Lecture notes on ridge regression

The linear regression model cannot be fitted to high-dimensional data, as the high-dimensionality brings about empirical non-identifiability. Penalized regression overcomes this non-identifiability by augmentation of the loss function by a penalty (i.e. a function of regression coefficients). The ridge penalty is the sum of squared regression coefficients, giving rise to ridge regression. Here many aspect of ridge regression are reviewed e.g. moments, mean squared error, its equivalence to constrained estimation, and its relation to Bayesian regression. Finally, its behaviour and use are illustrated in simulation and on omics data. Subsequently, ridge regression is generalized to allow for a more general penalty. The ridge penalization framework is then translated to logistic regression and its properties are shown to carry over. To contrast ridge penalized estimation, the final chapter introduces its lasso counterpart

Ridge Estimation of Inverse Covariance Matrices from High-Dimensional Data

We study ridge estimation of the precision matrix in the high-dimensional setting where the number of variables is large relative to the sample size. We first review two archetypal ridge estimators and note that their utilized penalties do not coincide with common ridge penalties. Subsequently, starting from a common ridge penalty, analytic expressions are derived for two alternative ridge estimators of the precision matrix. The alternative estimators are compared to the archetypes with regard to eigenvalue shrinkage and risk. The alternatives are also compared to the graphical lasso within the context of graphical modeling. The comparisons may give reason to prefer the proposed alternative estimators

Bose-Glass Phases in Disordered Quantum Magnets

In disordered spin systems with antiferromagnetic Heisenberg exchange, transitions into and out of a magnetic-field-induced ordered phase pass through a unique regime. Using quantum Monte Carlo simulations to study the zero-temperature behavior, these intermediate regions are determined to be a Bose-Glass phase. The localization of field-induced triplons causes a finite compressibility and hence glassiness in the disordered phase.Comment: 4 pages, 4 figure

Progress in the development of an 88-mm bore 10 Tn3Sn dipole magnet

A 10 T, 2-layer cos(&thetas;)-dipole model magnet with an 88 mm clear bore utilizing an advanced powder-in-tube Nb3Sn conductor is being developed for the LHC. A dedicated conductor development program has resulted in a well performing Rutherford cable containing strands that uniquely exhibit both an overall current density of 600 A/mm2 @ 11 T and filaments with a diameter of 20 ¿m. The resistance between crossing strands amounts to 30-70 ¿¿ by insertion of a stainless steel core. After being exposed to a transverse pressure of 200 MPa identical cables show negligible permanent degradation of the critical current. The mechanical support structure is further optimized in order to reduce the peak stress in the mid-plane to below 130 MPa at full excitation and to control the pre-stress build-up during system assembly. Prior to the manufacturing of the final coils a dummy 2-layer pole is wound, heat-treated at 675°C and vacuum resin impregnated. This paper presents the current status of the magnet development program and highlights in particular the successful conductor developmen

Constraining neutrino masses with the ISW-galaxy correlation function

Temperature anisotropies in the Cosmic Microwave Background (CMB) are affected by the late Integrated Sachs-Wolfe (lISW) effect caused by any time-variation of the gravitational potential on linear scales. Dark energy is not the only source of lISW, since massive neutrinos induce a small decay of the potential on small scales during both matter and dark energy domination. In this work, we study the prospect of using the cross-correlation between CMB and galaxy density maps as a tool for constraining the neutrino mass. On the one hand massive neutrinos reduce the cross-correlation spectrum because free-streaming slows down structure formation; on the other hand, they enhance it through their change in the effective linear growth. We show that in the observable range of scales and redshifts, the first effect dominates, but the second one is not negligible. We carry out an error forecast analysis by fitting some mock data inspired by the Planck satellite, Dark Energy Survey (DES) and Large Synoptic Survey Telescope (LSST). The inclusion of the cross-correlation data from Planck and LSST increases the sensitivity to the neutrino mass m_nu by 38% (and to the dark energy equation of state w by 83%) with respect to Planck alone. The correlation between Planck and DES brings a far less significant improvement. This method is not potentially as good for detecting m_nu as the measurement of galaxy, cluster or cosmic shear power spectra, but since it is independent and affected by different systematics, it remains potentially interesting if the total neutrino mass is of the order of 0.2 eV; if instead it is close to the lower bound from atmospheric oscillations, m_nu ~ 0.05 eV, we do not expect the ISW-galaxy correlation to be ever sensitive to m_nu.Comment: 10 pages, 8 figures. References added. Accepted for publication in Phys.Rev.

Wang-Landau sampling for quantum systems: algorithms to overcome tunneling problems and calculate the free energy

We present a generalization of the classical Wang-Landau algorithm [Phys. Rev. Lett. 86, 2050 (2001)] to quantum systems. The algorithm proceeds by stochastically evaluating the coefficients of a high temperature series expansion or a finite temperature perturbation expansion to arbitrary order. Similar to their classical counterpart, the algorithms are efficient at thermal and quantum phase transitions, greatly reducing the tunneling problem at first order phase transitions, and allow the direct calculation of the free energy and entropy.Comment: Added a plot showing the efficiency at first order phase transition

Modeling association between DNA copy number and gene expression with constrained piecewise linear regression splines

DNA copy number and mRNA expression are widely used data types in cancer studies, which combined provide more insight than separately. Whereas in existing literature the form of the relationship between these two types of markers is fixed a priori, in this paper we model their association. We employ piecewise linear regression splines (PLRS), which combine good interpretation with sufficient flexibility to identify any plausible type of relationship. The specification of the model leads to estimation and model selection in a constrained, nonstandard setting. We provide methodology for testing the effect of DNA on mRNA and choosing the appropriate model. Furthermore, we present a novel approach to obtain reliable confidence bands for constrained PLRS, which incorporates model uncertainty. The procedures are applied to colorectal and breast cancer data. Common assumptions are found to be potentially misleading for biologically relevant genes. More flexible models may bring more insight in the interaction between the two markers.Comment: Published in at http://dx.doi.org/10.1214/12-AOAS605 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

High yield fusion in a Staged Z-pinch

We simulate fusion in a Z-pinch; where the load is a xenon-plasma liner imploding onto a deuterium-tritium plasma target and the driver is a 2 MJ, 17 MA, 95 ns risetime pulser. The implosion system is modeled using the dynamic, 2-1/2 D, radiation-MHD code, MACH2. During implosion a shock forms in the Xe liner, transporting current and energy radially inward. After collision with the DT, a secondary shock forms pre-heating the DT to several hundred eV. Adiabatic compression leads subsequently to a fusion burn, as the target is surrounded by a flux-compressed, intense, azimuthal-magnetic field. The intense-magnetic field confines fusion $\alpha$-particles, providing an additional source of ion heating that leads to target ignition. The target remains stable up to the time of ignition. Predictions are for a neutron yield of $3.0\times 10^{19}$ and a thermonuclear energy of 84 MJ, that is, 42 times greater than the initial, capacitor-stored energy