Are Discoveries Spurious? Distributions of Maximum Spurious Correlations and Their Applications

Over the last two decades, many exciting variable selection methods have been developed for finding a small group of covariates that are associated with the response from a large pool. Can the discoveries from these data mining approaches be spurious due to high dimensionality and limited sample size? Can our fundamental assumptions about the exogeneity of the covariates needed for such variable selection be validated with the data? To answer these questions, we need to derive the distributions of the maximum spurious correlations given a certain number of predictors, namely, the distribution of the correlation of a response variable $Y$ with the best $s$ linear combinations of $p$ covariates $\mathbf{X}$, even when $\mathbf{X}$ and $Y$ are independent. When the covariance matrix of $\mathbf{X}$ possesses the restricted eigenvalue property, we derive such distributions for both a finite $s$ and a diverging $s$, using Gaussian approximation and empirical process techniques. However, such a distribution depends on the unknown covariance matrix of $\mathbf{X}$. Hence, we use the multiplier bootstrap procedure to approximate the unknown distributions and establish the consistency of such a simple bootstrap approach. The results are further extended to the situation where the residuals are from regularized fits. Our approach is then used to construct the upper confidence limit for the maximum spurious correlation and to test the exogeneity of the covariates. The former provides a baseline for guarding against false discoveries and the latter tests whether our fundamental assumptions for high-dimensional model selection are statistically valid. Our techniques and results are illustrated with both numerical examples and real data analysis

5-Sulfanyl­idene-2H,5H-1,3-dithiolo[4,5-d][1,3]dithiol-2-one

The title mol­ecule, C4OS5, is essentially planar, with an r.m.s. deviation of 0.032 (3) Å. All the C—S single bonds are shorter than the standard Csp 3—S single-bond length, showing the π-conjugated nature of the molecule. In the crystal, molecules lie parallel to one another and pack in columns along the a axis. Short inter­molecular S⋯S contacts [3.314 (3), 3.482 (2) and 3.501 (2) Å] are observed between the columns. The angle between the two mol­ecular dipole moments in the unit cell is 39.3 (1)° and the macro-polarization vector is along the [1 0 − 1.41] direction. As a result of the high polarization and π-conjugation of the structure, the crystalline powder exhibits a second harmonic generating intensity, which is as strong as that of the urea standard powder crystals, when irradiated by a 1053 nm laser beam. The diffraction space of the crystal showed a nonmerohedral twinning

Nematic topological superconducting phase in Nb-doped Bi2Se3

A nematic topological superconductor has an order parameter symmetry, which spontaneously breaks the crystalline symmetry in its superconducting state. This state can be observed, for example, by thermodynamic or upper critical field experiments in which a magnetic field is rotated with respect to the crystalline axes. The corresponding physical quantity then directly reflects the symmetry of the order parameter. We present a study on the superconducting upper critical field of the Nb-doped topological insulator NbxBi2Se3 for various magnetic field orientations parallel and perpendicular to the basal plane of the Bi2Se3 layers. The data were obtained by two complementary experimental techniques, magnetoresistance and DC magnetization, on three different single crystalline samples of the same batch. Both methods and all samples show with perfect agreement that the in-plane upper critical fields clearly demonstrate a two-fold symmetry that breaks the three-fold crystal symmetry. The two-fold symmetry is also found in the absolute value of the magnetization of the initial zero-field-cooled branch of the hysteresis loop and in the value of the thermodynamic contribution above the irreversibility field, but also in the irreversible properties such as the value of the characteristic irreversibility field and in the width of the hysteresis loop. This provides strong experimental evidence that Nb-doped Bi2Se3 is a nematic topological superconductor similar to the Cu- and Sr-doped Bi2Se3