Learning curves for Gaussian process regression: Approximations and bounds

We consider the problem of calculating learning curves (i.e., average generalization performance) of Gaussian processes used for regression. On the basis of a simple expression for the generalization error, in terms of the eigenvalue decomposition of the covariance function, we derive a number of approximation schemes. We identify where these become exact, and compare with existing bounds on learning curves; the new approximations, which can be used for any input space dimension, generally get substantially closer to the truth. We also study possible improvements to our approximations. Finally, we use a simple exactly solvable learning scenario to show that there are limits of principle on the quality of approximations and bounds expressible solely in terms of the eigenvalue spectrum of the covariance function.Comment: 25 pages, 10 figure

A generalized Fellner-Schall method for smoothing parameter estimation with application to Tweedie location, scale and shape models

We consider the estimation of smoothing parameters and variance components in models with a regular log likelihood subject to quadratic penalization of the model coefficients, via a generalization of the method of Fellner (1986) and Schall (1991). In particular: (i) we generalize the original method to the case of penalties that are linear in several smoothing parameters, thereby covering the important cases of tensor product and adaptive smoothers; (ii) we show why the method's steps increase the restricted marginal likelihood of the model, that it tends to converge faster than the EM algorithm, or obvious accelerations of this, and investigate its relation to Newton optimization; (iii) we generalize the method to any Fisher regular likelihood. The method represents a considerable simplification over existing methods of estimating smoothing parameters in the context of regular likelihoods, without sacrificing generality: for example, it is only necessary to compute with the same first and second derivatives of the log-likelihood required for coefficient estimation, and not with the third or fourth order derivatives required by alternative approaches. Examples are provided which would have been impossible or impractical with pre-existing Fellner-Schall methods, along with an example of a Tweedie location, scale and shape model which would be a challenge for alternative methods

Comparison of Material Properties and Microstructure of Specimens Built Using the 3D Systems Vanguard HS and Vanguard HiQ+HSSLS Systems

The HiQ upgrade to the 3D Systems Vanguard selective laser sintering (SLS) machine incorporates a revised thermal calibration system and new control software. The paper compares the tensile modulus, tensile strength, elongation at break, flexural modulus, Izod impact resistance and microstructure of two batteries of standard specimens built from recycled Duraform PA (Nylon 12). The first set is built on a Vanguard HS system and the second on the same system with the HiQ upgrade installed. The upgrade reduces user intervention, decreases total build time and improves surface finish. However, using the default processing parameters, tensile, flexure and impact properties are all found to decline after the upgrade is installed.Mechanical Engineerin

Multidimensional optical fractionation with holographic verification

The trajectories of colloidal particles driven through a periodic potential energy landscape can become kinetically locked in to directions dictated by the landscape's symmetries. When the landscape is realized with forces exerted by a structured light field, the path a given particle follows has been predicted to depend exquisitely sensitively on such properties as the particle's size and refractive index These predictions, however, have not been tested experimentally. Here, we describe measurements of colloidal silica spheres' transport through arrays of holographic optical traps that use holographic video microscopy to track individual spheres' motions in three dimensions and simultaneously to measure each sphere's radius and refractive index with part-per-thousand resolution. These measurements confirm previously untested predictions for the threshold of kinetically locked-in transport, and demonstrate the ability of optical fractionation to sort colloidal spheres with part-per-thousand resolution on multiple characteristics simultaneously.Comment: 4 pages, 2 figures. Accepted for publication in Physical Review Letter

The Effects of Negative Legacies on the Adjustment of Parentally Bereaved Children and Adolescents

This is a report of a qualitative analysis of a sample of bereaved families in which one parent died and in which children scored in the clinical range on the Child Behavior Check List. The purpose of this analysis was to learn more about the lives of these children. They were considered to be at risk of developing emotional and behavioral problems associated with the death. We discovered that many of these “high risk” children had a continuing bond with the deceased that was primarily negative and troubling for them in contrast to a comparison group of children not at risk from the same study. Five types of legacies, not mutually exclusive, were identified: health related, role related, personal qualities, legacy of blame, and an emotional legacy. Coping behavior on the part of the surviving parent seemed to make a difference in whether or not a legacy was experienced as negative

Seasonal predictability of the winter North Atlantic Oscillation from a jet stream perspective

The winter North Atlantic Oscillation (NAO) has varied on interannual and decadal timescales over the last century, associated with variations in the speed and latitude of the eddy-driven jet stream. This paper uses hindcasts from two operational seasonal forecast systems (the European Centre for Medium-range Weather Forecasts's seasonal forecast system, and the U.K. Met Office global seasonal forecast system) and a century-long atmosphere-only experiment (using the European Centre for Medium-range Weather Forecasts's Integrated Forecasting System model) to relate seasonal prediction skill in the NAO to these aspects of jet variability. This shows that the NAO skill realized so far arises from interannual variations in the jet, largely associated with its latitude rather than speed. There likely remains further potential for predictability on longer, decadal timescales. In the small sample of models analyzed here, improved representation of the structure of jet variability does not translate to enhanced seasonal forecast skill

Methods for detection and characterization of signals in noisy data with the Hilbert-Huang Transform

The Hilbert-Huang Transform is a novel, adaptive approach to time series analysis that does not make assumptions about the data form. Its adaptive, local character allows the decomposition of non-stationary signals with hightime-frequency resolution but also renders it susceptible to degradation from noise. We show that complementing the HHT with techniques such as zero-phase filtering, kernel density estimation and Fourier analysis allows it to be used effectively to detect and characterize signals with low signal to noise ratio.Comment: submitted to PRD, 10 pages, 9 figures in colo

The Distribution of Nearby Stars in Velocity Space Inferred from Hipparcos Data

(abridged) The velocity distribution f(v) of nearby stars is estimated, via a maximum- likelihood algorithm, from the positions and tangential velocities of a kinematically unbiased sample of 14369 stars observed by the HIPPARCOS satellite. f(v) shows rich structure in the radial and azimuthal motions, v_R and v_phi, but not in the vertical velocity, v_z: there are four prominent and many smaller maxima, many of which correspond to well known moving groups. While samples of early-type stars are dominated by these maxima, also up to 25% of red main-sequence stars are associated with them. These moving groups are responsible for the vertex deviation measured even for samples of late-type stars; they appear more frequently for ever redder samples; and as a whole they follow an asymmetric-drift relation, in the sense that those only present in red samples predominantly have large |v_R| and lag in v_phi w.r.t. the local standard of rest (LSR). The question arise, how these old moving groups got on their eccentric orbits. A plausible mechanism, known from solar system dynamics, which is able to manage a shift in orbit space involves locking into an orbital resonance. Apart from these moving groups, there is a smooth background distribution, akin to Schwarzschild's ellipsoidal model, with axis ratio of about 1:0.6:0.35 in v_R, v_phi, and v_z. The contours are aligned with the $v_r$ direction, but not w.r.t. the v_phi and v_z axes: the mean v_z increases for stars rotating faster than the LSR. This effect can be explained by the stellar warp of the Galactic disk. If this explanation is correct, the warp's inner edge must not be within the solar circle, while its pattern rotates with frequency of about 13 km/s/kpc or more retrograde w.r.t. the stellar orbits.Comment: 16 pages LaTeX (aas2pp4.sty), 6 figures, accepted by A

Iso-singlet Down Quark Mixing And CP Violation Experiments

We confront the new physics models with extra iso-singlet down quarks in the new CP violation experimental era with $\sin{(2\beta)}$ and $\epsilon'/\epsilon$ measurements, $K^+ \to \pi^+ \nu \bar{\nu}$ events, and $x_s$ limits. The closeness of the new experimental results to the standard model theory requires us to include full SM amplitudes in the analysis. In models allowing mixing to a new isosinglet down quark, as in E$_6$, flavor changing neutral currents are induced that allow a $Z^0$ mediated contribution to $B-\bar B$ mixing and which bring in new phases. In $(\rho,\eta)$, $(x_s,\sin{(\gamma)})$, and $(x_s, \sin{(2\phi_s)})$ plots we still find much larger regions in the four down quark model than in the SM, reaching down to $\eta \approx 0$, $0 \leq \sin{(\gamma)} \leq 1$, $-.75 \leq \sin{(2\alpha)} \leq 0.15$, and $\sin{(2\phi_s)}$ down to zero, all at 1$\sigma$. We elucidate the nature of the cancellation in an order $\lambda^5$ four down quark mixing matrix element which satisfies the experiments and reduces the number of independent angles and phases. We also evaluate tests of unitarity for the $3\times3$ CKM submatrix.Comment: 14 pages, 16 figures, REVTeX