We present a systematic analysis of statistical properties of turbulent
current and vorticity structures at a given time using cluster analysis. The
data stems from numerical simulations of decaying three-dimensional (3D)
magnetohydrodynamic turbulence in the absence of an imposed uniform magnetic
field; the magnetic Prandtl number is taken equal to unity, and we use a
periodic box with grids of up to 1536^3 points, and with Taylor Reynolds
numbers up to 1100. The initial conditions are either an X-point configuration
embedded in 3D, the so-called Orszag-Tang vortex, or an
Arn'old-Beltrami-Childress configuration with a fully helical velocity and
magnetic field. In each case two snapshots are analyzed, separated by one
turn-over time, starting just after the peak of dissipation. We show that the
algorithm is able to select a large number of structures (in excess of 8,000)
for each snapshot and that the statistical properties of these clusters are
remarkably similar for the two snapshots as well as for the two flows under
study in terms of scaling laws for the cluster characteristics, with the
structures in the vorticity and in the current behaving in the same way. We
also study the effect of Reynolds number on cluster statistics, and we finally
analyze the properties of these clusters in terms of their velocity-magnetic
field correlation. Self-organized criticality features have been identified in
the dissipative range of scales. A different scaling arises in the inertial
range, which cannot be identified for the moment with a known self-organized
criticality class consistent with MHD. We suggest that this range can be
governed by turbulence dynamics as opposed to criticality, and propose an
interpretation of intermittency in terms of propagation of local instabilities.

本研究通过簇分析系统地分析了湍流电流和涡流结构的统计特性，同时研究了雷诺数对簇统计的影响以及如何解释间歇性，并发现自组织临界性特征在可耗散尺度范围内。

磁流体力学湍流中的结构：检测和缩放

Structures in magnetohydrodynamic turbulence: detection and scaling

The k-means algorithm is a well-known method for partitioning n points that
lie in the d-dimensional space into k clusters. Its main features are
simplicity and speed in practice. Theoretically, however, the best known upper
bound on its running time (i.e. O(n^{kd})) can be exponential in the number of
points. Recently, Arthur and Vassilvitskii [3] showed a super-polynomial
worst-case analysis, improving the best known lower bound from \Omega(n) to
2^{\Omega(\sqrt{n})} with a construction in d=\Omega(\sqrt{n}) dimensions. In
[3] they also conjectured the existence of superpolynomial lower bounds for any
d >= 2.
Our contribution is twofold: we prove this conjecture and we improve the
lower bound, by presenting a simple construction in the plane that leads to the
exponential lower bound 2^{\Omega(n)}.

证明了 k-means 算法存在关于维数的指数下界，并以平面上的简单构造为例给出了指数下界的证明。

k-means 在平面上甚至需要指数次迭代

k-means requires exponentially many iterations even in the plane

Estimation of structure, such as in variable selection, graphical modelling
or cluster analysis is notoriously difficult, especially for high-dimensional
data. We introduce stability selection. It is based on subsampling in
combination with (high-dimensional) selection algorithms. As such, the method
is extremely general and has a very wide range of applicability. Stability
selection provides finite sample control for some error rates of false
discoveries and hence a transparent principle to choose a proper amount of
regularisation for structure estimation. Variable selection and structure
estimation improve markedly for a range of selection methods if stability
selection is applied. We prove for randomised Lasso that stability selection
will be variable selection consistent even if the necessary conditions needed
for consistency of the original Lasso method are violated. We demonstrate
stability selection for variable selection and Gaussian graphical modelling,
using real and simulated data.

介绍一种名为稳定性选择的高维数据结构估计方法，结合子抽样和选择算法，提供透明的原则以选择适当的正则化量来改善结构估计的可变性选择和图形建模。