PCA consistency in high dimension, low sample size context

Principal Component Analysis (PCA) is an important tool of dimension reduction especially when the dimension (or the number of variables) is very high. Asymptotic studies where the sample size is fixed, and the dimension grows [i.e., High Dimension, Low Sample Size (HDLSS)] are becoming increasingly relevant. We investigate the asymptotic behavior of the Principal Component (PC) directions. HDLSS asymptotics are used to study consistency, strong inconsistency and subspace consistency. We show that if the first few eigenvalues of a population covariance matrix are large enough compared to the others, then the corresponding estimated PC directions are consistent or converge to the appropriate subspace (subspace consistency) and most other PC directions are strongly inconsistent. Broad sets of sufficient conditions for each of these cases are specified and the main theorem gives a catalogue of possible combinations. In preparation for these results, we show that the geometric representation of HDLSS data holds under general conditions, which includes a $\rho$-mixing condition and a broad range of sphericity measures of the covariance matrix.Comment: Published in at http://dx.doi.org/10.1214/09-AOS709 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Principal arc analysis on direct product manifolds

We propose a new approach to analyze data that naturally lie on manifolds. We focus on a special class of manifolds, called direct product manifolds, whose intrinsic dimension could be very high. Our method finds a low-dimensional representation of the manifold that can be used to find and visualize the principal modes of variation of the data, as Principal Component Analysis (PCA) does in linear spaces. The proposed method improves upon earlier manifold extensions of PCA by more concisely capturing important nonlinear modes. For the special case of data on a sphere, variation following nongeodesic arcs is captured in a single mode, compared to the two modes needed by previous methods. Several computational and statistical challenges are resolved. The development on spheres forms the basis of principal arc analysis on more complicated manifolds. The benefits of the method are illustrated by a data example using medial representations in image analysis.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS370 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Pomeron intercept from BFKL gluon dynamics in deep inelastic charm production at HERA

In the framework of semihard (k$_T$ factorization) QCD approach, we consider the differential cross sections of $D^{*\pm}$ meson production at HERA. The consideration is based on BFKL and CCFM gluon distributions. We find that in the case of BFKL LO gluon distribution the theoretical results are sensitive to the Pomeron intercept parameter $\Delta$. We present a comparison of the theoretical results with available ZEUS experimental data.Comment: to be published in proceedings of Diffraction 200

Gas Dynamics of the Nickel-56 Decay Heating in Pair-Instability Supernovae

Very massive 140-260 Msun stars can die as highly-energetic pair-instability supernovae (PI SNe) with energies of up to 100 times those of core-collapse SNe that can completely destroy the star, leaving no compact remnant behind. These explosions can synthesize $0.1-30$ Msun of radioactive Ni56, which can cause them to rebrighten at later times when photons due to Ni56 decay diffuse out of the ejecta. However, heat from the decay of such large masses of Ni56 could also drive important dynamical effects deep in the ejecta that are capable of mixing elements and affecting the observational signatures of these events. We have now investigated the dynamical effect of Ni56 heating on PI SN ejecta with high-resolution two-dimensional hydrodynamic simulations performed with the CASTRO code. We find that expansion of the hot Ni56 bubble forms a shell at the base of the silicon layer of the ejecta about 200 days after the explosion but that no hydrodynamical instabilities develop that would mix Ni56 with the Si/O-rich ejecta. However, while the dynamical effects of Ni56 heating may be weak they could affect the observational signatures of some PI SNe by diverting decay energy into internal expansion of the ejecta at the expense of rebrightening at later times.Comment: Accepted to ApJ, 14 page

Temperature and light requirements for growth of two diatom species (Bacillariophyceae) isolated from an Arctic macroalga

In the present study, two abundant epiphyticdiatom taxa were isolated from the assimilation hairs ofthe brown macroalga Chordaria flagelliformis collected inthe Arctic Kongsfjorden (Spitsbergen, Norway), establishedas unialgal cultures and their growth rates determinedunder controlled photon fluence rate andtemperature conditions. Using morphological (light andscanning electron microscopy) and SSU rRNA gene databoth isolates (ROS D99 and ROS D125) were identifiedas members of a FragilariaSynedropsis clade. Themolecular data of ROS D99 and ROS D125 were notidentical to any other published sequence. While ROSD99 has been identified as Fragilaria barbararum mainlydue to the SEM characteristics, ROS D125 could not bedefinitely identified although morphological data speakfor Fragilaria striatula. Both diatom species showedsimilar growth rates at all temperatures and photon fluencerates tested. They grew well between 0 and 15Cwithoptimum temperatures of 1214C, but did not survive20C. Therefore, compared to Antarctic diatoms bothtaxa from Kongsfjorden can be characterised as eurythermalorganisms. Increasing photon fluence rates between2 and 15 lmol m2 s1 were accompanied with analmost twofold increase in growth rates, but photon fluencerates >15 lmol m2 s1 did not further enhancegrowth pointing to low light requirements. From thesedata optimum, minimum and maximum photon fluencerates and temperatures for growth can be assessed indicatingthat both diatoms are well acclimated to the fluctuatingenvironmental conditions in the Arctic habitat

The Effect of Inkjet Ink Composition on Rheology And Jetting Behaviour

This work presents recent results on the way linear and non linear viscoelastic properties of the fluids affect the jetting mechanism. Recent progress on quantitative characterising both high frequency linear (LVE) and non-linear (NLVE) viscoelasticity of fluids allows fluids to be assessed for their jettability before using such materials in a DoD print head. In term of linear viscoelastic measurements, the Piezo Axial Vibrator (PAV) was used to probe the rheology of the fluids on a frequency range between 10Hz and 10000Hz. A filament stretching apparatus, called the “Cambridge Trimaster”, was used in combination with high speed cinematography, to characterize the fluids high speed stretching and break-up behaviour. The series of fluids investigated here consist in dilutions of mono disperse polystyrene with different molecular weight (110, 210, 306 and 488 kg/mol respectively) diluted in diethyl phthalate. The choice of polymer weights and concentrations were chosen to match both the complex viscosity and the LVE. However, non linear rheological data experiments exhibit differences in the fluid relaxation time and filament break-up mechanism. Ultra-high speed cinematography of DoD jetting events were correlated with filament break-up experiments and demonstrated that fluid rheology provides valuable information on the jetting quality of the fluids