Don't Fall for Tuning Parameters: Tuning-Free Variable Selection in High Dimensions With the TREX

Lasso is a seminal contribution to high-dimensional statistics, but it hinges on a tuning parameter that is difficult to calibrate in practice. A partial remedy for this problem is Square-Root Lasso, because it inherently calibrates to the noise variance. However, Square-Root Lasso still requires the calibration of a tuning parameter to all other aspects of the model. In this study, we introduce TREX, an alternative to Lasso with an inherent calibration to all aspects of the model. This adaptation to the entire model renders TREX an estimator that does not require any calibration of tuning parameters. We show that TREX can outperform cross-validated Lasso in terms of variable selection and computational efficiency. We also introduce a bootstrapped version of TREX that can further improve variable selection. We illustrate the promising performance of TREX both on synthetic data and on a recent high-dimensional biological data set that considers riboflavin production in B. subtilis

Long-term gait measurements in daily life: Results from the Berlin Aging Study II (BASE-II)

BACKGROUND: Walking ability is an important prerequisite for activity, social participation and independent living. While in most healthy adults, this ability can be assumed as given, limitations in walking ability occur with increasing age. Furthermore, slow walking speed is linked to several chronic conditions and overall morbidity. Measurements of gait parameters can be used as a proxy to detect functional decline and onset of chronic conditions. Up to now, gait characteristics used for this purpose are measured in standardized laboratory settings. There is some evidence, however, that long-term measurements of gait parameters in the living environment have some advantages over short-term laboratory measurements. METHODS: We evaluated cross-sectional data from an accelerometric sensor worn in a subgroup of 554 participants of the Berlin Aging Study II (BASE-II). Data from the two BASE-II age groups (age between 22-36 years and 60-79 years) were used for the current analysis of accelerometric data for a minimum of two days and a maximum of ten days were available. Real world walking speed, number of steps, maximum coherent distance and total distance were derived as average data per day. Linear regression analyses were performed on the different gait parameters in order to identify significant determinants. Additionally, Mann-Whitney-U-tests were performed to detect sex-specific differences. RESULTS: Age showed to be significantly associated with real world walking speed and with the total distance covered per day, while BMI contributed negatively to the number of walking steps, maximum coherent distance and total distance walked. Additionally, sex was associated with walking steps. However, R2-values for all models were low. Overall, women had significantly more walking steps and a larger coherent distance per day when compared to men. When separated by age group, this difference was significant only in the older participants. Additionally, walking speed was significantly higher in women compared to men in the subgroup of older people. CONCLUSIONS: Age- and sex-specific differences have to be considered when objective gait parameters are measured, e.g. in the context of clinical risk assessment. For this purpose normative data, differentiating for age and sex would have to be established to allow reliable classification of long-term measurements of gait

Plasmonic modes of extreme subwavelength nanocavities

We study the physics of a new type of subwavelength nanocavities. They are based on U-shaped metal-insulator-metal waveguides supporting the excitation of surface plasmon polaritons. The waveguides are simultaneously excited from both sides of the U by incident plane waves. Due to their finite length discrete modes emerge within the nanocavity. We show that the excitation symmetry with respect to the cavity ends permits the observation of even and odd modes. Our investigations include near and far field simulations and predict a strong spectral far field response of the comparable small nanoresonators. The strong near field enhancement observed in the cavity at resonance might be suitable to increase the efficiency of nonlinear optical effects, quantum analogies and might facilitate the development of active optical elements, such as active plasmonic elements

Gradient-robust hybrid DG discretizations for the compressible Stokes equations

This paper studies two hybrid discontinuous Galerkin (HDG) discretizations for the velocity-density formulation of the compressible Stokes equations with respect to several desired structural properties, namely provable convergence, the preservation of non-negativity and mass constraints for the density, and gradient-robustness. The later property dramatically enhances the accuracy in well-balanced situations, such as the hydrostatic balance where the pressure gradient balances the gravity force. One of the studied schemes employs an H(div)-conforming velocity ansatz space which ensures all mentioned properties, while a fully discontinuous method is shown to satisfy all properties but the gradient-robustness. Also higher-order schemes for both variants are presented and compared in three numerical benchmark problems. The final example shows the importance also for non-hydrostatic well-balanced states for the compressible Navier-Stokes equations

Guaranteed upper bounds for the velocity error of pressure-robust Stokes discretisations

This paper improves guaranteed error control for the Stokes problem with a focus on pressure-robustness, i.e. for discretisations that compute a discrete velocity that is independent of the exact pressure. A Prager-Synge type result relates the errors of divergence-free primal and H(div)-conforming dual mixed methods (for the velocity gradient) with an equilibration constraint that needs special care when discretised. To relax the constraints on the primal and dual method, a more general result is derived that enables the use of a recently developed mass conserving mixed stress discretisation to design equilibrated fluxes that yield pressure-independent guaranteed upper bounds for any pressure-robust (but not necessarily divergence-free) primal discretisation. Moreover, a provably efficient local design of the equilibrated fluxes is presented that reduces the numerical costs of the error estimator. All theoretical findings are verified by numerical examples which also show that the efficiency indices of our novel guaranteed upper bounds for the velocity error are close to 1

Guaranteed upper bounds for the velocity error of pressure-robust Stokes discretisations

This paper improves guaranteed error control for the Stokes problem with a focus on pressure-robustness, i.e. for discretisations that compute a discrete velocity that is independent of the exact pressure. A Prager-Synge type result relates the errors of divergence-free primal and H(div)-conforming dual mixed methods (for the velocity gradient) with an equilibration constraint that needs special care when discretised. To relax the constraints on the primal and dual method, a more general result is derived that enables the use of a recently developed mass conserving mixed stress discretisation to design equilibrated fluxes that yield pressure-independent guaranteed upper bounds for any pressure-robust (but not necessarily divergence-free) primal discretisation. Moreover, a provably efficient local design of the equilibrated fluxes is presented that reduces the numerical costs of the error estimator. All theoretical findings are verified by numerical examples which also show that the efficiency indices of our novel guaranteed upper bounds for the velocity error are close to 1

Gradient-robust hybrid DG discretizations for the compressible Stokes equations

This paper studies two hybrid discontinuous Galerkin (HDG) discretizations for the velocity-density formulation of the compressible Stokes equations with respect to several desired structural properties, namely provable convergence, the preservation of non-negativity and mass constraints for the density, and gradient-robustness. The later property dramatically enhances the accuracy in well-balanced situations, such as the hydrostatic balance where the pressure gradient balances the gravity force. One of the studied schemes employs an H(div)-conforming velocity ansatz space which ensures all mentioned properties, while a fully discontinuous method is shown to satisfy all properties but the gradient-robustness. Also higher-order schemes for both variants are presented and compared in three numerical benchmark problems. The final example shows the importance also for non-hydrostatic well-balanced states for the compressible Navier-Stokes equations

Refined a posteriori error estimation for classical and pressure-robust Stokes finite element methods

Recent works showed that pressure-robust modifications of mixed finite element methods for the Stokes equations outperform their standard versions in many cases. This is achieved by divergence-free reconstruction operators and results in pressure-independent velocity error estimates which are robust with respect to small viscosities. In this paper we develop a posteriori error control which reflects this robustness. The main difficulty lies in the volume contribution of the standard residual-based approach that includes the L2-norm of the right-hand side. However, the velocity is only steered by the divergence-free part of this source term. An efficient error estimator must approximate this divergence-free part in a proper manner, otherwise it can be dominated by the pressure error. To overcome this difficulty a novel approach is suggested that uses arguments from the stream function and vorticity formulation of the NavierStokes equations. The novel error estimators only take the curl of the righthand side into account and so lead to provably reliable, efficient and pressure-independent upper bounds in case of a pressure-robust method in particular in pressure-dominant situations. This is also confirmed by some numerical examples with the novel pressure-robust modifications of the TaylorHood and mini finite element methods

SuperWASP Observations of the 2007 Outburst of Comet 17P/Holmes

We present wide-field imaging of the 2007 outburst of Comet 17P/Holmes obtained serendipitously by SuperWASP-North on 17 nights over a 42-night period beginning on the night (2007 October 22-23) immediately prior to the outburst. Photometry of 17P's unresolved coma in SuperWASP data taken on the first night of the outburst is consistent with exponential brightening, suggesting that the rapid increase in the scattering cross-section of the coma could be largely due to the progressive fragmentation of ejected material produced on a very short timescale at the time of the initial outburst, with fragmentation timescales decreasing from t(frag)~2x10^3 s to t(frag)~1x10^3 s over our observing period. Analysis of the expansion of 17P's coma reveals a velocity gradient suggesting that the outer coma was dominated by material ejected in an instantaneous, explosive manner. We find an expansion velocity at the edge of the dust coma of v(exp) = 0.55+/-0.02 km/s and a likely outburst date of t_0=2007 October 23.3+/-0.3, consistent with our finding that the comet remained below SuperWASP's detection limit of m(V)~15 mag until at least 2007 October 23.3. Modelling of 17P's gas coma indicates that its outer edge, which was observed to extend past the outer dust coma, is best explained with a single pulse of gas production, consistent with our conclusions concerning the production of the outer dust coma.Comment: 36 pages, 8 figures, accepted for publication in MNRA