM is inherently Shape-preserving without any complicated filter or limiter, and it is linear. This fact contributes to the ability to preserve the sum of multiple tracers exactly on the parcels in LASM. A newly proposed terminator "toy"-chemistry test is used to test LASM, which shows that LASM can preserve the weighted sum of two reactive species precisely. The physics–dynamics coupling (i.e., tendency evaluation type) is also discussed. A flow generated by a WRF large-eddy simulation is also used to test the 3-D extension of LASM.
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lagrangian advection scheme with Shape Matrix lasm v0 2 interparcel mixing physics dynamics coupling and 3 d extension
Geoscientific Model Development, 2015Co-Authors: L. Dong, B. Wang, L. Liu, Yongjie HuangAbstract:Abstract. The interparcel mixing algorithm in the Lagrangian advection scheme with Shape Matrix (LASM) is updated to make the scheme more robust. The linear degeneration criterion is replaced by the maximum deviation of the skeleton points so that the new algorithm is more effective in controlling the Shape of parcels, which is vital for long time simulation. LASM is inherently Shape-preserving without any complicated filter or limiter, and it is linear. This fact contributes to the ability to preserve the sum of multiple tracers exactly on the parcels in LASM. A newly proposed terminator "toy"-chemistry test is used to test LASM, which shows that LASM can preserve the weighted sum of two reactive species precisely. The physics–dynamics coupling (i.e., tendency evaluation type) is also discussed. A flow generated by a WRF large-eddy simulation is also used to test the 3-D extension of LASM.
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An updated interparcel mixing algorithm in the Lagrangian advection scheme with Shape Matrix (LASM) v0.2
2015Co-Authors: L. Dong, B. Wang, L. LiuAbstract:Abstract. The interparcel mixing algorithm in the Lagrangian advection scheme with Shape Matrix (LASM) is updated to make the scheme more robust. The linear degeneration criterion is replaced by the maximum deviation of the skeleton points so that the new algorithm is more effective in controlling the Shape of parcels, which is vital for long time simulation. LASM is inherently Shape-preserving without any complicated filter or limiter, so it is linear. This fact contributes to the ability of LASM of preserving the sum of multiple tracers exactly. A newly proposed terminator "toy"-chemistry test is also used to test LASM, which shows that LASM can preserve the weighted sum of two reactive chlorine-like species precisely.
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A Lagrangian advection scheme with Shape Matrix (LASM) for solving advection problems
Geoscientific Model Development, 2014Co-Authors: L. Dong, B. WangAbstract:Abstract. A new Lagrangian advection scheme with Shape Matrix (LASM) is proposed to take advantage of the extreme low numerical diffusion of the Lagrangian methods. The tracer is discretized into finite parcels, which move along the downstream trajectories. Different from other Lagrangian schemes, the parcel Shape is simulated explicitly by a linear transformation Matrix. By doing so, the aliasing error in the Lagrangian schemes is largely reduced without introducing substantial interparcel mixing in the pure advection stage, because the flow information will be respected when remapping tracer density onto the fixed model grids. An adaptive interparcel mixing algorithm is constructed to ensure the validity of the linear approximation of the parcel Shape, where the mixing is only triggered when it is necessary and resembles the physical mixing. The total tracer mass on the parcels is conserved exactly. The new scheme is validated by using several test cases.
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A Lagrangian Advection scheme with Shape Matrix (LASM) for solving advection problems
Geoscientific Model Development Discussions, 2014Co-Authors: L. Dong, B. WangAbstract:Abstract. A new Lagrangian advection scheme with Shape Matrix (LASM) is proposed to take advantage of extreme low numerical diffusion of Lagrangian methods. The tracer is discretized into finite parcels, which move along the downstream trajectories. Different from other Lagrangian schemes, the parcel Shape is simulated explicitly by a linear transformation Matrix. By doing so, the aliasing error in the Lagrangian schemes is largely reduced without introducing substantial interparcel mixing in the pure advection stage, because the flow information will be respected when remapping tracer density onto the fixed model grids. An adaptive interparcel mixing algorithm is constructed to ensure the validity of the linear approximation of the parcel Shape, where the mixing is only triggered when it is necessary and resembles the physical mixing. The total tracer mass on the parcels is conserved exactly. The new scheme is validated by using several test cases.
Davy Paindaveine - One of the best experts on this subject based on the ideXlab platform.
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Tyler Shape Depth
Biometrika, 2019Co-Authors: Davy Paindaveine, Germain Van BeverAbstract:SummaryIn many problems from multivariate analysis, the parameter of interest is a Shape Matrix: a normalized version of the corresponding scatter or dispersion Matrix. In this article we propose a notion of depth for Shape matrices that involves data points only through their directions from the centre of the distribution. We refer to this concept as Tyler Shape depth since the resulting estimator of Shape, namely the deepest Shape Matrix, is the median-based counterpart of the M-estimator of Shape due to Tyler (1987). Besides estimation, Shape depth, like its Tyler antecedent, also allows hypothesis testing on Shape. Its main benefit, however, lies in the ranking of the Shape matrices it provides, the practical relevance of which is illustrated by applications to principal component analysis and Shape-based outlier detection. We study the invariance, quasi-concavity and continuity properties of Tyler Shape depth, the topological and boundedness properties of the corresponding depth regions, and the existence of a deepest Shape Matrix, and we prove Fisher consistency in the elliptical case. Finally, we derive a Glivenko–Cantelli-type result and establish almost sure consistency of the deepest Shape Matrix estimator.
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Tyler Shape depth
arXiv: Statistics Theory, 2017Co-Authors: Davy Paindaveine, Germain Van BeverAbstract:In many problems from multivariate analysis, the parameter of interest is a Shape Matrix, that is, a normalized version of the corresponding scatter or dispersion Matrix. In this paper, we propose a depth concept for Shape matrices that involves data points only through their directions from the center of the distribution. We use the terminology Tyler Shape depth since the resulting estimator of Shape, namely the deepest Shape Matrix, is the median-based counterpart of the M-estimator of Shape of Tyler (1987). Beyond estimation, Shape depth, like its Tyler antecedent, also allows hypothesis testing on Shape. Its main benefit, however, lies in the ranking of Shape matrices it provides, whose practical relevance is illustrated in principal component analysis and in Shape-based outlier detection. We study the invariance, quasi-concavity and continuity properties of Tyler Shape depth, the topological and boundedness properties of the corresponding depth regions, existence of a deepest Shape Matrix and prove Fisher consistency in the elliptical case. Finally, we derive a Glivenko-Cantelli-type result and establish almost sure consistency of the deepest Shape Matrix estimator.
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Tyler Shape Depth
Research Papers in Economics, 2017Co-Authors: Davy Paindaveine, Germain Van BeverAbstract:In many problems from multivariate analysis (principal component analysis, testing for sphericity, etc.), the parameter of interest is a Shape Matrix, that is, a normalised version of the corresponding scatter or dispersion Matrix. In this paper, we propose a depth concept for Shape matrices which is of a sign nature, in the sense that it involves data points only through their directions from the center of the distribution. We use the terminology Tyler Shape depth since the resulting estimator of Shape — namely, the deepest Shape Matrix — is the depth-based counterpart of the celebrated M-estimator of Shape from Tyler (1987). We in- vestigate the invariance, quasi-concavity and continuity properties of Tyler Shape depth, as well as the topological and boundedness properties of the corresponding depth regions. We study existence of a deepest Shape Matrix and prove Fisher consistency in the elliptical case. We derive a Glivenko-Cantelli-type result and establish the almost sure consistency of the deepest Shape Matrix estimator. We also consider depth-based tests for Shape and investigate their finite-sample per- formances through simulations. Finally, we illustrate the practical relevance of the proposed depth concept on a real data example.
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A canonical definition of Shape
Statistics & Probability Letters, 2008Co-Authors: Davy PaindaveineAbstract:Very general concepts of scatter, extending the traditional notion of covariance matrices, have become classical tools in robust multivariate analysis. In many problems of practical importance (principal components, canonical correlation, testing for sphericity), only homogeneous functions of the scatter Matrix are of interest. In line with this fact, scatter functionals often are only defined up to a positive scalar factor, yielding a family of scatter matrices rather than a uniquely defined one. In such families, it is natural to single out one representative by imposing a normalization constraint: this normalized scatter is called a Shape Matrix. In the particular case of elliptical families, this constraint in turn induces a concept of scale; along with a location center and a standardized radial density, the Shape and scale parameters entirely characterize an elliptical density. In this paper, we show that one and only one normalization has the additional properties that (i) the resulting Fisher information matrices for Shape and scale, in locally asymptotically normal (LAN) elliptical families, are block-diagonal, and that (ii) the semiparametric elliptical families indexed by location, Shape, and completely unspecified radial densities are adaptive. This particular normalization, which imposes the condition that the determinant of the Shape Matrix should be equal to one, therefore can be considered canonical.
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SEMIPARAMETRICALLY EFFICIENT RANK-BASED INFERENCE FOR Shape I. OPTIMAL RANK-BASED TESTS FOR SPHERICITY
The Annals of Statistics, 2006Co-Authors: Marc Hallin, Davy PaindaveineAbstract:We propose a class of rank-based procedures for testing that the Shape Matrix V of an elliptical distribution (with unspecified center of symmetry, scale and radial density) has some fixed value V 0 ; this includes, for V 0 = I k , the problem of testing for sphericity as an important particular case. The proposed tests are invariant under translations, monotone radial transformations, rotations and reflections with respect to the estimated center of symmetry. They are valid without any moment assumption. For adequately chosen scores, they are locally asymptotically maximin (in the Le Cam sense) at given radial densities. They are strictly distribution-free when the center of symmetry is specified, and asymptotically so when it must be estimated. The multivariate ranks used throughout are those of the distances-in the metric associated with the null value V 0 of the Shape Matrix-between the observations and the (estimated) center of the distribution. Local powers (against elliptical alternatives) and asymptotic relative efficiencies (AREs) are derived with respect to the adjusted Mauchly test (a modified version of the Gaussian likelihood ratio procedure proposed by Muirhead and Waternaux [Biometrika 67 (1980) 31-43]) or, equivalently, with respect to (an extension of) the test for sphericity introduced by John [Biometrika 59 (1972) 169-173]. For Gaussian scores, these AREs are uniformly larger than one, irrespective of the actual radial density. Necessary and/or sufficient conditions for consistency under nonlocal, possibly nonelliptical alternatives are given. Finite sample performance is investigated via a Monte Carlo study.
L. Dong - One of the best experts on this subject based on the ideXlab platform.
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Lagrangian advection scheme with Shape Matrix (LASM) v0.2: interparcel mixing, physics–dynamics coupling and 3-D extension
Geoscientific Model Development, 2015Co-Authors: L. Dong, B. Wang, L. Liu, Yongjie HuangAbstract:Abstract. The interparcel mixing algorithm in the Lagrangian advection scheme with Shape Matrix (LASM) is updated to make the scheme more robust. The linear degeneration criterion is replaced by the maximum deviation of the skeleton points so that the new algorithm is more effective in controlling the Shape of parcels, which is vital for long time simulation. LASM is inherently Shape-preserving without any complicated filter or limiter, and it is linear. This fact contributes to the ability to preserve the sum of multiple tracers exactly on the parcels in LASM. A newly proposed terminator "toy"-chemistry test is used to test LASM, which shows that LASM can preserve the weighted sum of two reactive species precisely. The physics–dynamics coupling (i.e., tendency evaluation type) is also discussed. A flow generated by a WRF large-eddy simulation is also used to test the 3-D extension of LASM.
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lagrangian advection scheme with Shape Matrix lasm v0 2 interparcel mixing physics dynamics coupling and 3 d extension
Geoscientific Model Development, 2015Co-Authors: L. Dong, B. Wang, L. Liu, Yongjie HuangAbstract:Abstract. The interparcel mixing algorithm in the Lagrangian advection scheme with Shape Matrix (LASM) is updated to make the scheme more robust. The linear degeneration criterion is replaced by the maximum deviation of the skeleton points so that the new algorithm is more effective in controlling the Shape of parcels, which is vital for long time simulation. LASM is inherently Shape-preserving without any complicated filter or limiter, and it is linear. This fact contributes to the ability to preserve the sum of multiple tracers exactly on the parcels in LASM. A newly proposed terminator "toy"-chemistry test is used to test LASM, which shows that LASM can preserve the weighted sum of two reactive species precisely. The physics–dynamics coupling (i.e., tendency evaluation type) is also discussed. A flow generated by a WRF large-eddy simulation is also used to test the 3-D extension of LASM.
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An updated interparcel mixing algorithm in the Lagrangian advection scheme with Shape Matrix (LASM) v0.2
2015Co-Authors: L. Dong, B. Wang, L. LiuAbstract:Abstract. The interparcel mixing algorithm in the Lagrangian advection scheme with Shape Matrix (LASM) is updated to make the scheme more robust. The linear degeneration criterion is replaced by the maximum deviation of the skeleton points so that the new algorithm is more effective in controlling the Shape of parcels, which is vital for long time simulation. LASM is inherently Shape-preserving without any complicated filter or limiter, so it is linear. This fact contributes to the ability of LASM of preserving the sum of multiple tracers exactly. A newly proposed terminator "toy"-chemistry test is also used to test LASM, which shows that LASM can preserve the weighted sum of two reactive chlorine-like species precisely.
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A Lagrangian advection scheme with Shape Matrix (LASM) for solving advection problems
Geoscientific Model Development, 2014Co-Authors: L. Dong, B. WangAbstract:Abstract. A new Lagrangian advection scheme with Shape Matrix (LASM) is proposed to take advantage of the extreme low numerical diffusion of the Lagrangian methods. The tracer is discretized into finite parcels, which move along the downstream trajectories. Different from other Lagrangian schemes, the parcel Shape is simulated explicitly by a linear transformation Matrix. By doing so, the aliasing error in the Lagrangian schemes is largely reduced without introducing substantial interparcel mixing in the pure advection stage, because the flow information will be respected when remapping tracer density onto the fixed model grids. An adaptive interparcel mixing algorithm is constructed to ensure the validity of the linear approximation of the parcel Shape, where the mixing is only triggered when it is necessary and resembles the physical mixing. The total tracer mass on the parcels is conserved exactly. The new scheme is validated by using several test cases.
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A Lagrangian Advection scheme with Shape Matrix (LASM) for solving advection problems
Geoscientific Model Development Discussions, 2014Co-Authors: L. Dong, B. WangAbstract:Abstract. A new Lagrangian advection scheme with Shape Matrix (LASM) is proposed to take advantage of extreme low numerical diffusion of Lagrangian methods. The tracer is discretized into finite parcels, which move along the downstream trajectories. Different from other Lagrangian schemes, the parcel Shape is simulated explicitly by a linear transformation Matrix. By doing so, the aliasing error in the Lagrangian schemes is largely reduced without introducing substantial interparcel mixing in the pure advection stage, because the flow information will be respected when remapping tracer density onto the fixed model grids. An adaptive interparcel mixing algorithm is constructed to ensure the validity of the linear approximation of the parcel Shape, where the mixing is only triggered when it is necessary and resembles the physical mixing. The total tracer mass on the parcels is conserved exactly. The new scheme is validated by using several test cases.
Boaz Nadler - One of the best experts on this subject based on the ideXlab platform.
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Robust sparse covariance estimation by thresholding Tyler’s M-estimator
The Annals of Statistics, 2020Co-Authors: John Goes, Gilad Lerman, Boaz NadlerAbstract:Estimating a high-dimensional sparse covariance Matrix from a limited number of samples is a fundamental problem in contemporary data analysis. Most proposals to date, however, are not robust to outliers or heavy tails. Towards bridging this gap, in this work we consider estimating a sparse Shape Matrix from $n$ samples following a possibly heavy tailed elliptical distribution. We propose estimators based on thresholding either Tyler's M-estimator or its regularized variant. We derive bounds on the difference in spectral norm between our estimators and the Shape Matrix in the joint limit as the dimension $p$ and sample size $n$ tend to infinity with $p/n\to\gamma>0$. These bounds are minimax rate-optimal. Results on simulated data support our theoretical analysis.
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robust sparse covariance estimation by thresholding tyler s m estimator
arXiv: Statistics Theory, 2017Co-Authors: John Goes, Gilad Lerman, Boaz NadlerAbstract:Estimating a high-dimensional sparse covariance Matrix from a limited number of samples is a fundamental problem in contemporary data analysis. Most proposals to date, however, are not robust to outliers or heavy tails. Towards bridging this gap, in this work we consider estimating a sparse Shape Matrix from $n$ samples following a possibly heavy tailed elliptical distribution. We propose estimators based on thresholding either Tyler's M-estimator or its regularized variant. We derive bounds on the difference in spectral norm between our estimators and the Shape Matrix in the joint limit as the dimension $p$ and sample size $n$ tend to infinity with $p/n\to\gamma>0$. These bounds are minimax rate-optimal. Results on simulated data support our theoretical analysis.
John Goes - One of the best experts on this subject based on the ideXlab platform.
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Robust sparse covariance estimation by thresholding Tyler’s M-estimator
The Annals of Statistics, 2020Co-Authors: John Goes, Gilad Lerman, Boaz NadlerAbstract:Estimating a high-dimensional sparse covariance Matrix from a limited number of samples is a fundamental problem in contemporary data analysis. Most proposals to date, however, are not robust to outliers or heavy tails. Towards bridging this gap, in this work we consider estimating a sparse Shape Matrix from $n$ samples following a possibly heavy tailed elliptical distribution. We propose estimators based on thresholding either Tyler's M-estimator or its regularized variant. We derive bounds on the difference in spectral norm between our estimators and the Shape Matrix in the joint limit as the dimension $p$ and sample size $n$ tend to infinity with $p/n\to\gamma>0$. These bounds are minimax rate-optimal. Results on simulated data support our theoretical analysis.
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robust sparse covariance estimation by thresholding tyler s m estimator
arXiv: Statistics Theory, 2017Co-Authors: John Goes, Gilad Lerman, Boaz NadlerAbstract:Estimating a high-dimensional sparse covariance Matrix from a limited number of samples is a fundamental problem in contemporary data analysis. Most proposals to date, however, are not robust to outliers or heavy tails. Towards bridging this gap, in this work we consider estimating a sparse Shape Matrix from $n$ samples following a possibly heavy tailed elliptical distribution. We propose estimators based on thresholding either Tyler's M-estimator or its regularized variant. We derive bounds on the difference in spectral norm between our estimators and the Shape Matrix in the joint limit as the dimension $p$ and sample size $n$ tend to infinity with $p/n\to\gamma>0$. These bounds are minimax rate-optimal. Results on simulated data support our theoretical analysis.