The Degrees of Freedom of Partial Least Squares Regression

The derivation of statistical properties for Partial Least Squares regression can be a challenging task. The reason is that the construction of latent components from the predictor variables also depends on the response variable. While this typically leads to good performance and interpretable models in practice, it makes the statistical analysis more involved. In this work, we study the intrinsic complexity of Partial Least Squares Regression. Our contribution is an unbiased estimate of its Degrees of Freedom. It is defined as the trace of the first derivative of the fitted values, seen as a function of the response. We establish two equivalent representations that rely on the close connection of Partial Least Squares to matrix decompositions and Krylov subspace techniques. We show that the Degrees of Freedom depend on the collinearity of the predictor variables: The lower the collinearity is, the higher the Degrees of Freedom are. In particular, they are typically higher than the naive approach that defines the Degrees of Freedom as the number of components. Further, we illustrate how the Degrees of Freedom approach can be used for the comparison of different regression methods. In the experimental section, we show that our Degrees of Freedom estimate in combination with information criteria is useful for model selection.Comment: to appear in the Journal of the American Statistical Associatio

Quantum Gravitational Contributions to the CMB Anisotropy Spectrum

We derive the primordial power spectrum of density fluctuations in the framework of quantum cosmology. For this purpose we perform a Born-Oppenheimer approximation to the Wheeler-DeWitt equation for an inflationary universe with a scalar field. In this way we first recover the scale-invariant power spectrum that is found as an approximation in the simplest inflationary models. We then obtain quantum gravitational corrections to this spectrum and discuss whether they lead to measurable signatures in the CMB anisotropy spectrum. The non-observation so far of such corrections translates into an upper bound on the energy scale of inflation.Comment: 4 pages, v3: sign error in Eq. (5) and its consequences correcte

Spin Configuration in the 1/3 Magnetization Plateau of Azurite Determined by NMR

High magnetic field $^{63,65}$Cu NMR spectra were used to determine the local spin polarization in the 1/3 magnetization plateau of azurite, Cu$_3$(CO$_3$)$_2$(OH)$_2$, which is a model system for the distorted diamond antiferromagnetic spin-1/2 chain. The spin part of the hyperfine field of the Cu2 (dimer) sites is found to be field independent, negative and strongly anisotropic, corresponding to $\approx$10 % of fully polarized spin in a $d$-orbital. This is close to the expected configuration of the "quantum" plateau, where a singlet state is stabilized on the dimer. However, the observed non-zero spin polarization points to some triplet admixture, induced by strong asymmetry of the diamond bonds $J_1$ and $J_3$.Comment: Phys. Rev. Lett. 102, in press (2009

Dimensional reduction by pressure in the magnetic framework material CuF2_{2}(D2_{2}O)2_{2}pyz: from spin-wave to spinon excitations

Metal organic magnets have enormous potential to host a variety of electronic and magnetic phases that originate from a strong interplay between the spin, orbital and lattice degrees of freedom. We control this interplay in the quantum magnet CuF$_2$(D$_2$O)$_2$pyz by using high pressure to drive the system through a structural and magnetic phase transition. Using neutron scattering, we show that the low pressure state, which hosts a two-dimensional square lattice with spin-wave excitations and a dominant exchange coupling of 0.89 meV, transforms at high pressure into a one-dimensional spin-chain hallmarked by a spinon continuum and a reduced exchange interaction of 0.43 meV. This direct microscopic observation of a magnetic dimensional crossover as a function of pressure opens up new possibilities for studying the evolution of fractionalised excitations in low dimensional quantum magnets and eventually pressure-controlled metal--insulator transitions

Spatially Resolved Magnetization in the Bose-Einstein Condensed State of BaCuSi2O6: Evidence for Imperfect Frustration

In order to understand the nature of the two-dimensional Bose-Einstein condensed (BEC) phase in BaCuSi2O6, we performed detailed 63Cu and 29Si NMR above the critical magnetic field, Hc1= 23.4 T. The two different alternating layers present in the system have very different local magnetizations close to Hc1; one is very weak, and its size and field dependence are highly sensitive to the nature of inter-layer coupling. Its precise value could only be determined by "on-site" 63Cu NMR, and the data are fully reproduced by a model of interacting hard-core bosons in which the perfect frustration associated to tetragonal symmetry is slightly lifted, leading to the conclusion that the population of the less populated layers is not fully incoherent but must be partially condensed

Dependence of the BEC transition temperature on interaction strength: a perturbative analysis

We compute the critical temperature T_c of a weakly interacting uniform Bose gas in the canonical ensemble, extending the criterion of condensation provided by the counting statistics for the uniform ideal gas. Using ordinary perturbation theory, we find in first order $(T_c-T_c^0)/T_c^0 = -0.93 a\rho^{1/3}$, where T_c^0 is the transition temperature of the corresponding ideal Bose gas, a is the scattering length, and $\rho$ is the particle number density.Comment: 14 pages (RevTeX

The Shorter the Better? Effects of Privacy Policy Length on Online Privacy Decision-Making

Privacy policies provide Internet users with the possibility to inform themselves about websites’ usage of their disclosed personal data. Strikingly, however, most people tend not to read privacy policies because they are long and cumbersome, indicating that people do not wish to expend much (cognitive) effort on reading such policies. The present study aimed to examine whether shorter privacy policies can be beneficial in informing users about a social networking site’s (SNS) privacy practices, and to investigate associations between variables relevant for privacy decision-making using one theory-based integrative model. In an online experiment, participants (N = 305) were asked to create a personal account on an SNS after being given the option to read the privacy policy. Privacy policy length and the SNS’s level of privacy were varied, creating a 2 (policy length) x 2 (level of privacy) between-subjects design. The results revealed that participants who saw short policies spent less time on reading but gained higher knowledge about the SNS’s privacy practices - due to the fact that they spent more reading time per word. Factual privacy policy knowledge was found to be an indicator for participants’ subjective privacy perception. The perception and evaluation of the specific SNS´s privacy level influenced the assessment of privacy costs and benefits. Particularly when benefits were perceived as high, self-disclosure was increased