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"Mathematical minima"
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HIGHER ORDER ELICITABILITY AND OSBAND'S PRINCIPLE
A statistical functional, such as the mean or the median, is called elicitable if there is a scoring function or loss function such that the correct forecast of the functional is the unique minimizer of the expected score. Such scoring functions are called strictly consistent for the functional. The elicitability of a functional opens the possibility to compare competing forecasts and to rank them in terms of their realized scores. In this paper, we explore the notion of elicitability for multi-dimensional functionals and give both necessary and sufficient conditions for strictly consistent scoring functions. We cover the case of functionals with elicitable components, but we also show that one-dimensional functionals that are not elicitable can be a component of a higher order elicitable functional. In the case of the variance, this is a known result. However, an important result of this paper is that spectral risk measures with a spectral measure with finite support are jointly elicitable if one adds the \"correct\" quantiles. A direct consequence of applied interest is that the pair (Value at Risk, Expected Shortfall) is jointly elicitable under mild conditions that are usually fulfilled in risk management applications.
Journal Article
Control Profiles of Complex Networks
Studying the control properties of complex networks provides insight into how designers and engineers can influence these systems to achieve a desired behavior. Topology of a network has been shown to strongly correlate with certain control properties; here we uncover the fundamental structures that explain the basis of this correlation. We develop the control profile, a statistic that quantifies the different proportions of control-inducing structures present in a network. We find that standard random network models do not reproduce the kinds of control profiles that are observed in real-world networks. The profiles of real networks form three well-defined clusters that provide insight into the high-level organization and function of complex systems.
Journal Article
Interaction phenomena in graphene seen through quantum capacitance
Capacitance measurements provide a powerful means of probing the density of states. The technique has proved particularly successful in studying 2D electron systems, revealing a number of interesting many-body effects. Here, we use large-area high-quality graphene capacitors to study behavior of the density of states in this material in zero and high magnetic fields. Clear renormalization of the linear spectrum due to electron–electron interactions is observed in zero field. Quantizing fields lead to splitting of the spin- and valley-degenerate Landau levels into quartets separated by interaction-enhanced energy gaps. These many-body states exhibit negative compressibility but the compressibility returns to positive in ultrahigh B. The reentrant behavior is attributed to a competition between field-enhanced interactions and nascent fractional states.
Journal Article
Multistability and metastability: understanding dynamic coordination in the brain
Multistable coordination dynamics exists at many levels, from multifunctional neural circuits in vertebrates and invertebrates to large-scale neural circuitry in humans. Moreover, multistability spans (at least) the domains of action and perception, and has been found to place constraints upon, even dictating the nature of, intentional change and the skill-learning process. This paper reviews some of the key evidence for multistability in the aforementioned areas, and illustrates how it has been measured, modelled and theoretically understood. It then suggests how multistability—when combined with essential aspects of coordination dynamics such as instability, transitions and (especially) metastability—provides a platform for understanding coupling and the creative dynamics of complex goal-directed systems, including the brain and the brain–behaviour relation.
Journal Article
Glacial-Interglacial Indian Summer Monsoon Dynamics
The modern Indian summer monsoon (ISM) is characterized by exceptionally strong interhemispheric transport, indicating the importance of both Northern and Southern Hemisphere processes driving monsoon variability. Here, we present a high-resolution continental record from southwestern China that demonstrates the importance of interhemispheric forcing in driving ISM variability at the glacial-interglacial time scale as well. Interglacial ISM maxima are dominated by an enhanced Indian low associated with global ice volume minima. In contrast, the glacial ISM reaches a minimum, and actually begins to increase, before global ice volume reaches a maximum. We attribute this early strengthening to an increased cross-equatorial pressure gradient derived from Southern Hemisphere high-latitude cooling. This mechanism explains much of the nonorbital scale variance in the Pleistocene ISM record.
Journal Article
Multivariate empirical mode decomposition
Despite empirical mode decomposition (EMD) becoming a de facto standard for time-frequency analysis of nonlinear and non-stationary signals, its multivariate extensions are only emerging; yet, they are a prerequisite for direct multichannel data analysis. An important step in this direction is the computation of the local mean, as the concept of local extrema is not well defined for multivariate signals. To this end, we propose to use real-valued projections along multiple directions on hyperspheres (n-spheres) in order to calculate the envelopes and the local mean of multivariate signals, leading to multivariate extension of EMD. To generate a suitable set of direction vectors, unit hyperspheres (n-spheres) are sampled based on both uniform angular sampling methods and quasi-Monte Carlo-based low-discrepancy sequences. The potential of the proposed algorithm to find common oscillatory modes within multivariate data is demonstrated by simulations performed on both hexavariate synthetic and real-world human motion signals.
Journal Article
CONVERGENCE IN LAW OF THE MINIMUM OF A BRANCHING RANDOM WALK
We consider the minimum of a super-critical branching random walk. Addario-Berry and Reed [Ann. Probab. 37 (2009) 1044—1079] proved the tightness of the minimum centered around its mean value. We show that a convergence in law holds, giving the analog of a well-known result of Bramson [Mem. Amer. Math. Soc. 44 (1983) iv+190] in the case of the branching Brownian motion.
Journal Article
SparseNet: Coordinate Descent With Nonconvex Penalties
We address the problem of sparse selection in linear models. A number of nonconvex penalties have been proposed in the literature for this purpose, along with a variety of convex-relaxation algorithms for finding good solutions. In this article we pursue a coordinate-descent approach for optimization, and study its convergence properties. We characterize the properties of penalties suitable for this approach, study their corresponding threshold functions, and describe a df-standardizing reparametrization that assists our pathwise algorithm. The MC+ penalty is ideally suited to this task, and we use it to demonstrate the performance of our algorithm. Certain technical derivations and experiments related to this article are included in the Supplementary Materials section.
Journal Article
Tunable fractional quantum Hall phases in bilayer graphene
Symmetry-breaking in a quantum system often leads to complex emergent
behavior. In bilayer graphene (BLG), an electric field applied
perpendicular to the basal plane breaks the inversion symmetry of the
lattice, opening a band gap at the charge neutrality point. In a
quantizing magnetic field, electron interactions can cause spontaneous
symmetry-breaking within the spin and valley degrees of freedom,
resulting in quantum Hall effect (QHE) states with complex order. Here,
we report fractional QHE states in BLG that show phase transitions that
can be tuned by a transverse electric field. This result provides a
model platform with which to study the role of symmetry-breaking in
emergent states with topological order.
Journal Article